investigating and understanding the mechanical response of linked structures of hard and soft

i

INVESTIGATING AND UNDERSTANDING THE MECHANICAL RESPONSE OF

LINKED STRUCTURES OF HARD AND SOFT METALS

USING CONSTANT DISPLACEMENT APPROACH: A NUMERICAL STUDY

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science

Prashant Pawan Gargh

August, 2016

ii

INVESTIGATING AND UNDERSTANDING THE MECHANICAL RESPONSE OF

LINKED STRUCTURES OF HARD AND SOFT METALS

USING CONSTANT DISPLACEMENT APPROACH: A NUMERICAL STUDY

Prashant Pawan Gargh

Thesis

Approved: Accepted:

____________________________ ____________________________ Advisor Department Chair Dr. T.S. Srivatsan Dr. Sergio Felicelli

____________________________ ____________________________ Co-Advisor Interim Dean of the College Dr. Shivakumar Sastry Dr. Eric Amis

____________________________ ____________________________ Faculty Reader Dean of the Graduate School Dr. Xiaosheng Gao Dr. Chand Midha

____________________________ ____________________________ Faculty Reader Date Dr. Craig Menzemer

iii

ABSTRACT

A progressive increase in interest in the use of linked structures, or perforated metal

sheets/plates, has become increasingly evident in the time period spanning the last three decades,

since the early 1990s. These structures are gaining increasing attention for use in a spectrum of

both performance-critical and non-performance-critical applications. Two different sizes of the

perforations in a metal sheet were chosen resulting essentially in a structure that was held together

by a network of links of varying thickness. The two designs of the perforated metal sheet that

form the very essence of this research study were made possible using ABAQUS [version 6.13.2].

The specific metals chosen for this study belong to the families of both ferrous alloys and non-

ferrous alloys. The two ferrous alloys chosen were alloy steel 4140 and carbon steel 1018; both

metals known for their high strength. The two non-ferrous alloys chosen were aluminum alloy

6061 and pure copper; both metals known for their good ductility and popular choices for a

spectrum of lightweight applications. The method of finite elements in synergism with a numerical

approach was put to use to study the mechanical response of linked metal structures when subjected

to the influence of an external mechanical stimulus. The mechanical stimulus chosen in this study

was a tensile load. Five different load levels, as fractions of yield stress of the chosen metal, and

spanning the domains of both elastic and plastic deformation were chosen. The finite element

approach was used for determining the deformation or displacement experienced by the centroidal

nodes and the link elements. The results were also used to establish the variation of stress with

strain for linked metal structures under conditions of plane stress. For each metal chosen, i.e., thin

links and thick links, the response kinetics under the influence of an external load was determined

for the case of both symmetric loading and asymmetric loading. The mechanical response,

iv

quantified by displacement experienced by the centroid nodes was recorded and compared with the

aid of 3D bar graphs for the five levels of load chosen. This formulation is overall useful for purpose

of studying and rationalizing the mechanical response of linked metal structures when under the

influence of an external mechanical stimulus.

v

ACKNOWLEDGEMENTS

It is with immense gratitude that I acknowledge the support and help of my advisor, Dr. T.

S. Srivatsan and Co-advisor Dr. S. Sastry for their guidance throughout my research studies. They

were inspirational to me as individuals and their direction, drive and dedication to ensure diligent

articulation of my energy and efforts through my research endeavor was certainly inspiring,

intellectual and invaluable. This makes me extend ‘valued’ gratefulness for their patience,

enthusiasm and sustained support extended to me during my precious two years through graduate

school at the University of Akron.

I would like to extend my sincere thanks to Dr. Xiaosheng Gao and Dr. Craig C. Menzemer

for serving on my thesis committee. Additional, I utilize the opportunity to both express and extend

my sincere thanks and appreciation to the following individuals for their ‘valued’ contribution, by

way understanding and extension of knowledge and assistance that did enable in successful

completion of this research endeavor:

(i) Dr. Sergio Felicelli (Chair, Department of Mechanical Engineering) for awarding

me with a Teaching Assistant which helped me to complete my Master of Science degree

in the Department of Mechanical Engineering.

(ii) Dr. Atef Saleeb (Professor, Department of Civil Engineering) for instruction and

timely assistance through technical intricacies and guiding me for the use of ‘ABAQUS’

software.

vi

(iii) Mr. Clifford Bailey (Senior Engineering Technician, Department of Mechanical

Engineering), for instruction and timely assistance related to use of computers and

software.

Above all I want to extend my gratitude to my parents, and dear friends for their love,

encouragement and sustained support through the years of my schooling and valued moments

through college while pursuing undergraduate and graduate education in engineering.

vii

TABLE OF CONTENTS

Page

LIST OF TABLES .......................................................................................................................... x

LIST OF FIGURES ....................................................................................................................... xii

CHAPTER

I Introduction ......................................................................................................................... 1

1.1 Overview: Interest in use of perforated sheet metals .................................................. 1

1.2 Variation of Stress with Strain .................................................................................... 2

1.3 Elastic-Plastic Mechanics with Respect to Metals ...................................................... 5

1.4 Objectives of this Research Study .............................................................................. 5

II Review of the Published Literature ..................................................................................... 8

2.1 What is Perforated Metal Sheet .................................................................................. 8

2.2 Types of Perforations in a Metal ................................................................................. 9

2.3 A Review of Research done on Perforated Metal Sheets.......................................... 10

2.4 The Tension Test ....................................................................................................... 16

2.5 A Brief Theory Pertinent to Plane Stress .................................................................. 17

III The Materials Chosen ....................................................................................................... 19

3.1 The Ferrous Alloys [Alloy Steel 4140 and Carbon Steel 1018] ............................... 19

3.2 The Non-Ferrous Alloys [Aluminum Alloy 6061-T6 and Copper] .......................... 20

IV Design of Test Specimen .................................................................................................. 24

V Formulation of the Problem .............................................................................................. 29

5.1 Description ................................................................................................................ 29

5.2 Material Properties .................................................................................................... 32

viii

5.3 Deformation and Support .......................................................................................... 32

VI Finite Element Analysis: Numerical Procedure ................................................................ 33

6.1 Finite Element Formulation ...................................................................................... 33

6.2 Finite Element Simulation ........................................................................................ 36

VII Two Dimensional Finite Element Model .......................................................................... 39

7.1 Modelling .................................................................................................................. 39

