Development and Modeling of Functionally Graded Porous Structures and … › bitstream › 1807 ›...

Development and Modeling of Functionally Graded Porous Structures and Composites

by

Farooq Al Jahwari

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Mechanical and Industrial Engineering University of Toronto

© Copyright by Farooq Al Jahwari 2016

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Development and Modeling of Functionally Graded Porous

Structures and Composites

Farooq Al Jahwari

Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

2016

Abstract

Functionally graded (FG) materials provide the solution to spatial and temporal control of

material properties with smooth transition at different length scales of the material structure.

Tailoring material properties is becoming a necessity to the evolving products that have certain

design requirements. Smooth transition at different length scales is important to eliminate abrupt

change of material properties and microstructure which avoids stress jumps and other structural

problems like delamination. Two challenges are associated with FG materials, i) fabrication

procedures with correlation to the microstructure and mechanical properties, and ii) the

numerical treatment of FG materials that accounts for the microstructural details and gradient.

This research addresses both challenges with enough depth to design FG porous polymeric

materials. The fabrication process focused on developing FG porous structures and stitched

composites. Different processing parameters were successfully correlated to the microstructure

and showed the potential to produce structures with desired features. The microstructure was

further correlated to mechanical properties like creep compliance and impact energy absorption.

This motivated the development of numerical homogenization procedures that help in

conducting numerical experiments with FG structures. A statistical based homogenization model

was developed, which accounted for the microstructure gradient. The model was implemented to

higher order plate theory with stretching terms, and provided accurate results compared to

experimental data. To alleviate the dependency on experimental data which was the drawback of

the statistical model, a purely numerical procedure was developed which is based on 3D

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reconstruction of the microstructure to statistically reduce the system but preserve the same main

features of the parent structure. The model was in good agreement with experimental results.

This makes the homogenization procedure independent in providing details about mechanical

properties for an assumed graded porous structure or composite. Combined with the processing-

microstructure graphs, the numerical tool can be used in an inverse-homogenization procedure to

make graded porous structures with desired mechanical properties.

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Acknowledgments

I owe my deepest gratitude to my parents, Khalfan and Aysha, for their endless love, support and

encouragement. To them, I dedicate this thesis. I would like to express my warm and sincere

thanks to my Love of Life for her unconditional support. Her love and support made the hard

times easier and the good times wonderful. Furthermore, I’m extending my gratitude and

appreciation to my kids for their understanding to my situation as a student. I couldn’t allocate

much of my time for the last four years to them, because of my busy schedule with research and

study. They were very understanding, supportive, and provided me with the peace I needed to

excel in my studies.

I really don’t know where to start when it comes to telling about my supervisor, Professor Hani

Naguib. I’m approaching 40ies of my age and have been working with many people from

academia and industry, but never I met a person like Professor Naguib. The truth to be said is

that he is unique and one of his kind only. I can say with confidence that I’m not graduating with

one doctoral degree but in fact two. One doctoral degree in mechanical engineering science, and

the second doctoral degree is in life and humanity. His guidance and continuous follow up was

an essential element in the successful completion of this thesis work. The weekly research

discussions were always very fruitful and making me moving forward with solid steps.

It is an honor for me to have Professor Kamran Behdinan and Professor Chandra Veer Singh in

my PhD committee. I’m deeply thankful for their insightful comments and guidance to improve

the coherence of the thesis. I would like also to thank Dr. Jason Lo for his fruitful discussions

throughout the research.

Also, I would like to acknowledge gratefully my colleagues from the Smart Polymers &

Composites Lab (SAPL) for providing very friendly environment for research. Special thanks to

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Ahmed Anwer, I am deeply grateful to his help and support in the development of experimental

procedures and moulds. His help was significant and saved a lot of time for my research

progress. I am also very thankful to the faculty and staff at the Department of Mechanical &

Industrial Engineering at the University of Toronto for providing a stimulating and peaceful

environment in which to learn and grow.

Natural Sciences and Engineering Research Council (NSERC) of Canada, the Canada Research

Chairs Program, the Canada Foundation of Innovation, Sultan Qaboos University, Oman, and

CanmetMATERIALS, Canada, are all greatly acknowledged for their support.

Farooq Al Jahwari

Toronto, Canada

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Table of Contents

Acknowledgments.......................................................................................................................... iv

Table of Contents ........................................................................................................................... vi

List of Tables ...................................................................................................................................x

List of Figures ................................................................................................................................ xi

Chapter 1

Introduction ..................................................................................................................................1

1.1 Preamble ..............................................................................................................................1

1.2 Definition .............................................................................................................................1

1.3 Design of Functionally Graded Porous Composite Materials .............................................2

1.4 Motivation and Objectives .................................................................................................11

1.5 Organization of the Thesis .................................................................................................18

1.6 Contributions......................................................................................................................22

1.7 References ..........................................................................................................................23

Chapter 2

Fabrication and Microstructural Characterization of Functionally Graded Porous

Structures with Correlation to Creep Behaviour ...................................................................28

2.1 Introduction ........................................................................................................................29

2.2 Material and Processing .....................................................................................................32

2.3 Characterization of the Functionally Graded Porous Structures ........................................33

2.3.1 Microstructural Characterization ...........................................................................33

2.3.2 Creep Test and Correlation ....................................................................................36

2.4 Results and Discussion ......................................................................................................36

2.4.1 Porosity and Pore Diameter ...................................................................................36

2.4.2 Creep Behaviour ....................................................................................................41

2.5 Conclusion .........................................................................................................................44

2.6 References ..........................................................................................................................44

vii

Chapter 3

Relation of Impact Strength to the Microstructure of Functionally Graded Porous

Structures of Acrylonitrile Butadiene Styrene (ABS) Foamed by Thermally Activated

Microspheres ...............................................................................................................................47

3.1 Introduction ........................................................................................................................48

3.2 Materials and Testing .........................................................................................................51

3.3 Fabrication Procedure ........................................................................................................53

3.4 Morphological Characterization ........................................................................................56


3.5.1 Relationship between processing conditions and the microstructure ....................59

3.5.2 Propagation of the porous structure .......................................................................60

3.5.3 Relationship between porosity and expansion ratio...............................................63

3.5.4 Relationship between the relative density, ∆∅, and porosity ................................64

3.5.5 Effect of Microspheres Loading ............................................................................65

3.5.6 Effect of the thermal gradient on diameter difference ...........................................66

3.5.7 Impact Energy ........................................................................................................67

3.6 Conclusions ........................................................................................................................72

3.7 References ..........................................................................................................................73

Chapter 4

Experimental Evaluation of Impact Load Transfer of Through-thickness Stitched

Composite Structures with Graded Syntactic Foams ..............................................................76

4.1 Introduction ........................................................................................................................77

4.2 Material Structure and Processing .....................................................................................79

4.3 Experimental Set-up and Testing .......................................................................................82


4.4.1 Composite Structure and Microspheres Dispersion ...............................................83

4.4.2 Compressive Modulus ...........................................................................................85

4.4.3 Transferred Impact Load........................................................................................86

4.4.4 Relaxation Modulus and Correlation Analysis ......................................................88

4.5 Conclusions ........................................................................................................................90

4.6 References ..........................................................................................................................91

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Chapter 5

Analysis and Homogenization of Functionally Graded Viscoelastic Porous Structures

with a Higher Order Plate Theory and Statistical Based Model of Cellular Distribution ...93

5.1 Introduction ........................................................................................................................95

5.1.1 Fabrication of functionally Graded Porous Structures ...........................................96

5.1.2 Analysis of Functionally Graded Plates (FGPs) ....................................................96

5.2 Experimental Procedure .....................................................................................................99

5.2.1 Material and Processing .........................................................................................99

5.2.2 Characterization ...................................................................................................100

5.3 Numerical Procedure .......................................................................................................100

5.3.1 The Novel Plate Theory .......................................................................................100

5.3.2 Linear Viscoelastic Constitutive Law ..................................................................103

5.3.3 Homogenization Model .......................................................................................107

5.3.4 Finite Element Formulation and Element Selection ............................................110

5.4 Results and Discussion ....................................................................................................114

5.4.1 Fabrication of Functionally Graded PLA ............................................................114

5.4.2 Microstructure Characterization ..........................................................................117

5.4.3 Relaxation Modulus of Functionally Graded Porous Structures .........................118

5.4.4 Numerical Results ................................................................................................120

5.5 Conclusions ......................................................................................................................123

5.6 References ........................................................................................................................124

Chapter 6

Finite Element Creep Prediction of Polymeric Voided Composites with 3D Statistical-

based Equivalent Microstructure Reconstruction ................................................................130

6.1 Introduction ......................................................................................................................131

6.2 Materials and Processing .................................................................................................134

6.3 Microstructural Characterization and Creep Test ............................................................135

6.4 Construction of the Equivalent Microstructure................................................................136

6.5 Periodic Boundary Conditions and Calculations of Macroscopic Strains .......................141

6.6 Linear Viscoelastic Material Model .................................................................................143

6.7 Results and Discussion ....................................................................................................145

6.8 Conclusion .......................................................................................................................149

6.9 References ........................................................................................................................149

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Chapter 7

Conclusions and Recommendations .......................................................................................153

7.1 Conclusions ......................................................................................................................153

7.2 Recommendations for Future Work.................................................................................157

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List of Tables

Table 2.1. Effect of annealing temperature and time on the pores diameter ................................ 39

Table 2.2. Correlation coefficients for porosity and diameter against creep strain and compliance

....................................................................................................................................................... 42

Table 3.1. Foaming conditions of ABS repeated for 10, 20 and 30 wt% of Expancel

Microspheres ................................................................................................................................. 56

Table 3.2. Overall correlation coefficients between impact energy and microstructure.............. 68

Table 4.1. The fabricated specimens and lay-up order ................................................................ 81

Table 4.2. Correlation coefficients between relaxation modulus (𝐸𝑅), compressive modulus

(𝐸𝑐), and transferred impact load (𝐹). .......................................................................................... 90

Table 5.1. Prony series parameters for solid PLA...................................................................... 107

Table 5.2. Burr distribution parameters for the TH side of FG PLA (Reference to Fig 5.8) ...... 117

Table 5.3. The fitting parameters for equation (18) with two terms .......................................... 119

Table 6.1. Geometrical features of the equivalent structure in Fig 6.2c. ................................... 138

xi

List of Figures

Fig 1.1. Illustration of the basic types of morphologies associated with porous structures: (a)

randomly dispersed pores; (b) more dense pores’ dispersion; (c) interconnected porous structure;

and (d) functionally graded porous structure. ([6, 33]) ................................................................... 6

Fig 1.2. Porous cell representation of Gibson-Ashby and Kelvin ([43]) ........................................ 9

Fig 1.3. Experimental set-up for impact testing and sample load output. .................................... 14

Fig 1.4. Transmitted impact load for different polymer systems. ................................................ 15

Fig 1.5. Relaxation modulus of the solid, homogeneous and FG porous PLA. ........................... 16

Fig 1.6. Transferred acceleration from impact of rubber and porous base polymeric systems. ... 17

Fig 2.1. Schematic of the foaming technique of functionally graded ABS. ................................. 32

Fig 2.2. Image processing steps for calculations of porosity and pores’ diameters. .................... 35

Fig 2.3. Effect of TH on average porosity for different values of TL and TStage-1 with pressure of

2000 psi and 2 minutes annealing (𝜎𝑎𝑣𝑒 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛). ..................... 37

Fig 2.4. The cellular distribution of a functionally graded ABS under TH = 125 oC, TL = 52.5

oC,

tannealing = 5 minutes, P = 2000 psi, and TStage-1 = 52.5 oC. ............................................................ 38

Fig 2.5. Cellular structure of the case: TH = 135 oC, TL = 70

oC, tannealing = 2 minutes, P = 2000

psi, and TStage-1 = 70 oC. ................................................................................................................ 39

Fig 2.6. The effect of stage-1 temperature on the cellular structure at 2000 psi pressure. ........... 40

Fig 2.7. The effect of pressure at stage-1 on the cellular structure at 70 oC. ................................ 40

Fig 2.8. The effect of average porosity and diameter on creep strain for tannealing = 2 minutes, P =

2000 psi, TStage-1 = 78.75 oC and different hot side temperature (TH) ........................................... 42

xii

Fig 2.9a. Steady state creep strain and compliance for different porosities, pore diameters and . 43

TStage-1 = 52.5 oC ........................................................................................................................... 43

Fig 2.9b. Steady state creep strain and compliance for different porosities, pore diameters and 43

TStage-1 = 70 oC ............................................................................................................................... 43

Fig 2.9c. Steady state creep strain and compliance for different porosities, pore diameters and . 43

TStage-1 = 78.75 oC .......................................................................................................................... 43

Fig 3.1. FTIR analysis of ABS ..................................................................................................... 53

Fig 3.2. Schematic of the fabrication process of impact specimens. ............................................ 54

Fig 3.3. Effect of thermal distribution on the microstructure. ...................................................... 55

Fig 3.4. The mould with thermal insulation placed on the inner walls. ....................................... 55

Fig 3.5. Explanatory graph for the impact and pores’ gradient direction for the case of 20%

microspheres and 175/215 oC TL/TH ............................................................................................. 57

Fig 3.6. Circularity analysis of the cells for the case of 190/235 ºC TL/TH and 10% microspheres

loading........................................................................................................................................... 58

Fig 3.7. Dependence of porosity on the processing temperatures ................................................ 59

Fig 3.8. Cell collapse at higher values of processing temperatures for the case of 10%

microspheres and 190 oC/215

oC TL/TH processing temperatures. ................................................ 60

Fig 3.9. The microstructure of ABS with 10% microspheres at TL of 160 oC and different TH ... 60

Fig 3.10. The microstructure of ABS with different loading of Microsheres at TL of 160 oC and

TH of 190 oC .................................................................................................................................. 61

Fig 3.11. Viscosity change of ABS with increasing temperature. ................................................ 63

Fig 3.12. Terminal velocity change with viscosity of ABS melt.................................................. 63

xiii

Fig 3.13. The relationship between porosity and expansion ratio. ............................................... 64

Fig 3.14. Relationship between relative density and pores’ diameter gradient (∆∅) ................... 65

Fig 3.15. The relationship between microspheres loading, expansion ratio and relative density. 66

Fig 3.16. The relationship between microspheres loading, expansion ratio and diameter gradient.

....................................................................................................................................................... 66

Fig 3.17. The effect of thermal gradient on pores’ diameters for different TL values and 10%

microspheres loading. ................................................................................................................... 67

Fig 3.18. Pores’ diameter gradient vs. expansion ratio for different loadings of microspheres. .. 69

Fig 3.19. Effect of pores’ diameter gradient and relative density on impact strength with 10%,

20%, and 30% loading of microspheres. ...................................................................................... 70

Fig 3.20. Images of the failing specimens of solid ABS and functionally graded porous structure

having the same volume................................................................................................................ 72

Fig 4.1. Processing procedure for polyurethane composites ........................................................ 80

Fig 4.2. Schematic of the graded stitched foamed composite structure ....................................... 81

Fig 4.3. Sensing system for the experimental set-up of impact test. ............................................ 83

Fig 4.4. Sample of the impact load results for the case of stitched composite with 3% loading of

the microspheres. .......................................................................................................................... 83

Fig 4.5. The structure of the stitched graded foam composite for the case of 1%-2%-3%

microspheres loading. ................................................................................................................... 84

Fig 4.6. Compressive modulus of the specimens with different loadings of the microspheres. .. 85

Fig 4.7. SEM image showing the de-bonding between polyurethane and the microspheres ....... 86

Fig 4.8. The transferred impact load by each composite structure. .............................................. 87

Fig 4.9. Relaxation modulus of the fibrous and non-fibrous composite structures. ..................... 88

xiv

Fig 4.10. Relaxation modulus of the graded foamed composite structures. ................................. 90

Fig 5.1. Arbitrary region in real space ........................................................................................ 105

Fig 5.2 Stress relaxation function of solid PLA fitted to Prony series. ...................................... 106

Fig 5.3 Elements convergence to the exact solution of central transverse deflection of a simply

supported plate subjected to uniform pressure. ........................................................................... 113

Fig 5.4 A conforming quadrilateral element with four nodes .................................................... 114

Fig 5.5. Functionally graded PLA for TH = 130 oC, TL = 80

oC, and 5 minutes annealing ........ 115

Fig 5.6. Comparing homogeneous to functionally graded cellular distribution ......................... 116


oC, and 10 minutes annealing ..... 116

Fig 5.8. Microstructure characterization and fitting to Burr distribution ................................... 118

Fig 5.9. Relaxation modulus of the solid, homogeneous and FG porous PLA. ......................... 119

Fig 5.10. Comparing the experimental and predicted relaxation modulus of FG 130-80 oC ..... 120

Fig 5.11. Comparing exact transverse deflection of SiC FG plate to HSDT and CPT for a

uniform pressure and simply supported condition ...................................................................... 121

Fig 5.12. Comparing HSDT with experimental results for a relaxation test of FG 130-80 oC PLA

..................................................................................................................................................... 122

Fig 6.1. Experimental set up for the high pressure/temperature foaming chamber. ................... 134

Fig 6.2. The effect of seeding temperature on the microstructure at 1750 psi pressure (∅ is voids

fraction and d is average voids diameter). .................................................................................. 136

Fig 6.3. Image thresholding and microstructure recognition of voided ABS with 1750 psi

pressure and seeding at 78.75 ºC. ............................................................................................... 137

Fig 6.4. Voids diameters and area fractions of the actual and equivalent systems in Fig 6.2c. . 138

xv

Fig 6.5. The 3D reconstructed voided structure of SEM image in Fig 6.2c with 0.4223 voids

fraction. ....................................................................................................................................... 141

Fig 6.6. Normalized creep compliance of solid ABS ................................................................. 145

Fig 6.7. Sample mesh for the periodic box (1003206 four-node tetrahedral (C3D4) elements). 147

Fig 6.8. Deformation of the RUC that conforms to the periodicity of opposite faces (Gray:

undeformed shape). ..................................................................................................................... 147

Fig 6.9. Microstructure at different fracturing planes for the case of 78.75 oC and 1750 psi

seeding temperature and pressure. .............................................................................................. 147

Fig 6.10. Comparing FEM results to experimental data for creep compliance of the voided

structures. .................................................................................................................................... 148

Chapter 1

Introduction

1.1 Preamble

Materials have always played an essential role in human development. The fast industrial

revolution brought out the need to new engineered materials suiting certain applications.

Functionally graded materials (FGMs) are new class of advanced materials in the family of

composites. Functionally graded (FG) materials provide the solution to spatial and temporal

control of material properties with smooth transition at different length scales of the material

structure. Tailoring material properties is becoming a necessity for the evolving products that

have certain design requirements for an intended application. Smooth transition at different

length scales is important to eliminate abrupt change of material properties and microstructure

which avoids stress jumps and other structural problems like disassociation of different material

systems. Two challenges are mainly associated with the development of FG materials. The first

challenge is related to the fabrication procedures and the relation to microstructure. The second

challenge is concerned with the numerical treatment of FG materials, which will help in

understanding and accelerating the design cycle for corresponding structures. This research

addresses both challenges with considerable depth to design FG materials.

Chapter 1 starts by introducing the history of FGMS. A comprehensive review is then provided

to cover both the fabrication and modeling along with potential limitations associated with

FGMs. The chapter concludes by establishing the motivation and objectives of the research that

target to overcome some of the limitations for implementing FGMs in a wider range. Finally, the

thesis layout and flow of chapters will be explained.

1.2 Definition

The concept of functionally graded materials was first applied in 1984 when Japanese engineers

and scientist decided to use them for the design of a thermal barrier during the space plane

2

project [1]. The thermal barrier was required to withstand a temperature of 2000 K at one side

and gradually reducing to 1000 K at the other side within a thickness of 10 mm. Functionally

graded materials were the solution to such design requirement, due the continuous microstructure

that overcome the mismatch in thermal expansion between different material systems. With the

continuous need to improve structural performance, FGMs are being developed to tailor the

material architecture at microscopic scales to optimize certain functional properties of structures

[2]. Functionally graded materials can be classified as advanced composite materials at which the

composition and/or microstructure are gradually changing to serve the purpose of enhancing a

certain property.

A large focus has been on the metallic-ceramic functionally graded materials. In contrast, very

little attention was given to the analysis and fabrication of functionally graded porous media.

Polymeric porous materials (PPM) were first introduced by Martini et al. [3] as means to reduce

the density of solid polymers and thus saving cost where applications don’t require mechanical

strength like packaging industry. Nevertheless, cellular materials still show interesting

mechanical properties. The fatigue life of cellular polycarbonate (PC) of above 90% relative

density was experimentally reported to be as much as four times that of a solid PC [4], and a

greater impact strength with relative density over 60% [5]. Functionally graded porous materials

provide the advantage of making structures that are comparable in strength to the solid precursor

but with significant weight reduction [6].

1.3 Design of Functionally Graded Porous Composite Materials

A major problem in the design of FGMs, aside from that of materials selection, lies in

determining the optimum spatial dependence for the composition and/or microstructure. This can

be regarded as the composition/microstructure profile which best accomplishes the intended

purpose of the material while maintaining other thermal, physical, and mechanical properties

within limits that ensure acceptable performance. Another problem lies in predicting the

properties and behaviour of FGMs, for a given composition/microstructure profile, during

fabrication and under in-service conditions. After deciding about the spatial distribution and

composition, fabrication is another challenge to make the structure as designed with minimum

deviation. The following two sections outline some of the efforts to address the fabrication,

3

homogenization, and modeling challenges associated with FGMs with more focus to functionally

graded porous materials.

Fabrication

The aim of fabricating a functionally graded material is to achieve a gradual change of properties

with position to enhance a certain property while maintaining other properties within acceptable

limits. The property gradient may result as a consequence of position-dependent chemical

composition and microstructure or atomic order [7]. The advantages of FGMs was already

proven theoretically in 1972 by Bever, Duwez and Shen [7] but it took about 15 years of research

in manufacturing to bring those materials to application. FGMs made of metals and ceramics can

be classified based on processing techniques to constructive processes and those produced by

transport-based processes [8, 9]. In constructive processes, the material is placed in the

appropriate locations by some techniques such as vapor deposition or powder metallurgy.

However, transport-based processes such as centrifugal separation, utilize the presence of a steep

force gradient to promote mass transport from one location to another resulting from the

processing conditions. FGMs composed of metals and ceramics are intensively reviewed in

literature while the focus here will be on polymer based and especially porous functionally

graded materials.

Polymeric based porous FGMs (PPFGMs) have generated the least attention in research despite

their application potential is large especially in the biomedical and automotive industry. A

controlled morphology of a porous structure would be desired in many applications due to the

significant weight saving while optimizing the other properties as to suite a certain application

requirements. Fabrication techniques of cellular materials include, but are not limited to, solvent

particulate leaching [10, 11], gas foaming [11-13], co-continuous melt blending [14-19], and

rapid prototyping [20-22]. However, the combination of those techniques leads to more control

of the growth of pores [12, 18], and better mechanical properties [11]. Nadella et al. [12],

fabricated a closed-cell functionally graded PLA by annealing gas foamed samples with a

temperature gradient rather than quenching in a bath of uniform temperature. Zhang et al. [18],

4

fabricated an open-cell functionally graded PLA by mixing with polystyrene (PS), moulding at a

temperature above the melting point of both polymers, and finally annealing at temperatures

above the glass transition but below the melting point. PS was removed in a later stage by

dissolving in cyclohexane. Harris et al. [11], made disks of PLA and salt by compression

moulding, gas foaming, and finally leaching out the salt which made a mechanically stronger

foam than those made by traditional leach-out techniques. It is worth mentioning that processing

parameters to control the pores’ growth rate and size are not fully understood and optimized. The

addition of nano-fillers like chitin were reported to improve the foaming process [23]. Other

nano-fillers like carbon nanotubes and nanofibers were even reported to also provide a control

over porosity [24] which was attributed to the enhancement of nucleation sites.

Functionally graded syntactic foam sheets were fabricated by El-Hadek and Tippur [25] from

epoxy and microspheres to study the dynamic fracture behavior under low velocity impact

loading. The volume fraction of the microspheres was graded linearly over the width of the

sheets. Their study suggested that there is an increasing fracture toughness if the crack initiated

at the soft side of the functionally graded sheet and propagated to the stiff side. Gupta [26]

demonstrated the ability to control the compressive modulus, strength, and total energy

absorption of functionally graded syntactic foams made of epoxy and glass microballoons, by

controlling the volume fraction, type and distribution of the microballoons. The fabricated

functionally graded foams could withstand 60-75% compression ratio without significant loss of

strength. The fabrication process was optimized in a previous study by Caeti et al. [27].

With all those emerging fabrication techniques of PPFGMs, relating properties to the processing

conditions and microstructure becomes the deciding factor for optimization and usefulness of

such materials for specific applications. Imaging of the microstructure and studying different

cellular characteristics like cell sizes and growth rate can be used to optimize the fabrication

parameters. The microstructure is then related to the mechanical properties. This correlation

across processing conditions, microstructure, and mechanical properties provides a procedure to

design FGMs starting from a required mechanical property. It was already proven by many

researchers that the processing temperatures can be related to cell growth rate and size [12, 14,

5

15, 18, 19]. However, the relation is not yet clear and more research is required in this area.

Having control over the microstructure necessitates investigating the effect on properties leading

to a clear path for material design to target specific applications. There are studies where porous

structures were part of the investigated material, but the microstructure was not discussed. For

example, Daniel and Cho [28] studied experimentally the multi-axial behavior of polymeric

foams under static and dynamic loading conditions providing the mechanical properties at

different strain rates but without discussing the microstructure. Another study by Avalle et al.

[29] investigated energy absorption characteristics and mechanical properties of selected foams

under static and impact loading without correlating the findings to the microstructure of the

studied foams. Conversely, it was observed by other researchers [30, 31] that the microstructure

morphology gives variability in the effective properties. This emphasizes the importance of

accurate microstructural analysis when studying the mechanical properties of porous structures

in general. The importance of microstructural analysis becomes more significant when studying

microstructure-dependent structures like functionally graded materials.

Homogenization and Modeling

The use of theoretical models to aid in the design of FGMs is of crucial importance due to the

large dispersity of possible compositions and distributions to achieve a certain property. Once

established, a model can readily be used to conduct a wide range of computer “experiments” in

which the effect of changing input parameters, such as thermo-mechanical properties of the

constituent phases, or the composition/microstructure profile along the graded direction, are

systematically evaluated. The fact that the composition of FGMs can vary over such a wide

range means that a variety of fundamentally different microstructures can exist across the graded

direction. This, in turn, means that the thermo-mechanical properties, which are generally

strongly dependent on the microstructure, will also vary with position within the material. A

realistic model must appropriately account for this fact. The problem of calculating effective

properties of heterogeneous materials is an old one, dating back to Maxwell and Rayleigh.

As already noted, the effective properties of heterogeneous materials are related to the

microstructure of the material. Three basic types can be identified of geometric morphologies

6

associated with porous structures as inspired from the classification of Markworth et al. [32], (a)

a dispersed grain structure for which the volume fraction of the pores is low, and is discretely

and randomly distributed within the polymer (Fig 1.1a); (b) a structure in which the volume

fraction of the pores is somewhat higher, but the pores remain discretely and uniformly dispersed

(Fig 1.1b); (c) an interconnected skeletal structure, where the volume fraction of the pores is

increased further, reaching a critical value at which there exists a continuous random network of

struts (Fig 1.1c). A functionally graded porous structure may contain all these types of

morphologies as shown in Fig 1.1d.

a b

c d

Fig 1.1. Illustration of the basic types of morphologies associated with porous structures: (a)

randomly dispersed pores; (b) more dense pores’ dispersion; (c) interconnected porous structure;

and (d) functionally graded porous structure. ([6, 33])

7

Often, precise information about the size, shape, and distribution of the particles/pores may not

be available, and the effective moduli of the graded material must be evaluated based only on the

volume fraction distributions and the approximate shape of the dispersed phase or pores

including randomness. A recent approach which includes the actual statistics of the porous

structure into numerical procedures was developed for accurate prediction of the effective

properties [34]. Several micromechanics models have been developed over the years to obtain

the effective properties of macroscopically homogeneous composite/porous materials. The idea

of using these models with functionally graded composites can be divided to two approaches; the

first approach is based on extracting the distribution function and shape of the pores/particles

from the micrographs or processing procedures. The second approach, implements the

micromechanics models discretely in a point-wise manner to predict the effective properties as a

function of spatial coordinates according to the local microstructural details. Homogenization of

the local porous structure in the above mentioned approaches can broadly be classified to

microstructural detail-free estimates and to procedures that include the microstructure details.

The first category considers pores as inclusions with zero shear and bulk modulus. Some of them

account for the interacting fields between the pores while others assume a diluted state (e.g. rule

of mixture, self-consistent and generalized self-consistent schemes, Mori–Tanaka scheme and

others [35, 36]). The second category considers the pore as a single cell being part of an infinite

periodic structure [36-39]. In the following; a brief summary of these micromechanics methods

and their application to functionally graded materials is provided.

