MODELING OF MECHANICAL PROPERTIES OF POLYMER...

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CHAPTER 4

MODELING OF MECHANICAL PROPERTIES OF

POLYMER COMPOSITES

4.1 Introduction

Fillers added to polymer matrices as additives are generally intended for decreasing

the cost (by increase in bulk) of the in addition to improving some of the physical and

mechanical properties of the material. Filler, a chemically inert material can decrease

properties like linear expansion coefficient, lubricancy, toughness and elongation percent,

while properties like viscosity, thermal conductivity, stiffness, hardness, flame resistance

and yield strength increase. Fillers can be organic and inorganic. Non-metallic materials

like talc, calcium carbonate, metallic fibers and silica belong to inorganic category while

natural fibers (organic origin), wood flour and synthetic fibers are organic fillers.

Geometrically, fillers may be spherical, sheet, needle form and fibrous. Filler shape, type

and its concentration can influence the mechanical properties of the polymer to which it is

added. Particles with non-spherical shape lead to have more viscosity and manifest in

several problems during composite shaping (Valavala and Odegard, 2005).

Traditionally, composites were reinforced with micronsized inclusions. Recently,

processing techniques have been developed to allow the size of inclusions to go down to

nanoscale (Jordan, et al., 2005). Although experimental based research can ideally be used

to determine structure-property relationships of nanostructured composites, experimental

synthesis and characterization of nanostructured composites demands the use of

sophisticated processing methods and testing equipment; which could result in exorbitant

costs. To this end, computational modeling techniques for the determination of mechanical

properties of nanocomposites have proven to be very effective (Liu & Chen, 2003; Sheng,

et al., 2004; Chen & Liu, 2004). Computational modeling (CM) for predicting the

mechanical properties of composites is highly desirable. Computational modeling of the

mechanical properties of polymeric nano-composites for design and development of nano-

composite structures for engineering applications has been suggested by Valavala and

Odegard, (2005). The modeling of polymer-based nano-composites has become an

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important topic in recent times because of the need for the development of these materials

for engineering applications. It was found that deal with modeling and offering appropriate

formulations for the estimation of mechanical properties of polymeric composites.

Different modeling theories and equations for describing the reinforcing mechanisms of

polymer matrix by additives (like fillers) have been suggested that are discussed in this

chapter.

4.2 Overview of the Modeling Method

The importance of modeling in understanding of the behavior of matter is

illustrated in Figure 4.1. The earliest attempt to understanding material behavior is through

observation via experiments. Careful measurements of observed data are subsequently

used for the development of models that predict the observed behavior under the

corresponding conditions. The models are necessary to develop the theory. The theory is

then used to compare predicted behavior to experiments via simulation. This comparison

serves to either validate the theory, or to provide a feedback loop to improve the theory

using modeling data. Therefore, the development of a realistic theory of describing the

structure and behavior of materials is highly dependent on accurate modeling and

simulation techniques. Valavala and Odegard (2005) have presented a review of modeling

techniques for predicting the mechanical behavior of polymeric nano-composites. It is

proposed that mechanical properties of nano-structured materials can be determined by a

select set of computational methods. Details Computational Chemistry and Computational

Mechanics modeling are presented. Computational Chemistry techniques are primarily

used to predict atomic structure using first-principles theory, while Computational

Mechanics is used to predict the mechanical behavior of materials and engineering

structures.

To facilitate the development of nano-structured polyamide composite materials

for this purpose, constitutive relationships must be developed that predict the bulk

mechanical properties of the materials as a function of the molecular structure of the

polyamide and reinforcement. These constitutive relationships can be used to influence the

design of these materials before they are synthesized. Even though it has been shown that

these materials have the potential to have excellent mechanical properties, the relatively

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high costs of development and manufacturing of nano-polymer composites has been

prohibitive. A lower cost approach is the use of particles in the polymer.

Figure 4.1 The schematic of developing theory and validation of experimental data

through simulation.