7.2 Material Selection and Type of Analysis .................................................................. 40

7.3 Size of Mesh and Configuration ............................................................................... 40

7.4 Boundary Conditions ................................................................................................ 42

7.5 Loading ................................................................................................................ 43

7.5.1 Symmetric Loading ........................................................................................... 44

7.5.2 Asymmetric Loading ......................................................................................... 45

VIII Results and Discussions .................................................................................................... 51

8.1 A Comparison between Thin Structures of Alloy Steel 4140 And Carbon Steel 1018 ......................................................................................................................... 52

8.1.1 Symmetric Loading of Alloy Steel 4140 and Carbon Steel 1018 ..................... 52

8.1.2 Asymmetric Loading of Alloy Steel 4140 and Carbon Steel 1018 ................... 63

8.2 A Comparison between Thin Structures of Aluminum Alloy 6061 and Copper C 10-200 ............................................................................................................... 74

8.2.1 Symmetric Loading of Aluminum Alloy 6061-T6 and Copper C 10-200 ........ 74

8.2.2 Asymmetric loading of Aluminum Alloy 6061-T6 and Copper C10-200 ........ 85

8.3 A Comparison between thick structures of Alloy Steel 4140 and Carbon Steel 1018 ............................................................................................................. 95

8.3.1 Symmetric Loading Alloy Steel 4140 and Carbon Steel 1018.......................... 95

8.3.2 Asymmetric loading: Alloy Steel 4140 and Carbon Steel 1018...................... 105

8.4 A Comparison between Thick Structures of Aluminum Alloy 6061-T6 and Copper C 10-200 ............................................................................................................. 116

ix

8.4.1 Symmetric loading of Aluminum Alloy 6061 –T6 and Copper 10200 ........... 116

8.4.2 Asymmetric loading of Aluminum Alloy 6061-T6’and Copper 10200 .......... 126

IX Conclusions ..................................................................................................................... 137

References ................................................................................................................................. 140

x

LIST OF TABLES

Table Page

3.1 Nominal chemical composition of the two hard metals chosen for purpose of analysis. (In weight percent.) .............................................19

3.2 Nominal chemical composition of the two non-ferrous metals chosen for purpose of analysis ...................................................................21

3.3 Uniaxial tensile properties of the two hard metals chosen for this study ...........................................................................................................19

3.4 Uniaxial tensile properties of the two non-ferrous metals chosen for this study ...........................................................................................................21

4.1 Dimensions of the structure containing thin links and thick links .............27

4.2 Dimensions of the perforations chosen to form the linked metal structures used in this study ........................................................................................27

7.1 The boundary conditions for two-dimensional FEM for uniaxial tension ........................................................................................................43

7.2 Node locations chosen for application of load for both symmetric and asymmetric loading for both thin link and thick link metal structure of the four chosen metals belonging to the ferrous alloy family and non-ferrous family .........................................................................................................48

8.1 A comparison of the values of displacements occurring at the internal nodes of the linked metal structure containing a network of thin links upon being subject to 100 pct. of the yield stress for symmetric loading ...........55

8.2 A comparison of the displacements obtained by different internal nodes of the thin linked structure, when subjected to 100 pct. of the yield stress for asymmetric loading ....................................................................................71

8.3 A comparison of the displacements obtained by different internal nodes of the thin linked metal structure, when subjected to 100 pct. of the yield stress and for the case of symmetric loading .............................................78

xi

8.4 A comparison of the vale of displacements experienced by different internal nodes of the thin linked structure, when subjected to a load that was 100 pct. of the yield stress for the case of asymmetric loading ..........93

8.5 A comparison of the displacements experienced by the internal nodes of a thick linked metal structure, when subjected to 100 pct. of the yield stress under symmetric conditions .....................................................................103

8.6 A comparison of the displacements obtained by different internal nodes of the thick linked structure, when subjected to 100 pct. of the yield stress for asymmetric loading ..................................................................................114

8.7 A comparison of the displacements obtained by different internal nodes of the thick linked structure, when subjected to 100 pct. of the yield stress for symmetric loading ....................................................................................119

8.8 A comparison of the displacements obtained by different internal nodes of the thick linked structure, when subjected to 100 pct. of the yield stress for asymmetric loading ..................................................................................135

xii

LIST OF FIGURES

Figure Page

3.1 Optical micrographs showing microstructure of alloy steel 4140 at two different magnifications and the two key micro-constituents: pearlite and ferrite ..........................................................................................................22

3.2 Optical micrographs showing microstructure of carbon steel 1018 at two different magnifications and the two key micro-constituents: cementite and ferrite ...................................................................................................22

3.3 Optical micrographs showing microstructure of aluminum alloy 6061 at two different magnifications showing grains of varying size and a non-uniform distribution of both the coarse and intermediate-sixe second phase particles through the microstructure ..........................................................23

4.1 The perforated metal plate comprising of a network of thin links or thick link elements ..............................................................................................26 (a) Three-Dimensional view, (b) Two-dimensional view of the metal plate

4.2 The perforated metal plate comprising of a network of thick links or thick link elements ..............................................................................................27 (a) Three-Dimensional view (b) Top-dimensional view of the metal plate.

4.3 Dimensions of the links in the perforated metal plate ...............................28 (a) Plate with thin links or link elements, and (b) Plate with thick links or link elements

5.1 Isometric view & XY plane view of the thin linked structure ...................30

5.2 Isometric view and XY plane view of the thick linked structure ..............30

5.3 A schematic of the metal structure containing a network of links with identification of the different nodes ...........................................................31

7.1 Size of mesh for a structure containing a network of thin links ................41

xiii

7.2 Size of mesh for a structure containing a network of thick links ...............42

7.3 Pictorial view for the boundary conditions applied on the linked structure......................................................................................................43

7.4 Structural Chart depicting the work performed for purpose of analyzing the thin linked and thick linked metal structures of the four chosen metals .........................................................................................................44

7.5 Pictorial representation of metal structure containing a network of thin links and subject to “Symmetric” Loading” ..............................................46

7.6 Pictorial representation of metal structure containing a network of thin links and subject to “Asymmetric” Loading” ............................................47

7.7 Methodology used for location of the Nodes in the linked metal structure (Thin link structure and Thick link structure) ............................................49

8.1 Profile showing contours of the Von Mises stress for alloy steel 4140 that was subjected to symmetric loading at Node (2,6) and Node (4,2) for the metal structure containing a network of thin links at the yield stress ( σYS ) of the material ............................................................................................56