The rule of mixture does not consider the interactions among the constituents [40] which makes

it suitable of porous structures of type (a), Fig 1.1a. Mori-Tanaka [41] and the self-consistent

[42] methods are more suitable for the estimation of the effective properties of dense porous

structures. Mori–Tanaka scheme for estimating the effective moduli is applicable to regions of

the graded microstructure that have a well-defined continuous matrix and a discontinuous

particulate/pore phase. Let’s assume that the matrix phase, denoted by the subscript 1, is

reinforced by spherical particles of a particulate phase or just dispersed pores, denoted by the

subscript 2. In this notation, K1 and μ1 are the bulk and shear moduli, and V1 is the volume

8

fraction of the matrix phase. K2, μ2, and V2 are the corresponding material properties and volume

fraction of the particulate phase or the pore respectively. The equations (1.1) and (1.2) estimate

the effective bulk modulus K and shear modulus μ which are useful for a random distribution of

isotropic particles in an isotropic matrix.

𝐾 − 𝐾1

𝐾2 − 𝐾1=

𝑉2

1 + (1 − 𝑉2)𝐾2 − 𝐾1

𝐾1 +43𝜇1

(1.1)

𝜇 − 𝜇1

𝜇2 − 𝜇1=

𝑉2

1 + (1 − 𝑉2)𝜇2 − 𝜇1

𝜇1 +𝜇1(9𝐾1 + 8𝜇1)6(𝐾1 + 2𝜇1)

(1.2)

Note that V1 + V2 = 1. The Lamé constant λ¸ is related to the bulk and the shear moduli by λ=K-

2μ/3.

The Mori-Tanaka model takes into account the interaction of the elastic fields between

neighboring inclusions or pores. The self-consistent method assumes that each reinforcement

inclusion or pore is embedded in a continuum material whose effective properties are those of

the composite. This method does not distinguish between matrix and reinforcement phases, and

the same overall moduli is predicted in another composite in which the roles of the phases are

interchanged. This makes it particularly suitable for determining the effective moduli in those

regions that have an interconnected skeletal microstructure as shown in Fig 1.1c. The effective

properties are given by equations (1.3) and (1.4).

1

𝐾 +43𝜇

=𝑉1

𝐾 − 𝐾2+

𝑉2

𝐾 − 𝐾1 (1.3)

9

𝐾

𝜇 (𝐾 +43𝜇)

=𝑉1

𝜇 − 𝜇2+

𝑉2

𝜇 − 𝜇1 (1.4)

The model proposed by Gibson and Ashby [37] is one of the first models in the second category

for homogenization of porous media. A regular polyhedral-cell model is assumed to be

representative of the whole cellular structure. It is based on the concept of a representative unit

cell (RUC) with the assumption that the deformation pattern repeats periodically at its

boundaries. The unit cell of Gibson and Ashby for open cells is shown in Fig 1.2a. Many

statistical and mechanics related questions are raised about Gibson-Ashby cell, which led

researchers to create other models. One of each is the Kelvin cell model shown in Fig 1.2b. Most

researchers assumed rules for cell generation to generate many examples of a cellular structure

with similar overall porosity. The predicted mechanical properties vary from example to

example, so averages must be taken. The effective properties for open-cell porous media as

proposed by Gibson-Ashby model are given in equations (1.3). There are equivalent formulas for

the closed-cells and for predicting other properties like the critical buckling load, the elastic limit

before plateau region, and the densification strain.

a. Gibson Ashby cubic cell model b. Kelvin cell model

Fig 1.2. Porous cell representation of Gibson-Ashby and Kelvin ([43])

10

𝐸∗

𝐸𝑠= (

𝜌∗

𝜌𝑠)2

𝐺∗

𝐸𝑠=

3

8(𝜌∗

𝜌𝑠)2

(1.3)

where 𝐸𝑠 and 𝜌𝑠 are the solid material modulus of elasticity and density. 𝐸∗ and 𝜌∗ are the

corresponding properties of the porous media. 𝐺∗ is the shear modulus of porous media.

In the following, a brief review of some work in the area using one of the above mentioned

homogenization techniques. Lee et al. [44] developed a finite element analysis based

micromechanical method for understanding the fracture behavior of functionally graded foams.

A model similar to Gibson-Ashby unit cell was assumed in this work. The domain was

discretized into regions of different cell sizes where the material in each element is homogeneous

and of constant cellular structure. Wang [45] developed a unit truss element for the analysis of

porous structures including geometrical non-uniformity in the cellular structure. A Voronoi

meshing technique was used by Jones [46] to study the stress concentration around circular holes

for functionally graded porous structures using truss elements also. Babaee [47] developed

analytical relationships to predict the mechanical properties and response of open three-

dimensional Kelvin cell. The cell edge material was assumed to be elastic-perfectly plastic and

the effective mechanical properties of the cellular structure were related to the cell edge material

properties, and the relative density of the cellular structure. Cellular randomness was considered

also and its effect was shown to be significant on the effective mechanical properties. The

author used a finite element analysis to validate the developed analytical models. Poveda and

Gupta [48] demonstrated the weight-saving potential of functionally graded foams in designing

damage-tolerant structures through finite element analysis. A tetrakaidecahedron-shaped cell

(Fig 1.2b) was used to create open and closed-cell foam models. The crack tip constraint effects

on fracture behavior under low velocity impact loading of functionally graded syntactic foams

made of epoxy and microspheres was studied by El-Hadek and Tippur [25] using finite element

simulations. Their analysis suggested a strong link between in-plane constraint and fracture

toughness in graded materials.

11

Despite of all the work at the continuum level, little work at the atomistic level has been

accomplished for porous structures. Atomistic simulations of the mechanical behavior of metallic

nano porous materials with molecular dynamics (MD) simulations are reported in literature but

no published work for polymeric nano porous materials. Zhao et al. [49] presented results for the

stress-strain behavior and incipient yield surface of nano porous single crystal copper. They

modeled the nano pore as a periodic unit cell subjected to multi-axial loading. MD simulations

were used by Murillo et al. [50] to generate nano porous silica aerogels through direct expansion

of crystalline samples of β-cristobalite along with thermal processing. The mechanical properties

of the generated structure were then investigated by MD. The effect of porous defects on

monocrystalline gold specimens under mechanical loads at different temperatures was studied

using molecular dynamics analysis by Lin et al. [51]. They obtained the stress-strain curves for

such nano-defected material and the stress concentration factor around nano pores. Li et al. [52]

developed a scaling constitutive law, by using MD simulations, which accounts for the effects of

disordered Zn atoms and nano pores on the mechanical properties of β-Zn4Sb3. The effect of

nano pores on the elastic modulus, tensile strength, and fracture toughness were investigated.

1.4 Motivation and Objectives

Functionally graded porous composite structures have high strength-to-weight ratio compared to

their solid precursors. They can be a good alternative to solid materials in structural applications

when designed carefully. One of the challenges in designing FG polymeric materials with porous

structures is the absence of accurate and robust homogenization techniques that take into account

the microstructural details. Another challenge lies in constructing the porous structure for

numerical homogenization. Once homogenization issues are resolved, the last challenge is

fabricating the desired microstructure with the aid of established microstructure-processing

relations. This PhD research focuses on the aspects of modeling and fabrication of FG polymeric

porous composite structures to aid in the design of this class of materials.

The research hypothesis is formulated as follows, “A modeling technique based on a numerical

homogenization approach with the aid of microstructure-processing-properties relations, enables

12

the successful and efficient design and fabrication of functionally graded porous polymeric

materials”. To study this hypothesis, the research was divided to two main components,

fabrication and modeling. In the fabrication component of the research, new fabrication

techniques and procedures were developed to produce the FG porous structures with

considerable control over the microstructure. The resulting microstructures were then

characterized for morphological features and mechanical properties (creep/relaxation tests, and

impact strength). Two correlation analyses were conducted at this stage; the first analysis studied

the relationship between fabrication parameters and the resulting morphological features. The

second analysis focused on investigating the relationship between morphological features and the

mechanical properties. The results of both analyses revealed the existence of strong correlation

between fabrication parameters and mechanical properties, to the corresponding microstructure.

This finding proved the first component of the hypothesis.

In the second component of the research, accurate homogenization and modeling techniques

were developed for the analysis of FG porous structures and composites. Homogenization of the

porous structures was a challenge because of the continuous gradient of pore sizes and the

complexity of the microstructure. As mentioned earlier, most of the work in homogenization of

porous structures was either neglecting the microstructural details, or only dealing with the local

details of the microstructure. A statistical homogenization model was developed for the purpose

of predicting the mechanical properties of FG porous structures, taking into account the gradient

of pore sizes. The model featured the inclusion of control points across the structure where each

point represents a local morphological feature, and then all the control points were connected in

the form of Prony series. The developed statistical model was then implemented in a new higher

order shear deformation plate theory to test the accuracy and suitability for FG porous structures.

The plate theory was formulated with stretching element across the thickness of the material

structure to account for out-of plane deformations. The results of this numerical homogenization

and modeling agreed well with the experimental data. The next step was to develop a purely

numerical local homogenization tool which can be integrated with the statistical model, and

hence make the design process entirely numerical before proceeding to fabrication of the actual

microstructure. This goal was achieved by developing a three-dimensional reconstruction

procedure of the local microstructure, with the aid of granular mechanics which makes it

13

possible to achieve high porosity values. The reconstruction procedure yielded accurate

prediction of the mechanical properties when compared to the experimental results. This

accuracy of the procedure makes it a reliable source, in replacement to experimental data, for

providing the local mechanical properties at each control point of the statistical homogenization

model.

The results of the two components, combined together proved the hypothesis which was

formulated at the beginning of the research. The fabrication and modeling procedures developed

in this thesis provide a valuable tool for designing FG porous structures and composites. The

design starts from a required mechanical property and ends with fabrication of the microstructure

which satisfies this mechanical property in addition to other design requirements. In the

following, two preliminary studies are presented which were conducted to explore the potential

of functionally graded materials and porous structures. These two studies motivated the work on

fabrication and modeling in this work.

Solid Polymeric Blends for Impact Energy Absorption

The shortlisted polymers for this study were polycarbonate (PC), acrylonitrile butadiene styrene

(ABS), and thermoplastic polyurethane (TPU). Each of the three polymers has certain desirable

property for both maximizing impact resistance and reducing momentum transfer. For instance,

PC has very high impact resistance but gives poor performance in reducing momentum transfer

and is prone to irreversible plastic deformation. In contrast, TPU is a highly viscoelastic polymer

that can significantly dissipate impact load with large reversible deformation but at the expense

of maximized momentum transfer depending on the thickness. All polymer systems were

prepared with shear pulverization for scalability and enhanced impact strength [53]. The

different segments of the thickness were prepared with compression moulding at a unified

temperature and pressure for ease of processing.

14

The experimental set up shown in Fig 1.3 was developed to measure the residual impact load at

the receiver side rather than at the impacted surface. The setup is discussed in greater details in

Chapter 4. The purpose of this setup was to pay attention to the impact force transmitted across

the material towards the receiver. The material is then designed to minimize the transferred force

while keeping reasonable level of deformation. Fig 1.4 shows some results of dropping 3.63 Kg

(8 Ib) from height of 0.635 m (25 inches). The base line material is PC which transmitted about

14 kN of the load to the receiver when blended with 5% TPU. The triple blend of 90%-PC, 5%-

ABS, and 5%-TPU reduces this load transfer to less than 10 kN. The thickness of each polymer

is another factor that affects impact and momentum quantities. For example, 7 mm thickness of

TPU transmitted 10% less load than the 5 mm thick TPU. A functionally graded solid blend was

fabricated from the three polymers, which could reduce the transmitted impact load from 14 kN

to about 4 kN only (Fig 1.4). Multiple impacts on the same spot showed hardening behavior for

the load transfer with a rate of 0.187 kN/impact event (Fig 1.4). This problem can be alleviated

by the introduction of a functionally graded porous core away from the impacted surface.

Starting with small pores of the cellular structure near the impact surface and gradually

increasing to a threshold cell size value will significantly enhance energy absorption and reduce

the hardening effect resulting from multiple impacts. This behavior is further discussed in

Chapter 3.

a. Sensing system of the experimental set-up b. Load output for 95% PC + 5%

TPU blend.

Fig 1.3. Experimental set-up for impact testing and sample load output.

15

Fig 1.4. Transmitted impact load for different polymer systems.

Graded Porous Structures

Functionally graded porous structures of polylactic acid (PLA) were fabricated in a one-stage

foaming process. Annealing occurred under a thermal gradient after seeding the pores in the first

stage. Details of the fabrication process are presented in Chapter 5. The relaxation modulus was

obtained experimentally from the relaxation test for solid PLA, homogeneous cellular PLA, and

functionally graded porous PLA (Fig 1.5). Functionally graded PLA showed higher values for

the relaxation modulus compared to the homogeneous cellular PLA plates which is believed to

be due to the gradient of cells’ sizes. The drop in relaxation modulus for the FG PLA was 8.44%

only while it was 46.72% for the homogeneous case. The density of solid PLA is 1.24 g/cm3

while it was measured to be 0.905 g/cm3 for the FG PLA. The drop in density for the FG PLA

was 27% while the drop in relaxation modulus was 8.44% only. However, for the homogeneous

cellular PLA, the density drop was 29.06% only compared to 46.72% drop in the modulus. This

evident the potential of FG cellular materials as light weight but strong structures that can find a

wide range of applications in many industries like automotive and aerospace load bearing

members. This also motives the efforts done in this research to model such materials for tailoring

their properties to suite certain applications.

16

Fig 1.5. Relaxation modulus of the solid, homogeneous and FG porous PLA.

Another preliminary study was conducted to compare the effect of porous structures and rubber

materials in damping impact loading using the same setup as shown in Fig 1.3. Composites of

Green High Density Polyethylene (GHDPE) and Ground Tire Rubber (GTR) were fabricated by

compression moulding. Thermally activated microspheres were used as the mean to introduce a

porous structure in the composite. The fabrication process was similar to the one presented in

Chapter 3. A strike-face (the face of the material structure which faces the striker) made of

polyurethane and 5% silicon carbide was used to protect the porous structure from damage by

the striker. The polyurethane/silicon carbide sheet was attached to all the three samples in order

to compare the effect of rubber to porous structure in terms of energy absorption. The difference

between pure GHDPE and the corresponding composite with 10% GTR was negligible as can be

seen in Fig 1.6. This could be due to the strike-face that is a rubber-like material which hindered

the effect of rubber particles. In contrast, the porous structure with 10% microspheres loading

reduced the transferred acceleration by about 28% even with the strike-face. In a functionally

graded porous structure, the strike-face can be integrated as part of the system which eliminates

17

the possibility of delamination as well as ease of processing. Chapter 3 is devoted to similar

structures.

Fig 1.6. Transferred acceleration from impact of rubber and porous base polymeric systems.

Objectives

The main objective of the research is to develop a design tool for FG polymeric porous structures

and composites. This class of viscoelastic materials has generated the least attention in literature,

despite their potential to replace metals and ceramics for structural applications, which motivated

the formulation of the main objective. The main objective was divided to three sub-objectives at

which each one of them serves a certain aspect of the design tool;

1. Fabricate FG porous structures and composites with physical and chemical blowing

agents. This sub-objective was formulated to establish fabrication procedures with

accurate control over the microstructure and gradient of the pore sizes. New mould

designs and procedures were developed in the scope of achieving this sub-objective.

2. Establish the relationships between fabrication, microstructure morphology, and

mechanical properties. This sub-objective involved the development of deterministic

techniques for analyzing the microstructure morphology. Viscoelastic properties like

18

creep and relaxation functions were measured in addition to impact strength of the FG

structures.

3. Develop fully numerical homogenization procedure for the analysis of FG structures.

This sub-objective constitutes the stage of numerical design before fabrication of the

actual graded structure. Control points were established across the structure where each

point represents the local morphological features. A statistical homogenization model

was developed to fit the control points to a modified form of Prony series. The model

accurately represented the structure and property gradient. Finally, a three-dimensional

reconstruction of the local microstructure morphology was developed to predict the local

mechanical properties. The three-dimensional reconstruction alleviated the dependency

on experimental data for providing the overall properties of the FG structure.

1.5 Organization of the Thesis

The research was divided in two main parts, namely, fabrication (Chapters 2, 3, and 4) and

modeling (Chapters 5 and 6). The focus of the fabrication part was on developing fabrication

procedures for functionally graded porous structures and composites. Processing-microstructure

relationships were established for the purpose of providing guidelines to control the

microstructure to a desired morphology. The microstructure was then correlated to mechanical

properties like creep compliance and impact strength. This completes the fabrication part in the

design of FGMs once the desired microstructure is known. The problem of knowing the

microstructure which will give the desired mechanical properties can be alleviated by numerical

modeling. This was the focus of the second part of this thesis. Modeling was accomplished by

local and global homogenization of the functionally graded structures prior to macroscale

analysis. A homogenization step is important due to the different length scales between the

characteristic length of the microstructure and the actual dimensions of the part which are

different by an order of magnitude. This is particularly important when dealing with porous

structures that usually have very high degree of microstructural complexity. This complexity

arises from the wide range of pore sizes, walls thicknesses, shape, and distribution that all exist

in a single microstructure. Initially, a statistical based homogenization model was developed

which accounts for the gradient in mechanical properties along one or more spatial coordinates.

19

The model was implemented to a new higher order plate theory which was developed especially

for functionally graded porous structures. Next step was to make the statistical model

independent of the experimental data in predicting the desired microstructure. A Finite element

based analysis was developed with a three dimensional reconstruction procedure of the local

porous structure. The developed numerical procedures were in good agreement with

experimental data which proved their validity to conduct accurate numerical experiments. In

conclusion, the design strategy can be summarized as follows;

The overall performance of the part at macroscale can be analyzed by integrating the

developed homogenization models into a finite element analysis with an assumed

gradient of the microstructure and base material properties. Adjustments can be made to

the model until convergence is reached to the required microstructure and pores’ size

gradient which will give the set of required mechanical properties.

The desired microstructure which will give a set of required mechanical properties can be

also mapped from microstructure-property graphs; or alternatively, it can be obtained

from the developed homogenization procedures.

The fabrication procedures and parameters to fabricate the microstructure can be obtained

from the microstructure-processing graphs.

Chapter 2

The research started by investigating the processing-microstructure-property relationships of

functionally grade porous structures. The investigation included the study of the potential of

tailoring mechanical properties by controlling the porous gradient. Functionally graded structures

of Acrylonitrile Butadiene Styrene (ABS) were fabricated with one-stage foaming process to

seed the initial uniform porous structure. The pores’ size gradient was induced by exposing the

specimen to a thermal gradient across one spatial coordinate. The microstructure was analyzed

with a deterministic microstructural characterization technique which was developed especially

for this purpose. The microstructure was then correlated to fabrication parameters and then to

creep compliance. The results showed a strong correlation between the microstructure and creep

20

compliance. Processing-microstructure relations were also established which makes it possible to

fabricate a desired microstructure. These findings motivated the challenge to apply functionally

graded porous structures to demanding applications in industry like impact absorption and

damping. This will be the topic of next chapter.

Chapter 3

This chapter utilized the already developed techniques from the previous chapter to fabricate

functionally graded porous structures of Acrylonitrile Butadiene Styrene (ABS) for impact

energy absorption. The porous structure was induced by thermally activated microspheres

subjected to a thermal gradient. Different processing parameters were studied to correlate the

microstructure and gradient to the fabrication procedures. The developed procedures were

targeted to be scalable to large structures. The microstructure and gradient were then correlated

to impact strength. The processing, microstructure, and impact strength showed strong

correlation. This result proved the potential to tailor impact strength of FG porous structures to a

desired value. The fabricated FG porous structures showed their superiority for energy

absorption over the solid precursors. This high damping performance of FG porous structures is

devalued by the overall weakness of the structure to withstand external loads which make them

prone to damage. This motivated the efforts to include reinforcement in the porous structure to

enhance structural strength and hence get use of the high damping performance without exposing

the FG structure to damage. This is the topic of next chapter.

Chapter 4

The overall weakness of functionally graded porous structures can be alleviated by reinforcement

with strong fibers to produce a FG porous composite structure. This was the focus of Chapter 4.

Polyurethane was selected as the matrix because of high damping performance. Hybrid fabrics of

Kevlar and carbon fibers were used as the reinforcement across planar directions, and ultrahigh

molecular weight polyethylene braids were used to reinforce the out-of plane direction.

Microspheres were used to induce the porous structure within the composite. The effect of

21

microsphere loading and gradient on impact absorption was investigated while keeping the

reinforcement consistent among all the samples. This produced light-weight functionally graded

porous composite structures that have the damping performance of porous structures and the

strength of high performance fibers. This work on fabrication and correlation to microstructure in

Chapters 2, 3, and 4, paved the way to develop numerical procedures that can accurately

homogenize and model functionally graded porous composite structures. This will be the topic

for Chapters 5 and 6.

Chapter 5

A statistical model was proposed in this chapter which accounts for the microstructure

complexity and gradient through control points across the graded direction. A new higher order

plate theory was developed to implement the homogenization model. A higher order plate theory

was derived with stretching effect across the gradient direction. This was necessary for

functionally graded porous structures because of large compressive deformations. A specific

numerical scheme with C1 continuity was developed to account for the gradient across the

thickness. The proposed plate theory and homogenization model agreed well with the

experimental data. The numerical tool proved its validity for conducting accurate computer

experiments for plate-like functionally graded porous structures. The statistical homogenization

model depends on experimental data and needs a wide database in order to work independently

for the design of FG porous composite structures. In order to alleviate this problem, a purely

numerical homogenization procedure was developed which is the topic of next chapter.

Chapter 6

The focus of this chapter was to develop a purely numerical procedure to predict the effective

properties of local porous structures, which in turn can be used in the homogenization model that

was developed in Chapter 5. This step eliminated the need for experimental data which was the

main drawback of the statistical model. The developed numerical technique proposed a reduced

three dimensional reconstruction procedure of the porous/composite structure that is equivalent

22

to the actual structure. Randomness and porosity are the main two parameters of the porous

structure which were preserved. The procedure produced local porous structures that were

subjected to numerical experiments to predict the mechanical properties like the relaxation

modulus. The results of this technique agreed well with experimental data and proving its

validity to conduct accurate numerical experiments. The output from this numerical procedure

can replace the experimental data for the homogenization model which was developed in

Chapter 5. This makes the homogenization model independent of experimental database. The

model can therefore directly be integrated with a macroscale analysis of functionally graded

porous composite materials.

1.6 Contributions

The main contribution of this thesis is the development of an accurate tool for guiding the design

of functionally graded porous structures and composites. This includes both, the fabrication and

modeling of these materials. New fabrication procedures were developed and correlated to the

resulting microstructure. This enables the fabrication of a desired microstructure with pre-

defined features like porosity, cell sizes, and gradient. The fabrication methodology was

supported by new numerical procedures, which are able to conduct accurate numerical

experiments. The developed homogenization model accounts for gradients in properties with a

tolerable number of control points, which is decided based upon the gradient nature in both, the

material properties and microstructure. The main contributions can be summarized as follows:

Development of fabrication procedures and guidelines for functionally graded porous

structures. The microstructure was correlated to process parameters which facilitates the

production of structures with desired features.

Developing and validating a statistical homogenization model that accounts for

microstructural and property gradient with tolerable accuracy.

Development of a higher order plate theory with stretching effect that accounts for the

through-thickness deformation of porous structures.

23

Correlating microstructure to creep compliance and impact energy absorption of

functionally graded structures.

Development of graded stitched composites that provide the damping of graded

structures and the structural strength of high performance fibres.

Adapting a local thresholding technique for microstructure characterization of porous

structures that is deterministic and independent of the analyst.

Developing and validating three-dimensional reconstruction of the reduced

microstructure that accurately predicted the mechanical properties of functionally graded

porous materials.

1.7 References

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24

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(1) The content of this chapter has been published in the Journal of Polymer Science Part B:

Polymer Physics;

F. Al Jahwari, A. A. W. Anwer, and H. E. Naguib, "Fabrication and microstructural characterization of functionally

graded porous acrylonitrile butadiene styrene and the effect of cellular morphology on creep behavior," Journal of

Polymer Science Part B: Polymer Physics, vol. 53, pp. 795-803, 2015. DOI: 10.1002/polb.23698. Reproduced by

permission from John Wiley and Sons (License # 3902061268629).

Chapter 2

Fabrication and Microstructural Characterization of Functionally

Graded Porous Structures with Correlation to Creep Behaviour (1)

This Chapter starts the research by investigating the processing-microstructure-property

relationships of functionally grade porous structures. The investigation included the study of the

potential of tailoring mechanical properties by controlling the porous gradient. Functionally

graded porous structures of Acrylonitrile butadiene styrene (ABS) were fabricated by a solid-

state constrained foaming process. Correlating the microstructure to material properties requires

a deterministic analysis of the cellular structure. This was accomplished by analysing SEM

images with a locally adaptive image threshold technique based on variational energy

minimization. This characterization technique of the cellular morphology is analyst independent

and is suited for porous structures. Inferences were drawn from the effect of processing on

microstructure and then correlated to creep strain and creep compliance. Creep was strongly

correlated to porosity and pore sizes but more associated to the size than to porosity. The results

showed the potential of controlling the cellular morphology and hence tailoring creep

strain/compliance of ABS to desired values.

29

Nomenclature

FGM Functionally graded Material

FGPS Functionally graded porous structures

∅ Average porosity

d Average pores diameter

TH Higher platen’s temperature

TL Lower platen’s temperature

TStage-1 Temperature of the foaming vessel

tannealing Annealing time between the platens

P Pressure of stage-1 foaming

2.1 Introduction

Functionally graded materials are advantageous over composite materials with the continuous

microstructure and gradual change of properties. This continuity eliminates problems like stress

jumps and delamination which are major issues with composite materials. When representing the

material properties with continuous functions in space and/or time, FGMs become more

favorable for numerical procedures than conventional composites. Most of the focus had been on

metallic-ceramic functionally graded materials and particulate-base functionally graded

composites. However, little attention had been paid by the scientific community to functionally

graded porous structures which is the focus of this paper. In addition to preserving the advantage

of continuous microstructure and gradual change of properties that FGPS provide, it offers a

significant weight reduction comparable to the fully porous structures, and reasonable strength as

compared to the precursor foaming material. This excellent weight-to-strength ratio is due to the

balance between porous structure and solid phase of the material that is only achievable by

functionally graded materials. In a previous work of the authors [1], functionally graded porous

polylactic acid shows higher values for the relaxation modulus compared to the uniform porous

structure of PLA plates which is due to the larger portion of solid base material. The drop in

relaxation modulus for the functionally graded PLA is 8.44% only while it is 46.72% for the

uniform porous structure. The drop in density for the FG PLA is 27% while the drop in

30

relaxation modulus is 8.44% only. In comparison for the uniform porous PLA, the density drop

is 29.06% compared to 46.72% drop in the modulus.

Acrylonitrile butadiene styrene is an amorphous copolymer alloy where acrylonitrile provides

the chemical resistance and thermal stability, butadiene for toughness, and styrene for ease of

processing [2]. ABS has been mass-produced with wide range of applications since 1960s

because of its interesting properties, chemical resistance, ease of processing and foaming. Many

reviews had been published about ABS [3] and its foam structures [4]. High density cellular

structures of ABS can be used as a replacement of solid ABS as a weight saving technique while

keeping the overall effective properties within acceptable range. Fabrication techniques of

cellular materials include solvent particulate leaching [5, 6], gas foaming [6-8], co-continuous

melt blending [9-14], and rapid prototyping [15-17]. Murray et al. [2] produced microcellular

ABS using CO2 as the physical blowing agent in a solid-state foaming process. Closed-cell ABS

foams have been produced with a wide range of densities and cell sizes to investigate the effect

of different processing parameters. Increasing the concentration of CO2 in ABS is a determining

factor for the attainable densities with solid-state foaming process. Gandhi et al. [18] used

ultrasound-induced nucleation technique as an alternative to high pressure vessels for enhancing

the concentration due to its incurred high cost. They could prove the significance of sonication

on the foaming process. However, the combination of those techniques lead to more control of

the pores growth [7, 13], and better mechanical properties [6]. Nadella et al. [7], fabricated a

closed-cell functionally graded polylactic acid by annealing gas foamed samples with a

temperature gradient rather than quenching in a bath of uniform temperature. Lin et al. [19]

foamed ABS with a skin layer near the surface by controlling the foaming temperature.

The behaviour of porous structures can be related to the cellular morphology [20] which

necessitates a deterministic analysis of the microstructure. Analysing the images of scanning

electron microscopy (SEM) is one the common means of analysing cellular morphology. SEM

images can effectively be analysed by thresholding and converting to binary format.