Vollenberg and Heikens (1989) explained that if there is a strong interaction

between the polymer and the particle, the polymer layer in the immediate proximity of the

particle will have a higher density. For most systems, density is proportional to elastic

modulus, so the region directly surrounding the inclusions will be a region of high

modulus. The polymer right outside this high modulus region will have a lower density due

to the polymer chains that are moved towards the particle. For large particles, the size of

the low density region will be relatively large, and the contribution of the high modulus

filler will be diminished. For nano-particles, the number of particles for a given volume

fraction is much larger, thus the particles will be much closer to one another. If the

particles are densely packed, the boundary layer of polymer at the interface will comprise a

large percentage of the matrix and can create a system where there is no space for a low

modulus region to form. This results in the elastic moduli of composites with smaller

particle (nano) size being greater than the moduli of composites with larger inclusions

(Jordan, et al., 2005). The small inter-particle distance in nano-composites was used as

another parameter to explain the changes in the elastic modulus and strength of these

materials compared to the composites with micron-sized particles.

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The same parameter also plays a role in the glass transition temperature changes

observed in nano-composites versus composites with micron-sized reinforcement. Ash, et

al. (2002) found that for their system the glass transition temperature was constant until

around 0.5% weight fraction of particles, and then had a sharp drop, and then it remained

constant for weight fractions above 1%. When there is little or no interfacial interaction

between the filler and matrix and the inter-particle distance is small enough, the polymer

between two particles acts as a thin film, and for thin films, the glass transition temperature

decreases with the drop of film thickness. The distance between particles in a composite

with the filler weight fraction below 0.5% is relatively large, and in such case the polymer

between each particle is not considered to belong to the thin film regime. As the filler

concentration increases, the inter-particle distance and the resulting thickness of the film

decrease. This theory, however, does not explain why the glass transition temperature

levels off rather than continues to drop as a function of increasing weight fraction of the

filler. The behavior of polymeric nano-composite systems are shown in the Table 4.1.

From the viewpoint of fundamental laws, reinforcing effect of nano-particles is dependent

on Aspect Ratio and also on the interactions between fillers particles and polymeric matrix

(Jordan, et al., 2005). Figure 4.2 shows typical tensile stress–displacement curves for PVC

and its nano-composites. Pure PVC is brittle, however, when CaCO3 nano-particles are

added in the PVC matrix, the composites show ductile behaviors, such as stress whitening

and necking. Their Young’s modulus, tensile yield strength and elongation at break are

calculated and plotted in Figures 4.3 to 4.5.

In Figure 4.3, Young’s modulus of the PVC/CaCO3 nano-composites is observed

to increase with the loading of CaCO3 nano-particles up to 5 wt% and then decrease

marginally at 7.5 wt%. These results confirm that the CaCO2 nano-particles do stiffen

PVC. The yield strength of the nano-composites is plotted in Figure 4.5 and is broadly

independent of nano-particle loading (Xie, et al., 2004).

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Table 4.1 Behavior of polymer nano-composite (Jordan, et al., 2005).

Attribute Crystalline State Amorphus State Elastic modulus Increase w/volume fraction

Increase or no change with decrease of size

Increase w/volume fraction

Increase w/decrease size

Greater increase than for good interaction





Yield

stress/strain



Decrease with addition of particles

N/A


Ultimate

stress/strain

Increase w/ decrease size

No unfilled result for change in Vf

Lower than pure for small volume fraction

Nano>micro after 20% weight


Density/volume Increased volume as size decreases

N/A

Increased volume as size decreases

N/A

Strain-to failure Decrease with addition of particles


increase with addition of particles

increase with addition of particles

Tg Decrease with addition of particles

N/A


Level until 10%, drops off level 1-10

Crystallinity No major effect

No major effect

N/A

N/A

Viscoalasticity Increase w/volume fraction


N/A

Increase w/volume fraction nano less


Figure 4.2 Tensile stress - displacement curves for PVC and nano-composites

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Figure 4.3 Variation of Young’s modulus of nano-composite with nano-particle loading.

Figure 4.4 Variation of tensile yield strength of nano-composite with nano-particle

loading.

4.3 Theoretical Developments

4.3.1 Guth’s Equation and Nicolais-Narkis Theory

The modulus and yield strength of particle-filled composites can be predicted by

Guth’s equation (Equation 4.1) and Nicolais-Narkis theory (Equation 4.2), respectively

(Bliznakov, et al., 2000):

𝐸𝐶 = 𝐸𝑚 1 + 2.5 𝜙𝑓 + 14.1 𝜙𝑓2 (4.1)

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𝜎𝑦𝑐 = 𝜎𝑦𝑚 1 − 1.21 𝜙𝑓2/3 (4.2)

Where, E and σy are Young’s modulus and yield strength, respectively; subscripts

m, f, and c denote matrix, filler and composite, and Φf is volume fraction of particles. It is

quite obvious, that the experimental modulus (except at 7.5 wt% loading) and yield

strength values are larger than the predicted values, as shown in Figures 4.3 and 4.4. These

results indicate the limitations of the theories when applied to nano-composites. The strong

interaction between CaCO3 nano-particles and PVC matrix caused by the large interfacial

areas has led to much higher elongations-at-break, with a maximum at 5 wt% nano-

particles, as shown in Figure 4.5 (Xie, et al., 2004).