8.2 Profile showing the contours of the Von Mises stress for carbon steel 1018 that was subjected to symmetric loading at Node (2,6) and Node (4,2) for the structure containing a network of thin links at a load equal to yield stress of the material ..................................................................................56

8.3 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of alloy steel 4140 upon being subject to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..................................................................................................57

8.4 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..................................................................................................57

8.5 Profile showing the displacement experienced by the Different nodes of the “thin” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 50 percent of the yield stress, applied at Node (2,6) and Node (4,2) ...................................................................................58

8.6 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 100 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..................................................................................................58

xiv

8.7 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140 containing a network of thin links, when subjected to symmetric loading, that is 102 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..............................................59

8.8 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140,containing a network of thin links, when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2) ........................................................59

8.9 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140, containing a network of thin links, when subjected to symmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,2) ........................................................60

8.10 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of alloy steel 4140 containing a network of thin links when subjected to symmetric loading, that is 50 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..............................................60

8.11 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140, containing a network of think links, when subjected to symmetric loading, that is 100 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..............................................61

8.12 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140, containing a network of thin links, when subjected to symmetric loading, that is 102 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..............................................61

8.13 A contour profile showing the magnitude of displacements experienced by linked structure of alloy steel 4140, containing a network of thin link elements, when subjected to symmetric loading, 102 pct. of the yield stress ...........................................................................................................62

8.14 A contour profile showing the magnitude of displacements experienced by linked structure of carbon steel 1018, containing a network of thin links, when subjected to symmetric loading, 102 pct. of the yield stress ............62

8.15 Profile showing the contours of the Von Mises stress for alloy steel 4140 that was subjected to asymmetric loading at Node (2,6) and Node (4,1) for the structure containing a network of thin links and at load corresponding to yield stress of the metal .........................................................................64

8.16 Profile showing the contours of the Von Mises stress for carbon steel 1018 that was subjected to asymmetric loading at Node (2,6) and Node (4,1) for the structure containing a network of thin links at a load corresponding to yield stress of the metal ..............................................................................64

xv

8.17 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140, containing a network of thin links, when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1) ..............................................66









xvi


8.27 A contour profile showing the magnitude of displacements experienced by linked structure of alloy steel 4140, containing a network of thin links, when subjected to asymmetric loading, at 102 pct. of the yield stress ......72

8.28 A contour profile showing the magnitude of displacements experienced by linked structure of carbon steel 1018, containing a network of thin links, when subjected to symmetric loading at 102 pct. Of the yield stress ........72

8.29 Profile showing the contours of the Von Mises stress for aluminum alloy 6061 that was subjected to symmetric loading for Node (2,6) and Node (4,2) of the metal plate containing a network of thin links and at a load corresponding to yield stress of the metal ..................................................77

8.30 Profile showing the contours of the Von Mises stress for pure copper C- 10200 that was subjected to symmetric loading at Node (2,6) and Node (4,2) of the structure containing a network of thin links and corresponding to a load that is equal to yield stress of the chosen metal ..........................77

8.31 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of aluminum alloy 6061 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2) ...................................................................................79





xvii

8.36 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of copper C-10200 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..................................................................................................81




8.40 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of copper C-10200 when subjected to symmetric loading, that is 110 percent of the yield stress, applied at Node (2,6) and Node (4,2....................................................................................................83

8.41 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, 110 pct. of the yield stress ........................................84

8.42 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of copper C 10200 when subjected to asymmetric loading, 110 pct. of the yield stress ............................................................84

8.43 Profile showing the contours of the Von Mises stress for aluminum alloy 6061 that was subject to asymmetric loading at Node (2,6) and Node (4,1) of the thin linked structure at a load corresponding to yield stress of the metal ...........................................................................................................87

8.44 Profile showing the contours of the Von Mises stress for copper C 10200 that was subjected to asymmetric loading at Node (2,6) and Node (4,1) of the thin linked structure at a load corresponding to yield stress of the metal ...........................................................................................................87

8.45 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1) ...................................................................................88

xviii





8.50 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of copper C-10200 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1) ..................................................................................................90





8.55 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, 110 pct. of the yield stress ........................................94

xix

8.56 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of copper C 10200 when subjected to asymmetric loading, 110 pct. of the yield stress ............................................................94

8.57 Profile showing the contours of the Von Mises stress for alloy steel 4140 that was subjected to symmetric loading for Node (2,6) and Node (4,2) of the structure containing a network of thick links at a loads equal to the yield stress of the metal ..............................................................................97

8.58 Profile showing the contours of the Von Mises stress for carbon steel 1018 that was subjected to symmetric loading for node (2,6) and node (4,2) of the thick linked structure at the elastic limit ..............................................97

8.59 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2) ..................................................................................................98




8.63 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 102 percent of the yield stress, applied at Node (2,6) and Node (4,2) ................................................................................................100

8.64 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of carbon steel 1018 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2) ................................................................................................100


xx




8.69 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of alloy steel 4140 when subjected to symmetric loading, 102 pct. of the yield stress ..........................................................104

8.70 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of carbon steel 1018 when subjected to symmetric loading, 102 pct. of the yield stress ..........................................................104

8.71 Profile showing the contours of the Von Mises stress for alloy steel 4140 that was subjected to asymmetric loading for Node (2,6) and Node (4,1) of the thick linked structure at load corresponding to yield stress of the chosen metal.............................................................................................108

8.72 Profile showing the contours of the Von Mises stress for carbon steel 1018 that was subjected to asymmetric loading for Node (2,6) and Node (4,1) of the thick linked structure at a load corresponding to the yield stress ......108

8.73 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of alloy steel 4140 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1) ................................................................................................109



xxi



8.78 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of carbon steel 1018 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1) .................................................................................111





8.83 A contour profile showing the magnitude of displacements experienced by “thick” linked structure of alloy steel 4140 when subjected to asymmetric loading, 102 pct. of the yield stress ..........................................................115

8.84 A contour profile showing the magnitude of displacements experienced by “thick” linked structure of carbon steel 1018 when subjected to symmetric loading, 102 pct. of the yield stress ..........................................................115

8.85 Profile showing the contours of the Von Mises stress for aluminum alloy 6061 that was subjected to symmetric loading for Node (2,6) and Node (4,2) of the thick linked structure at load corresponding to yield stress .........................................................................................................118

xxii

8.86 Profile showing the contours of the Von Mises stress for pure copper C-10200 that was subjected to symmetric loading for Node (2,6) and Node (4,2) of the thick linked structure at a load corresponding to yield stress .........................................................................................................118