Thresholding is classified into global and local techniques. The global thresholding technique

binarizes an image based on a global threshold value where pixels of greater value are assigned 1

31

and others as 0. The most popular technique of this class is Otsu where the optimal global

threshold value is determined by maximizing the between–class variance with an exhaustive

search [21]. However, the global thresholding technique fails to capture the cellular morphology

because of poor illumination and overlapping pixels of SEM images. On the other hand, local

thresholding generates a surface that is a function of the image domain and depends on the local

image characteristics which makes it superior over global thresholding [22]. Different local

thresholding algorithms are available in literature but most of them need some tuning parameters.

This makes the morphological analysis of porous structures very dependent on analyst judgment

about those parameters which induces a lot of variability in characterizing the same porous

structure. This in turn reduces the possibility of an effective utilization of cellular structure for

analysis purpose. Accurate representation of the microstructure can be utilized to develop a

representative unit cell (RUC). The RUC can then be used to predict the mechanical properties

from the knowledge of solid foaming material only based on mechanics homogenization

approach. Ray et al. [22] proposed an automated locally adaptive image thresholding technique.

This technique is based on minimizing an energy functional of the thresholding surface through a

variational Minimax algorithm. The authors proposed a non-linear convex combination of the

data fidelity and the regularization terms in the energy functional and seek the optimum

threshold surface via minimax principle. This makes the method works in automatic manner

without any tuning parameters unlike most of the available methods to date. This technique is

adopted in this work.

In this chapter, functionally graded porous structures of ABS are fabricated with a solid-state

constrained foaming process. The cellular morphology is analysed with the proposed technique

in [22] and inferences are drawn to link it with the processing parameters and creep

strain/compliance. The results show the possibility of tailoring the cellular structure and hence

tailoring the creep compliance.

32

2.2 Material and Processing

Pure acrylonitrile butadiene styrene is in sheets from McMaster-Carr with thickness of 1.59 mm

(1/16 inches). The pre-foamed samples are made with dimensions of 11.3 mm x 32 mm to fit the

high pressure/temperature foaming chamber. The foaming is done in two stages. The aim of first

stage is to seed small pores at temperatures below the glass transition of ABS (Tg = 105 oC). The

pores are being expanded in the second stage with variable sizes through the thickness direction

by exposure to a thermal gradient between high temperature side (TH) and low temperature side

(TL). In first stage, the samples are subjected to high pressure, P, 2000 psi (13.80 MPa), 1800 psi

(12.41 MPa), 1600 psi (11.03 MPa), 1400 psi (9.65 MPa), and 1200 psi (8.27 MPa) and

temperatures, TStage-1 , below Tg (1/2 Tg, 2/3 Tg, and ¾ Tg) then rapidly release the pressure to

nucleate pores. In the second stage, the samples are annealed for a time, tannealing, (1 minute, 2

minutes, 5 minutes, and 10 minutes) between the two hot platens to establish a thermal gradient

(From 0 oC to 72.5

oC with an increment of 5

oC) and allow pores to expand with different

ratios. This procedure allows for a porous ABS with changing cellular structure through the

direction of thermal gradient. Fig 2.1 shows a schematic of the two stages.

Fig 2.1. Schematic of the foaming technique of functionally graded ABS.

33

2.3 Characterization of the Functionally Graded Porous Structures

2.3.1 Microstructural Characterization

Characterizing the microstructure aims to identify cellular distribution and porosity. That is

achieved by analyzing the SEM images of foamed ABS samples across the thickness. Two

samples are made for each experiment. One of the two samples is frozen in liquid Nitrogen for

some time, fracture by shearing, and finally sputter coated with platinum before imaging under

the scanning electron microscope (JEOL JSM 6060). The samples were fractured at three

different planar locations for averaging purpose. Based on a previous study by the authors [1],

three locations are selected across the thickness as control points to capture the gradually

changing microstructure. One of each is near the lower temperature platen (TL), the second is at

the middle region, and last point is near the platen of higher temperature (TH).

The identification of cell boundaries is done with the local thresholding method proposed by Ray

et al. [22]. The authors proposed a variational energy functional consisting of a non-linear

combination of a data and regularization term. The energy functional is a function of the

threshold surface, the image, as well as a weighting parameter, which makes a balance between

the data and the regularization terms so it prevents one of them dominating the other. A minimax

solution of the proposed energy functional is obtained iteratively by alternating minimization and

maximization of the energy functional respectively with regard to the threshold surface and the

weighting parameter.

If the grayscale SEM image and threshold surface function are denoted by I(x,y) and T(x,y)

respectively. Then the energy functional can be defined as [22]:

𝐸(𝑇; 𝛼) = √1 − 𝛼2𝐸1(𝑇) + 𝛼𝐸2(𝑇) (2.1)

34

where 𝛼 ∈ [0,1] is the weighting parameter, E1 is edge sensitive energy term that encourages the

threshold surface T to intersect the image surface I, E2 is the regularization term with L2 norm

that enforces smoothness in the threshold surface.

The contrast of raw grayscale SEM image is enhanced by the Matlab function “adapthisteq”

which transforms the values using contrast-limited adaptive histogram equalization. This contrast

enhancement technique eliminates artificially induced boundaries and avoid amplifying any

noise that might be present in the image. The adjusted image is then binarized with the algorithm

proposed by Ray et al. . The image is ready now for pores recognition which is accomplished by

the Matlab function “bwboundaries”. This function traces the exterior boundaries of objects, as

well as boundaries of holes inside these objects, in the binarized image. The option “hole” in this

function is selected to do the cleaning process for internal impurities of the identified pores.

Porosity (∅) is defined as the ratio between pores volume to the total volume of the considered

sub-volume. In this analysis, the white and black areas represent pores and boundaries

respectively. Consequently, porosity is obtained by calculating the ratio of white pixels to the

total image size. This process is repeated for three different random locations at the same planar

surface. The reported porosity is then the average value of the three porosities as will be

explained later in this section. Calculation of average pores’ diameter (𝑑) is accomplished by

obtaining the area of each pore with the Matlab function “regionprops” that measures the area of

each connected component (pore) in the binarized image. The pore diameter is then calculated

with a suitable conversion scale factor and assuming a circular pore. As for porosity, the

diameter is extracted for the three locations and averaged.

Each sample is fractured at three different locations along the length. For each fractured surface,

the porosity is measured as mentioned above. The porosities are first averaged at planar sections

and then over the thickness. The accuracy of this averaging procedure was discussed in a

previous work of the authors for porous structures made of polylactic acid (PLA) [1]. The

effective density of porous structure (𝜌∗) can be predicted by (1- ∅)𝜌𝑠, where 𝜌𝑠 is the density of

base foaming material (1.24 g/cm3 for PLA). The SEM images were analyzed for porosity as

35

explained which reveals a value of 0.696 for ∅ and hence 𝜌∗ is 0.863 g/cm3. The porous structure

density was determined experimentally with the “Density Determination Kit” by Denver

Instruments and it gave a value of 0.880 g/cm3. This gives an error of 1.94% only which is very

small compared to the complexity of microstructure. The accuracy will increase as the number of

sampling points in ∅ increase. The same microstructural characterization technique was

implemented in a homogenization model for FG porous PLA structures that predicts the

relaxation modulus [1]. The homogenization model is a modified formula of Prony series for

viscoelastic materials and heavily depends on parameters that are extracted from microstructural

analysis. The formula predicted the relaxation modulus with 1.6% error compared to

experimental data which is primarily due to the accuracy of microstructural characterization

technique.

The different image processing steps are shown in Fig 2.2. At the first step, the image is

binarized with the proposed procedure as shown in Fig 2.2b. Then the Matlab functions

“bwboundaries” and “regionprops” do the edges identification (Fig 2.2b) and further cleaning of

the identified cells (Fig 2.2c) respectively. This algorithm works very well in identifying the

cellular structure without any tuning parameters which eliminates any source of variability.

a. Original SEM image b. Binarized image with the

algorithm proposed in [22]. c. Clean SEM image.

Fig 2.2. Image processing steps for calculations of porosity and pores’ diameters.

36

2.3.2 Creep Test and Correlation

Instron Micro-tester (Model 5848) is used to run creep tests. A constant force of 50 N is applied

and then measuring deformation with time for 300 seconds. The creep results presented in this

work are not meant to characterize the foamed samples but are rather utilized to compare the

effect of cellular structure on creep behaviour. Following ASTM D 2990 and the Test Method D

638 in terms of dimensions was very difficult due to size limitation. However, all other

requirements are satisfied as per the aforementioned ASTM standards.

Linear correlation coefficient is calculated between porosity and pore diameter to the creep strain

and creep compliance. The correlation coefficient can determine the strength and nature of the

relationship among all the four quantities. This coefficient will be used to draw inferences about

the strength of relationship and introduce a new parameter for the correlation that combines both,

porosity and pores sizes. The equation for the correlation coefficient is [23]:

𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 (𝑥, 𝑦) =∑(𝑥 − �̅�)(𝑦 − �̅�)

√∑(𝑥 − �̅�)2 ∑(𝑦 − �̅�)2 (2.2)

where x and y represent the two quantities for which the correlation coefficient to be calculated,

�̅� and �̅� are the averages of x and y respectively.

2.4 Results and Discussion

2.4.1 Porosity and Pore Diameter

The gradually changing porous structure is achieved successfully with the abovementioned

procedure in section 2.2. The average porosity and average pores diameter show increasing trend

when fixing TL but increasing TH from 105 oC to 125

oC with an increment of 5

oC as shown in

Fig 2.3. That is observed for all the three temperatures used for pore seeding at stage 1.

37

Fig 2.3. Effect of TH on average porosity for different values of TL and TStage-1 with pressure of

2000 psi and 2 minutes annealing (𝜎𝑎𝑣𝑒 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛).

The standard deviations are relatively large for pores diameter due to the disparity of cells across

each planar dimension since foaming is constrained along the thickness direction only. The

increasing porosity with increasing TH is gradual when TL is the same as TStage-1 but tends to be

sharp when TL equals to the glass transition of ABS. The same trend is observed for pores

average diameter. The gaps between the three curves are due to the temperature of low side TL.

The difference in TL between the TStage-1 70 oC and 78.75

oC is only 8.75

oC but it is 17.5

oC for

the case of 52.5 oC. The difference manifests more at higher values of TH where cells expand

more between the hot platens which are both equal or above glass transition of ABS.

Fig 2.4 shows SEM images for the case of TH = 125 oC, TL = 52.5

oC, tannealing = 5 minutes, P =

2000 psi, and TStage-1 = 52.5 oC. The porosity is gradually increasing in this case from the low to

high temperature platens. This is not the case with all samples. However, the pores diameter is

always smaller at the lower temperature side (TL) compared to the diameter at the other side (TH).

In Table 2.1, a sample of numerical data is presented for different annealing durations. This

result is expected since the cells near hot platen can expand more than those near the low

temperature platen. The cellular structure at TL side is always uniform with well-defined cells.

38

However, the cells at the side of TH are more irregular with some of them collapsing and merge

with neighbouring cells. Fig 2.5 shows this for the case of TH = 135 oC, TL = 70

oC, tannealing = 2

minutes, P = 2000 psi, and TStage-1 = 70 oC.

∅ = 50.422

d = 3.54 𝜇𝑚

∅ = 54.337

d = 4.43 𝜇𝑚

∅ = 55.848

d = 4.44 𝜇𝑚

Fig 2.4. The cellular distribution of a functionally graded ABS under TH = 125 oC, TL = 52.5

oC,

tannealing = 5 minutes, P = 2000 psi, and TStage-1 = 52.5 oC.

39

Table 2.1. Effect of annealing temperature and time on the pores diameter

Stage 1 Stage 2 Pore Diameter

(𝝁𝒎)

T

(oC)

P

(psi)

tannealing

(minutes)

TH

(oC)

TL

(oC)

TL

Side

Middle

region

TH

Side

52.5 2000 2 110 105 4.02 4.41 4.62

125 105 3.68 5.241 6.14

5 105 52.5 5.04 5.30 5.41

10 125 52.5 3.42 5.02 6.11

70 2000 2 110 70 3.78 3.93 4.27

125 70 3.94 4.91 5.0

a) Low temperature region

∅ = 52.030

d = 3.51 𝜇𝑚

b) Middle region

∅ = 59.542

d = 4.40 𝜇𝑚

c) High temperature region

∅ = 61.397

d = 5.11 𝜇𝑚

Fig 2.5. Cellular structure of the case: TH = 135 oC, TL = 70

oC, tannealing = 2 minutes, P = 2000

psi, and TStage-1 = 70 oC.

The temperature at stage-1 has a significant effect on the initial porosity and hence the overall

functionally graded structure, but very minor effect on pore sizes. Fig 2.6 shows the

microstructure of the three TStage-1 temperatures. The increase in porosity is 77% when seeding at

40

70 oC compared to 52.5

oC while it is only decreasing by 9% if seeding at 78.75

oC at the same

pressure. This further affects the functionally graded structure at the annealing stage.

a) 52.5 oC

∅ = 12.282

d = 1.97 𝜇𝑚

b) 70 oC

∅ = 53.065

d = 1.73 𝜇𝑚

c) 78.75 oC

∅ = 58.451

d = 1.70 𝜇𝑚

Fig 2.6. The effect of stage-1 temperature on the cellular structure at 2000 psi pressure.

However, the effect of stage-1 pressure is very significant on the initial cellular structure.

Porosity reduces from 53.065 for 2000 psi (Fig 2.6b) to only 13.148 for 1400 psi (Fig 2.7c). No

porous structure is observed at 1200 psi except few large pores at the boundaries. For the case of

1400 psi, the middle region has a porosity of 8.761 only with pores average diameter of 35.93

𝜇𝑚 and far distant from each other.

a) 1800 psi

∅ = 51.942

d = 5.18 𝜇𝑚

b) 1600 psi

∅ = 23.873

d = 15.21 𝜇𝑚

c) 1400 psi

∅ = 13.148

d = 26.35 𝜇𝑚

Fig 2.7. The effect of pressure at stage-1 on the cellular structure at 70 oC.

41

2.4.2 Creep Behaviour

Creep tests were performed for all the samples as described in section 2.3.2. The output of

interest from creep test is the creep strain and creep compliance. The cellular structure is

characterized by porosity and average cellular size which can be related to creep behaviour.

Functionally graded porous structures provide a control over the porosity and pores diameter

which in turn can control creep behaviour. Correlation coefficients are calculated to measure the

degree of association between porosity, pore diameter, creep strain, and creep compliance which

are presented in table 2.2 for different combinations. Creep properties correlate better to pore

diameter than to porosity. A new parameter, ∅𝑑2, was introduced which is a representation of

porous area of the solid. The objective of introducing this parameter is combining porosity and

pore diameter together as a design parameter since both of them can be controlled.

The results in section 2.4.1 prove the ability to control ∅ and d and hence the creep properties

can be tailored. Fig 2.8 shows the effect of average porosity and average pore diameter of the

functionally graded structure on creep strain. Creep strain is increasing with increasing porosity

and pore diameter. Similar observation is made by the authors [1] but with decreasing relaxation

modulus for increasing porosity of functionally graded porous structures of polylactic acid.

Having smaller pores and low porosity tends to preserve more of the material strength but

damping is achieved with high porosity. Controlling the amount of solid material, porosity, and

pore diameter provides a control over the material properties. Controlling porosity and pore sizes

in position is even more advantageous. For example, in the case of designing a structure for

impact loading, it is required to keep a solid layer of the material facing the load which is

followed by a porous layer of the same material. The solid layer receives and transfers the load to

the underneath porous layer which acts as a damper. This provides the damping characteristic of

functionally graded structures. Higher porosities of the porous layer can absorb more energy by

allowing more deformation as indicated in Fig 2.8.

42

Table 2.2. Correlation coefficients for porosity and diameter against creep strain and compliance

Steady state creep strain Steady state creep compliance

∅ 0.895 0.896

𝑑 0.951 0.943

∅𝑑2 0.960 0.948

A numerical procedure for the analysis of plate-like functionally graded structures had been

developed by the authors in their previous work [1]. The numerical procedure was customized

for polymeric functionally graded porous structures and compares very closely to experimental

findings. It can be implemented in an inverse design procedure where the required creep strain or

creep compliance can be mapped for porosity and pore diameter from Figs 2.9a, 2.9b, or 2.9c.

Using the mapped value jointly with Fig 2.3, one can predict the required processing parameters

for ABS in this case. Thus, it can be decided about the required material properties first from the

numerical tool, then map the equivalent porosity and pore diameter from design curves as in Fig

2.9, finally the processing parameters can be obtained from Fig 2.3. From the results in section

2.4.1, it can be inferred that there are different paths which can be followed for fabricating a

functionally graded porous structures with required average porosity and/or average diameter.

Developing those processing curves as in Fig 2.3 and material property curves as in Fig 2.9 will

open the doors for tailored material design of functionally graded porous structures.

Fig 2.8. The effect of average porosity and diameter on creep strain for tannealing = 2 minutes, P =

2000 psi, TStage-1 = 78.75 oC and different hot side temperature (TH)

43

Steady state creep strain Steady state creep compliance (1/MPa)

Fig 2.9a. Steady state creep strain and compliance for different porosities, pore diameters and

TStage-1 = 52.5 oC


Fig 2.9b. Steady state creep strain and compliance for different porosities, pore diameters and

TStage-1 = 70 oC


Fig 2.9c. Steady state creep strain and compliance for different porosities, pore diameters and

TStage-1 = 78.75 oC

Porosity

Po

re d

iam

ete

r (

m)

0.56 0.57 0.58 0.59 0.6 0.614.2

4.4

4.6

4.8

5

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Porosity

Pore

dia

mete

r (

m)

0.56 0.57 0.58 0.59 0.6 0.614.2

4.4

4.6

4.8

5

0.03

0.035

0.04

0.045

0.05

0.055

Porosity

Po

re d

iam

ete

r (

m)

0.535 0.54 0.545 0.55 0.5553.9

4

4.1

4.2

4.3

4.4

4.5

4.6

0.012

0.014

0.016

0.018

0.02

Porosity

Pore

dia

mete

r (

m)

0.535 0.54 0.545 0.55 0.555

4

4.2

4.4

4.6

0.012

0.013

0.014

0.015

0.016

0.017

Porosity

Pore

dia

mete

r (

m)

0.545 0.55 0.555 0.56

3.6

3.8

4

4.2

0.0125

0.013

0.0135

Porosity

Po

re d

iam

ete

r (

m)

0.545 0.55 0.555 0.56

3.6

3.8

4

4.2

0.0114

0.0116

0.0118

0.012

0.0122

0.0124

44

2.5 Conclusion

Functionally graded porous structures of acrylonitrile butadiene styrene ABS were fabricated

with a constrained solid-state foaming process. The ABS samples are first seeded with pores in a

one-stage foaming process and then annealed between two hot platens of different temperatures

to induce a thermal gradient. The induced thermal gradient across thickness makes the pores to

expand with different ratios. Controlling the processing parameters (TStage-1, pressure, annealing

temperatures and duration) leads to controlling the cellular morphology. Since linking the

microstructure to material properties necessitates an effective tool for analysing the cellular

structure, a locally adaptive image threshold technique was used which is based on variational

energy minimization. The thresholding technique works very well in capturing the cells

boundaries and hence calculating porosity and pore sizes. Creep tests were run for the different

microstructures and correlated to creep strain and creep compliance. Porosity and pores sizes are

both strongly correlated to the creep properties. However, pore sizes are more correlated than

porosity. Combining the fabrication and characterization in this work to a previously developed

numerical procedure for similar structures, provides a design tool for tailoring functionally

graded porous structures to different design requirements.

2.6 References

[1] F. A. Jahwari and H. E. Naguib, "Linear viscoelastic modeling and validation of

functionally graded heterogeneous porous PLA structures with a C1-continuous plate

theory and novel homogenization," presented at the Foams, New Jersey, USA, 2014.

[2] R. E. Murray, J. E. Weller, and V. Kumar, "Solid-state microcellular acrylonitrile-

butadiene-styrene foams," Cellular Polymers, vol. 19, pp. 413-425, 2000.

[3] J. D. Moore, "Acrylonitrile-butadiene-styrene (ABS) - a review," Composites, vol. 4, pp.

118-130, 1973.

[4] J. Zappala, "Structural foam ABS," Journal of Cellular Plastics, vol. 7, pp. 309-312,

1971.



2000.

45













Reviews, vol. 43, pp. 87-141, 2003.



2611-2617, 2004.



pp. 1954-1964, 2010.



vol. 25, pp. 2161-70, 2004.











2005.

[18] A. Gandhi, N. Asija, H. Chauhan, and N. Bhatnagar, "Ultrasound‐induced nucleation in

microcellular polymers," Journal of Applied Polymer Science, vol. 131, pp. 9076-9080,

2014.

46

[19] G.-G. Lin, D.-J. Lin, L.-J. Wang, and T.-W. Kuo, "Absorption and foaming of plastics

using carbon dioxide," Research on Chemical Intermediates, vol. 40, pp. 2259-2268,

2014/07/01 2014.



[21] N. Otsu, "A threshold selection method from gray-level histogram," IEEE Transactions

on Systems, Man, and Cybernetics, vol. 8, pp. 62-66, 1978.

[22] N. Ray and B. N. Saha, "Edge sensitive variational image thresholding," presented at the

Image Processing, IEEE International Conference, 2007.

[23] M. Office, "Microsoft excel help," ed, 2010.

(1) The content of this chapter has been published in the Journal of Polymer;

F. Al Jahwari, Y. Huang, H. E. Naguib, and J. Lo, "Relation of impact strength to the microstructure of functionally

graded porous structures of acrylonitrile butadiene styrene (ABS) foamed by thermally activated microspheres,"

Polymer, vol. 98, pp. 270-281, 2016. DOI: 10.1016/j.polymer.2016.06.045. Reproduced by permission from

Elsevier (License # 3902080016968).

Chapter 3

Relation of Impact Strength to the Microstructure of Functionally

Graded Porous Structures of Acrylonitrile Butadiene Styrene

(ABS) Foamed by Thermally Activated Microspheres (1)

In Chapter 2, the hypothesis of the existence of relationship between fabrication, morphology,

and mechanical properties was proved. This motivated the challenge to apply functionally graded

porous structures to demanding applications in industry like impact absorption and damping.

This Chapter utilized the already developed techniques in Chapter 2 to fabricate functionally

graded porous structures of Acrylonitrile Butadiene Styrene for impact energy absorption. The

functionally graded porous structures of ABS were fabricated with thermally activated

microspheres. One-dimensional heat flow was introduced across the thickness with different

terminal temperatures to induce a thermal gradient. Different compositions and processing

conditions were carried out to investigate the relationship between impact energy and

microstructure. Impact energy showed a stronger correlation to the pore diameter gradient than to

porosity. The correlation strength between diameter gradient, permitted expansion ratio, and

porosity indicates the potential to control the microstructure and hence impact energy absorption.

Functionally graded porous structures of ABS demonstrated their superiority for impact

absorption with a strength-to-weight ratio of 46.02 J.cm3/g compared to 25.71 J.cm

3/g for solid

ABS. This work provides processing guidelines to fabricate FG porous structures of ABS in

relation to impact energy.

48

3.1 Introduction

Functionally graded porous structures can offer significant weight reduction comparable to that

offered by homogeneous porous structures while maintaining tailorable strength compared to the

solid precursors [1]. One of the attractive applications of polymers is impact resistance and

energy absorption due to their viscoelastic nature that helps in dissipating energy in addition to

their light weight compared to metals and ceramics. Introducing pores to the material system,

helps in further weight reduction and energy absorption through large compressive deformation.

Moreover, fabricating porous structures of polymers is easier and cost effective compared to

metals and ceramics.

The porous structures of polymers are fabricated with physical or chemical blowing agents. Both

techniques induce gases such as CO2 as a second phase in the polymer or encapsulated in a shell.

In the case of physical blowing agents, the gas is induced by means of pressure and temperature.

However, chemical blowing agents (CBAs) induce the gas by decomposition. Foaming with

chemical blowing agents is either done with direct contact between the produced gases and the

polymer or the CBA is encapsulated in microspheres that expand up on heating. In a previous

work of the authors [1], functionally graded porous structures of ABS were fabricated by

physical blowing agent, and the microstructure was successfully correlated to creep strength.

However, scalability of the foaming process with physical blowing agents is an issue because of

infrastructure requirement in addition to the long time needed for the foaming process. Chemical

blowing agents are an ideal solution for scalability but compatibility with the precursor polymer

limits generalization of such technique. Microspheres that encapsulate the CBA are very

promising technology and produce more uniform cellular structure than those of the other type of

chemical blowing agents.

Polymeric based materials for impact resistance can be classified to five main categories;

polymeric blends [2], Multiscale composites [3], conventional composites [4], graded porous

structures [5], and sandwich structures with porous core [6]. Porous structures perform well for

high impact resistance and energy absorption due to the large compressive deformation [7]. The

49

foamed polycarbonate exhibited dramatic increase in impact strength compared to the unfoamed

polymer [8]. Miller and Kumar [9] showed experimentally that the nano-porous structures of

polyetherimide provide an improvement in impact strength by up to 350% compared to micro-

scale porous structures. Isotropy and oriented cellular structures of polystyrene were fabricated

by Jin-Biao et al. [10] to study the effect of morphology on impact strength. Cell morphologies

oriented perpendicular to the impact direction could significantly enhance the toughness of

polystyrene foams. The impact strength of grafted nano-SiO2/polypropylene composites was

enhanced by introducing pores into the microstructure [11]. The addition of nanofillers like

carbon nano-fibers could enhance the formation of pores but make the porous structure more

rigid which reduces impact strength [12].

Exposing porous structure directly to the impact load causes unrecoverable damage to the cell

walls. Including the porous structure between solid sheets is one of the solutions to this problem.

Sachse et al. [13] fabricated sandwich panels with nanoclay-filled polyurethane foams as the

core; and glass fiber reinforced polyamide and polypropylenes as face sheets. The addition of

nanoclay in the polyurethane foam core improved both energy absorption and maximal

deflection during impact. The improvement in energy absorption was between 66% and 92% for

polypropylenes face sheet sandwiches and between 23% and 34% for the polyamide face sheet

sandwiches. An increase in the compression modulus of 20–37% was recorded for the sandwich

panels with polyamide face sheets. Rajaneesh et al. [14] fabricated sandwich plates with cores

made of aluminum alloy foam and polyvinyl chloride foam while the face sheets were made of

aluminum only. The impact absorption of the polymeric core foam made of polyvinyl chloride

exceeded the sandwich made of aluminum alloy foam by 45%.

Graded porous structures are found to provide additional advantage of retaining comparable

strength to the solid precursors [15] which is lacking in fully porous structures. Functionally

graded aluminum foam with linear decreasing density gradient was found to possess excellent

performance in energy absorption [16]. A range of linear PVC, crosslinked PVC and PEI foams

bonded together to produce a three layer core were fabricated by Zhou et al. [6]. It was

concluded that the graded structures out-perform a single type foam structure. One important

50

result from their work suggested placing the high density foam core against the top surface skin

for an improved perforation resistance. Zeng et al. [17] developed sandwich plates with face

sheets made of aluminum alloy while the cores were made of graded polymeric hollow sphere

composite with various density gradients. The authors concluded that the best gradient profile in

terms of the energy absorption and the non-sheet breaking criterion should be a weaker first layer

and a progressively enhanced core. The two contradicting findings may suggest that the impact

behavior of graded sandwich porous structures is system dependent and need more investigation.

Thermally activated microspheres have attracted foaming researchers for the last decade, and

being already applied in shoe industry [18]. Functionally graded polyethylene foams were

developed by Barzegari et al. [19-21] with a thermal gradient. By comparing different

temperatures, and different grades of the microspheres, the authors concluded that the increase of

processing temperatures of linear medium density polyethylene (LMDPE) leaded to decrease in

density. The thermal gradient was also reported to decrease density [19, 20]. The lowest density

of foamed LMDPE was as small as 50% of the solid samples. In addition, graded porous

LMDPE specimens showed a slight decrease in tensile strength compared to the solid material.

However, the decrease in density of the foamed samples was also significant [21].

Polymeric blends are excellent for impact resistance but the weight is an obstacle for further

applications where strength-to-weight ratio has to be very high for industries like automotive and

aerospace. Conventional composites have limitations such as delamination which causes

catastrophic failures. Multiscale composites, graded porous structures, and sandwich structures

are the future materials for ultra-light weight systems with very high impact strength and

toughness. The focus of this chapter is on fabrication and impact characterization of functionally

graded porous structures of ABS with thermally activated microspheres. The study investigated

different processing conditions and compositions. Important conclusions about the functionally

graded porous structures for impact strength are drawn from the results. The correlation between

microstructure and processing parameters was established and linked to impact strength.

Contribution of the work is attributed to the process guidelines and procedures to fabricate FG

51

porous structures in relation to impact strength. The microstructure was correlated to processing

conditions and then to impact energy.