Figure 4.5 Variation of Charpy notched impact energy of nano-composites with nano-

particle loading (Xie, et al., 2004).

The specific surface area of filler gives important information about the filler

reinforcement properties. In his review paper on filler-elastomeric interactions, Wang

(1998) notes that the specific surface area of carbon black directly affects dynamic

mechanical properties.

When the surface area for silica increases, there is a higher compound viscosity

which requires more energy (torque) for mixing and also contributes to the buildup of heat

in the mixture. If too much heat is generated the compound will cure prematurely. At the

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same time, as the filler surface areas increases the tensile and tear strength of the rubber

increases. Much is still unknown concerning the mechanisms of how these properties

directly relate to reinforcement and what the exact mechanisms of reinforcement are

(Jordan, et al., 2005).

4.3.2 The Mixture Rule

The field of composite material behavior can be studied from two perspectives:

micromechanics and macromechanics. The goal of most micromechanics approaches is to

determine the elastic moduli or stiffness of a composite material in terms of the elastic

properties of the constituent materials. Most of the analytical models presented presume

the idealization that there is perfect adhesion between the phases and that the particles are

spherical and evenly dispersed. Some of the earlier attempts in modeling composites were

performed by Einstein and Guth. Guth and Smallwood extended Einstein’s theory to

explain rubber reinforcement. Both of these attempts have proved to be applicable, but

only at low concentrations of particulate. Thus, the focus will be on the newer works,

separated into two approaches, defined as either a mechanics of materials or an elasticity

approach. In the mechanics of materials approach some simplifying assumptions are made,

the most significant of which is that the strain in the matrix is equal to the strain in the

particulate. With this assumption the most simplistic of all methods of predicting the

moduli of a composite, known as the rule of mixtures, can be obtained (Haghighat, et al.,

2005):

𝐸𝑐 = 𝐸𝑓𝜑𝑓 + 𝐸𝑝𝜑𝑝 (4.3)

Ec, Ef, and Ep, are the elastic modulus of compound, filler and polymer matrix,

respectively. φf is the volume fraction of filler and φp is the volume fraction of polymer

matrix. The mixing law represents the linear relationship of elastic modulus in which the

effects of size, shape and particle distribution has been neglected. Generally, the law of

mixtures has been considered as the upper limit of elastic modulus. The absolute lower

bound on elastic modulus can be obtained, assuming equal stress in the matrix and

particulate (Haghighat, et al., 2005; Zhang, et al., 2003):

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𝐸𝑐 =1

𝜙𝑓

𝐸𝑓+

𝜙𝑝

𝐸𝑝

(4.4)

The upper and lower limits on elastic moduli represent the most widely used

relationships derived through material mechanics approach. Although other expressions

have also been proposed using various assumptions, the elasticity approach has received

maximum attention (Haghighat, et al., 2005).

Two of the most important models which are now applied for the nano-composites

are proposed by Hashin and Halpin-Tsai model (Haghighat, et al., 2005). Although, the

calculated values resulting from these models do not accurately predict the mechanical

properties, these may be considered as an estimation of elastic modulus for nano-

composite samples. In the following, the related equations for these models have been

discussed.