8.87 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of aluminum alloy 6061 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2) .................................................................................120





8.92 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of copper C-10200 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2) ................................................................................................122



xxiii



8.97 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of aluminum alloy 6061 when subjected to symmetric loading, 110 pct. of the elastic limit .......................................125

8.98 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of copper C 10200 when subjected to symmetric loading, 110 pct. of the elastic limit .........................................................125

8.99 Profile showing the contours of the Von Mises stress for aluminum alloy 6061 that was subjected to asymmetric loading for Node (2,6) and Node (4,1) of the thick linked structure at load commensurate with yield stress of the metal ..............................................................................................129

8.100 Profile showing the contours of the Von Mises stress for copper C 10200 that was subjected to asymmetric loading at Node (2,6) and Node (4,1) of the thick linked structure at load commensurate with yield stress of the metal .........................................................................................................129

8.101 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6), and Node (4,1) ................................................................................130

8.102 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,1) .................................................................................130



xxiv


8.106 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of copper C-10200 when . subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1) ................................................................................................132

8.107 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of copper C-10200 when subjected to asymmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,1) ................................................................................................133




8.111 A contour profile showing the magnitude of displacements experienced by “thick” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, 110 pct. of the yield stress of the chosen metal ......136

8.112 A contour profile showing the magnitude of displacements experienced by “thick” linked structure of copper C 10200 when subjected to asymmetric loading, 110 pct. of the yield stress of the chosen metal .........................136

1

CHAPTER I

INTRODUCTION

1.1 Overview: Interest in the use of perforated sheet metals

Perforated metal can at best be visualized as metal sheet or metal plate that is usually made

using various styles of perforating, embossing, slotting, and even checkered plates [1] . The process

of creating perforations in a metal sheet or metal plate is often referred to as perforating, which

often involves puncturing the workpiece (usually a thin sheet of metal) using a tool. Slotted

perforated metal is a sheet or coil of material made from metal that contains holes that are punched

using a die. Embossing is considered to be a metal forming process for producing raised or sunken

designs and even relief in sheet material [2]. Checkered plate, also known as diamond plate, is

typically a light-weight metal sheet having a regular pattern of raised diamonds or lines. Various

types of metal sheet can be manufactured using a variety of perforations. A perforated metal sheet

is rigid, and is often chosen for use in a wide variety of applications primarily because of ability to

offer consistent performance. Over the last two decades, the views on the use of perforated sheets

in the manufacturing industry have observably changed. In the early years, the major causes for

concern were the perceived difficulty of generating the perforated sheets of high quality and it often

lead to a question mark on accuracy of the results. Currently, using potentially feasible and

available advanced computational techniques in synergism with computer-aided design (CAD) and

computer aided manufacturing (CAM) this has been made possible.

Perforated metal sheets are of interest when it comes to the traditional methods for

optimum use of a complete sheet of metal when compared to a perforated metal sheet. Using

2

conventional technologies, it is possible to manufacture perforated metal sheets on a production

basis. In the prevailing era a number of perforations are available in the market, which both forms

and shows a continuous trend with specific reference to advances in design by providing not only

an impressive look but also a neatly build structure [3]. Various features a perforated metal sheet

possesses besides an overall attractive appearance are the following: (i) light in weight, (ii) resistant

to corrosion, and (iii) a large open area, which is conducive for the passage of air. These plates can

be manufactured in a variety of gage lengths, using prevailing technologies currently in widespread

use. The overall strength - to- weight ratio [σ / ] is relatively good. The perforated metal sheet

offers numerous applications where it can be put to effective use.

Pioneering research on the design and analysis of perforated metal sheets has been done by

only a few researchers [4]–[7]. Most of the research that is available in the published literature is

primarily focused on circular perforations in a metal sheet and uniformly distributed through the

thickness. Perforations in a metal sheet can be of two types (i) uniform perforations, and (ii) non

uniform perforations.

(i) Uniform perforations

Previous researchers have proposed a yield criterion using the continuum approach for the

plane stress condition and an assumption of isotropic metal sheet, which describes plastic

flow behavior of a perforated metal sheet.

(ii) Non Uniform Perforations

When the perforations are non-uniform, the stress through the thickness or z- direction

varies. Hence, it represents a condition of plane strain. Also, to obtain better results, a

3dimensional geometry is preferred for both a study and understanding of the deformation

behavior of perforated metal sheets.

1.2 Variation of Stress with Strain

3

In today’s era, materials like carbon steel, alloy steel, aluminum and copper, to name a few, are

gaining widespread use in a spectrum of engineering applications, which has enhanced both their

selection and use in the commercial market pertinent to engineering products. Therefore, the

characterization of a material under variety of conditions is made possible by the stress versus strain

curve, thereby providing an overview of both the linear and nonlinear regions pertinent to

deformation prior to failure by fracture. The stress versus strain curve for every specific material is

available in the published literature and does provide an understanding on the suitability and/or

applicability of a material for selection and use in a specific engineering application [8]. The stress

versus strain curve is equally important in estimating fracture strength of the material. The stress

versus strain curve for a material is generally obtained from doing a tensile test on a test specimen

of the material. Alloy steels are a class of versatile ferrous-based metal that offers a combination

of high strength, acceptable fracture toughness, good ductility, good cyclic fatigue resistance,

moderate resistance to corrosion, good resistance to wear and an overall acceptable to good

combination of mechanical properties. One such alloy steel that is preferentially chosen for use in

a spectrum of engineering products and applications is Alloy steel 4140. However, its nonlinear

behavior makes it quite different from plain carbon steel, such as AISI 1018.

For ductile metals, such as aluminum alloy and copper, the mechanical properties, such as:

yield stress, ultimate tensile stress, ductility, fracture toughness and even cyclic fatigue resistance

play an important role in their selection for use in engineering products. The engineering stress is

defined as the ratio of applied load (P) to the cross section area of the test specimen along the gage

length (Ao). The engineering strain is defined as the ratio of extension in the gauge length (l – lo)

to the original chosen gage length (lo). The maximum engineering stress is defined as the ratio of

maximum load (Pmax.) to the cross section area of the test specimen along the gage length (Ao), also

referred to or known as ultimate tensile strength (UTS). These properties are also used to predict

the work hardening behavior of the material. At low values of stress and strain, the variation of

stress wit strain follows the ‘Hooks law’, which ensures that stress is directly proportional to strain.