3.2 Materials and Testing

The precursor polymer used in this work is acrylonitrile butadiene styrene pellets from

Filaments, Canada (Magnum 3404 from Dow Chemical Company). The density is 1.04 g/cm3

and softens at 101 oC. The thermally activated microspheres are Expancel grade 980 DU 120

from Akzonobel, USA. Grade 980 DU 120 is supplied as light yellow powder with an initial

particle size range of 25-40 μm, and density up to 14 Kg/m3. The activation point of this grade

ranges from 158 to 173 oC and maximum yield at 215 to 235

oC. These microspheres comprise a

thermoplastic shell encapsulating low boiling point liquid hydrocarbon. When heated to a

temperature high enough to soften the thermoplastic shell, the increasing pressure of the

hydrocarbon will cause the microsphere to expand. Generally, the volume of those microspheres

can increase from 60 to 80 times.

There are various types of impact tests available for finding the impact energy of different

materials. The drop weigh impact test is adopted for this study according to ASTM D5420-10. In

the drop weight impact testing, a mass is raised to a known height and released to impact the

specimen. Twenty specimens were tested for each material. The dropped mass, dropping height,

and specimen condition after impact were recorded for each specimen. The specimen was

considered failing if cracks occur after impact at the top or bottom faces only. The mean failure

energy or impact energy was calculated by E = h*w*f, where h = mean failure height, w =

constant mass, Kg (or Ib), and f = factor for conversion to joules (0.11299). The constant mass

used for testing was 3.628 Kg (8 Ib).

Samples density was measured by the “Density Determination Kit” from Denver Instruments.

The operational manual followed is version “902216.1 Rev B” for “301377.1” model. The

52

procedure is based on Archimedean principle by immersing the sample in water and measuring

the specific gravity of the sample using the following formula;

𝑝 =𝑊(𝑎) ∙ (𝑝(𝑓𝑙) − 𝑝(𝑎))

0.99983 (𝑊(𝑎) − 𝑊(𝑓𝑙))+ 𝑝(𝑎) (3.1)

where;

p(a): Density of Air

p: Specific gravity of the sample

p(fl): Density of water

W(a): Weight of the sample in air

W(fl): Weight of the sample in water

Fourier Transform Infrared Spectroscopy (FTIR) analysis was used to determine the ratios of

acrylonitrile, butadiene, and styrene to further identify the ABS used in this work. This is

because different suppliers may have different ratios depending on the application. The

procedure proposed by Fischer et al. [22] was followed to identify the ratios. Five samples of

ABS were analysed and averaged (Fig 3.1).

53

Fig 3.1. FTIR analysis of ABS

The procedure is based on the ratios of absorbance at four different wavenumbers, 1602 cm-1

(the

monomer specific absorbance of styrene), 1659 cm-1

, 2242 cm-1

(the monomer specific

absorbance of acrylonitrile) and 3065 cm-1

. From Fig 3.1, the absorbance values are 0.053,

0.035, 0.025, and 0.025 at 1602 cm-1

, 1659 cm-1

, 2242 cm-1

and 3065 cm-1

respectively. So the

ratios 1602/1659 and 2242/3065 are 1.513 and 0.987 respectively. Fischer et al. [22] provided

experimental data where the ratios of acrylonitrile, butadiene, and styrene can be mapped from

the absorbance ratios 1602/1659 and 2242/3065. Mapping from those data revealed 39.9% for

butadiene, 23.9% for styrene, and 36.2% for acrylonitrile.

3.3 Fabrication Procedure

The required specimen for impact test was fabricated according to ASTM D5420-10 with

compression moulding at temperatures below the activation point of the microspheres. The

specimen was then exposed to thermal gradient for 3 minutes while restraining expansion at this

stage. The 3 minutes heating prior to expansion was selected to preserve the thermal gradient

across thickness but also guaranteeing enough time to melt ABS. Once the 3 minutes passed, the

54

compression mould was released which allowed the specimen to expand. A special mould was

designed for this purpose that is able to control expansion. In this study, a constant force is

applied on the top surface of 3.5 N by mean of weights without putting restriction on the amount

of expansion. The purpose of this is to study the effect of thermal gradient and microspheres

percentage aside from expansion restriction. Fig 3.2 shows a schematic of the fabrication

process.

Prepare the plate of

ABS/Microspheres

under pressure of 0.4

metric ton and

temperature of 155 oC

for 15 minutes

Apply Pressure of 0.4

metric ton and

temperature gradient for

3 minutes

Allow Expansion at room

temperature for 1 minute

Fig 3.2. Schematic of the fabrication process of impact specimens.

The mould configuration in Fig 3.2 causes multi-dimensional heat flow to the specimen which

makes it difficult to define the thermal gradient in addition to random expansion of the spheres.

Fig 3.3 shows the resulted microstructure where pores growth and distribution do not have a

clear trend. Cells excessive expansion and collapse, dominated the microstructure. Thermally

insulating sheets made of semi-rigid polyimide were placed as shown in Fig 3.4 to stop heat flow

from the sides and make it one dimensional from bottom and top surfaces only. The

microstructure improved significantly as will be discussed in section 3.5.

55

Fig 3.3. Effect of thermal distribution on the microstructure.

Fig 3.4. The mould with thermal

insulation placed on the inner

walls.

ABS pellets were pulverized by dipping in liquid nitrogen for five minutes and then grinded.

ABS with pre-defined weight percentage of microspheres was dry-mixed at room temperature

for 5 minutes. An equivalent mixture of 5 mm thick sample was then prepared by compression

moulding at 155 oC for 15 minutes after the temperature stabilized under 0.4 metric ton. That

produced a solid layer of ABS and unexpanded microspheres. Solid layers of 1.58 mm (1/16

inches) made of ABS were placed on top and bottom of the ABS/microspheres plate. The mould

was then placed between the platens with different or equal temperatures to induce the thermal

gradient for 3 minutes once the temperature stabilized and compressed to 0.4 metric tons. Finally

the mould was removed and left outside the compression mould for one minute to allow

expansion of the microspheres before quenching in cold water to stop the expansion process and

stabilize pore sizes. The top platen temperature is designated as TL (lower value temperature) and

the bottom platen as TH (higher value temperature). A total of 27 experiments (Table 3.1) were

performed to study the effect of thermal gradient and microspheres loading on the microstructure

and impact strength. The selected temperatures for TL were 160 oC, 175

oC, and 190

oC while for

TH were 190 oC, 215

oC, and 235

oC. Those temperatures cover the whole range starting from

activation point up to the maximum expansion limit. Microspheres loadings were 10, 20, and 30

weight percentages (wt%).

56

Table 3.1. Foaming conditions of ABS repeated for 10, 20 and 30 wt% of Expancel

Microspheres

TL

(oC)

TH

(oC)

Experiment

#

TL

(oC)

TH

(oC)

Experiment

#

TL

(oC)

TH

(oC)

Experiment

#

160

190 1

175

190 4

190

190 7

215 2 215 5 215 8

235 3 235 6 235 9

3.4 Morphological Characterization

The fabricated specimens were completely immersed in liquid nitrogen for 7 minutes in order to

fully freeze the cellular structure and make it brittle. The specimens were fractured inside the

liquid nitrogen after the 7 minutes. Fractured specimens were then placed in vacuum oven at

room temperature for 8 hours in order to remove all the remaining gas for better coating quality.

After that, the specimens were put in a sputter coater connected to argon gas. During the coating

procedure, the current was controlled within a range of 3 mA to 5 mA for 540 seconds to fully

coat the specimens with platinum. Finally, the microstructure of the coated specimens was

imaged by scanning electron microscope (SEM, JEOL JSM-6060) at a suitable magnification.

The porosity of each specimen was measured from SEM images and confirmed with 𝜌∗ = 𝛤𝜌𝑠

where 𝜌∗ is the density of porous material, 𝜌𝑠 is the density of base solid material, and 1 − 𝛤 is

the averaged porosity [23]. Densities of the porous structures were measured by the density

determination Kit as explained in section 3.2. SEM measurements for porosity were carried out

with local threshoulding that binarizes the SEM image and then calculate the ratio of white

regions (pores) to black region (solid material) [1]. Experimental data were favoured whenever

there was a mismatch with SEM images analysis and then pores’ diameters were measured

manually with ImageJ software.

57


Functionally graded structures were achieved by applying a thermal gradient across the thickness

of the ABS/microspheres plate. In other words, adjusting different temperatures of the

compression mould platens could successfully induce a gradual porous structure across the

thickness. The case of 20% microspheres loading and 175/215 oC TL/TH thermal gradient is

presented in Fig 3.5 to explain more about the gradient of the porous structure and direction of

impact. The average pore diameter in this case is 124.1 µm. To study the gradient in more

details, a new parameter, ∆∅, is introduced which is defined as (𝑑@ 𝑇𝐻− 𝑑@ 𝑇𝐿

) 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠⁄ ,

where 𝑑 is the pores’ average diameter at each side. For the sample in Fig 3.5, ∆∅ =

(146.4 − 103.4) 15⁄ = 2.9 μm mm⁄ . The difference in diameter from TH to TL is only about 43

μm. This difference is due to the one minute waiting time after releasing the compression mould

platens, which allowed for more uniform free expansion of the microspheres.

Fig 3.5. Explanatory graph for the impact and pores’ gradient direction for the case of 20%

microspheres and 175/215 oC TL/TH

58

The mould allows expansion in the thickness direction only while restricting expansion in the

planar directions. The purpose of that restricted expansion is to have a one dimensional heat

gradient across the specimen thickness. To study cells’ isotropy, the circularity of the pores was

analyzed. Circularity can be defined as the ratio between an actual cell’s area to the area of an

equivalent circular cell producing the same perimeter;

Circularity =4𝜋𝐴

𝑃2

; where A and P are the actual cell’s

area and perimeter (3.2)

A circularity value near unity indicates that the cell is more spherical with minor elongation. Fig

3.6 shows the circularity analysis for the case of 10% microspheres loading and 190/235 ºC

TL/TH. Circularity is mainly bounded between 0.98 and 1.02, which is a very good indication of

isotropy. This could be attributed to the free expansion after the 3 minutes compression when the

polymer melt viscosity is very low. Other cases were analysed with similar indication of good

circularity.

Fig 3.6. Circularity analysis of the cells for the case of 190/235 ºC TL/TH and 10% microspheres

loading.

59

3.5.1 Relationship between processing conditions and the microstructure

Fig 3.7 presents the summary of the 27 experiments relating the thermal gradient to porosity.

Higher porosity values are observed at higher thermal gradients (lower values of TL and high

values of TH). Lower porosity values are observed at low thermal gradient (both TL and TH are

high) which is assumed to be due to cell collapse. Fig 3.8 shows the cells’ collapse for the case

of 10% microspheres and 190 oC/215

oC TL/TH processing temperatures. Higher values of TH

induced more expansion to the microspheres on this side, which pushed with more force to the

side of TL where the microspheres already softened because of high TL values. This combination

of high temperature and high pressure due to the microspheres’ expansion caused collapse of the

cells and hence a decrease in porosity.

10% 20%

30%

Fig 3.7. Dependence of porosity on the processing temperatures

60

Fig 3.8. Cell collapse at higher values of processing temperatures for the case of 10%

microspheres and 190 oC/215

oC TL/TH processing temperatures.

3.5.2 Propagation of the porous structure

Fig 3.9 shows the case of ABS with 10% microspheres at TL of 160 oC and different TH values of

190 oC, 215

oC, and 235

oC respectively. There is a solid band at TL of 160

oC and TH of 190

oC.

The solid band gets smaller at TH of 215 oC and the thickness is completely porous for TH of 235

oC. For TL of 175

oC, the thickness is completely porous at TH of 215

oC. In general, it can be

observed that as TL increases, the width of the solid band decreases.

TH = 190 oC TH = 215

oC TH = 235

oC

Fig 3.9. The microstructure of ABS with 10% microspheres at TL of 160 oC and different TH

61

20% 30%

Fig 3.10. The microstructure of ABS with different loading of Microsheres at TL of 160 oC and

TH of 190 oC

The loading of microspheres increased the spread of the porous region even under the same

thermal gradient conditions. Comparing the two microstructures of Fig 3.10 and the one in Fig

3.9 at TH of 190 oC, it can be seen that the solid band is getting smaller as the percentage of

microspheres increases. The effect of temperature and microspheres loading can be understood

from Stoke’s law of spherical particles. Stoke’s law states that the drag force exerted by a liquid

on spherical particles can be expressed as;

𝐹𝑑 = 6𝜋𝜇𝑅𝑣 (3.3)

where; 𝐹𝑑: The frictional force (or Stokes' drag), 𝜇: Fluid viscosity, 𝑅: Particle radius, and 𝑣: The

flow velocity relative to the particle.

ARES Rheometer (TA Instruments) model 400401.901 with parallel plate geometry was used

with strain-controlled condition to measure the viscosity change of ABS melt with temperature

(Fig 3.11). There is a sharp drop of viscosity at 160 ºC to about 180 ºC with slope of 13.3 KPa-s;

above 180 ºC, the viscosity becomes more stable at a decreasing rate of 0.64 KPa-s. The drag

62

force 𝐹𝑑will be decreasing as well with the decreasing ABS melt viscosity. At terminal (or

settling) velocity (V), the excess force 𝐹𝑔 that results from the difference between the weight and

buoyancy of the particle is given by:

𝐹𝑔 = (𝜌𝑝 − 𝜌𝑓)𝑔4

3𝜋𝑅3 (3.4)

where; 𝜌𝑝: Density of the particle, 14 Kg m3⁄ and 𝜌𝑓: Density of the fluid, 1040 Kg m3⁄ .

Thus, the terminal velocity (V) can be calculated by setting 𝐹𝑑 = 𝐹𝑔;

𝑉 =2

9

𝜌𝑝 − 𝜌𝑓

𝜇𝑔𝑅2 (3.5)

It can be noticed from equations 3.3 and 3.5 that the drag force scales with R while the terminal

speed scales with R2. The microspheres are already activated at 160 ºC so R starts increasing and

hence the terminal velocity increases proportionally. Fig 3.12 shows the change of terminal

velocity with viscosity at an average microspheres’ diameter of 32.5 µm. At increasing terminal

velocity, the microspheres will start moving from the side of TH to the side of TL due to the effect

of buoyancy that results from the increasing diameter of the microspheres. This motion of the

microspheres results in the solid band. At high values of TL, the microspheres expand more while

accompanied with a significant reduction in ABS melt viscosity which puts less restriction on the

expansion. Those expanded microspheres at the side of TL with reduced polymer viscosity put

restriction on the microspheres floating from the side of TH. This process happens during the 3

minutes of constant volume compression before releasing the mould platens. Increasing the

loading of microspheres puts further restriction on the expanding microspheres from the TH side

since the same volume is preserved.

63

Fig 3.11. Viscosity change of ABS with

increasing temperature.

Fig 3.12. Terminal velocity change with

viscosity of ABS melt.

3.5.3 Relationship between porosity and expansion ratio

The expansion ratio was calculated as the ratio of the height difference before and after

expansion to the height before expansion. The planar dimensions of the specimens are the same

before and after expansion because of the mould design which allows expansion in the height

direction only. The relationship between porosity and expansion ratio can be derived as follows;

𝜌𝑝

𝜌𝑠= 1 − ∅ (3.6)

𝐸𝑅 =𝐻𝑎 − 𝐻𝑏

𝐻𝑏=

𝐻𝑎

𝐻𝑏− 1 → 𝐸𝑅 + 1 =

𝐻𝑎

𝐻𝑏 (3.7)

where, 𝜌𝑝 and 𝜌𝑠 are the densities of the porous structure and solid material respectively, and ∅

is porosity. 𝐸𝑅 is the expansion ratio & 𝐻𝑎 and 𝐻𝑏 are the heights before and after expansion.

Substituting for density in equation (3.6) knowing that the mass and planar dimensions (width,

w, and length, L) of the specimen are the same before and after expansion, the following

expression can be derived;

𝐸𝑅 =1

1 − ∅− 1 (3.8)

64

The relation between porosity and the expansion ratio in equation 3.8 is plotted in Fig 3.13. The

R2 and standard error of the estimate (S) values are fairly good. The standard error of the estimate

(S) was calculated as, √((∑(𝑌 − 𝑌′) 𝑁⁄ )), where 𝑌and 𝑌′ are the actual and predicted values

respectively while 𝑁 is the number of data points. From equation (3.8), it is intuitive that it is

only necessary to analyse the expansion ratio or porosity, rather than analysing both of them. The

same applies to the relative density (𝜌𝑝 𝜌𝑠⁄ ) as well. However, the 0.48 value for S is low enough

to make a conclusive decision that porosity or expansion ratio will provide the same inferences if

being used solely as a representative of the other measure. There is no standard value for S but

usually it should be very small compared to the predicted values. In this case, the smallest value

for ((1 (1 − ∅)⁄ ) − 1) is 0.52 which is very close to 0.48.

Fig 3.13. The relationship between porosity and expansion ratio.

3.5.4 Relationship between the relative density, ∆∅, and porosity

Relative density, porosity, and the pores’ diameter gradient (∆∅) are all related. However, those

three measures of the microstructure are not related with the same strength. Relative density and

porosity are basically the same measure as can be inferred from equation 3.6. However, ∆∅ is not

correlated with the same strength to relative density (Fig. 3.14). This is attributed to the strong

correlation between ∆∅ and expansion ratio as will be discussed in section 3.5.6; where

65

expansion ratio is not strongly correlated to porosity (Fig. 3.13). These relations hold very well

in foaming with physical blowing agents where the porous structure is mostly uniform. Partial

cells’ collapse in foaming with expandable microspheres may result in height that is not

proportional to the reduced porosity caused by the collapse.

Fig 3.14. Relationship between relative density and pores’ diameter gradient (∆∅)

3.5.5 Effect of Microspheres Loading

The effect of microspheres loading on ∆∅ and porosity (or relative density) was hindered

because of the uncontrolled expansion. Similar values of ∆∅ and porosity but for different

microspheres loading can be achieved by controlling the expansion ratio. Thus to study the effect

of microspheres loading; relative density and ∆∅ can be mapped for the same expansion ratios

but different microspheres loading from the experimental data (Figs 3.15 and 3.16). Increasing

microspheres loading yielded to decreased relative density (or increased porosity) as intuitively

expected. Microspheres loading of 20% offered the highest gradient in pores’ diameter over the

10% and 30% loadings. This could be the optimal loading of microspheres for the material and

processing conditions used in this work.

66

Fig 3.15. The relationship between

microspheres loading, expansion ratio and

relative density.

Fig 3.16. The relationship between

microspheres loading, expansion ratio and

diameter gradient.

3.5.6 Effect of the thermal gradient on diameter difference

Fig 3.17 shows the thermal gradient versus diameter difference between TL and TH sides for 10%

microspheres loading and different TL values. The trend of increasing diameter difference with

increasing thermal gradient was observed at TL of 160 ºC for all TH values. This increasing trend

held true up to TH of 215 ºC for TL values of 175 ºC and 190 ºC. Eventhough the thermal gradient

is higher at TH of 235 ºC but the diameter difference reduced to lower values than those at TH of

215 ºC. Lower values of TL gave higher and consistent gradient in pores’ diameters regardless of

TH values. Temperatures of 175 ºC and 190 ºC are at the upper end of minimum point required

for activation of the microspheres; this caused more expansion of the microspheres during the 3

minutes compression and 1 minute expansion stage. Significant reduction in ABS melt viscosity

could have contributed also to the easier expansion of the microspheres and hence reduction of

diameter difference.

67

Fig 3.17. The effect of thermal gradient on pores’ diameters for different TL values and 10%

microspheres loading.

3.5.7 Impact Energy

The functionally graded porous structures were fabricated and tested for impact strength as

outlined in the previous sections. Microstructural results were utilized in this section to study the

relationship between impact strength and cellular morphology. Table 3.2 summarizes the degree

of association between impact energy and microstructure through the correlation coefficient.

Porosity, pores’ diameter gradient, and expansion ratio are used as the numerical quantification

of microstructure for the correlation analysis. The correlation coefficient is calculated as follows

[24]:


√∑(𝑥 − �̅�)2 ∑(𝑦 − �̅�)2 (3.9)



68

Table 3.2. Overall correlation coefficients between impact energy and microstructure

Correlation Coefficients

∆∅

Expansion

Ratio Porosity

Impact Energy 0.78 -0.66 -0.49

Porosity -0.75 0.69 1.0

∆∅ 1.0 -0.91 -0.75

Pores’ diameter gradient and expansion ratio demonstrated the highest inverse correlation

coefficient. That is attributed to the one minute waiting time after releasing the compression

mould platens, which allowed for more free expansion of the microspheres. Higher expansion

ratios induce less resistance to the microspheres expansion and resulted in smaller gradient of

pores’ diameter (∆∅). Impact energy showed fairly strong correlation to the diameter gradient.

This result is in agreement with the work of Zhou et al. [6] where they also found that, the

structures with high diameter gradient offer higher impact resistance than those with smaller

gradient. The inverse correlation between impact energy and expansion ratio is very intuitive

since higher expansion ratios cause mass redistribution on a larger volume, and hence reducing

the specific strength of the structure. Higher expansion ratios cause higher porosities which

provide more damping to the impact load through large compressive strains. In contrast, higher

porosities weaken the structure due to high void fraction compared to the solid mass and make it

prone to damage. So there is a balance between weakening the microstructure for damping and

preserving enough strength to withstand the impact load.

Expansion ratio and ∆∅ are both correlated fairly strong to the impact energy but not the same

strength to porosity (or relative density). Eventhough the three measures (∆∅, expansion ratio,

and porosity) are correlated to each other but it is with different magnitudes. For example, ∆∅

and expansion ratio are 91% correlated while ∆∅ is correlated with 75% strength to porosity

which is still high. Thus the proposal in this analysis is that a better control on impact energy is

69

possible by controlling ∆∅. Changing ∆∅ is going to affect porosity and expansion ratio but with

the advantage of more control on impact energy.

10% 20%

30%

Fig 3.18. Pores’ diameter gradient vs. expansion ratio for different loadings of microspheres.

Fig 3.18 shows the correlation between expansion ratio and pores’ diameter gradient graphically

along with a sample before and after foaming. Fig 3.18 in combination with Fig 3.19 can be

considered as design graphs. The corresponding pores’ diameter gradient for a certain impact

energy can be mapped from Fig 3.19 and then the expansion ratio can be obtained from Fig 3.18

along with Fig 3.19 for certain weight requirement (relative density). Finally, the processing

parameters can be mapped from graphs similar to Fig 3.7 that can be easily generated from the

experimental database.

70

The relationship between impact energy, relative density (density of the porous structure/density

of the solid, 1.04 g/cm3), and pores’ diameter gradient is presented graphically in Fig 3.19. The

same marker type is used for equivalent points at the graphs of

∆∅ and relative density to study about which affects more the impact energy.

10%

20%

30%

Fig 3.19. Effect of pores’ diameter gradient and relative density on impact strength with 10%,

20%, and 30% loading of microspheres.

71

There is a trend of increasing impact strength with increasing diameter gradient but more

interestingly is that the dependence of impact strength on diameter gradient is more than that on

porosity. This interpretation can be concluded when comparing the same markers on left (relative

density) and right (pores’ diameter gradient) of Fig 3.19. As an example, let’s take the case of

20% microspheres loading for 160-190 oC, 190-215

oC, and 190-235

oC. The impact energy of

160-190 oC is 28 J which is higher than 21.4 J of 190-235

oC. That is a consequence of the higher

relative density of 0.51 and larger diameter gradient of 2.48 µm/mm in the case of 160-190 oC

compared to those of 0.38 and 1.62 µm/mm for 190-235 oC. When considering the case of 190-

215 oC, the impact energy is 24 J which is lower than that of 160-190

oC even though the relative

density is 0.53 which is higher. That is reasoned to the smaller diameter gradient of 1.74 µm/mm

in the case of 190-215 oC than that of 160-190

oC which is 2.48 µm/mm. This observation was

predicted earlier from the correlation coefficients in Table 3.2. The correlation coefficient

between impact energy and diameter gradient is 0.78. However, the correlation coefficient

between impact energy and porosity is very weak and not conclusive. This is in support to the

interpretation obtained from Fig 3.19. That is, the pores’ diameter gradient is more correlated to

impact energy than to the relative density.

All the previous results are for Solid ABS of the same mass (10.53 g) and (45 mm x 45 mm x 5

mm) dimensions before foaming. Solid ABS exhibited much lower impact energy compared to

the same mass of functionally graded porous structures. The impact energy of solid ABS is 7.14

J only while the highest impact energy recorded for the functionally graded porous structures

was 30 J which is more than four times. To clarify more about the effect of geometry and mass

on impact energy, a solid ABS of equivalent volume (45 mm x 45 mm x 15 mm) was fabricated

for the case of 10% microspheres and 160-215 oC annealing temperatures. Impact energy of the

15 mm thick solid ABS was measured 26.74 J while it was 25.31 J for the functionally graded

porous structure of same volume. The strength-to-weight ratio of solid ABS is 25.71 J.cm3/g

while it is 46.02 J.cm3/g (Fig 3.20) for the functionally graded porous structure which is almost

double. Fig 3.20 shows that the damping capability of impact load by the graded porous structure

is more than that of the solid sample even for the same volume. Both specimens have indentation

at the top face (impact face). However, the solid specimen showed a crack at the bottom surface

while the graded porous structure seemed to absorb the load where no crack was visible. This

72

result demonstrates the superiority of functionally graded porous structures for impact resistance.

It can be concluded that the whole process is an optimization between the portions of the solid

and that of the porous structure while maintaining high diameter gradient and small pores.

Completely solid or porous material always gives lower impact energy than that of the

functionally graded porous structures when compared to the weight of each.

Solid ABS Functionally Graded ABS

Top Face Bottom Face Top Face Bottom Face

Impact

Energy (J) 26.74 25.31

Relative

Density 1.0 0.55

Fig 3.20. Images of the failing specimens of solid ABS and functionally graded porous structure

having the same volume.

3.6 Conclusions

Functionally graded porous structures of Acrylonitrile Butadiene Styrene (ABS) were fabricated

by the mean of one dimensional thermal gradient across the thickness. Microspheres

encapsulating the chemical blowing agent were used as the mean of introducing pores to the

solid. A total of 27 experiments were conducted with the aim of studying the effect of thermal

gradient and microspheres loading on the microstructure and impact energy. High processing

73

temperatures (lower thermal gradient) caused cells collapse and reduced porosity. The increase

of microspheres loading or the processing temperatures spread the porous structure more across

the thickness. Controlling those two parameters leaded to graded porous structures that

outperform the solid even for the same volume. The correlation coefficient between impact

energy and diameter gradient is fairly strong. However, the correlation coefficient between

impact energy and porosity is very weak and not conclusive. The strength-to-weight ratio of

functionally graded porous structures was almost double than that of the solid polymer. This high

ratio can be tailored to some values of interest to certain applications since the impact strength is

correlated to the cellular morphology. The microstructure itself is correlated to the different

processing parameters with different magnitudes, which give a road map for fabricating some

desired microstructures to target certain impact strength.

3.7 References





[2] M. F. Diop, W. R. Burghardt, and J. M. Torkelson, "Well-mixed blends of HDPE and

ultrahigh molecular weight polyethylene with major improvements in impact strength

achieved via solid-state shear pulverization," Polymer, vol. 55, pp. 4948-4958, 9/15/

2014.

[3] T.-H. Hsieh and C.-H. Liang, "Improvement of impact absorbed energy of CFRPs on

adding the nanoparticles into epoxy resins," Journal of Chemistry and Chemical

Engineering, vol. 8, 2014.

[4] D. Chen, R. Matsumoto, K. Arakawa, and S. Yoon, "Energy absorption property of

CFRP under impact loadings," Applied Mechanics and Materials, vol. 566, pp. 219-224,

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[5] Liang Cui, Stephen Kiernan, and M. D. Gilchrist, "Designing the energy absorption

capacity of functionally graded foam materials," Materials Science and Engineering: A,

vol. 507, pp. 215 – 225, 2009.

[6] J. Zhou, Z. W. Guan, and W. J. Cantwell, "The impact response of graded foam sandwich

structures," Composite Structures, vol. 97, pp. 370-377, 2013.

74

[7] M. Avalle, G. Belingardi, and R. Montanini, "Characterization of polymeric structural

foams under compressive impact loading by means of energy-absorption diagram,"

International Journal of Impact Engineering, vol. 25, pp. 455-472, 5// 2001.

[8] D. I. Collias, D. G. Baird, and R. J. M. Borggreve, "Impact toughening of polycarbonate

by microcellular foaming," Polymer, vol. 35, pp. 3978-3983, 1994.

[9] D. Miller and V. Kumar, "Microcellular and nanocellular solid-state polyetherimide (PEI)

foams using sub-critical carbon dioxide II. Tensile and impact properties," Polymer, vol.

52, pp. 2910-2919, 2011.

[10] J.-B. Bao, T. Liu, L. Zhao, G.-H. Hu, X. Miao, and X. Li, "Oriented foaming of

polystyrene with supercritical carbon dioxide for toughening," Polymer, vol. 53, pp.

5982-5993, 2012.