4.3.3 Hashin-Shtrikman Model

The composite spheres model, introduced by Hashin, consists of a graduation of

sizes of spherical particles fixed in a continuous matrix phase. In line with this model,

Hashin and Shtrikman developed the bounds for the shear and bulk modulus (Hashin,

1962; Wang and Pyrz, 2004; Haghighat, et al., 2005). The resulting bounds on the Young’s

modulus are as follows:

Lower bound for the Bulk moduli, Kl:

𝐾𝑙 = 𝐾1 +𝜙2

1𝐾2 − 𝐾1

+3𝜙1

3𝐾1 + 4𝐺1

(4.5)

The upper bound on Ku is,

𝐾𝑢 = 𝐾2 +𝜙1

1𝐾1 − 𝐾2

+3𝜙2

3𝐾2 + 4𝐺2

(4.6)

Similarly, the lower bound of the Shear moduli, Gl:

𝐺𝑙 = 𝐺1 +𝜙2

1𝐺2 − 𝐺1

+6(𝐾1 + 2𝐺1)𝜙1

5𝐺1(3𝐾1 + 4𝐺1)

(4.7)

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and the corresponding upper bound:

𝐺𝑢 = 𝐺2 +𝜙1

1𝐺1 − 𝐺2

+6(𝐾2 + 2𝐺2)𝜙2

5𝐺2(3𝐾2 + 4𝐺2)

(4.8)

The resulting bounds on the Young’s modulus are following equations:

Lower bound:

𝐸𝑙 =9𝐾𝑙𝐺𝑙

3𝐾𝑙 + 𝐺𝑙

=

9 𝑘1 +𝜙2

1𝑘2 − 𝑘1

+3𝜙1

3𝑘1 + 4𝐺1

𝐺1 +𝜙2

1𝐺2 − 𝐺1

+6(𝑘1 + 2𝐺1)𝜙1

5𝐺1(3𝑘1 + 4𝐺1)

3 𝑘1 +𝜙2

1𝑘2 − 𝑘1

+3𝜙1

3𝑘1 + 4𝐺1

+ 𝐺1 +𝜙2

1𝐺2 − 𝐺1

+6(𝑘1 + 2𝐺1)𝜙1

5𝐺1(3𝑘1 + 4𝐺1)

(4.9)

Upper bound:

𝐸𝑢 =9𝐾𝑢𝐺𝑢

3𝐾𝑢 + 𝐺𝑢

=

9 𝑘2 +𝜙1

1𝑘1 − 𝑘2

+3𝜙2

3𝑘2 + 4𝐺2

𝐺2 +𝜙1

1𝐺1 − 𝐺2

+6(𝑘1 + 2𝐺1)𝜙1

5𝐺2(3𝑘2 + 4𝐺2)

3 𝑘2 +𝜙1

1𝑘1 − 𝑘2

+3𝜙2

3𝑘2 + 4𝐺2

+ 𝐺2 +𝜙1

1𝐺1 − 𝐺2

+6(𝑘1 + 2𝐺1)𝜙1

5𝐺2(3𝑘2 + 4𝐺2)

(4.10)

In the above equation, subscripts 1 and 2 refer to the polymer and the filler

respectively. It may be worth mentioning that the above equations are applicable when

K1<K2 and G1<G2 (Wang and Pyrz, 2004; Haghighat, et al., 2005).

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4.3.4 Halpin-Tsai Model

The semi-empirical equations suggested by Halpin-Tsai are yet another way for

predicting the composite properties, equations 4.11 and 4.12:

𝐸 =𝐸1(1 + 𝜁𝜂𝜙2)

1 − 𝜂𝜙2 (4.11)

Where

𝜂 =

𝐸2𝐸1

− 1

𝐸2𝐸1

+ 𝜁 (4.12)

These equations result from simplification and approximation of micromechanical

complex models. The importance of this method lies in its simplicity and its ability for

generalization. The only problem in this method is related to the parameter of ζ, which is

best determined experimentally. Generally, this parameter for particulate composites has

been approximated to have a value of 2 for matter properties (Haghighat, et al., 2005).

Some other mechanical relationships for the Young’s modulus and Poisson’s ratio

of composites have been given by Budiansky (1965) as follows:

𝐸 =9𝐾𝐺

3𝐾 + 𝐺 (4.13)

𝜐 =3𝐾 − 2𝐺

6𝐾 + 2𝐺 (4.14)

and for each constituent

𝐾𝑛 =𝐸𝑛

3 − 6𝜐𝑛 (4.15)

𝐺𝑛 =𝐸𝑛

2 + 2𝜐𝑛 (4.16)

Where, n = 1 or 2 (denoting continuous and filler phases respectively).