4

Elastic modulus (E) is the slope of the elastic portion of the stress versus strain curve. Elastic

Modulus can be safely categorized to be of three types depending upon the nature of loading and

strain induced. Depending on the nature of stress and strain, i.e., tensile, shear and compression,

the elastic modulus is referred to as Young’s Modulus, Shear Modulus and Bulk Modulus. In

several branches of the industry, mechanical tests spanning tensile, compression, hardness to

include both macroscopic hardness and microscopic hardness, shear, bending and even fatigue, to

name a few are often performed. Both the machines and their components often fail by fracture

often caused as a consequence of excessive loading. Hence, a design engineer estimates the

anticipated stress, which the component or structure can withstand using a specific material. This

can be done using experimental analysis or finite element (FE) simulations to check quality of the

product coupled with ability to bear a load based on which selection of an appropriate material can

be made. In the prevailing era the high cost of machinery coupled with the time taken by

manufacturing techniques, has led to rapid advances in field of finite element simulation, which

can and has been used even during the early stages of product development so as to ensure an

improved product and its applicability for selection and use in the industry specific to

engineering[9]. Based on data made public in the published literature the proper choice of a

material during the preliminary stages of design is mandatory. During the early stages, the changes

are relatively easy to make and this often lead to substantial savings during the later stages of

product development and commercialization. Several researchers have developed models to

simulate the stress versus strain response of structures to predict the deformation behavior of the

family of steels when subjected to loading [8][10]. During these years, the materials are often

described using the stress versus strain curve. A description of the stress versus strain behavior of

a material is important for the purpose of studying numerical modelling and real life applications

under different conditions of loading [8]. Initially Ramberg and Osgood[11][12] provided a simple

formula to describe the stress versus strain curve using three parameters and compared them with

the tensile data for both carbon steel and an aluminum alloy. In their technical report, they

5

concluded that apart from the young’s modulus and yield strength, one more parameter, i.e. ‘n’ is

required to show different regions of the stress versus strain curve. It also helps in determining or

establishing the non-linearity of the curve, resulting in a perfect elastic-plastic behavior when n

becomes infinite.

Mirambell and Real [12] proposed a two-stage model for a material established from the

work of Ramberg and Osgood . Since the experimental stress versus strain curve and the curve

obtained using the formula put forth by Ramberg-Osgood were in good agreement for stress levels

up until the yield point, whereas at the higher stress levels, the correlation showed lack of a fit good.

1.3 Elastic-Plastic Mechanics with Respect to Metals

In the fascinating world of material science, a material has the capability to deform

elastically and plastically by a number of mechanisms. When the stress exceeds the proportional

limit, a small portion of deformation remains upon removal of the load. The deformation remaining

after removal of the applied load is referred to as elastic plastic deformation. There are other

possibilities for a material to flow plastically, such as, (i) slip, (ii) twinning, and (iii) a combination

of other mechanisms.

Elastic plastic mechanics can be done using a real stress versus strain curve, which helps

us to understand deformation behavior of the component to be simulated and the load carrying

capacity, which can be estimated using finite element analysis. A variety of modelling techniques

have been put forth in the published literature in an attempt to both represent and explain the

nonlinear elastic behavior of metals. The elastic plastic behavior of a material or metal eventually

leads to the initiation of one or more fine microscopic cracks that grow with time and eventually

culminate in failure of both the material and structure.

1.4 Objectives of this Research Study

A perforated sheet of metal resulting in a structure that is essentially held together by a

network of fine links was subjected to mechanical deformation by way of application of load

application in tension and is the focus of this research study. Two different sizes of the perforations

6

in a metal sheet were chosen resulting essentially in a structure that was held together by a network

of links of varying thickness. A perforated sheet of the chosen metal was held together by a

network of thin links, while another was held together by a network of thick links. The two designs

of the perforated metal sheet were made possible using ABAQUS [version 6.13.2][13]. The metals

chosen for this research study were: (a) two steels having varying degrees of strength, i.e., an alloy

steel and a carbon steel, and (b) two non-ferrous alloys having varying strength and ductility. The

method of finite element analysis was used to analyze the mechanical response, or behavior, of the

perforated metal sheets when subjected to the influence of a load that is applied in tension. For five

chosen levels of the applied load, as a function of yield load of the chosen metal, the displacements

induced in the links and at the nodes or nodal points were determined. For each situation, i.e., thin

links and thick links, the response kinetics under the influence of an external load was determined

for two loading conditions, referred to henceforth through this document as: (i) symmetric loading,

and (ii) asymmetric loading.

A sustained Interest in the use of linked metal structures has shown noticeable growth

during recent years. This lead to an interesting study on the mechanical response of grid structures

when subjected to loading. A finite element approach for purpose of analyzing the mechanical

behavior of linked metal structures was adopted. Finite element simulations were carried out to

study the deformation of structures containing thin link and structure containing thick links. The

approach was also applied to a commercially used aluminum alloy and pure copper for five levels

of loading as a fraction of yield strength of the chosen metal. The analysis was carried out to obtain

the stress versus strain curve under conditions of plane stress and the displacement experienced by

the link elements and nodes in the linked metal structure when subjected to loading. The mechanical

response of the centroid nodes was recorded and the values obtained were plotted using 3D bar

graphs for five chosen levels of load. The load levels chosen represent both the elastic and inelastic

regime of the stress versus strain curve of the chosen metal. This formulation and analysis is useful

for the study of mechanical response of linked-metal structures.

7

This thesis document or report is organized into nine chapters with a bibliography of the

references cited provided towards the end.

• Chapter-1 provides a very brief overview of perforated metal sheet, which is referred to in

this study as a linked metal structure / perforated sheet metals. In this chapter towards the

end is listed the objectives of this research study to understand the behavior of linked metal

structures held together by a network of links of varying thickness.

• Chapter-2 provides a brief review of the work done on linked structures having different

types of perforations. The various theories proposed on the basis of displacement and stress

acting on perforated metal sheets, or linked structures, and the methods of approach used

are briefly highlighted.

• Chapter 3 presents a brief overview on the materials, i.e., metals, chosen for this study and

for purpose of comparison the basic mechanical properties of the four metals chosen for

this study

• Chapter 4 provides the design of the specimen modelled via modelling software and

Chapter 5 provide a summary of the research efforts on modeling using the tensile test on

linked structures of the chosen metals.