[11] L. F. Cai, X. B. Huang, M. Z. Rong, W. H. Ruan, and M. Q. Zhang, "Effect of grafted

polymeric foaming agent on the structure and properties of nano-silica/polypropylene

composites," Polymer, vol. 47, pp. 7043-7050, 2006.

[12] D. J. VanHouten and D. G. Baird, "Generation and characterization of carbon nano-fiber–

poly(arylene ether sulfone) nanocomposite foams," Polymer, vol. 50, pp. 1868-1876,

2009.

[13] S. Sachse, M. Poruri, F. Silva, S. Michalowski, K. Pielichowski, and J. Njuguna, "Effect

of nanofillers on low energy impact performance of sandwich structures with

nanoreinforced polyurethane foam cores," Journal of Sandwich Structures and Materials,

vol. 16, pp. 173-194, March 1, 2014 2014.

[14] A. Rajaneesh, I. Sridhar, and S. Rajendran, "Relative performance of metal and

polymeric foam sandwich plates under low velocity impact," International Journal of

Impact Engineering, vol. 65, pp. 126-136.

[15] F. A. Jahwari and H. E. Naguib, "Linear viscoelastic modeling and validation of

functionally graded heterogeneous porous PLA structures with a C1-continuous plate

theory and novel homogenization," presented at the Foams, New Jersey, USA, 2014.

[16] X. Zhang and H. Zhang, "Optimal design of functionally graded foam material under

impact loading," International Journal of Mechanical Sciences, vol. 68, pp. 199-211,

2013.

[17] H. B. Zeng, S. Pattofatto, H. Zhao, Y. Girard, and V. Fascio, "Perforation of sandwich

plates with graded hollow sphere cores under impact loading," International Journal of

Impact Engineering, vol. 37, pp. 1083-1091, 11// 2010.




75

[19] M. R. Barzegari, D. Rodrigue, and J. Yao, "Polyethylene foams produced under a

temperature gradient with Expancel and blends thereof," Cellular Polymers, vol. 29, pp.

259-281, 2010.

[20] M. R. Barzegari, D. Rodrigue, and J. Yao, "Density graded linear medium density

polyethylene foams produced under a temperature gradient with Expancel microbeads,"

Cellular Polymers, vol. 30, pp. 157-185, 2011.

[21] M. R. Barzegari, J. Yao, and D. Rodrigue, "Mechanical properties of density graded

foams: tensile properties," Cellular Polymers, vol. 32, pp. 323-341, 2013.

[22] G. Fischer, I. Lüderwald, and P. Ottenbreit, "Characterization of acrylonitrile-butadiene-

styrene terpolymers by computer aided FT-IR spectroscopy," Die Angewandte

Makromolekulare Chemie, vol. 149, pp. 179-187, 1987.




[24] J. Lee Rodgers and W. A. Nicewander, "Thirteen ways to look at the correlation

coefficient," The American Statistician, vol. 42, pp. 59-66, 1988/02/01 1988.

(1) The content of this chapter has been submitted for publication to the Journal of Composite

Structures;

F. Al Jahwari, A. A. W. Anwer, and H. E. Naguib, " Experimental Evaluation of Impact Load Transfer of Through-

thickness Stitched Composite Structures with Graded Syntactic Foams," Journal of Composite Structures

Chapter 4 Experimental Evaluation of Impact Load Transfer of Through-

thickness Stitched Composite Structures with Graded Syntactic Foams

(1)

The high damping performance of FG porous structures which was shown in Chapter 3 is

devalued by the overall weakness of the porous structures to withstand external loads which

makes them prone to damage. This makes the topic of this Chapter to develop a reinforcement in

the porous structure to enhance structural strength; and hence get use of the high damping

performance without exposing the FG structure to damage. Through-thickness stitched

composite structures were fabricated with graded syntactic foamed layers. Polyurethane resin

was used as the matrix with microspheres to make the syntactic foam with different

concentrations. Kevlar/carbon fibres hybrid fabrics were embedded between three layers of the

matrix for planar reinforcement. Ultrahigh molecular weight polyethylene braids provided the

out-of-plane reinforcement by stitching them through the thickness of the composite structure.

The effect of fibrous reinforcement and foam gradient on the transferred load was investigated

with customized drop weight set up for this purpose. The fibrous reinforcement and increased

loading of the microspheres caused reduction in load transfer. Compressive and relaxation

moduli of the composite structures were measured and correlated to the amount of load transfer.

Both of the moduli showed fairly strong inverse correlation to the transferred load and

proportional correlation to each other. This is indicative of the potential to tailor such composite

structures to specific values of load transfer during impact event.

77

4.1 Introduction

Impact performance of materials is assessed by two main quantities; the ability of the material to

withstand the impact load without damage and the amount of load transferred to the other side of

impact. The transferred impact load to the supporting apparatus/human would address safety-

critical issues since structural damage to the apparatus or serious injuries to the human are a

direct result of the forces experienced during impact. Even though the existing personal

protection equipment like helmets and bullet-proof systems can absorb impact energy without

being damaged, but mostly they don’t sufficiently mitigate the load which constitutes critical risk

on humans using those products on the long run and even to machines. The focus of this work is

on the transferred load from impact to the receiver which can be humans or machines.

Much of the recent guidelines and inspiration for the development of high performance

structures comes from the designs existing in nature itself. Widespread examples include the

clubs of mantis shrimps, nacre-like structure in sea shells, and bone. A characteristic feature in

all these structures is high stiffness but also high deformation at large stresses. Stiffness and

toughness of a material are normally exclusive properties. The reason is that stiffer materials

have lower ability to flow and relax internal stress during impact. Intrinsically, it is difficult to

find high strength, toughness, and low density all in one material. For obtaining the most optimal

combination of these characteristics, extrinsic reinforcement systems must be employed. It is

such intrinsic and extrinsic reinforcement mechanisms which results in effective properties of

materials far above the individual components. Khayer et al. [1] concluded very important result

which states that the interface in nacre-like composite does not need to be tough; the extensibility

or ductility of the interfaces may be more important than their strength and toughness to produce

toughness at the macroscale.

Composites are the solution to make these high performance structures similar to those in nature.

Composite materials combine different material components to form a multi-constituent structure

that outperforms the individual components. The interface between different components was

and still one of the major problems associated with composite material. Weak interfaces result in

78

delamination of the structure and put limitation on load transfer between the different

components. Weak interfaces can be advantageous in energy dissipation and overall enhanced

strength of the composite through frictional effect at the interface [1, 2]. However, delamination

is the disadvantage of weak interfaces. Stitching the composite in the out-of-plane direction

alleviates the problem of delamination and reduces the requirement of strong adhesion between

different components. This is in addition to the extra reinforcement in the out-of-plane direction.

Quek et al. [3] fabricated composite tubes made of E-glass fibres and polyurethane resin. The E-

glass fibres were used in the form of continuous fabrics and braids to study the energy absorption

during compressive crushing of each reinforcement type. The tubes made of fabrics exhibited

lateral cracks which was not observed in the braided tubes. Through thickness stitched sandwich

structures were fabricated by Dawood et al. [4] to study their bending behaviour. The impact

response of stitched sandwich structures was significantly improved in comparison to their

unstitched counterparts with limited delamination of the structure [5]. Stitched sandwich

structures were also developed by Potluri et al. [6] to address the delamination problem that is

typically associated with such structures. Through thickness stitching of sandwich structures was

shown to increase the maximal cracking width and penetration depth by 67% and 4%

respectively [7].

In this chapter, through thickness graded syntactic foamed composite structures were fabricated

and tested for impact load transfer and stress relaxation. Including strong fibres in a soft matrix

gives the benefit of damping and structural integrity [8]. Polyurethane resin with dispersed

microspheres was used as the soft matrix in this work. Polyurethane and its foams were reported

by many researchers for enhanced energy absorption applications [9, 10]. The composite

structure composed of three 4 mm thickness layers of polyurethane/microspheres with

Kevlar/carbon fibres hybrid fabrics embedded between them. Polyurethane resin was used as the

adhesive agent between the different components. Braids made of ultrahigh molecular weight

polyethylene (UHMWPE) were stitched through the thickness of the structure at the edges to

limit delamination and provide global bending response of the whole composite during impact.

UHMWPE fibres can provide structural strength and damping at the same time [11]. The

polyurethane/microspheres layers with different concentrations of the microspheres were placed

with different sequences along the impact direction to study the effect of gradient on the

79

transferred load. Previous study by the authors showed significant influence of gradient on

impact energy absorption [12] which motivated the study of gradient effect. Other researchers

also reported similar effect of the gradient on energy absorption [13]. Compressive and

relaxation moduli of the fabricated structures were measured and correlated to the transferred

impact load.

4.2 Material Structure and Processing

Polyurethane was selected as the matrix material due to its high damping performance and tear

resistant. The polyurethane is 60 Shore A, grade 3360 from Fibre Glast, USA. A 50/50 hybrid

fabric of Kevlar and carbon fibres from Fibre Glast was used to provide the in-plane

reinforcement to the composite structure. Ultrahigh molecular weight polyethylene (UHMWPE)

braids with 0.76 mm in diameter were used to provide the out-of-plane reinforcement. The

UHMWPE braids are grade Spectra-200 from Twinline, USA. The tensile strength of these

braids is 2048 MPa which is about five times stronger than structural steel. The syntactic

structures of polyurethane were made by dispersing Expancel microspheres grade 980 DU 120

from Akzonobel, USA. These microspheres are polymeric based and supplied as light yellow

powder with an initial particle size range of 25-40 μm, and density of 14 kg/m3. These

microspheres comprise a thermoplastic shell encapsulating low boiling point liquid hydrocarbon.

The microspheres loading considered in this study are 1%, 2% and 3%. Uniform dispersion of

the microspheres was achieved by sonication.

The processing steps can be divided into two main stages. At the first stage, 4 mm thickness

sheets with 45 mm x 45 mm planar dimensions were fabricated. Different sheets of polyurethane

were made with 0%, 1%, 2%, and 3% loadings of the microspheres. The fabrication process is

explained schematically in Fig 4.1. The mixing ratio of polyurethane parts A and B is 100:55 by

weight. The microspheres were dispersed in the resin (part A) by sonication at 10% amplitude

for 5 minutes using s-4000 Sonicator from Misonix. The resin/microspheres composite was then

cooled by ice bath to avoid instant setting with part B upon mixing. Part B was then mixed

manually with the resin/microspheres for one minute. The mixture was then degassed at -30 in-

80

Hg (-101592 Pa) in a vacuum chamber until no more air bubbles were observed which took

about seven minutes. The mixture was finally poured slowly in the desired mould and cured at

50 oC for 12 hours.

Fig 4.1. Processing procedure for polyurethane composites

At the second stage of the processing, the graded foamed composites were made by laying up

three layers of the 4 mm thickness polyurethane composites, and placing the Kevlar/carbon

fibres hybrid fabric between each two layers. Polyurethane was used as the wetting agent

between different layers and the fabrics. Finally, the composite structures were stitched in the

out-of-plane direction with UHMWPE braids after 12 hours curing of the composites lay-up at

50 oC. This produced specimens with 12 mm in thickness and planar dimensions of 45 mm x 45

mm. Exception from this process was made to the reference samples which were made directly

by moulding 12 mm thick specimens. The reference specimens are polyurethane, polyurethane

with 1% and 3% microspheres loading. A schematic of the stitched graded composites is shown

in Fig 4.2.

The specimens were fabricated to test the effect of microspheres loading and gradient across the

thickness. Table 4.1 shows the fabricated specimens. The first three specimens were meant as

81

reference samples to compare the effect of fabrics and foam gradient. Specimens 4,5, and 6 were

made to study the effect of fabrics and lay-up without gradient in the microspheres

concentration. The last four specimens were intended to investigate the effect of foam gradient

on impact energy absorption. UHMWPE braids were stitched manually by industrial type needle

for all the specimens except the first three.

Fig 4.2. Schematic of the graded stitched foamed composite structure

Table 4.1. The fabricated specimens and lay-up order

Experiment # Microspheres %

Pure PU + Microspheres

1 0

2 1

3 3

Fibrous PU + Microspheres

4 0

5 1

6 3

Microspheres Gradient %

(Top-Middle-Bottom)

Fibrous PU + FG Microspheres

7 1-1-3

8 1-3-1

9 1-3-3

10 3-1-3

11 1-2-3

82

4.3 Experimental Set-up and Testing

The standard drop weight test set up (according to ASTM D5420-10) was instrumented with

sensing system to measure the residual impact load at the receiver side rather than at the

impacted surface as conventionally being done (Fig 4.3). The purpose of this set up was to

measure the impact force transmitted across the material towards the receiver. This test helps to

evaluate the effective material damping by measuring the actual force transferred. The load cell

is from Dytran, model 1061V6 with dynamic force capacity of 50000 Ibf (222.41 kN). The

impact specimen dimensions conform to ASTM D5420-10 with drop mass of 3.628 Kg (8 Ib)

and drop height of 0.33 m (13 inches). A sample of the force output is presented in Fig 4.4 for

the case of stitched composite with 3% loading of the microspheres. There is a peak of the force

at the time of impact which is considered as the transferred load to the receiver. The compressive

modulus was measured according to ISO-604 which conforms well with the fabricated

specimens dimensions. A stress relaxation test was also conducted for all the specimens to study

the effect of foam gradient and fibrous reinforcement to relax the applied impact loads. A

constant strain of 0.08 was applied and the stress was measured for three minutes using Instron

Micro-tester (Model 5848). Three specimens were tested for each configuration for averaging.

For morphological characterization, the fabricated specimens were completely immersed in

liquid nitrogen for 7 minutes in order to fully freeze the microstructure and make it brittle. The

specimens were fractured inside the liquid nitrogen after the 7 minutes. Fractured specimens

were then placed in vacuum oven at room temperature for 2 hours in order to remove all the

remaining gas for better coating quality. After that, the specimens were put in a sputter coater

connected to argon gas. During the coating procedure, the current was controlled within a range

of 1 mA to 3 mA for 180 seconds to fully coat the specimens with platinum. Finally, the

microstructure of the coated specimens was imaged by scanning electron microscope (SEM,

JEOL JSM-6060) at a suitable magnification.

83

Fig 4.3. Sensing system for the experimental set-up of impact test.

Fig 4.4. Sample of the impact load results for the case of stitched composite with 3% loading of

the microspheres.


4.4.1 Composite Structure and Microspheres Dispersion

The eleven specimens were fabricated and tested as outlined in sections 4.2 and 4.3. It was

observed that, polyurethane is not a good wetting agent between the Kevlar/carbon fabric and

84

polyurethane itself. The UHMWPE fibres provided the out-of-plane mechanical reinforcement

which alleviated the weak adhesion between the three layers. As discussed earlier, weak

interface is advantageous in energy dissipation. It is the main mechanism of high toughness of

nacre-like composites in nature [1]. Fig 4.5 shows the structure of the stitched graded foam

composite for the case of 1%-2%-3% loadings of the microspheres. SEM images showed good

dispersion of the microspheres by sonication. No sign of delamination of the composite

structures was observed after impact.

Fig 4.5. The structure of the stitched graded foam composite for the case of 1%-2%-3%

microspheres loading.

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4.4.2 Compressive Modulus

The compressive modulus increased for the increased loading of the microspheres as shown in

Fig 4.6 with and without the fibre reinforcement. The first three specimens were fabricated as

one piece with 12 mm thickness as indicated in section 4.2. This increase in the compressive

modulus is attributed to the microspheres loading. The weak interface with polyurethane allowed

for independent response of the microspheres to the compressive load which might cause this

increase in the modulus. The SEM image shown in Fig 4.7 provided an evidence of the weak

interface and hence de-bonding between polyurethane and the microspheres which allowed for

different deformation regimes to each phase. Compressive moduli for the layered fibrous

structures were lower than their non-fibrous equivalent specimens. This could be due to the

frictional losses at the weak interfaces between the layers of polyurethane and fabrics which

allowed for sliding. The reduction of compressive modulus is advantages in damping at the event

of impact. The graded syntactic foamed structures showed variability in the compressive

modulus depending on the layering arrangements according to the microspheres loading.

Including more layers of the 3% loading gave rise to the compressive modulus in comparison to

other structures with less 3% layers. The lowest value of the compressive modulus was reported

for the case of placing two layers of 1% microspheres loading on top of one layer of 3% loading.

Fig 4.6. Compressive modulus of the specimens with different loadings of the microspheres.

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Fig 4.7. SEM image showing the de-bonding between polyurethane and the microspheres

4.4.3 Transferred Impact Load

The transferred impact load by each specimen is shown in Fig 4.8. The graded foamed structures

were impacted from both sides to study the effect of gradient. The transferred impact load

decreased with increasing load of the microspheres. This is attributed to the frictional dissipation

between the microspheres and polyurethane matrix due to the weak interface between them.

Debonding and independent deformation contribute also to the reduction in transferred load.

Placing the 3% microspheres loading layer at the side of impact, always resulted in reduced load

transfer. The lowest load transfer was reported for the case of 3%-3%-1% configuration of the

layers. In comparison, the reverse configuration, 1%-1%-3%, resulted in the highest load transfer

among all the graded foamed structures. This result suggests that placing larger portion of the

composite structure with higher loading of the microspheres, followed by smaller portion with

lower loading of the microspheres provides better damping performance. Referring to the

discussion of Fig 4.6 about compressive modulus, a structure with enhanced damping

performance should start with stiffer portion of the graded composite facing the striker and

placing a reduced stiffness layer at the receiver side. The first two layers of 3% loading

dissipated most of the impact load through the mechanisms discussed earlier, and the 1% layer

87

damped the remaining of the load stopping it from propagating to the receiver. The UHMWPE

braids at the edges of each specimen facilitated independent bending of each layer away from the

edges while restricting the bending to be global to all layers at the edges. This bending

mechanism breaks the continuity in load transfer across the three layers and enhances frictional

dissipation through the relative motion between each two layers away from the edges. The

bottom layer works in synergy with the other two layers and Kevlar/carbon fibres hybrid fabrics

to limit the deformation of the residual impact load. This is possible because of the strong

connection at the edges made by UHMWPE braids. The gradual decrease or increase in

microspheres loading along the impact direction did not help with the same magnitude in

enhancing load reduction as compared to the cases of bias portion of one concentration to the

other. The number of transferred G’s scales with the transferred impact load. It was calculated as

the ratio between the transferred impact load and striker mass, all divided by 9.81 m/s2. The

maximum recommended number of G’s is 300 [14]. All the fabricated composite structures

showed values below 300 with maximum value of 95.08 for the case of pure polyurethane and

minimum value of 69.88 for the case of 3%-3%-1%.

Fig 4.8. The transferred impact load by each composite structure.

88

4.4.4 Relaxation Modulus and Correlation Analysis

The ability to relax imposed stresses during the impact event was studied by stress-relaxation

experiments as outlined in section 4.3. Relaxation modulus (𝐸𝑅) is calculated as 𝜎(𝑡) 휀0⁄ , where

𝜎(𝑡) is the measured stress with time, and 휀0 is the constant applied strain which is 0.08 in this

work. The non-fibrous specimens showed lower relaxation moduli compared to the fibrous

specimens (Fig 4.9). The restriction imposed by the Kevlar/carbon fibres hybrid fabrics for

planar deformation might cause the increase of stresses for the same magnitude of strain. It can

be noticed that, small relaxation moduli correspond to higher values of the load transfer by

comparing the results in Fig 4.9 and Fig 4.8. This result suggests that higher values of the

relaxation modulus induced more sharing of the impact load to the fabrics and UHMWPE braids.

This in turn resulted in reduced overall transferred loads.

Fig 4.9. Relaxation modulus of the fibrous and non-fibrous composite structures.

Fig 4.10 shows the relaxation moduli of the graded foamed structures. The gradient has a

significant effect on the relaxation moduli and hence on the load transfer as discussed earlier.

Alternating layers with different microspheres loading resulted in the highest values for

relaxation moduli. A gradient or consistent layers of microspheres loading provided lower values

for the relaxation moduli. To elucidate more about the relationship between relaxation modulus

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(𝐸𝑅), compressive modulus (𝐸𝑐), and transferred impact load (𝐹); a correlation analysis was

performed. The correlation coefficients were calculated as follows [15]:


√∑(𝑥 − �̅�)2 ∑(𝑦 − �̅�)2 (4.1)



The correlations coefficients shown in table 4.2 are fairly strong in general which indicates the

potential to tailor impact load transfer by controlling the compressive and relaxation modulus.

The transferred impact load is inversely correlated to both, the compressive and relaxation

moduli. The inverse correlation coefficient of 0.649 between compressive modulus and

transferred load is counterintuitive to the general understanding that stiff materials transmit more

load with minimal damping. However, the restriction on planar deformation that is imposed by

the Kevlar/carbon fibres hybrid fabrics might resulted in stiffer response to the impact load even

for soft layers of polyurethane syntactic foams. In this context, more damping to impact load will

be provided by softer materials, since the structural strength is provided by the fabrics in addition

to the global boundary restriction by UHMWPE braids. The earlier discussion about the effect of

stress relaxation of the material in load transfer was proved by the strong correlation coefficient

of 0.744 between 𝐸𝑅 and the transferred impact load. There is more sharing of the impact loads

for higher values of the relaxation moduli which makes the response to impact event more global

rather than being taken by the layer facing the striker only. This resulted in reduced values of

transferred load. The strong direct correlation between compressive and relaxation moduli is

indicative to the possibility of controlling both at the same time and hence controlling the

amount of transferred impact load.

90

Fig 4.10. Relaxation modulus of the graded foamed composite structures.

Table 4.2. Correlation coefficients between relaxation modulus (𝐸𝑅), compressive modulus (𝐸𝑐),

and transferred impact load (𝐹).

𝑬𝒄 𝑭

𝑬𝒄 1 -0.649

𝑭 -0.649 1

𝑬𝑹 0.899 -0.744

4.5 Conclusions

Stitched composite structures with graded syntactic foams were fabricated from polyurethane,

Kevlar/carbon fibres hybrid fabrics, and UHMWPE braids to minimize the load transfer from

impact event. Microspheres were dispersed by sonication in polyurethane to make syntactic

foams with different concentrations. In addition to the instrumented drop weight impact testing,

the compressive and relaxation moduli of the fabricated specimens were measured. The impact

set up measured the transferred load from impact at the receiver side rather than measuring the

impact load at the impacted surface. Increasing load of the microspheres increased the

compressive modulus but significantly reduced the transferred impact load. This is attributed to

the frictional dissipation between the microspheres and polyurethane due to weak interface

91

between them. The reduction in transferred loads was even more when the syntactic

polyurethane layers were reinforced by Kevlar/carbon fibres hybrid fabrics and UHMWPE

braids at the edges. This composite structure allowed for more frictional dissipation between the

layers while keeping compatibility of the system global deformation with the stitched UHMWPE

braids at the edges. Correlation analysis was performed to study the relationship between

transferred impact loads to the compressive and relaxation moduli. The transferred load inversely

correlated to both moduli with fairly strong correlation coefficients. Higher values of relaxation

modulus induced more sharing of the impact load which reduced the overall transferred load.

The compressive and relaxation moduli were strongly correlated which is indicative to the

potential of tailoring transferred impact load to some desired values by controlling both moduli

at the same time.

4.6 References

[1] A. Khayer Dastjerdi, R. Rabiei, and F. Barthelat, "The weak interfaces within tough

natural composites: Experiments on three types of nacre," Journal of the Mechanical

Behavior of Biomedical Materials, vol. 19, pp. 50-60, 2013.

[2] R. G. Hoagland, J. P. Hirth, and A. Misra, "On the role of weak interfaces in blocking

slip in nanoscale layered composites," Philosophical Magazine, vol. 86, pp. 3537-3558,

2006.

[3] S. Ching Quek, A. M. Waas, J. Hoffman, and V. Agaram, "The crushing response of

braided and CSM glass reinforced composite tubes," Composite Structures, vol. 52, pp.

103-112, 2001.

[4] M. Dawood, E. Taylor, and S. Rizkalla, "Two-way bending behavior of 3-D GFRP

sandwich panels with through-thickness fiber insertions," Composite Structures, vol. 92,

pp. 950-963, 2010.

[5] B. Lascoup, Z. Aboura, K. Khellil, and M. Benzeggagh, "Impact response of three-

dimensional stitched sandwich composite," Composite Structures, vol. 92, pp. 347-353,

2010.

[6] P. Potluri, E. Kusak, and T. Y. Reddy, "Novel stitch-bonded sandwich composite


[7] F. Xia and X.-q. Wu, "Study on impact properties of through-thickness stitched foam

sandwich composites," Composite Structures, vol. 92, pp. 412-421, 2010.

92

[8] P. Fratzl, I. Burgert, and H. S. Gupta, "On the role of interface polymers for the

mechanics of natural polymeric composites," Physical Chemistry Chemical Physics, vol.

6, pp. 5575-5579, 2004.

[9] G. Zhang, B. Wang, L. Ma, L. Wu, S. Pan, and J. Yang, "Energy absorption and low

velocity impact response of polyurethane foam filled pyramidal lattice core sandwich

panels," Composite Structures, vol. 108, pp. 304-310, 2014.

[10] J. Zhou, X. Deng, Y. Yan, X. Chen, and Y. Liu, "Superelasticity and reversible energy

absorption of polyurethane cellular structures with sand filler," Composite Structures,

vol. 131, pp. 966-974, 2015.

[11] T. K. Ćwik, L. Iannucci, P. Curtis, and D. Pope, "Investigation of the ballistic

performance of ultra high molecular weight polyethylene composite panels," Composite


[12] F. Al Jahwari, Y. Huang, H. E. Naguib, and J. Lo, "Relation of impact strength to the

microstructure of functionally graded porous structures of acrylonitrile butadiene styrene

(ABS) foamed by thermally activated microspheres," Polymer, vol. 98, pp. 270-281,

2016.



[14] H. Azhar, A. A. Hafeez, S. M. Syazwan, I. M. Hafzi, and Y. Ahmad, "Comparative study

of motorcycle helmets impact performance," Applied Mechanics and Materials, vol. 575,

pp. 306 - 310, 2014.

[15] J. Lee Rodgers and W. A. Nicewander, "Thirteen ways to look at the correlation

coefficient," The American Statistician, vol. 42, pp. 59-66, 1988/02/01 1988.

(1) The content of this chapter has been published in the Journal of Applied Mathematical

Modelling;

F. A. Jahwari and H. E. Naguib, "Analysis and homogenization of functionally graded viscoelastic porous structures

with a higher order plate theory and statistical based model of cellular distribution," Applied Mathematical

Modelling, vol. 40, pp. 2190–2205, 2016. DOI: 10.1016/j.apm.2015.09.038. Reproduced by permission from

Elsevier (License # 3902071287801).

Chapter 5

Analysis and Homogenization of Functionally Graded Viscoelastic

Porous Structures with a Higher Order Plate Theory and

Statistical Based Model of Cellular Distribution (1)

The fabrication procedures with accurate control over the microstructure were developed in

Chapters 2, 3, and 4. The relationships between fabrication, microstructure, and mechanical

properties were also established in the previous chapters at which if the required mechanical

property is known then the corresponding microstructure can be fabricated. This completed the

fabrication component of the research. In this chapter, the first stage of numerical design of FG

structures is established by developing a statistical homogenization law and procedure which

account for the gradient of properties and microstructure. Plate-like structures of Polylactic Acid

were fabricated with a constrained foaming process. The fabricated structures have a gradual

porosity with variable cellular size throughout the thickness but homogeneous on-average at

parallel planar sections. SEM characterization of the microstructure is statistically analyzed with

Burr distribution and used as an input to a new homogenization model that accounts for cellular

distribution. A higher order plate theory is developed for the analysis that satisfies the free

traction condition a priori to the consistency of transverse shear strain energy which had been

rarely considered by similar theories in the area. Implementing the theory in finite element

procedure necessitates the curvature continuity across elements. This was resolved by

implementing the penalty enforcement technique and the use of conforming/nonconforming

finite elements with derivative based nodal degrees of freedom. The conforming element showed

94

a better convergence and thus selected for further analysis. The linear viscoelastic behaviour of

PLA is assumed to obey Boltzmann superposition principle with hereditary integrals. The stress

relaxation functions are determined experimentally for property characterization and from the

homogenization model. The proposed plate theory and homogenization model agree very well

with the work done by other researchers and experimental data. The numerical tool proved its

validity for conducting accurate computer experiments with the minimal input about material

properties and microstructure.