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4.3.5 Micro-mechanics Models

It is well known that most composite materials are anisotropic/transversely

isotropic. Many composite analyses are performed using a macroscopic approach where

the properties of the composites are homogenized to produce an anisotropic, yet

homogeneous continuum before the analysis is conducted. However, the micromechanical

approach to analyzing composites considers the filler and the matrix separately and applies

the loading and the boundary conditions at the individual filler and matrix level. The

overall properties of the composite are developed by relating the average stresses and

strains. Thus the micro-mechanical approach is expected to provide much more detail into

the true interactions between the filler and matrix, potentially leading to a more accurate

model of the composite behavior (Gardner, 1994). Two continuum-based micro-mechanic

models are briefly described in the following sub-sections.

4.3.5.1 Mori-Tanaka Model

The Mori-Tanaka approach, which is based on the Eshelby Tensor, can be used to

predict the elastic properties of two-phase composites (matrix and effective particle

phases) as a function of the effective particle volume fraction and geometry and they may

be perfectly bonded to each other (Mori & Tanaka, 1973; Benveniste, 1987). The Mori-

Tanaka approach has been used to accurately predict overall properties of composites

when the reinforcements are on the micrometer-scale level, or higher (Odegard, et al.,

2005). At these higher length scales, the assumption of the existence of two phases is

apparently acceptable. However, for nanometer-sized reinforcement, it has been shown

that the molecular structure of the polymer matrix is significantly perturbed at the

reinforcement/polymer interface, and this perturbed region is on a length scale that is the

same at that of the nanometer-sized reinforcement. Therefore, at the nanometer level, the

reinforcement and adjacent polymer region is not accurately described as consisting of just

two phases, thus the Mori-Tanaka model is not expected to perform well for nano-

structured reinforcements.

4.3.5.2 Effective Interface Model

Because of the aforementioned drawbacks to the Mori-Tanaka approach, another

modeling approach was developed. The effective interface model can be used to predict

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the elastic properties of a composite with effective particles that have an interface of the

same spherical shape as the effective particle. The effective interface has a finite size and

models the region immediately surrounding the spherical reinforcement, which is

commonly referred to as an interphase or an interaction zone (Dunn & Ledbetter, 1995).

Odegard, et al. (2005) reported that unlike the Mori-Tanaka model, the effective interface

model should be applicable to both nanometer-sized and larger-sized reinforcement.

4.4 Final Remarks on Models

A critical analysis of the brief literature on the modeling of the mechanical properties

of polymer composites presented above suggests that there is no general appropriate and

correct model applicable for all nano-composites for estimating their properties. The

conclusions from these researches and studies may be summarized as follows:

1. Considering the dimensions of polymer chains and their crystalline assemblies, it

can be said that all polymers have structure on the nanometer size scale and,

further, that the mechanical properties of polymers are governed by the interactions

of these nanostructures with one another. Therefore, to influence the interactions

that govern the mechanical properties of polymers, specific nano-scale

reinforcement is efficient and beneficial.

2. Where the particle/matrix interface has a high energy, a very stiff network can be

formed. In such situations, the stiffness of the matrix is much lower than the

network stiffness and the mechanical behavior of the composite material can be

rather well predicted on the basis of a structural modeling.

3. From a very general point of view, the elastic modulus of a filled polymer is

affected by (i) the elastic properties of its constitutive phases (i.e., modulus and

Poisson’s ratio), (ii) the volume fraction of filler (iii) the morphology (i.e., shape,

aspect ratio, and distribution of the filler into the polymeric matrix), and (iv) the

interactions between fillers. Various models are proposed in the literature to

understand the complex interplay between these parameters and to predict of the

elastic modulus of polymer composites.

4. The available theoretical models may be used to predict elastic modulus as a first

approximation or initial estimation.

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5. Micromechanical approach to analyzing composites considers the filler and the

matrix separately and applies the loading and the boundary conditions at the

individual filler and matrix level.

4.5 Summary

To facilitate the development of nano-structured composites, constitutive

relationships are developed that predict the bulk mechanical properties of the materials as a

function of the molecular structure of the polyamide and reinforcement. These constitutive

relationships can be used to influence the design of these materials before they are

synthesized. In this study the following theoretical models obtained from literature were

used to predict Young’s modulus of micro and nano-composite samples synthesized.

1. The Rule of Mixtures

2. Nicolais-Narkis Theory and the Guth’s equation

3. The composite sphere model of Hashin-Shtrikman.

4. The semi-empirical micromechanical model of Halpin-Tsai.

Results obtained from these computations and their predictive capability is

discussed later in Chapter-6.

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