• Chapter 6 provides a summary of approach taken for purpose of simulation and analysis is

performed using the method of finite elements (FEA).

• Chapter-7 provides a detailed summary of the two dimensional finite element modelling

performed on the ABAQUS software.

• Chapter 8 presents a comparison of results obtained using the finite element method in

synergism with numerical computation of the linked structures of the chosen metals when

subject to loading. A comparison of displacements experienced by both the link elements

and nodes in the four chosen metals, containing a network of either thin links or thick links,

is made.

• Chapter 9 provides a highlight of the key findings obtained from this research study.

8

CHAPTER II

REVIEW OF THE PUBLISHED LITERATURE

2.1 What is Perforated Metal Sheet?

A perforated metal sheet is obtained when a metal sheet is punched to provide a

desired shape. Perforated metal sheets are gaining increasing interest and widespread use

in a spectrum of engineering-related applications spanning both performance-critical and

non-performance-critical. A perforated metal sheet is fast gaining interest in both selection

and use as a viable alternative to a whole sheet of metal. In the industries spanning

manufacturing, production and even mechanical, the punching process is often referred to

as ‘blanking’ or piercing. Blanking refers to cutting of a sheet metal along well-defined

path to separate the piece from the surrounding sheet metal using a punch and die assembly.

The part that is extracted from the sheet metal is the desired product that is referred to as

blank. In punching, the blank is the discarded product while the sheet metal with a hole is

the desired end product subsequent to the operation. In the early days, a perforated metal

sheet was manufactured using a hydraulic press where movements of both the punch and

the die were judiciously used to manufacture a perforated metal sheet. Currently, use of

conventional technologies, made possible by a healthy synergism of computer-aided design

(CAD) and computer-aided manufacturing (CAM) have been used to create designs in a

metal sheet or metal plate thereby reducing or minimizing human effort, cost and even

enhancing productivity. A perforated metal sheet metal can have a wide variety of

9

perforations, which can be observed in daily used products, making it convenient for use

in a number of ways for purpose of human life. In essence, perforated metal sheets are used

in a variety of applications for several reasons. A few examples where perforated metal

sheets are chosen for use in commercial products are the following: (i) clothes washer, (ii)

dryer drums, (iii) speaker covers, (iv) automotive grills, (v) exhaust components, (vi)

engine silencers, (vii) filters used in the water industry, (viii) air diffusers in HVAC market,

(ix) acoustic panels for noise control, and even (x) architectural elements in building

construction, to name just a few. In fact, the use of perforated metal sheets is unlimited.

The multipurpose use of the perforated metal sheets has both inspired and motivated

researchers spanning the domains of mechanical, electrical and manufacturing engineering

to study their behavior [1][14][15] [16] . In common practice, using perforated metal sheet

in a machinery enclosure does certainly help in providing an easy passage for air or any

other medium while concurrently lightening the weight of the structure.

2.2 Types of Perforations in a Metal

For any perforated sheet of metal sheet, there are several design for perforations

available using conventional technology. A few examples are discussed in this section for

the study of a wide variety of use of perforated metal sheet. Due to a variety of

configuration for holes in a sheet metal, both tensile strength and the yield strength depend

entirely on the pattern and orientation of the perforation in the metal sheet. Depending upon

the requirement of engineering design the shape of perforations made in the metal sheet

using the technique of CAM include the following:

1 Square perforations in a sheet

2 Circular perforation or Round Holes

10

3 Hexagonal perforations.

4 Slot Holes

5 Diamond shape perforation and Cross-shaped perforations.

2.3 A Review of Research done on Perforated Metal Sheets

O’Donnell and Langer (1962) [3] from the United States put forth a method for

calculating stresses and deflections induced in a metal plate having triangular-shaped

perforations. Their study did include the method, which was used for calculating the

effective elastic constants under conditions of plane stress and during bending. The method

used by them for purpose of analysis of a perforated metal plate was using an equivalent

plate that was similar to a solid plate, in which the elastic constants, namely elastic modulus

(E) and poisson’s ratio (υ), were contrived instead of the actual elastic constants (E) and

poisson’s ratio (υ). They also described the role and/or influence of distance between two

holes, due to which stress limits in a perforated metal sheet should not appreciably increase.

There was a more exhaustive analytical method for perforated sheets, which are chosen for

use both in boilers and heat exchangers. This method did produce more detailed results

when compared to results produced by ligament theory. Complicated problems in plane

stress of the perforated metal sheet, comprising of a uniform distribution of circular

perforations was investigated by Goldberg & Jabbour [4]. In 1966, Slot and Yalch [17]

studied and solved the problem of (i) perforated sheet under uniform edge loading, (ii)

hollow hexagon under the influence of internal pressure loading, and (iii) a hollow square

under uniform shear loading. They obtained numerical solutions for the three loading

conditions, with the help of which the appropriate boundary conditions were discovered by

examining displacement, stresses and resulting forces adjacent to the boundary. Yettram &

Awadalla [18] [19] proposed a method for evaluating the critical load of a metal plate under

11

the following conditions: (a) simply supported plate under compression and shear, and (b)

rectangular plate tapered in the thickness direction and undergoing compression through

the thickness.

A preliminary study on behavior and use of perforated metal sheets and plates was

initially proposed by Goldberg & Jabbour[4] using an analytical method for the purpose of

investigating displacements in a perforated metal plate using the field equations related to

classical elastic theory. Another important result put forth by these researchers is that the

principal stress and principal strain for each and every point in the plate can be obtained

using the general solution for a stress field when subjected to loading. The values that

were obtained using the general solution were observed to converge faster to values of the

key elastic constants. Yettram and Awadalla [18] presented a method for determining the

elastic stability of plates having a rectangular pattern, using the finite element technique.

Their method helped in providing an understanding for the prediction of buckling load.

The computational method used could be considered to be easy, effective and reliable. One

other important concern with respect to application is the occurrence of bending of a

perforated metal plate having a collection of square pattern, the elastic constants were

obtained under conditions of bending. In one other independent study, Fort [5] justified

the solution provided was capable of determining the stress distribution at all nodes of a

plate that was subjected to both symmetric and asymmetric bending. The metal plate, on

account of ‘local’ stress concentration arising from the presence of perforated regions, does

develop transverse shear stress that exert an influence on the following: (i) Influencing

stress components in the element, and (ii) Affecting overall deformation of an element in

the plate.