Nomenclature

PLA Polylactide Acid

FGP Functionally graded plate

HSDT Higher Order Shear Deformation

Plate Theory

FGMs Functionally graded materials

CPT Classical Plate Theory

a Width of the plate

b Length of the plate

h Thickness of the plate

u Displacement along x-axis

v Displacement along y-axis

w Displacement along z-axis

𝜏 The history variable

D The stress relaxation matrix

Ω The region being analyzed

𝒙 The position vector

q The number of time steps

∆𝑑𝑞 The difference of nodal

displacements between two

successive time points

𝑘𝑞 The size of the time step

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𝑩 The strain-displacement matrix

𝑨 Time-dependent viscoelastic

stiffness matrix

𝐸∞ Long term modulus

𝛼𝑗 The relaxation time constants

𝐸∗(𝑡) Relaxation modulus of the

porous structure

𝐸𝑠(𝑡) Relaxation modulus of the solid

polymer

1 − 𝛤 Average porosity

𝜆 Statistical factor extracted from

pores distribution

5.1 Introduction

The concept of FGMs had been proven by Bever, Duwez and Shen since 1972 but it took about

15 years of research in manufacturing before bringing them to applications [1]. The continuous

microstructure of functionally graded materials makes them advantageous over conventional

composites for many applications. Thermal-barrier system is one of the successful applications

to metallic-ceramic FGMs due to the gradual and smooth transition of material properties. Many

reviews are published for conventional functionally graded materials covering analysis and their

fabrication [1, 2]. However, the focus of this work is on polymeric cellular FGMs. A controlled

morphology of the cellular structure would be desired in many applications due to the significant

weight saving while optimizing the other properties as to suite certain applications. A variable

stiffness pad and shock absorbers in high strain loading are some of the potential applications of

cellular FGMs. Two major problems in general are associated with FGMs apart from material

selection, namely, i) determining the optimal spatial distribution of constituents/cells that give

the required properties and hence the behaviour, ii) fabrication of the optimal microstructure.

Both problems are not well documented for functionally graded polymeric cellular materials. An

accurate analysis tool that can predict the behaviour based on microstructure and a set of loading

and boundary conditions will be of an importance in the area. However, fabrication processes

96

should also go along with the development of analysis tools. The first problem is addressed in

this work for functionally graded porous plate structures.

5.1.1 Fabrication of functionally Graded Porous Structures

Fabrication techniques of cellular materials include but not limited to solvent particulate leaching

[3, 4], gas foaming [4-6], co-continuous melt blending [7-12], and rapid prototyping [13-15].

However, the combination of those techniques lead to more control of the pores growth [5, 11],

and better mechanical properties [4]. Nadella et al. [5], fabricated a closed-cell functionally

graded PLA by annealing gas foamed samples with a temperature gradient rather than quenching

in a bath of uniform temperature. Zhang et al. [11], fabricated an open-cell functionally graded

PLA by mixing with polystyrene (PS), molding at a temperature above the melting point of both

polymers, and finally annealing at temperatures above the glass transition but below melting. PS

is removed in a later stage by dissolving in cyclohexane. Harris et al. [4], made disks of PLA and

salt by compression molding, gas foam it, and finally leach out salt which made a mechanically

stronger foam than those made by traditional leach-out techniques. It is worth mentioning that

processing parameters to control the pores’ growth rate and size are not fully understood and

optimized. The addition of nano-fillers like chitin had been reported to improve the foaming

process [16]. Other nano-fillers like carbon nanotubes and nanofibers are even reported to

provide a control over porosity [17] which was reasoned to the enhancement of nucleation sites.

5.1.2 Analysis of Functionally Graded Plates (FGPs)

Plate analysis is basically a three-dimensional problem that is simplified to a two dimensional

analysis for ease of calculations and less computational cost. The two-dimensional theories can

be derived by making suitable assumptions about the kinematics of displacements and/or the

distribution of stress through the thickness of the plate. There are many techniques available for

reducing the three-dimensional equations of the theory of elasticity to two-dimensional

equations. The aspects of this reduction problem are summarized by Gol’denveizer [18]. Since

1947 when Lekhnitski introduced anisotropic plates [19], considerable progress has been made

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in the analysis of plates. The theories of plate analysis maybe classified to mainly three

categories; 1) Classical plate theory (CPT), 2) First order shear deformation theory (FSDT), and

3) Higher order shear deformation theories (HSDTs). The classical laminate plate theory is based

on the Kirchhoff hypothesis that straight lines normal to the un-deformed mid-plane remain

straight and normal to the deformed mid-plane and do not undergo stretching in the thickness

direction. These assumptions imply vanishing of the transverse shear and normal strains. Several

modification were done after that by many researchers but Pagano [20] showed the inadequacy

of the classical plate theory for the analysis of thick laminated plates as the theory is neglecting

transverse shear deformations. The ratio difference of in-plane Young’s modulus to transverse

shear modulus of anisotropic plates dictates the need for calculating transverse shear stresses.

FSDT is the first try to calculate those stresses. In FSDT it is assumed that the normals to mid-

plane remain straight after deformation but does not necessarily perpendicular as assumed in

CPT. Thus by including transverse shear strains as constant through the thickness, it follows that

the transverse shear stresses will be also constant. FSDT was first proposed by Reissner in 1945

[21] and then further developed later by Mindlin [22]. These theories were extensively used by

researchers to estimate the effect of transverse shear on moderately thick to thick laminated

plates. FSDT can predict better deflections and in-plane stresses than CPT for thin (side-to-

thickness ratio > 30) and moderately thick (10 < side-to-thickness ratio < 30) plates. Moreover,

FSDT take into account the transverse shear strains and stresses whereas CPT neglects them.

However, the assumption of constant transverse shear stress through thickness is not the actual

case as obtained from three dimensional elasticity. Hence, there emerged a need to develop

theories to accurately predict transverse shear stresses. This is accomplished by using higher

order functions for approximating displacement fields through the thickness. Those higher order

shear deformation theories (HSDTs) relaxed the assumption of normality and straightness of

normal to the mid-surface. That implies parabolic or higher variation of transverse shear strains

through thickness. The consistency of transverse strain energy need to be considered when

developing HSDTs [23] which had been rarely considered by researchers. The free traction

condition needs to be satisfied also when no external surface loads are applied. These two points

will be discussed later in the numerical procedure section.

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Following is a brief to some of those HSDTs focusing on the topic of this work. Suresh et al.

[24], studied the non-linear bending of FGPs with an HSDT. However, both of the free traction

condition, and the consistency of transverse shear strain energy on the displacement field were

not satisfied. Mahi et al. [25] satisfied the free traction condition in their formulation for the

study of bending and free vibration problem of FG plates. They assumed a hyperbolic shape

function that accounts for parabolic variation of transverse shear strains and stresses across the

thickness. The hyperbolic function was replaced by trigonometric function in the HSDT by

Tounsi et al. [26] for the study of thermoelastic buckling of FG sandwich plates . The condition

of free traction was satisfied and the displacement in their work was split to bending and shear

components to reduce the number of unknowns. The same HSDT was used by Meziane et al.

[27] to study the buckling and free vibration of exponentially FG sandwich plates. The authors

compared trigonometric and polynomial shape functions for stratifying the free traction

condition in the assumed displacement filed where both give very close results. Zidi et al. [28]

implemented the same four parameters HSDT developed in [25-27] with polynomial shape

function to satisfy the free traction condition for the study of bending of FG plates under hygro-

thermo-mechanical loading. Mantari et al. [29] developed an analytical solution for the bending

behavior of FGPs using a trigonometric based HSDT. The properties were assumed to vary

exponentially through the plate thickness. The same authors applied their HSDT to curved

functionally graded shells [30], and it yields good results compared to the other existing theories.

Thai et al. [31], developed analytical solutions for the bending and free vibration analysis of

simply supported functionally graded plates with HSDT. The applicability of HSD theories for

viscoelastic behavior of plates was verified by Nguyen et al. [32], showing very good agreement

with the 3D solution. Tran et al. [33], presented a C1 continuity solution for HSDT equations

with the use of B-spline functions in the finite element formulation. Their approach provides

very accurate results compared to the C0 continuity solutions that most of researchers attempted.

Senthil et al. [34], obtained an exact solution for three-dimensional deformations of a simply

supported functionally graded rectangular plate subjected to mechanical and thermal loads on its

top and/or bottom surfaces. Mantari et al. [35], modified their previously developed

trigonometric based HSD theory to include the thickness stretching which most HSD theories

suffer from. They derived analytical solutions for the static analysis of functionally graded

plates. More discussion is devoted to this point in the section of numerical procedure. Many

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other HSD theories are being developed with different formulation for the displacement field

and/or the adopted element/geometry for analysis [36-38].

The porous structures have been accounted for in plate theories either with microstructural

detail-free estimates or with procedures that include the microstructure details. Microstructural

detail-free estimates include role of mixture, self-consistent and generalized self-consistent

schemes, Mori–Tanaka scheme and others. Most of the work in literature implements the role of

mixture to account for pores as voids. Some of the work in this context covers the wave

propagation analysis in FG plates with porosities by Yahia et al. [39] and bending behaviour of

FG porous viscoelastic plates by Altenbach and Eremeyev [40]. Kou and Tan [41] developed a

procedure for including details of the microstructure for graded porosities and pore distributions

with Voronoi diagram and B-Spline representation. Their procedure can be used to obtain the

effective properties of the graded porous structure and then incorporated in plate theories. Other

researchers assumed generalized functions of properties through the functionally graded porous

thickness like the work done by Magnucki and Stasiewicz [42] for the analysis of elastic

buckling of functionally graded porous beams. The present work belongs to the class of

techniques that include the microstructural details. The FG porous structure is projected to a

statistical function with three control points that accounts for pore sizes and distribution across

the graded direction.

5.2 Experimental Procedure

5.2.1 Material and Processing

The process for fabricating FG PLA porous structures is adopted from the work of Nadella et al.

[5] on constrained foaming. PLA raw material is grade 3052D from NatureWorks in pellets

form. Hydraulic compression mold was used at 185 oC and 3 tons pressure to make the plate-like

samples with dimensions of 35 mm x 13 mm x 2.5 mm (Length x Width x Thickness). The

samples are then placed in a pressurized chamber of carbon dioxide (CO2) at 400 psi for 48

hours. CO2 is the physical foaming agent in this process. The pressure and time of saturation are

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being selected based on previous experience with PLA to assure enough quantity of the physical

blowing agent being dissolved. Skin layer can be formed if the samples left at atmospheric

pressure for some time (desorption) after the 48 hours of pressurizing in CO2. However, the

samples in this work are annealed between two hot platens at different temperatures without

desorption immediately after the pressure release. The higher temperature is denoted as TH and

the lower temperature by TL in the remaining of the text. The minimum of platens temperature is

the glass transition point of PLA (56 oC). This causes a gradual distribution in cells’ sizes and

density because of the induced thermal gradient. The cells grow bigger in size from the platen of

lower temperature to the platen of higher temperature. Two samples are fabricated for each

thermal gradient while keeping the other processing parameters fixed (Dimensions, saturation at

400 psi for 48 hours).

5.2.2 Characterization

Scanning Electron Microscopy and relaxation test are used for the characterization purpose. One

of the two samples is frozen in liquid Nitrogen for some time, fracture by shearing, and finally

sputter coated with platinum before imaging under the scanning electron microscope (SEM)

(JEOL JSM 6060). The samples were fractured at different planar locations for averaging

purpose. ImageJ software is used for further analysis of SEM images to extract quantitative

details about the microstructure like cell sizes and distribution. The other sample is tested with

DMAQ800 dynamic analyzer to extract the viscoelastic material parameters (relaxation function

in bending mode).

5.3 Numerical Procedure

5.3.1 The Novel Plate Theory

There are many HSD based theories developed since 1970s and a detailed review can be found in

[43]. The HSD plate theory has basically the same assumptions of the CPT and FSDT but

relaxing the assumption of straightness and normality of the transverse normals after

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deformation. That is accomplished by expanding the in-plane displacements u (along x-axis) and

v (along y-axis) as cubic functions (or even higher) of the thickness coordinate (z-axis). Most of

those theories suffer from the inextensibility assumption of the normals to mid-surface by

assuming the normal displacement w (along z-axis) as a function of x and y only. However,

Simmonds [44] shows that even the accuracy of classical lamination plate theory improves if the

inextensibility assumption of transverse normal is released. The extensibility of transverse

normal has been incorporated in plate theories by modifying the assumed function of w to

include the thickness coordinate (z-axis). Neves et al. [45], Vidoli and Batra [46], Thai and Choi

[47] assumed polynomial expansion of w in the thickness coordinate. Trigonometric functions in

the thickness coordinate for w assumed by Fekrar et al. [48] and Houari et al. [49]. Hyperbolic

functions were considered by Belabed et al. [50] and Hebali et al. [51]. The extensibility is

considered in this work by assuming the transverse normal displacement to be a linear function

of thickness coordinate. Consider the displacement field initially proposed by Christensen and

Wu [52]:

𝑢(𝑥, 𝑦, 𝑧) = 𝑢0(𝑥, 𝑦) + 𝑧 𝜙𝑥(𝑥, 𝑦) + 𝑧2 𝜃𝑥(𝑥, 𝑦) + 𝑧3 𝜆𝑥(𝑥, 𝑦)

𝑣(𝑥, 𝑦, 𝑧) = 𝒗0(𝑥, 𝑦) + 𝑧 𝜙𝑦(𝑥, 𝑦) + 𝑧2 𝜃𝑦(𝑥, 𝑦) + 𝑧3 𝜆𝑦(𝑥, 𝑦)

𝑤(𝑥, 𝑦, 𝑧) = 𝑤0(𝑥, 𝑦) + 𝑧 𝜙𝑧(𝑥, 𝑦)

(5.1)

where 𝑢0 , 𝜙𝑥 , 𝜃𝑥 , 𝜆𝑥 , 𝑣0 , 𝜙𝑦 , 𝜃𝑦 , 𝜆𝑦 , 𝑤0 , 𝑎𝑛𝑑 𝜙𝑧 are functions to be determined.

The functions 𝑢0, 𝑣0 and 𝑤0 are specifically the mid-surface displacements along x-, y-, and z-

axis respectively.

The ten unknown functions of equation (5.1) can be reduced by imposing the free traction

condition on the upper and lower surfaces as expressed in equation (5.2);

102

𝜎𝑥𝑧 (𝑥, 𝑦, ±ℎ

2) = 0, and 𝜎𝑦𝑧 (𝑥, 𝑦, ±

ℎ

2) = 0 , where h is the plate thickness. (5.2)

The consistency of transverse shear strain energy is further imposed before presenting the

reduced displacement field. This criterion was proposed by Love [53] to check the accuracy of

the theories of thin elastic shells. The transverse shear strain energy of an isotropic or orthotropic

plate is only associated with the transverse shear stresses and strains [23]. The displacement field

of a plate theory can be constrained to satisfy the transverse shear strain energy of an equivalent

beam which is well-known from 3D elasticity solution of elastic plates. For an isotropic plate

with shear modulus G, the transverse shear stress resultants Qx and Qy can be expressed as [23];

𝑄𝑥 =5

6𝐺ℎ ( 𝜙𝑥(𝑥, 𝑦) +

𝜕𝑤0(𝑥,𝑦)

𝜕𝑥), and 𝑄𝑦 =

5

6𝐺ℎ ( 𝜙𝑦(𝑥, 𝑦) +

𝜕𝑤0(𝑥, 𝑦)

𝜕𝑦) (5.3)

Then the transverse shear strain energy 𝛱, can be calculated;

𝛱𝑥𝑧𝐵𝑒𝑎𝑚 =

5

12𝐺 ( 𝜙𝑥(𝑥, 𝑦) +

𝜕𝑤0(𝑥,𝑦)

𝜕𝑥)2ℎ , and 𝛱𝑦𝑧

𝐵𝑒𝑎𝑚 =5

12𝐺 ( 𝜙𝑦(𝑥, 𝑦) +

𝜕𝑤0(𝑥, 𝑦)

𝜕𝑦)

2

ℎ (5.4)

By reducing the displacement field in (5.1) with the free traction condition of (5.2), and

integrating over the thickness for Π, the following relations can be derived:

𝛱𝑥𝑧𝐻𝑆𝐷𝑇 =

4

15𝐺𝜉𝑥𝑧

2 ( 𝜙𝑥(𝑥, 𝑦) +𝜕𝑤0(𝑥,𝑦)

𝜕𝑥)2

ℎ , and 𝛱𝑦𝑧𝐻𝑆𝐷𝑇 =

4

15𝐺𝜉𝑦𝑧

2 ( 𝜙𝑦(𝑥, 𝑦) +𝜕𝑤0(𝑥, 𝑦)

𝜕𝑦)

2

ℎ (5.5)

103

where 𝜉𝑥𝑧 and 𝜉𝑦𝑧 are the parameters of equivalency that can be obtained by equating equations

(5.4) and (5.5) which reveals a value of 5/4. The constrained displacement field can be expressed

now as:

𝑢(𝑥, 𝑦) = 𝑢0(𝑥, 𝑦) + 𝑧 (1

4 𝜕𝑤0(𝑥, 𝑦)

𝜕𝑥+

5

4 𝜙𝑥(𝑥, 𝑦)) −

1

2𝑧2

𝜕 𝜙𝑧(𝑥, 𝑦)

𝜕𝑥−

5

3ℎ2𝑧3 (

𝜕𝑤0(𝑥, 𝑦)

𝜕𝑥+ 𝜙𝑥(𝑥, 𝑦))

(5.6) 𝑣(𝑥, 𝑦) = 𝑣0(𝑥, 𝑦) + 𝑧 (

1

4 𝜕𝑤0(𝑥, 𝑦)

𝜕𝑦+

5

4 𝜙𝑦(𝑥, 𝑦)) −

1

2𝑧2

𝜕 𝜙𝑧(𝑥, 𝑦)

𝜕𝑦−

5

3ℎ2𝑧3 (

𝜕𝑤0(𝑥, 𝑦)

𝜕𝑦+ 𝜙𝑦(𝑥, 𝑦))

𝑤(𝑥, 𝑦, 𝑧) = 𝑤0(𝑥, 𝑦) + 𝑧 𝜙𝑧(𝑥, 𝑦)

It is worth mentioning that a similar displacement field was derived by Murthy and the modified

field of Levinson which are both summarized in [23]. However, the two proposed fields do not

include thickness extensibility where the transverse displacement w was assumed to be a

function of planar dimensions only. The inclusion of extensibility is important in the analysis of

cellular materials at which different planar sections through the thickness may experience

different transverse deflection resulting from the gradual cellular structure. Thus the assumption

of rigid transverse deflections is not valid for polymeric cellular materials. However, it is still a

very good approximation for metallic-ceramic FGMs.

5.3.2 Linear Viscoelastic Constitutive Law

The constitutive law assumed for PLA is a linear viscoelastic model based on Boltzmann

superposition principle. Creep and stress relaxation are two phenomena characterizing

viscoelastic materials. Creep is an increase in deformation under constant load while relaxation is

a decrease in load with constant deformation. The stress at a point in time doesn’t depend on the

strain at that point only but rather depends on the whole strain history up to the current time

starting from an assumed virgin state of the material. Thus viscoelastic materials are said to have

a memory effect. The memory effect is neglected in linear viscoelasticity while the effect of

strain history being approximated by piecewise constant step functions and superimposed to the

104

elastic response [54]. In experimental tests on viscoelastic materials, it is observed that the

instantaneous changes in stress or strain are governed by Hooke’s law, that is, the instantaneous

response is elastic [55]. This yield the following constitutive law [54]:

𝝈(𝒙, 𝑡) = 𝑫(𝒙, 𝑡, 0)𝜺(𝒖(𝒙, 0)) + ∫ 𝑫(𝒙, 𝑡, 𝜏)𝜺 (𝑑𝒖(𝒙, 𝜏)

𝑑𝜏)

𝑡

0

d𝜏 (5.7)

where 𝜏 is the history variable and D is the stress relaxation matrix obtained experimentally.

Based on the law of conservation of momentum, and considering the quasistatic equations of

equilibrium:

∑𝜕𝜎𝑖𝑗

𝜕𝑥𝑗

(𝒙, 𝑡) = 𝑓𝑖(𝒙, 𝑡), 𝑖 = 1,2,3

3

𝑗=1

(5.8)

With the boundary conditions:

𝒖(𝒙, 𝑡) = 0, 𝒙 ∈ 𝜕Ω𝑔

(5.9) ∑𝜎𝑖𝑗�̂�𝑗(𝒙, 𝑡)

3

𝑗=1

= 𝒈𝑖(𝒙, 𝑡),

𝒙 ∈ 𝜕Ω𝑡

105

Fig 5.1. Arbitrary region in real space

where 𝜕Ω𝑔 and 𝜕Ω𝑡 are the regions of prescribed displacement 𝒖(𝒙, 𝑡) and external traction

𝒈𝑖(𝒙, 𝑡) traction as defined in Fig. 5.1.

Fully discretizing equation (5.8) in time and space yields:

∑∆𝑑𝑞

𝑘𝑞∫ 𝑨(𝑡𝑗 , 𝜏)𝑑𝜏 = 𝑭(𝑡)

𝑡𝑞

𝑡𝑞−1

𝑗

𝑞=1

(5.10)

where;

𝑨(𝑡𝑗 , 𝜏) = ∫ 𝑩𝑇(𝒙)𝑫(𝒙, 𝑡, 𝜏)𝑩(𝒙)𝑑ΩΩ

(5.11)

where q, ∆𝑑𝑞, 𝑘𝑞, and 𝑩(𝒙) are the number of time steps, difference of nodal displacements

between two successive time points, size of the time step, and the strain-displacement matrix

106

respectively. The derivation details can be found in a standard finite element textbook or referred

to [55] for brevity.

A two-terms Prony series is assumed for the relaxation function of PLA. The general form of

Prony series reduced to two terms is:

𝐸(𝑡) = 𝐸∞ + ∑ 𝐸𝑗𝑒−𝛼𝑗𝑡𝑁

𝑗=1 = 𝐸∞ + 𝐸1𝑒−𝛼1𝑡 + 𝐸2𝑒

−𝛼2𝑡 , for N = 2 (5.12)

where 𝐸∞ is the long term modulus and 𝛼𝑗′𝑠 are the relaxation time constants.

The stress relaxation function was obtained by DMAQ800 at bending mode for a solid PLA

samples made by the previously mentioned procedure. Fig. 5.2 below shows the curve-fitted to

two-terms Prony series.

Fig 5.2 Stress relaxation function of solid PLA fitted to Prony series.

107

The fitting results revealed the following material parameters for PLA:

Table 5.1. Prony series parameters for solid PLA

𝐸∞

(MPa)

𝐸1

(MPa) 𝛼1

𝐸2

(MPa) 𝛼2

8.81E+08 1.04E+08 3.233 2.92E+08 0.1393

Poisson’s ratio is assumed to be a function of time as derived from the correspondence principle.

The Poisson’s ratio for an isotropic elastic solid can be expressed in terms of the bulk

compliance K = 1/k where k is the bulk modulus. For simplicity of analysis, it is assumed that

bulk compliance K is constant over the time domain. By the use of correspondence principle and

the equivalent strain equation of (5.7), the following formula for Poisson’s ratio can be derived

[56]:

𝑣(𝑡) =1

2−

1

6K𝐸(𝑡) (5.13)

5.3.3 Homogenization Model

The approach for predicting and representing the effective properties of FG porous structures is a

modified version of Gibson-Ashby formula [57] and Ma et al. [58]. The approach is based on

adding a statistical factor, 𝜆, that represents cells’ distribution on average sense. The original

Gibson-Ashby formula for the effective elasticity modulus and [58]’s modification are given in

equations (5.14) and (5.15) respectively.

108

𝐸∗

𝐸𝑠= 𝐶1𝐶

2 (𝜌∗

𝜌𝑠)2

+ 𝐶1′(1 − 𝐶)

𝜌∗

𝜌𝑠 (5.14)

𝐸∗

𝐸𝑠= (𝜆𝐶)2 (

𝜌∗

𝜌𝑠)2

+ (1 − 𝜆𝐶)𝜌∗

𝜌𝑠 (5.15)

where C is the portion of solid in the cell edges, 𝐶1 and 𝐶1′ are constants, (*) denotes the porous

structure properties where (s) is for the foaming solid material with E is the modulus of elasticity

and 𝜌 is the density.

The procedure is based on curve fitting of the cells’ distribution to a cumulative probability

function. The authors in [58] tested several probability distribution functions and the best fit was

with Burr distribution. Burr distribution cumulative function is defined as;

𝐹(𝑥) = 1 − (1 + (𝑥/𝛽)𝛼)−𝛾 (5.16)

where 𝛽, 𝛼, and 𝛾 are fitting parameters. The independent variable, x, is the cell area in this work

which is a replacement to cell volume that was used in [58]. It is easier to extract the area from

SEM images than the volume from X-ray Microtomography which goes very complicated and

not practical. The factor 𝜆 is defined as;

𝜆 = 𝑒𝑎𝛾+𝑏𝛼+𝑐𝛽+𝑑 (5.17)

where a, b, and c are fitting parameters.

109

The formulation developed by Ma et al. [58] was further modified to eliminate density from the

equation and replace the modulus of elasticity with the relaxation modulus. The effort spent on

analyzing the microstructure should be enough to characterize the cellular structure and hence

predict the properties. Our hypothesis is based on that the knowledge of the properties of base

foaming material and the microstructure are enough to predict the properties of the porous

structure. Equation (5.15) is modified to:

𝐸∗(𝑡)

𝐸𝑠(𝑡)= Γ(1 + ∑𝑎𝑖𝑒

−(𝑏𝑖𝛾𝛼+𝑑𝑖𝛽)𝑡

𝑁

𝑖=1

) (5.18)

where 1 − 𝛤 is an averaged porosity taken at many sampling points through the thickness, N is

the number of terms in the series to fit experimental data, 𝑎𝑖, 𝑏𝑖, and 𝑑𝑖 are fitting parameters. Γ

is defined as:

𝛤 = n−(∑ 𝑐𝑖

𝑛𝑖=1 )

n , where n is the number of zones and ci is the porosity at the ith zone (5.19)

= 3−𝑐1−𝑐2−𝑐3

3 , for i = 3 (5.20)

The relative density assumes that 𝜌∗ = 𝛤𝜌𝑠 where 𝜌∗ is the effective density of the porous

material and 𝜌𝑠 is the density of base material. This assumption had been tested in the case of

homogeneous cellular structure with TH = 130 oC and TL = 130

oC. The SEM image in Fig 5.6b

was analyzed for porosity as explained later which reveals a value of 0.304. Noting that there are

no distinct zones as in the case for FG plate which implies that all ci’s are the same which gives a

value of 0.696 for Γ and hence ρ∗ is 0.863 g/cm3. The density of porous structure was determined

experimentally with the “Density Determination Kit” by Denver Instruments and it gave a value

110

of 0.880 g/cm3. This gives an error of 1.94% only which is very small compared to the simplicity

of analysis. The accuracy will increase as the number of sampling points in 𝛤 increase.

The disadvantage of the proposed formula is the dependency on curve fitting but it takes into

account the microstructural details and statistical distribution of cells. Nevertheless, this

disadvantage is alleviated when considering the curve fitting as tabulating material parameters

the way it is done for conventional materials as modulus of elasticity, Poisson’s ratio and so on.

Those parameters can then be used for different FG PLA with different microstructure. The only

effort that is required is an SEM image of the microstructure and few steps of image processing

that will be explained in section 5.4.2.

5.3.4 Finite Element Formulation and Element Selection

The general procedure for defining a finite element model is well known [59] and is only briefly

stated here. The finite element equations are obtained by discretizing the planar plate domain ‘A’

into ‘nelem’ elements. Each element ‘e’ has ‘enodes’ nodes, where each node is identified with a

number of degrees of freedom per node following the element formulation. The presence of first

derivatives of 𝑤0 and 𝜙𝑧 in the displacement field presented in (5.6) necessitates the existence

and continuity of their second derivatives since they will appear in the energy equation

(Zienkiewicz [59]). This problem can be alleviated with the implementation of

conforming/nonconforming finite elements or with the use of penalty formulation.

Four different elements have been tested for convergence and accuracy with the proposed HSDT.

The first is the typical 9-nodes Lagrange element. The six degrees of freedom indicated by

equation (5.6) are represented by Lagrange isoparametric shape functions where the

displacement vector at any arbitrary point within each element can be obtained as ∑ 𝑁𝑖𝑑𝑖𝑒𝑛𝑜𝑑𝑒𝑠𝑖=1 ,

where 𝑑𝑖 = [𝑢0 𝑣0 𝑤0 𝜙𝑥 𝜙𝑦 𝜙𝑧 ]𝑇. The shape functions Ni can be found in any standard finite

element textbook. With this approach, the derivatives are not continuous across elements’

111

boundaries. This element is indicated as “Q9-Lagrange” in Fig 5.3 and it shows an apparent

divergence even with increasing number of elements. As a solution to the derivatives continuity

of “Q9-Lagrange” element, “Q9-Penalty” was developed using penalty formulation to enforce

derivatives continuity at the nine nodes. Each node is having 10 degrees of freedom with 𝑑𝑖

defined as;

[𝑢0 𝑣0 𝑤0 𝑑𝑤0𝑥 𝑑𝑤0

𝑦 𝜙𝑥 𝜙𝑦 𝜙𝑧 𝑑 𝜙𝑧

𝑥 𝑑 𝜙𝑧

𝑦]𝑇

where the following conditions are enforced:

𝑑𝑤0𝑥 =

𝜕𝑤0

𝜕𝑥 , 𝑑𝑤0

𝑦=

𝜕𝑤0

𝜕𝑦 , 𝑑 𝜙𝑧

𝑥 =𝜕 𝜙𝑧

𝜕𝑥 , 𝑑 𝜙𝑧

𝑦=

𝜕 𝜙𝑧

𝜕𝑦 (5.21)

The basic constraint equation for the penalty formulation is:

[𝑪]{𝑫} − [𝑸] = {𝟎} (5.22)

where {D} is the assembled vector of system degrees of freedom, [C] and {Q} are constants

matrices to enforce the constraints.

From the constraint relation (5.22):

{𝒕} = [𝑪]{𝑫} − {𝑸} (5.23)

112

so that {t} = {0} defines satisfaction of the constraints. The potential energy function Πp can be

augmented by a penalty function {𝒕}𝑇[𝝑]{𝒕}/2 where [𝝑] is a diagonal matrix of penalty

numbers. Thus;

𝛱𝑝 =1

2{𝑫}𝑇[𝑲]{𝑫} − {𝑫}𝑇{𝑭} +

1

2{𝒕}𝑇[𝝑]{𝑡} (5.24)

The penalty of constraint violation becomes greater as [𝝑] increases. From equations (5.23),

(5.24), and the minimum condition {𝜕𝛱𝑝 𝜕𝑫⁄ } = {𝟎}:

([𝑲] + [𝑪]𝑇[𝝑][𝑪]){𝑫} = {𝑭} + [𝑪]𝑇[𝝑]{𝑸} (5.25)

Up to this point, the penalty formulation is complete. More details can be found in [60]. This

element converged in a similar trend to the nonconforming element as indicated in Fig 5.3 and

will be explained below.

The third element is the 4-nodes nonconforming quadrilateral element developed by Melosh [61]

and Zienkiewicz and Cheung [62]. This element doesn’t account for continuity of mixed

derivatives (𝜕 𝜕𝑥𝜕𝑦⁄ ) at which it is not considered as a degree of freedom for nodes. Thus the

nodal degrees of freedom vector is defined as [𝑢0 𝑣0 𝑤0 𝜕𝑤0

𝜕𝑥 𝜕𝑤0

𝜕𝑦 𝜙𝑥 𝜙𝑦 𝜙𝑧

𝜕 𝜙𝑧

𝜕𝑥 𝜕 𝜙𝑧

𝜕𝑦]𝑇

. The

convergence of this element is very similar to the Q9-Penalty since both of them in principle

account for derivatives continuity across element boundaries except the mixed derivative.

113

Fig 5.3 Elements convergence to the exact solution of central transverse deflection of a simply

supported plate subjected to uniform pressure.

However, the element adopted in this work is the 4-nodes conforming quadrilateral element

developed by Bogner, Fox, and Schmidt [63] which offer the best convergence over the other

three elements. Lagrange interpolation is added to the nodal degrees of freedom to account for

in-plane displacements while the Hermite representation is used for 𝑤0 and 𝜙𝑧 only. The

objective of the mixed approach is to reduce computational cost. Thus, there are 12 degrees of

freedom per node and a total of 48 degrees of freedom per element. The nodal displacement

vector “d” is then [𝑢0 𝑣0 𝑤0 𝜕𝑤0

𝜕𝑥 𝜕𝑤0

𝜕𝑦 𝜕2𝑤0

𝜕𝑥𝜕𝑦 𝜙𝑥 𝜙𝑦 𝜙𝑧

𝜕 𝜙𝑧

𝜕𝑥 𝜕 𝜙𝑧

𝜕𝑦 𝜕2 𝜙𝑧

𝜕𝑥𝜕𝑦]𝑇

. The Hermite

interpolation functions for this element are [64] (refer to Fig 5.4):

𝜑𝑖𝑒 = 𝑔𝑖1 (𝑖 = 1,5,9,13); 𝜑𝑖

𝑒 = 𝑔𝑖2 (𝑖 = 2,6,10,14);

(5.26)

𝜑𝑖𝑒 = 𝑔𝑖3 (𝑖 = 3,7,11,15); 𝜑𝑖

𝑒 = 𝑔𝑖4 (𝑖 = 4,8,12,16);

where:

114

𝑔𝑖1 =1

16(𝜉 + 𝜉𝑖)

2(𝜉0 − 2)(𝜇 + 𝜇𝑖)2(𝜇0 − 2)

(5.27)

𝑔𝑖2 =1

16𝜉𝑖(𝜉 + 𝜉𝑖)

2(1 − 𝜉0)(𝜇 + 𝜇𝑖)2(𝜇0 − 2)

𝑔𝑖3 =1

16𝜇𝑖(𝜉 + 𝜉𝑖)

2(𝜉0 − 2)(𝜇 + 𝜇𝑖)2(1 − 𝜇0)

𝑔𝑖4 =1

16𝜉𝑖𝜇𝑖(𝜉 + 𝜉𝑖)

2(1 − 𝜉0)(𝜇 + 𝜇𝑖)2(1 − 𝜇0)

Fig 5.4 A conforming quadrilateral element with four nodes


5.4.1 Fabrication of Functionally Graded PLA

The functionally graded PLA had been fabricated successfully with a gradually changing

porosity as shown in Fig 5.5. The microstructure is observed to be of three main zones starting

with large cells at the side of higher temperature (zone 1), then having a transition region with

cells’ sizes shifting to smaller pores (zone 2), then a sudden drop of porosity with cells

distributed in islands-like structure but with much smaller pores than zones 1 and 2. The SEM

images shown in Fig 5.5 are for TH = 130 oC, TL = 80

oC and 5 minutes annealing. Zone 3

115

dominates 77% of the thickness but 23% for the zones 1 and 2 combined. This depth of the zones

is believed to be a function of annealing time for fixed pressure and saturation time, TH and TL as

will be proved later.


oC, and 5 minutes annealing

The transition from continuous cellular structure to isolated cells distribution is very rapid (Fig

5.6a). This can be attributed to the non-uniform thermal gradient due to very low thermal

conductivity of PLA. The inclusion of conductive fillers through the thickness will probably

enhance a smoother transition of cells. A homogeneous cellular distribution is produced when

both the platens were set to 130 oC with an averaged cells’ sizes of 4.0631 𝜇𝑚2 as shown in Fig

5.6b. The ratio of the total thickness between the three zones changed when the annealing time

was increased to 10 minutes (Fig 5.7) and the transition is smoother. The whole thickness is

cellular but with increasing cells’ sizes and porosity from TH towards TL. Larger average cell size

116

(8.654 𝜇𝑚2) and porosity (0.321) at TH (130 oC) due to higher formation and expansion of cells.

At the other end of 80 oC, much smaller cells’ sizes and density is observed (Fig 5.7).

(a) Transition zone (b) Ave. cell area 4.0631𝜇𝑚2

Fig 5.6. Comparing homogeneous to functionally graded cellular distribution


oC, and 10 minutes annealing

117

5.4.2 Microstructure Characterization

The SEM images are obtained as indicated in section 5.2.2. ImageJ software is then used for

further processing. All images were converted to binary format first and adjusted for height

variability. The cells’ sizes are then extracted in tabulated form. A MatLAB code was written for

representing the data and fitting to Burr distribution. Fig 5.8 shows the different steps in image

processing procedure for the case of 130 oC (TH), 80

oC (TL), and 5 minutes annealing. The

fitting parameters of equation (5.16) for two annealing conditions are presented in Table 5.2.

Those two conditions will be used for further analysis in this work.

Table 5.2. Burr distribution parameters for the TH side of FG PLA (Reference to Fig 5.8)

TH

(oC)

TL

(oC)

𝜷 𝜶 𝜸

130 80 2.0 0.7094 1.937

130 130 2.0 0.7437 1.43

a) Original image b) Binary image

Fig 5.8.1. Image Processing with thresholding

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c) Frequency distribution of cells’ area d) Curve fitting with Burr distribution

Fig 5.8.2. Statics about the microstructure

Fig 5.8. Microstructure characterization and fitting to Burr distribution

5.4.3 Relaxation Modulus of Functionally Graded Porous Structures

The relaxation modulus was obtained experimentally from the relaxation test for solid PLA,

homogeneous cellular PLA, and 130-80 oC FG PLA (Fig 5.9). Functionally graded PLA shows

higher values for the relaxation modulus compared to the homogeneous cellular PLA plates

which is due to the gradient of cells distribution. The drop in relaxation modulus for the FG PLA

is 8.44% only while it is 46.72% for the homogeneous case. The density of solid PLA is 1.24

g/cm3 but it is measured to be 0.905 g/cm

3 for the FG PLA using the density kit. The drop in

density for the FG PLA is 27% while the drop in relaxation modulus is 8.44% only. However,

for the homogeneous cellular PLA, the density drop is 29.06% only compared to 46.72% drop in

the modulus. This evident the potential of FG cellular materials as light weight but strong

structures that can find a wide range of applications in many industries like automotive and

aerospace load bearing members. This also motives the efforts done in this work to model such

materials for tailoring their properties to suite certain applications.

119

Fig 5.9. Relaxation modulus of the solid, homogeneous and FG porous PLA.

As indicated before, the porosity of homogeneous porous PLA is 0.304 as obtained from the

processing of SEM images. The porosity was also calculated from SEM images of the 130-80 oC

FG PLA at 10 minutes annealing for the three zones to be 0.321, 0.185, and 0.074 respectively.

Two terms are selected for equation (5.18) along with the parameters in table 5.1 to perform non-

linear least squares regression between the 130-130 oC homogeneous case and the 130-80

oC FG

case with 10 minutes annealing. The fitting parameters are listed in table 5.3.

Table 5.3. The fitting parameters for equation (18) with two terms

a1 b1 c1 a2 b2 c2

-0.265 0.047 -0.035 0.124 -4.148 5.662

The selection of fitting cases is assumed to be between the FG case and a homogeneous case at

one the annealing temperatures of the FG case. The fitting of equation (5.18) is very good

compared to the experimental data for the 130-80 oC FG PLA as shown in Fig 5.10. Equation

120

(5.18) can be used directly for numerical analysis as a representation of relaxation modulus. The

fitting parameters may be listed for a material covering a range of porous structures which in turn

can be used to interpolate for other porous structure of the same material. Consequently, equation

(5.18) has the capability of predicting the relaxation modulus of a functionally graded structure

from the modulus of the base material and interpolated fitting parameters. Analyzing the

microstructure of that material will be then sufficient for predicting the relaxation modulus.

Fig 5.10. Comparing the experimental and predicted relaxation modulus of FG 130-80 oC

5.4.4 Numerical Results

The numerical procedure was first validated for linear elasticity of metallic-ceramic based FGMs

since most literature is devoted to this class of material [2]. The FG plate considered for

validation is composed of aluminum (Al) at the bottom surface and gradually changing to Silicon

carbide (SiC) at the top surface according to 1 2 ∗⁄ (½ + 𝑧/ℎ)𝑛 where n is the volume fraction

index and vary from 0 to ∞. The results of current procedure compare very well to the exact

solution given by Senthil and Batra [34] as presented in Fig 5.11 for the normalized central

transverse deflection (�̅� = 100 ∗ 𝐸𝐴𝐿 ∗ ℎ3 ∗ 𝑤 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 ∗ 𝑎4⁄ ) of a simply supported

functionally graded Al-SiC plate under a uniform pressure of -EAl*h4. The results of CPT are

way far from the exact solution for thick plates. The error in stresses is higher than those in

121

displacement [34]. This error of CPT in the transverse deflection can be attributed to the large

shear deformation that occurs in thick plates but it is ignored by CPT. The FGPs analyzed in this

work are thick (a/h ≈ 5) which necessitates a theoretical formulation that accounts for shear

deformations. The proposed HSDT accounts for shear deformation through the higher order

terms in the assumed displacement field.

Fig 5.11. Comparing exact transverse deflection of SiC FG plate to HSDT and CPT for a

uniform pressure and simply supported condition

The linear viscoelastic code was verified for predicting the time dependent response of

functionally graded PLA. Equation (5.18) is used as the representation of material properties

with coefficients from tables 5.2 and 5.3. Fig 5.12 shows the force response in a relaxation test of

130-80 oC functionally graded PLA. Both HSDT and experimental data matched very well at the

beginning of relaxation test which could be attributed to the linear elastic response of viscoelastic

materials to instantaneous load [55]. The proposed HSDT agrees very well in linear elastic

materials as already proved in Fig 5.11 also. There is a very small deviation with asymptotically

increasing error to below 2% as the times passes.

122

Fig 5.12. Comparing HSDT with experimental results for a relaxation test of FG 130-80 oC PLA

The proposed procedure accurately predicts the relaxation response of viscoelastic FG porous

thick plates. It includes a homogenization step prior to numerical analysis with the proposed

HSDT. Many HSDTs in the area are equivalently accurate as the present theory. They satisfy the

stretching and the free traction conditions. Some of them also satisfy the consistency of

transverse shear strain energy. More novelty of this work is attributed to the homogenization

procedure that accounts for the microstructural details including pore sizes and distribution

which is directly incorporated in the plate theory at implementation level. Most of the other

theories adopt microstructural detail-free estimates that might not be accurate for complex

microstructures. Statistical estimates like the one presented in this work are more accurate and

accounts for microstructural complexity. The microstructural details (pores sizes and

distribution) are extracted directly from the SEM images in an easy-to-implement methodology

which makes it very accurate and system-specific. The proposed formula for numerical

implementation of the homogenization model could accurately predict the relaxation modulus

from the knowledge of base material properties and microstructure only. Thus no further testing

is required once the microstructure is known of a certain viscoelastic material. Provided a

database for different homogeneous porous structures of the polymer, the homogenization model

can predict the properties for any FG porous structure in between. This in turn makes the

123

homogenization procedure combined with the accurate HSDT very valuable tool for conducting

numerical experiments to design FG porous structures. In a previous work of the authors [65],

the creep compliance of FG acrylonitrile butadiene styrene (ABS) was successfully correlated to

the microstructure. Property-structure and processing graphs have been established for the FG

porous ABS from experimental data. So there is a complete design cycle now of the FG porous

structure, the HSDT is used to conduct numerical experiments with a set of design loads and

boundary conditions implementing the proposed homogenization model. Then the parameters of

the converged homogenization model can be used to map all possible microstructures. Finally, at

least one of the microstructures will have processing graphs to map for the fabrication route. This

route for design saves time and cost for material development. Functionally graded porous

structures are very promising materials due to the potential of significant weight reduction while

maintaining high strength compared to the precursor polymers. One of the practical applications

of FG porous structures is designing materials for high impact absorption [66, 67]. A superior

polymeric-base material system for impact resistance should have gradual transition of material

properties and microstructure from highly viscoelastic at the surface of impact to balanced

hardness and ductility of the polymer system at the receiver side without discontinuity.

Functionally graded porous core with designed cellular structure away from the impact and

receiver surfaces acts as a large set of springs and dashpots that significantly dissipate the load.

The homogenization model and plate theory proposed in this work can handle the design of such

material with accuracy and tackle microstructure complexity.

5.5 Conclusions

A higher order plate theory had been developed for the analysis of functionally graded plates

with a novel homogenization model. The plate theory is based on higher order terms in the

assumed displacement field which were subjected to the condition of satisfying the constraint on

transverse strain energy. A conforming 4-nodes quadrilateral element is adopted for the analysis

based on a convergence test over several candidate elements. A novel homogenization model is

being proposed similar to the form of Prony serious. It is based on statistical fitting to the cells

distribution with Burr function. The results of fitting are used in the proposed model to further

relate the properties of functionally graded porous structure to the properties of base material.

124

The proposed model can predict the properties of the functionally graded porous structure from

the knowledge of base material and microstructure. A data base for the model is the next target

for the authors. Functionally graded plate-like porous structures were fabricated through a

constrained foaming process. The porous structure can be controlled by controlling the

processing parameters like annealing time/temperatures as in this study. The proposed HSDT

and homogenization model agree very well with the work done by other researchers and

experimental data. The numerical tool proved its validity for conducting accurate computer

experiments with the minimal input about material properties and microstructure. This makes it

possible to be utilized in a controlled fabrication process as an inverse-homogenization

procedure to fabricate materials possessing a required design strength.

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(1) The content of this chapter has been published in the Journal of Composites Part B:

Engineering;

F. Al Jahwari and H. E. Naguib, "Finite element creep prediction of polymeric voided composites with 3D

statistical-based equivalent microstructure reconstruction," Composites Part B: Engineering, vol. 99, pp. 416-424,

2016. DOI: 10.1016/j.compositesb.2016.06.042. Reproduced by permission from Elsevier (License #

3902080154469).

Chapter 6

Finite Element Creep Prediction of Polymeric Voided Composites

with 3D Statistical-based Equivalent Microstructure

Reconstruction (1)

The statistical homogenization model which was developed in Chapter 5 depends on

experimental data and needs a large database in order to work independently for the design of FG

porous composite structures. In order to alleviate this problem, a purely numerical

homogenization procedure was developed in this chapter, which can replace the experimental

data in providing the local mechanical properties at the control points. A reduced 3D

reconstruction procedure of voided composite structures based on statistical considerations and

granular mechanics was developed. Voids’ diameters and fractions of the actual microstructure

were extracted with a deterministic locally adaptive threshouling technique. The diameters were

categorized with Freedman-Diaconis method but preserving the overall voids’ fractions. The

simulation box was then created from the reduced voids’ diameters and fractions with granular

mechanics. Numerical experiments were conducted with periodic boundary conditions to the

walls of the simulation box. Voided structures of Acrylonitrile Butadiene Styrene (ABS) were

fabricated with physical foaming agent and tested for creep compliance to validate the proposed

procedure. The agreement with experimental results for creep compliance is very good with

maximum error of 8.62%. The contribution of the procedure is attributed to the simplicity and

131

accuracy in developing representative voided structures from SEM images which otherwise need

to be extracted from tedious processes like X-ray microtomography reconstruction.

6.1 Introduction

Homogenization is very efficient tool in modeling of complex phenomena in heterogeneous

media. That is due to different length scales between the characteristic length of the

microstructural features and the actual dimensions of the part of interest for macroscale analysis.

This is particularly important when dealing with voided structures which usually have very high

degree of microstructural complexity. The complexity arises from the wide range of void sizes,

walls thicknesses, shape, and distribution that all exist in a single microstructure. Viscoelasticity

puts additional temporal challenge for homogenization. An accurate homogenization tool that

can predict the effective properties for further macroscale analysis of such material would be of a

significant value in materials design of advanced structures.

Homogenization of voided structures can be broadly classified to microstructural detail-free

estimates and to procedures that include the microstructure details. Microstructural detail-free

estimates can also be classified to two main approaches based on the way the void is being

treated. The first approach considers the void as a single cell being part of an infinite periodic

structure [1-4]. The second approach considers voids as inclusions with zero shear and bulk

modulus. Some of them account for the interacting fields between the pores while others assume

the diluted state (e.g. role of mixture, self-consistent and generalized self-consistent schemes,

Mori–Tanaka scheme and others [4, 5]). Both approaches subject the idealized structure to

numerical experiments for obtaining the effective properties. Pioneer work in the area covers that

done by Gibson-Ashby [1], Gent and Thomas [2], Dement’ev and Tarakanov [3], Christensen

[4]. Many of the work in literature is expansion of the two microstructural detail-free approaches

to either more complex/stochastic geometrical configurations of the unit cell or microstructure

[6, 7] or to more representative constitutive models of the base material or numerical procedures

[8, 9]. Other researchers extended the existing analytical approaches to more complex

microstructures/materials and syntactic foams [10-13]. Both approaches of the microstructural

132

detail-free estimates are in fact identical as proven in the work by Christensen [4]. One major

disadvantage of both approaches is that they don’t consider the exact geometry of the voids or

distribution. However, it was observed by other researchers [14] that the irregularity of

microstructure has an effect on the mechanical properties. In finite deformation problems, it is

necessary to build more representative microstructure than just describing the volume fraction of

the voids [15].

More accurate approach is the second category of homogenization that accounts for

microstructural details. The process starts with imaging the microstructure to a scale equivalent

to the characteristic void size of the microstructure under consideration. Two routes are followed

after imaging; the first is to analyze the image and extract details about the microstructure like

void’s sizes and distribution which are included in statistical based homogenization models like

the one developed by Ma et al. [16]. Their approach is based on introducing a void volume

distribution coefficient that represents voids’ distribution and sizes which then included to

modified version of Gibson-Ashby formula [1]. The distribution coefficient was obtained by

reconstruction of the voided structure with X-ray microtomography and then curve fitting of the

voids’ volume distribution to a cumulative probability function which was found to best fit with

Burr distribution. The approach provided very accurate results compared to those by the original

Gibson-Ashby formula for linear elastic constants. Farooq and Naguib [17] proposed a similar

statistical approach for the relaxation modulus of viscoelastic solids but based on scanning

electron microscope (SEM) images which is easier, accurate, and more practical to use. The

second route is to replicate an idealized unit cell of the microstructure or directly incorporate the

processed image in numerical homogenization (i.e. finite element/volume analysis). The

processed image is an exact or statistical representation of the microstructure while the replicated

unit cell is based on the extracted geometrical details that are used to reconstruct an idealized but

more realistic model for the cell. Kumar et al. [18] developed analytical procedure and

correlation between porosity and cell geometry that allow reconstruction of representative unit

cell for finite element-based homogenization. The procedure was developed to predict the

effective thermal conductivity of open-cell porous structures. Microstructure topology is shown

to give variability in the effective properties which emphasizes the importance of accurate

microstructure representation.

133

The major drawback of single unit cell reconstruction approach in microstructural-detail

homogenization is the pre-assumed periodicity that doesn’t account for the nature of irregular

distribution that is common in porous/voided structures. That is in addition to the fact that the

whole microstructure is reduced to one void geometry which neglects a lot of the microstructure

details. This work presents an accurate finite element viscoelastic homogenization procedure for

polymeric voided structures by reconstructing equivalent 3D structure based on statistical

considerations. The voided structure was made of acrylonitrile butadiene styrene (ABS) by one-

step foaming process with CO2 as the physical blowing agent. Multiple samples were produced

for the same processing condition for averaging purpose. Some of the samples were imaged with

scanning electron microscopy (SEM) while the others used for creep test. SEM images were

processed with locally adaptive thresholding technique for accurate identification of the

microstructural features. The algorithm is automatic and based on minimizing an energy

functional of the thresholding surface through a variational Minimax algorithm. That makes it

deterministic in analyzing the microstructure and eliminates sources of variability. Deterministic

microstructural analysis is very essential for accurate and repeatable homogenization procedure.

The images were then binarized to extract statistics about void sizes and fractions. The extracted

information about voids’ sizes and area fractions are analyzed statistically with Freedman-

Diaconis method to produce an equivalent system with the same void’s volume fraction but with

much less number of voids. The equivalent structure was generated by the open source code

LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [19] which treats the

voids as granular particles being poured into the simulation box under the influence of gravity.

The equivalent structure was treated as the representative unit cell (RUC) of the local structure

and subjected to numerical experiments for extracting the effective creep compliance. Periodic

boundary conditions were applied to the RUC faces as appropriate. The proposed procedure is in

good agreement with experimental data. Particulate composites with high inclusions clustering

can be effectively studied with the proposed procedure as an alternative to theoretical

calculations like those adopted by Tszeng [20].

134

6.2 Materials and Processing

Strips of acrylonitrile butadiene styrene (ABS) were foamed with one stage physical foaming

process to fabricate the voided structures for validation of the numerical procedure. Commercial

ABS sheets with thickness of 1.59 mm (1/16 inches) from McMaster-Carr are used as the base

foaming material. Strips of ABS are prepared to fit the high pressure/temperature foaming

chamber (Fig 6.1) with planar dimensions of 32 mm by 11.3 mm. The specimens were subjected

to 1750 psi (12.07 MPa) pressure and three different temperatures below the glass transition

temperature (Tg) of ABS (105 oC) for saturation time of 1 hour and 30 minutes then rapidly

release the pressure to nucleate voids. The foaming temperatures selected to be 1/2 Tg, 2/3 Tg,

and ¾ Tg, namely, 52.5 oC, 70

oC, and 78.75

oC. The physical blowing agent considered in this

work is CO2. The rapid release of pressure induces the required instability of CO2 that produces

the voids with aid of softened polymer chains due to elevated temperature inside the foaming

chamber.

Fig 6.1. Experimental set up for the high pressure/temperature foaming chamber.

135

6.3 Microstructural Characterization and Creep Test

The specimens fabricated according to section 6.2 were characterized for the microstructural

morphology and creep test. Standard procedure followed for SEM imaging where each specimen

was frozen in liquid Nitrogen for some time, fractured by shearing, and finally sputter coated

with platinum before imaging under the scanning electron microscope (JEOL JSM 6060). Voids

identification from gray SEM images can be done automatically by thresholding or manually

identifying the voids’ boundaries with oval geometry construction. Automatic thresholding can

be classified to global and local techniques. Global thresholding techniques are based on a

constant value where pixels of greater value are assigned 1 and others as 0. The most popular

technique of this class is Otsu where the optimal global threshold value is determined by

maximizing the between–class variance with an exhaustive search [21]. Most of the work in

literature for thresholding is achieved with global techniques that are user-dependent and mostly

not accurate. For the microstructural analysis to be accurate and worth the numerical treatment, it

has to be deterministic and independent of the analyst. Variable void sizes and wall thicknesses

need a technique that can treat each feature locally due to poor illumination and planar leveling

of imaged surfaces of such structures. In contrast to global thresholding, local thresholding

techniques generate a surface that is a function of the image domain and depends on the local

image characteristics which makes it superior over global thresholding [22]. Even for local

thresholding, there are many algorithms available in literature but most of them need some

tuning parameters which make them user-dependent. However, Ray et al. [22] proposed an

automated locally adaptive image thresholding technique that does not need tuning. This

technique is adopted in the present work. This technique is based on minimizing an energy

functional of the thresholding surface through a variational Minimax algorithm. The authors in

[22] proposed a non-linear convex combination of the data fidelity and the regularization terms

in the energy functional and seek the optimum threshold surface via minimax principle. This

makes the method works in automatic manner without any tuning parameters unlike most of the

available methods to date. More details can be found in [23].

Instron Micro-tester (Model 5848) was used to run creep tests. A constant force of 50 N was

applied and then measuring deformation with time for 300 seconds. The creep results presented

in this work are not meant to characterize the foamed samples but are rather utilized to validate

136

the homogenization procedure. Following ASTM D 2990 and the Test Method D 638 for the

foamed samples in terms of dimensions was very difficult due to size limitation of the high

temperature/pressure foaming chamber. However, all other requirements were satisfied as per the

aforementioned ASTM standards. The creep compliance of solid ABS was obtained by exactly

following ASTM D 2990 and the Test Method D 638.

6.4 Construction of the Equivalent Microstructure

The three different seeding temperatures yields microstructures with different features as shown

in Fig 6.2. The first step in reconstructing the microstructure is to accurately characterize those

microstructural features. The algorithm of Ray et al. [22] was implemented as part of a further

tuning procedure to accurately capture the voided structure. The contrast of raw grayscale SEM

image was enhanced by the MatLAB function “adapthisteq” which transforms the values using

contrast-limited adaptive histogram equalization. This contrast enhancement technique

eliminates artificially induced boundaries and avoid amplifying any noise that might be present

in the image. The adjusted image was then binarized with the algorithm proposed by Ray et al.

[22]. The image is ready now for voids recognition which is accomplished by the MatLAB

function “bwboundaries”. This function traces the exterior boundaries of objects, as well as

boundaries of holes inside these objects, in the binarized image. The option “hole” in this

function was selected to do the cleaning process for internal impurities of the identified voids.

a) 52.5 oC

∅ = 0.114

d = 4.74 𝜇𝑚

b) 70 oC

∅ = 0.336

d = 4.05 𝜇𝑚

c) 78.75 oC

∅ = 0.422

d = 3.91 𝜇𝑚

Fig 6.2. The effect of seeding temperature on the microstructure at 1750 psi pressure (∅ is voids

fraction and d is average voids diameter).

137

These different thresholding and cleaning steps are shown in Fig 6.3. At the first step, the image

was adjusted for contrast (Fig 6.3b) and then binarized with the proposed procedure. Then the

MatLAB function “bwboundaries” was implemented for voids identification with the option

“hole” for further cleaning of the identified voids (Fig 6.3c). This algorithm works very well in

identifying the microstructure without any tuning parameters which eliminates any source of

variability.

a. Original SEM image. b. SEM image after contrast

adjustment.

c. Binarized and cleaned

image with the algorithm

proposed in [22] and

MatLAB.

Fig 6.3. Image thresholding and microstructure recognition of voided ABS with 1750 psi

pressure and seeding at 78.75 ºC.