12

Yettram & Brown [19] in their independent study presented and discussed the

effects of stress distribution, under conditions of plane stress, on buckling load. A

combination of symmetry and asymmetry shapes of the mode coupled with actual size of

a square perforation in a metal plate were important factors that need to be taken into

consideration. This method provided an understanding that depending upon size of the

perforation used in the metal sheet and prevailing boundary conditions, the mode shape

cannot be easily determined without a complete analysis. Hence, for the case of a

perforated metal sheet having a sizeable number of large perforations the buckling

coefficient decreases with an increase in size of the perforation in the metal sheet. In a

subsequent study, Sirkis and Lim [5] provided evidence that use of automated grid methods

did help in obtaining an accurate solution. They used an automated grid method for

purpose of analyzing both the displacements and strains there were developed in a

perforated specimen of a commercial aluminum alloy when subjected to loading. They

observed and even recorded plastic deformation to occur at locations, or regions,

surrounding the perforation pattern. Chen[10] investigated the plastic deformation

response of sheet metals by using a continuum approach for a perforated sheet of metal.

Results of their stress versus strain analysis obtained by the use of finite elements coupled

with actual experiments under condition of uniaxial tension in both the X and Y directions

did reveal elastically isotropic.

Under the influence of plastic deformation, an observable amount of anisotropy

was observed. For a simple uniaxial test in tension the yield stress and resultant strains

were compared between FEA simulation for the elastic-plastic condition and the

experimental results provided a reasonably good fit. Results for both stresses and strains

for the occurrence of ‘local’ deformation were consistent and showed a near similar trend.

13

Over the years, few attempts have been made to characterize the deformation behavior of

perforated metal sheets [20].

Garino and co-workers [21] presented and discussed the results obtained from

finite element simulation using several techniques and compared their numerical results

with experimental results for the case of a cylinder bar. From the results, they concluded

that use of the finite element technique in conjunction with a large strain elastic-plastic

model was both necessary and essential for simulating a simple tension test. Using finite

element simulation for a simple tension test the large strains could be computed and this

was used to determine the displacements, strains and stresses experienced by both the node

and link element and verified with the results obtained from experimental tests. Norris

and co-workers [22] performed finite element simulation for purpose of calculating both

the stress and strain at the time of fracture during elastic-plastic analysis. They computed

and found both the stresses and strains to have the following: (i) a maximum gradient at

the center where fracture occurred, and (ii) a minimum value both at and near the edges.

Contemporary materials are traditionally characterized by their complex/variety structures

at various scales. Another alternative to choose linked structures is that in addition to both

a reasonable and accurate design they are essentially light in weight, possess multi-

functional properties, such as: (i) high specific strength, (ii) high specific stiffness, (iii)

good load bearing capability, and (iv) acceptable heat dissipation characteristics. A

significant amount of research has been carried out on modelling of metal plates having

different designs for the perforation. Very few authors have proposed analytical results for

both stresses and displacements in perforated metal plates[19], as a consequence of interest

in the subject arising from its wide variety of applications, In a subsequent study by Slot

and co-workers [17] on perforated metal plates resulted in a solution for the perforated

14

plates for different variety of loadings, such as: (i) edge loading, (ii) internal pressure

loading, and (iii) uniform shear loading. Plane stress condition was studied for the chosen

types of loading. The solutions were also provided for an equibiaxial tension and shear,

and for square perforations, which showed good agreement with the work done by other

researchers. Yettram and co-workers [18] investigated the use of matrix stability method

for purpose of analyzing the elastic stability of metal plates. Also, this method helps in

predicting the buckling load, which is helpful. A brief review of the published literature

reveals only few attempts to have been made to both examine and characterize the behavior

of perforated metal sheets/ linked structures by proposing viable methods to study both the

stress and deflection response [5][19][22]–[26].Chen [10] developed a theoretical model

for yield to study both the deformation and flow behavior of perforated metal sheets using

finite element analysis. Their study provided a good agreement with the experimental

findings or results, considering the structure to be both isotropic and follows the plane

stress condition. Few other researchers [1][15], [27]–[33] did conduct both analytical and

experimental studies using the conventional mechanical modelling approach for purpose

of investigating the behavior of perforated metal sheets under conditions of uniaxial

loading for both 2-D and 3-D perforated metal sheets using the finite element method for

purpose of analyzing their behavior. The yield criterion for triangular sheets having circular

perforations and a low ligament ratio was proposed [30]. Kormi and co-workers [34] put

forth a method to compute the effective elastic constants for any pattern of the perforation

in a metal sheet or metal plate by replacing a unit module with an overall module having

the same dimension. The response of the material replaced must be identical to the overall

module. Loading using a concentrated force at a specific node often generated a near

uniform distribution of pressure. Also, the stress developed around the perforated area did

15

constitute or represent the stress intensity factor. Garino and co-workers [21]performed

finite element simulations for a tension test on a cylindrical bar of aluminum and concluded

that the tension test could be used to study both the stress and displacement field using

finite element simulation. They made this observation since the numerical simulations

were in good agreement with the experimental results provided by other researchers. The

finite element analysis has become a powerful engineering tool for the purpose of both

designing and solving problems comprising of both linear-elastic and elastic plastic

analysis. Dobrzański and co- workers [35] did provide a comprehensive review on the

existence of finite element methods that has enabled a revolution in the industry specific to

mechanical engineering. Using advanced computational technologies, the subject of FEM

has gradually developed and being increasingly used since experimental testing and

analysis is fast becoming obsolete as a viable means for quantifying the needed mechanical

properties during the prevailing time period. Besides, it also lowers the price of analysis

by eliminating the need for experimental testing.

The presence of ‘local’ stress concentration often occurs at the middle of a test

specimen due in essence to the formation and presence of voids. At fairly large values of

strain the specimen fails. The stress calculated tends to show a monotonic decreasing value

in the radial direction. In the present study, an analytical model is designed and simulated

for the purpose of determining the displacement obtained at the intersection of the link

elements of a perforated metal plate having a number of square perforations is formulated.

The displacements experienced by the link elements and the nodes or nodal points

for the case of both symmetric loading and asymmetric loading is calculated and then

graphically represented on a 3-D bar graph (3D). This was done for different values of

the applied load and the pattern obtained is compared among: (i) the two chosen metals of

16

the ferrous alloy family, and (ii) two chosen metals from the non-ferrous, light-weight

metal alloys.