In order to preserve voids fraction and randomness nature of the microstructure, statistical based

equivalent system is being proposed in this work. The voided structure in Fig 6.2c was analyzed

with the locally adaptive threshoulding algorithm. The output is the individual voids diameters

and area fraction of each void. The main idea of the equivalent microstructure is to categorize

voids diameters into smaller number of voids but keeping both the voids fraction and

randomness of the actual system. Voids diameters and area fractions of the SEM image in Fig

6.2c are categorized based on the Freedman-Diaconis method [24] which is based upon the inter-

quartile range and number of data as shown in Fig 6.4 and table 6.1. Other methods are also

possible like Scott's method [25] and Sturges' method [26].

138

Fig 6.4. Voids diameters and area fractions of the actual and equivalent

systems in Fig 6.2c.

Table 6.1. Geometrical features of the equivalent structure in Fig 6.2c.

Void Diameter

(µm) Area Fraction

1.33 0.0083

2.19 0.0290

3.05 0.0671

3.91 0.0937

4.76 0.1185

5.62 0.0675

6.48 0.0382

Overall Voids Fraction 0.4223

139

Number of voids in the actual microstructure is 105 voids which are binned to 50 categories in

Fig 6.4 for simplicity of representation only. After categorizing with the Freedman-Diaconis

method, the number of equivalent voids that can represent the microstructure is 7 voids only. The

overall voids fraction of 0.4223 is preserved by summing all area fractions within each bin width

together. A simulation box that is at least twice the largest void diameter is assumed for the

analysis which is 15 μm box sides in this case. Each void diameter was distributed in the box

according to the area fractions shown in table 6.1 with total of 75 voids for the 15 μm simulation

box. The voids dispersion was achieved by LAMMPS using the command “fix pour” under the

effect of gravity via a granular potential. Voids are interacting according to the two Hookean

styles force formula [19]. The force formula is:

𝐹ℎ𝑘 = (𝑘𝑛𝛿𝑛𝑖𝑗 − 𝑚𝑒𝑓𝑓𝛾𝑛𝑣𝑛) − (𝑘𝑡∆𝑠𝑡 + 𝑚𝑒𝑓𝑓𝛾𝑡𝑣𝑡) (6.1)

Where;

𝑘𝑛: Elastic constant for normal contact

𝑘𝑡 : Elastic constant for tangential contact

𝛾𝑛: Viscoelastic damping constant for normal contact

𝛾𝑡: Viscoelastic damping constant for tangential contact

𝑚𝑒𝑓𝑓: 𝑀𝑖𝑀𝑗 (𝑀𝑖 + 𝑀𝑗)⁄ = effective mass of 2 particles of mass 𝑀𝑖 and 𝑀𝑗

∆𝑠𝑡: Tangential displacement vector between 2 spherical particles which is truncated to satisfy

a frictional yield criterion.

𝑛𝑖𝑗: Unit vector along the line connecting the centers of the 2 particles

𝑣𝑛: Normal component of the relative velocity of the 2 particles

𝑣𝑡: Tangential component of the relative velocity of the 2 particles

140

The Hookean styles force formula accounts for frictional forces between the voids that are

assumed as granular particles during the energy minimization step. The first parenthesized term

in the equation is the normal force between the two voids and the second parenthesized term is

the tangential force. The normal force has two terms, a contact force and a damping force. The

tangential force has also two terms, a shear force and a damping force. The shear force is a

“history” effect that accounts for the tangential displacement between the particles for the

duration of the time they are in contact. In the Hookean model, the normal push-back force for

two overlapping particles is a linear function of the overlap distance and can be thought of as an

elastic spring of constant 𝑘𝑛. The parameters in Eq. (6.1) were adjusted in such a way to provide

reasonable repulsion between the voids for better dispersion after several trials. The assumed

values are 𝑘𝑛 = 200, 𝑘𝑡 = 57.14, 𝛾𝑛= 50, 𝛾𝑡= 25. All quantities are unitless for LJ units system.

The “fix pour” command inserts voids in the simulation box according to the fractions and

diameters shown in table 6.1. Few voids are inserted each time step from the top face that fit the

size of 15 µm x 15 µm planer dimension. The voids fall down with the effect of gravity and

assigned random locations at which each void is checked for overlapping. If overlapping

detected then the voids assigned new random locations and run for enough time to equilibrate the

system before inserting another set of voids. The process continued until filling the simulation

box with 75 non-overlapping voids positioned randomly. Finally, the simulation box was

subjected to energy minimization with Brownian dynamics for 1,000,000 time steps at room

temperature. The energy minimization step ensures that the system of voids is at a state of

minimum energy and not just randomly distributed in the simulation box.

Unlike other methods such as the random sequential adsorption (RSA) algorithm [27], the

granular pouring method doesn’t have limitation on the maximum possible volume fraction of

voids. Huang and Li [28] built 3D FE model for glass microballoons dispersed in epoxy with

similar approach to this work but following a log-normal distribution based on statistic

information of the microballoon. Kursun et al. [29] developed 3D homogenization procedure for

alumina reinforced aluminum matrix composites with remarkable accuracy. However, the

particles were randomly distributed in the simulation box without describing the distribution

algorithm. Most algorithms for random distribution are limited in the maximum volume fraction

of particles or voids [27]. Our work is advantageous in the sense that the number of voids is

141

reduced and can be dispersed more representatively when selecting the appropriate elastic

constants for the matrix. Moreover, the proposed method deals with variable voids’ sizes and

high void fractions. The re-constructed simulation box with this method is shown in Fig 6.5. Two

simulation boxes are constructed, one is enclosing the voids inside the box and the other is

periodic at the box walls. Both approaches of construction will be discussed in section 6.7.

a. Simulation box with enclosed voids b. Simulation box with periodic walls.

Fig 6.5. The 3D reconstructed voided structure of SEM image in Fig 6.2c with 0.4223 voids

fraction.

6.5 Periodic Boundary Conditions and Calculations of

Macroscopic Strains

Displacement-base periodic boundary conditions were assumed to the surfaces of the

representative unit cell (RUC). Each RUC is the re-constructed SEM image that is representative

of its local microstructure with enough details and assumed to be repeated over the whole

structure. The periodic boundary conditions are implemented in a way that the relative

142

displacement between equivalent nodes on two opposite surfaces is zero. Opposite surfaces are

separated by the characteristic length (L) that is the length of the RUC sides. This can be

expressed as a function of the displacement vector �⃗� ;

�⃗� (0, 𝑥2, 𝑥3) − �⃗� (𝐿, 𝑥2, 𝑥3) = 0

(6.2) �⃗� (𝑥1, 0, 𝑥3) − �⃗� (𝑥1, 𝐿, 𝑥3) = 0

�⃗� (𝑥1, 𝑥2, 0) − �⃗� (𝑥1, 𝑥2, 𝐿) = 0

where 𝑥1, 𝑥2, and 𝑥3 are the three Cartesian coordinates.

The periodic boundary condition was implemented in ABAQUS through linear constraint

equation that is scripted in the input file. The constrain equation is presented as [30];

𝐴1𝑢𝑖𝑃 + 𝐴2𝑢𝑗

𝑄 + ⋯+ 𝐴𝑁𝑢𝑘𝑅 = �̂� (6.3)

where R is node index, k is degree of freedom, i.e. 1, 2, and 3 which represent 𝑥1, 𝑥2, and 𝑥3

directions respectively, 𝐴𝑁 is a constant coefficient that defines the relative motion of nodes, �̂� is

the deformation of an arbitrary dummy node which is set to zero in this work since a creep load

of 50 N is applied (�̂� is set to the displacement value in relaxation test).

Traction continuity should be imposed on opposite edges/surfaces to have equal magnitude but

with opposite sign. However, it was shown in [31] that if the RUC is analyzed with

displacement-based finite element method, the application of only Eq. (6.3) can guarantee the

uniqueness of the solution and thus traction continuity is automatically satisfied.

Average strain theorem links local strains of the RUC to the macroscopic or effective strains of

the homogenized microstructure. The macroscopic strain, 휀�̅�𝑗, can be obtained by integrating

143

local strains, 휀𝑖𝑗, of the outer surfaces, 𝑆𝑅𝑈𝐶, of the simulation box (RUC) in the respected

direction and divide by the RUC volume, 𝑉𝑅𝑈𝐶, as follows [32];

휀�̅�𝑗 =1

𝑉𝑅𝑈𝐶∫ 휀𝑖𝑗 𝑑𝑉𝑅𝑈𝐶𝑉𝑅𝑈𝐶

=1

𝑉𝑅𝑈𝐶∫

1

2(𝑢𝑖𝑛𝑗 + 𝑢𝑗𝑛𝑖) 𝑑𝑆𝑅𝑈𝐶𝑆𝑅𝑈𝐶

(6.4)

In the case of loading along 𝑥1 direction:

휀1̅1 =1

𝑉𝑅𝑈𝐶∫ 𝑢1𝑛1 𝑑𝑆𝑅𝑈𝐶𝑆𝑅𝑈𝐶

(6.5)

Displacements at the outer surfaces of the RUC are printed from ABAQUS to text file and then

input to MatLAB. The integration is calculated by double trapezoidal integration along both

planar directions of each surface.

6.6 Linear Viscoelastic Material Model

One of the most interesting features about polymers is that for a given polymer can display all

the intermediate range of properties between an elastic solid and a viscous liquid depending on

the temperature and chosen timescale. This combination of liquid-like and solid-like features is

viscoelasticity that makes the properties of polymers depend on many parameters like time and

loading history. The stress at a point in time doesn’t depend on the strain at that point only but

rather depends on the whole strain history up to the current time starting from an assumed virgin

state of the material. Thus viscoelastic materials are said to have a memory effect. The memory

effect is neglected in linear viscoelasticity while the effect of strain history being approximated

by piecewise constant step functions and superimposed to the elastic response [33]. In

experimental tests on viscoelastic materials, it is observed that the instantaneous changes in

stress or strain are governed by Hooke’s law, that is, the instantaneous response is elastic [34].

However, the history part can be seen in creep and stress relaxation which characterize

viscoelastic materials. Creep is an increase in deformation under constant load while relaxation is

a decrease in load with constant deformation. Linear viscoelasticity is considered in this work.

Creep test is used also to verify the homogenization procedure.

144

Time domain viscoelasticity is available in ABAQUS for small-strain applications where the

rate-independent elastic response can be defined with a linear elastic material model which is the

case of this work. In relaxation test, a sudden strain 휀 is applied to the test specimen and then

held constant for long time. To account for loading history, the zero time is considered as the

moment of applying the sudden strain. The viscoelastic material model in ABAQUS defines the

varying stress with time, 𝜎(𝑡), [35] as:

𝜎(𝑡) = ∫ 𝐸𝑅(𝑡 − 𝑠) 휀̇(𝑠)𝑡

0

𝑑𝑠 (6.6)

where 𝐸𝑅(𝑡) is the time-dependent relaxation modulus that characterizes the material’s response.

The viscoelastic material model is “long-term elastic” in the sense that, after having been

subjected to constant strain for a very long time, the response settles down to a constant stress;

i.e., 𝐸𝑅(𝑡) → 𝐸∞ as 𝑡 → ∞ where 𝐸∞ is long-term modulus. The relaxation modulus can be

written in dimensionless form [35]; 𝑒𝑅(𝑡) = 𝐸𝑅(𝑡) 𝐸0⁄ , where 𝐸0 = 𝐸𝑅(0) is the instantaneous

relaxation modulus. So the expression for stress in ABAQUS takes the form;

𝜎(𝑡) = 𝐸0 ∫ 𝑒𝑅(𝑡 − 𝑠) 휀̇(𝑠)𝑡

0

𝑑𝑠 (6.7)

The dimensionless relaxation function has the limiting values 𝑒𝑅(0) = 1 and 𝑒𝑅(∞) = 𝐸∞ 𝐸0⁄ .

Either creep compliance or stress relaxation data can be used as time-dependent viscoelastic

properties in the ABAQUS FE solver. Creep compliance (J) measurements are converted to

stress relaxation using the convolution integral;

∫ 𝐸𝑅(𝑡 − 𝑠) 𝐽𝑅(𝑡)𝑡

0

𝑑𝑠 = 𝑡 (6.8)

In general, 𝐽𝑅 ≠ 1 𝐸𝑅⁄ but in the limits of 𝑡 → 0 then 𝐽𝑅 → 1 𝐸𝑅⁄ . For the FE analysis,

compliance data are normalised with respect to the instantaneous modulus of the material [36].

145

Hence, it is not strictly necessary to determine the creep compliance for ABAQUS input data. In

practice, normalisation can be carried out with respect to the initial elastic displacement, 𝛿0,

measured after applying the force and the measured displacement, 𝛿(𝑡). The normalized creep

compliance, 𝑗𝑅(𝑡), is calculated as [36]:

𝑗𝑅(𝑡) =𝛿(𝑡)

𝛿0 (6.9)

The times (t) are those after application of the load. 𝑗𝑅(𝑡) will be 1 at time = 0 and will increase

with time.

The measured 𝑗𝑅 values for solid pure ABS are shown in Fig 6.6 according to ASTM D 2990 and

the Test Method D 638. Density of solid ABS is 1.04 E-15 Kg μm3⁄ , while modulus of elasticity

and Poisson’s ratio are 594.55 MPa and 0.35 respectively.

Fig 6.6. Normalized creep compliance of solid ABS


The homogenization procedure was validated for three voided structures at 1750 psi (12.07 MPa)

and three different seeding temperatures for saturation time of 1 hour and 30 minutes. The

146

properties of solid ABS were used as described in section 6.6 since it is the base foaming

material. In order to keep voids fraction and randomness of the microstructure, and reduce

computational cost with reasonable accuracy, this work proposed creating another microstructure

that is statistically equivalent to the actual structure. Voids’ fraction is preserved in exact while

void’s sizes are preserved in statistical average sense. Categorizing void’s diameters with

Freedman-Diaconis method makes a statistical base of grouping void’s diameters into smaller

number of voids. The area fraction of each bin is then the sum of all fractions that fall within the

bin range which keeps the same value of overall voids’ fraction of the microstructure. In the case

of 78.75 oC seeding temperature, the number of voids reduces from 105 to 7 voids only as shown

in Fig 6.4 and table 6.1 but reproduced to 75 voids in the simulation box. The voids need to be

dispersed randomly in the simulation box while keeping the voids fraction at 0.4223.

GRANULAR package in LAMMPS does the random dispersion of voids as pouring of granular

particles into a container under the influence of gravity. At each timestep, particles are inserted

and placed randomly inside the simulation box. Each particle is tested for overlapping with

existing particles and if found overlapping, then another random insertion attempt is made. The

code keeps pouring voids until the target number achieved which is decided based on the area

fraction of each void and the total volume of the simulation box. To decide the number of voids

for each diameter, it was assumed that the volume and area fractions are equivalent, i.e.,

𝐴𝑖 𝐴𝑡⁄ = 𝑛𝑖𝑉𝑖 𝑉𝑡⁄ , where the terms from left are ith void’s area, total area, number of voids in the

simulation box for particular area fraction, void’s volume, and the volume of simulation box

respectively. The resulted simulation box based on this procedure is shown in Fig 6.5 for both

the enclosed case and periodic walls. After imposing the periodic boundary conditions,

macroscopic strain was calculated as described in section 6.5. The FE model was discretized

with four-node tetrahedral (C3D4) elements for simplicity and accuracy to capture the complex

geometry. Fig 6.7 shows the mesh in the case of periodic cell walls with 1003206 elements. The

deformation of simulation box is shown in Fig 6.8 for creep test in x-direction. The outer walls’

deformation is periodic as per the constrain condition prescribed in section 6.5. The one-stage

foaming process produced very uniform voided structures across the foamed sample. Because of

the microstructure uniformity, similar variation of voids’ diameters was observed across the

sample even when fractured at different planar directions. Fig 6.9 shows the microstructure for

the case of 78.75 oC and 1750 psi seeding temperature and pressure for sampling at different

147

planar directions. This uniformity of the microstructure alleviates the need for 3D

microstructural characterization and makes the 2D SEM images sufficient.

Fig 6.7. Sample mesh for the periodic box

(1003206 four-node tetrahedral (C3D4)

elements).

Fig 6.8. Deformation of the RUC that

conforms to the periodicity of opposite faces

(Gray: undeformed shape).

Fig 6.9. Microstructure at different fracturing planes for the case of 78.75 oC and 1750 psi

seeding temperature and pressure.

148

Creep compliance of the two reconstructed 3D microstructures is shown in Fig 6.10c in

comparison to the experimental data. Both approaches provide very close results with 0.6%

improvement in the error for the case of periodic simulation box and 78.75 oC seeding

temperature. The maximum error of FE prediction is 8.62% only for case of 78.75 oC. This error

is marginal compared to the simplicity in modeling and the reduction technique adopted in this

work which reduces computational cost significantly. It is worth mentioning that the enclosed

case of voided microstructure reconstruction is easier and faster. The small improvement of 0.6%

when using periodic microstructure suggests that the enclosed approach is justifiable. The cases

of 52.5 oC and 70

oC were constructed for the enclosed case only. The error increases with

increasing voids fraction but the increase is not significant even at 42.23% voids with value of

8.62%.

a. 52.5 oC b. 70

oC

c. 78.75 oC

Fig 6.10. Comparing FEM results to experimental data for creep compliance of the voided

structures.

149

6.8 Conclusion

Reduced three dimensional voided structure was reconstructed based on geometrical features of

the actual microstructure. The reconstructed structure has the same voids’ fraction as the actual

microstructure. Variability of void’s diameters and randomness were preserved on average sense

based on statistical analysis of the extracted information about the actual microstructure.

Computational cost is also less than that required for the actual microstructure because of

significant reduction in the number of voids. The contribution of the procedure is attributed to

the simplicity and accuracy in developing representative voided structures from 2D SEM images.

Otherwise the structure needs to be extracted from tedious processes like X-ray

microtomography reconstruction. The agreement with experimental results is very good with

maximum error of 8.62% which is very marginal compared to the simplicity of analysis and

reduction of the actual microstructure. The proposed reconstruction approach can be used to

conduct accurate numerical experiments of voided composite structures with wide range of

voids’ fraction and void’s diameters. Construction of the structure with LAMMPS’s granular

package preserves diameters range and fractions with no upper limit of maximum voids’

fractions unlike other algorithms. Easiness of construction of the microstructure combined with

the reduced computational cost makes this procedure valuable tool for design and accelerating

Lab-to-market cycle of voided composite materials.

6.9 References



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Chapter 7 Conclusions and Recommendations

7.1 Conclusions

The focus of this research was on the fabrication and modeling of functionally graded porous

structures and composites. Brief introduction about the topic was given in Chapter 1 along with

the motivation and flow of chapters to address both, fabrication and modeling. The thesis work

was divided in two parts. Part 1 focused on the development of procedures and guidelines for

fabrication of the material structures. It was further divided into three chapters. Chapter 2 started

by developing procedures for fabrication and their correlation to the microstructure and

mechanical properties. The developed procedures and analysis from Chapter 2 were

implemented in Chapter 3 to develop functionally graded porous structures for impact

applications. Fabrication procedures and morphological features were correlated to the impact

strength. Chapter 4 focused on alleviating the overall structural weakness of graded porous

microstructures by incorporating them in a composite structure with in-plane and out-of-plane

reinforcements. Part 2 of the thesis focused on modeling of the functionally graded porous

structures. The start was in Chapter 5 by developing a statistical based homogenization model

that accounted for microstructural details and gradient by seeding control points across the

gradient direction. The model was then implemented to higher order plate theory with stretching

effect to account for through-thickness deformations of porous structures. The model yielded

very accurate results compared to the experimental data. Chapter 6 established numerical

homogenization procedure based on finite element method for local porous structures. This

numerical model can provide the input to the statistical model that was developed in Chapter 5

and alleviates the dependence on experimental data. So part 2 of the thesis developed an accurate

and independent homogenization procedure for functionally graded porous structures and

composites.

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In Chapter 2, functionally graded porous structures of acrylonitrile butadiene styrene (ABS) were

fabricated with a one-step but two-stages constrained solid-state foaming process. Strips of ABS

were saturated with carbon dioxide (CO2) at different temperatures and pressures in a closed

pressure vessel. Sudden release of the pressure caused seeding of the pores with different sizes

and distributions depending on the seeding temperatures. The seeded specimens were annealed

between platens of different temperatures to induce a thermal gradient. This produced graded

porous structures with different morphological features depending on the overall processing

conditions. The microstructures were characterized from scanning electron microscope (SEM)

images with locally adaptive thresholding technique. The result of this characterization was

pores’ sizes and distribution. The adopted microstructural characterization technique is

deterministic in the sense that it is automatic and does not depend on the analyst. This

determinacy in characterizing the microstructure was important because next step was

correlating morphological features to the processing and creep compliance. The correlation

analysis revealed strong relationships between processing parameters, microstructural details,

and creep compliance. Design graphs were developed that enable the design of functionally

graded porous structures with desired creep compliance.

After establishing the relationships between processing, microstructure, and creep compliance in

Chapter 2, the focus of Chapter 3 was to apply these tools in developing graded porous structures

of ABS for impact energy absorption. The fabrication procedure was adopted to work with

encapsulated chemical blowing agents after it was based on physical blowing agents in Chapter

2. The effect of different concentrations of the chemical blowing agent was investigated. The

fabrication technique was based on inducing a thermal gradient of different values across the

thickness of each specimen. The thermal gradient caused the microspheres to expand more at the

side of higher temperature of the gradient while expanding less at the other side. This different

expansion of the microspheres is due to the temperature spectrum of the chemical blowing agent

across activation and maximum yield points. The functionally graded porous structures were

fabricated successfully. The objective of changing to chemical blowing agents was scalability to

industrial applications of making large parts of functionally graded structures. This was tested by

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making large customized mould of 200 mm by 200 mm in planar dimensions and 25 mm in

thickness. The specimen was fabricated successfully with graded structure. The relationships

between processing parameters, microstructure, and impact strength were established in this

chapter. The relationships were studied graphically and also by correlation analysis. Pores’

diameter gradient and expansion ratio demonstrated the highest inverse correlation coefficient.

This strong correlation was indicative to the possibility of controlling the microstructure to some

morphological features which produce graded porous structures with desired values for impact

strength. Impact energy showed fairly strong correlation to the diameter gradient and was

inversely correlated in the expansion ratio. The inverse correlation between impact energy and

expansion ratio is very intuitive since higher expansion ratios cause mass redistribution on a

larger volume, and hence reducing the specific strength of the structure. Higher expansion ratios

cause higher porosities which provide more damping to the impact load through large

compressive strains. In contrast, higher porosities weaken the structure due to high void fraction

compared to the solid mass and make it prone to damage. So there is a balance between

weakening the microstructure for damping and preserving enough strength to withstand the

impact load. Eventhough the three measures (diameter gradient, expansion ratio, and porosity)

were correlated but it was with different magnitudes to each other and to the impact strength.

Pores’ diameter gradient was more correlated to impact energy than the other microstructural

measures. Thus the proposal in Chapter 3 was that, a better control on impact energy is possible

by controlling the diameter gradient. Changing the diameter gradient is going to affect porosity

and expansion ratio but with the advantage of more control over impact energy. It is worth

mentioning that the graded porous structures were compared to their counter solid parts of the

same volume for impact absorption. The graded porous structure was superior with almost

double the specific energy absorption compared to the solid precursor.

Eventhough the graded porous structures are superior in energy absorption when compared to

their solid precursors but they lack the structural strength to receive the load directly. In Chapter

3, a solid sheet of ABS was placed on top of the graded porous structure to protect the

microstructure. This problem was alleviated in Chapter 4 by developing graded syntactic foamed

composites. The composites were stitched through the thickness with ultrahigh molecular weight

polyethylene braids to avoid delamination and provided out-of-plane strength. Kevlar/carbon

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fibres hybrid fabrics were embedded in the composite to provide in-plane reinforcement.

Polyurethane resin with uniformly dispersed microspheres composed the damping matrix of the

graded composite. Different concentrations of the microspheres were used to induce gradient

along the impact direction. The fabricated graded composites were structurally strong and

provided effective damping to impact loads. The amount of load transfer to the other side of

impact was considered as a measure of damping. A customized experimental set up was

developed for this purpose. Correlation analysis was performed to study the relationship between

transferred impact loads to the compressive and relaxation moduli. The transferred load inversely

correlated to both moduli with fairly strong correlation coefficients. Higher values of relaxation

modulus induced more sharing of the impact load which reduced the overall transferred load.

The compressive and relaxation moduli were strongly correlated which is indicative to the

potential of tailoring transferred impact load to some desired values by controlling both moduli

at the same time.

Numerical modeling was going in parallel to the development of fabrication procedures for

functionally graded porous structures. It was realized that homogenization step is required prior

to modeling at macroscale of the fabricated structures. This was the topic of Chapter 5. A

statistical based homogenization model was developed in Chapter 5 to predict the relaxation

modulus of graded porous structures. The model was based on a formula similar to Prony series

with control points across the graded direction. The microstructure was analysed at each control

point to extract details like porosity and pores’ area distribution. The model depends on fitting

the graded porous structure to multiple equivalent uniform structures where each representing

local morphological features. Higher order plate theory was developed to test the

homogenization model for macroscale analysis. The plate theory was formulated with stretching

terms in the out-of-plane directing to allow for independent deformation across the graded

direction. The model predictions were very accurate compared to experimental data. The

numerical tool proved its validity for conducting accurate computer experiments with the

minimal input about material properties and microstructure. This makes it possible to be utilized

in a controlled fabrication process as an inverse-homogenization procedure to fabricate materials

possessing a required design strength.

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The only problem with the homogenization model that was developed in Chapter 5 is the

dependency on experimental data for the details about the microstructure at each control point. It

was the task of Chapter 6 to develop purely numerical procedure for the prediction of mechanical

properties of a given porous structure. The numerical procedure was based on generating an

equivalent three dimensional porous structure of the parent microstructure with known porosity

and pore’s diameter distribution. Alternatively, the average pores’ diameter with standard

deviation can be provided to the model. Dispersion of the pores with their respective fractions

and diameters was achieved by granular mechanics with the assumption that the pores are

interacting and non-overlapping spheres. Unlike other dispersion techniques in literature, the

granular mechanics technique does not have a limitation on the maximum value of porosity. The

generated microstructures were subjected to numerical experiments with finite element method.

A script was written to assign periodic boundary conditions to the surfaces of the microstructure.

Average strain theorem was used to calculate the macroscopic strains. The model was in good

agreement with experimental data. This makes it an effective replacement to experimental data

for the model which was developed in Chapter 5. Consequently, the homogenization procedure

becomes independent in providing details about mechanical properties for an assumed graded

porous structure or composite. Combined with the processing-microstructure graphs which were

developed in Chapters 3 and 4, the numerical tool can be used in an inverse-homogenization

procedure to make graded porous structures with desired mechanical properties.

7.2 Recommendations for Future Work

Fabrication and modeling of functionally graded porous structures and composites were

addressed in this research. The work has shed light on an area that still needs an intensive

research and a lot of concepts are not yet fully understood. Most of the fabrication work on

functionally graded materials was limited to combinations of metals and ceramics. Fabrication

procedures were developed for polymeric functionally graded porous materials with correlation

to microstructure and mechanical properties. An area that still needs more research is the

fabrication of graded porous structures with chemical blowing agents. Encapsulated chemical

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blowing agents work very well with compression moulding as proved in Chapter 3. However,

non-encapsulated chemical blowing agents did not work effectively for compression moulding

even there was some success. Careful designs of the mould with proper selection of the suitable

chemical blowing agent for each polymer will make it possible to fabricate graded porous

structures with good control over the microstructure. One of the considerations for the mould is

effective cooling system to stabilize cells’ growth at a certain stage. Another consideration is

smooth interfaces between different components to allow repeatable expansion of the polymer

melt.

The modeling approach should account for the microstructural details in a continuum frame of

finite element analysis. This necessitates a homogenization step prior to the macroscale analysis

in a similar approach to the one presented in Chapters 5 and 6. Linear viscoelastic material

model was adopted in this work but it is encouraged to use more representative material models

like nonlinear viscoelasticity. Porous structures within the continuum scale (>100 nm) can be

homogenized with approaches similar the procedure developed in Chapter 6. However, porous

structures which are less than 100 nm in size should be modeled with approaches similar to

molecular dynamics to extract the mechanical properties. The homogenized local material

properties can then be used as an input to the statistical homogenization model which was

developed in Chapter 5. The homogenization model can directly be included as an additional

subroutine to the macroscale finite element analysis. The whole process can be programmed in a

single work frame as an optimization problem with desired mechanical properties as the

objective function. The design process should start with an assumed graded porous structure and

then iterate the numerical procedure until converging to the required mechanical properties.

Processing-microstructure graphs will be useful to map the right processing parameters to

fabricate the converged graded microstructure. Examples to these graphs were developed in

Chapters 3 and 4 for acrylonitrile butadiene styrene (ABS) with physical and chemical blowing

agents. Similar graphs should be developed for each polymer but the general guidelines

developed in this thesis can be used.

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