2.4 The Tension Test

The simple tension test can be safely categorized to be both a viable and useful

method for purpose of evaluating the strength of a material. Strength in this context refers

to mechanical properties of the material. This is done with the primary purpose of

determining the various prospects of selecting or choosing a material for a specific design

and resultant engineering application. Tension test helps in determining the mechanical

behavior of a metal while concurrently establishing the quality of the product. With rapid

advances in technology, the tension test is an easy way to determine and compare different

materials with the primary objective of selecting or choosing the application that requires

use over a prolonged period of time. Strength of a material is the primary factor that needs

to be considered when choosing a material for a specific application. The tension test also

helps in determining the stress versus strain relationship, yield stress, tensile stress, changes

in length or area of cross section, and modulus of elasticity[8], [36]–[40] . Use of the tensile

test can help in both certifying and comparing different materials for the purpose of

determining and establishing their capacity to resist load without failure by fracture. This

test is also referred to as ‘Pull Test’.

In a tensile test the ends of the workpiece are connected in to the grips, whose one

end is attached to load measuring device installed on tensile machine while the other end

to the straining device. Hence, the specimen is progressively deformed up until fracture

using a gradually increasing tensile load that is applied uniaxially along the axis of the

specimen. The strain is applied using the crosshead, which is generally driven by either an

electro-mechanical or servo-hydraulic motor. The elongation of the specimen occurs as a

17

consequence of the relative movement of the crosshead. The output values are recorded as

a load versus elongation curve, which is to a large extent dependent on dimensions of the

specimen. In current practice, the load is recorded on a Data Acquisition System [DAS]

that is integral with the test machine.

2.5 A Brief Theory Pertinent to Plane Stress.

For an important problem of appreciable intensity, it is recommended to make

certain approximations for purpose of simplifying the 3-D stress array. It may involve that

one surface be free from stresses can also be considered of vital importance. The

approximations are made by analyzing a thin plate having minimal thickness (t). Hence,

the stress acts only on two other faces. The end structure experiences both normal stress

and shear stress on the X and Y axis. The shear stress on the X axis is represented by , in

which the first subscript refers to the plane on which the stress acts and second subscripts

refers to the direction in which the shear stress acts. The normal stress is represented by ,

the subscript representing the direction of action of the normal stress. Baik and co-workers

[41] analyzed the deformation behavior of perforated metal sheet having uniform holes and

under uniaxial tension for the case of plane stress and a three-dimensional condition for

purpose of elastic-plastic finite element analysis. They concluded from their study that

deformation behavior of a perforated sheet, having uniform holes and under conditions of

plane stress, were in good agreement. Khatam and Pindera [42] proposed a

homogenization theory to study the deformation behavior of metal sheets under conditions

of plane stress having a variety of punched holes and compared the results of the numerical

technique with experimental values under conditions of plane stress. These researchers

concluded that due to the presence of perforations in a metal sheet, it does cause the

18

presence of ‘local’ plasticity near the region of the hole. The macroscopic stress is different

due to the presence of ‘local’ plasticity at several locations through the perforated metal

sheet. This occurs due to the presence of ‘local’ stress concentration, which causes an

anisotropic condition for a material that was considered to be essentially isotropic.

Yettram and Brown[43] proposed a matrix method for perforated metal sheets

having an array of square perforations and under conditions of biaxial loading. They used

the conjugate load/displacement method for obtaining the solution for the Eigen- value

problem. They considered the non-symmetric matrix to be a function of plane stress

condition for the deformation of a plate containing nodes, which are essentially at the

intersection of the link elements.

Chen and Lee[1] proposed a method for studying the deformation behavior of a

circular perforated metal sheet comprising of few uniform perforations. They compared

the analysis using two metal sheets one having a uniform perforation and the other having

a non-uniform perforation to study the yield criterion for both plates. The perforated sheet

was considered to be isotropic for plastic behavior and under conditions of plane stress.

They concluded using the finite element technique that the apparent stress and strain for a

circular perforated sheet having uniform perforations to be close when compared with a

sheet having non-uniform circular sheet perforations.

19

CHAPTER III

THE MATERIALS CHOSEN

3.1 The Ferrous Alloys [Alloy Steel 4140 and Carbon Steel 1018]

It is fairly well known from the principles of Materials Science that microstructure plays a

significant role in influencing the elastic - plastic behavior of a metal. The two metals chosen for

this study were the following:

(a) Carbon steel (i.e. AISI 1018) having a low carbon content, i.e., 0.18 pct., and

(b) Alloy steel (i.e., AISI 4140) having a carbon content of 0.40 pct.

The nominal chemical composition of the two steels is provided in Table 3.1[44]. The

basic mechanical properties of the two chosen steels are summarized in Table 3.2 [45].

Table 3.1 Nominal chemical composition of the two hard metals chosen for purpose of analysis. (In weight percent.)

Material Fe Cr. Mn C Si Mo S P 4140 Balance 0.80-

1.10 0.75- 1.0

0.380- 0.430

0.15- 0.30

0.15- 0.25

0.040 0.035

1018 Balance - 0.6- 0.9

0.14- 0.20

- - 0.040 0.050

Table 3.3 Uniaxial tensile properties of the two hard metals chosen for this study.

Materials Used Density Elastic

Modulus

Tensile Strength Yield Strength

Elongati on in

50 mm (2 in.)

Poisson’s ratio

g/cm3 GPa Ksi MPa Ksi MPa Ksi (%)

4140 7.85 205 29732.7

855 124 415 60.19 15 0.29 3

1018 7.87 230 33358.6 7

400 58 604 87.60 25 0.29

20

The alloy steel, commensurate with its high strength, revealed a microstructure comprising

predominantly of dark regions, which is the pearlite micro-constituent inter-dispersed at random

locations with pockets of white or precipitate-free region, namely the ferrite micro-constituent.

Overall, the microstructure of this alloy steel was a combination of pearlite and ferrite [Figure 3.1].

The carbide particles, of varying size, were randomly distributed through the microstructure. The

carbon steel, i.e., 1018[46] revealed a sizeable fraction of white regions, or ferrite micro constituent,

inter-dispersed with isolated traces of the dark region, or cementite. In the white or precipitate-

free region, the grains were non-uniform in size with a near needle-shape morphology [Figure 3.2].

3.2 The Non-Ferrous Alloys [Aluminum alloy 6061-T6 and Copper]

Selection of an appropriate light-weight material is both an important and crucial step in

the prevalent era of modern technology, which is largely dependent on our ability to put these

materials to effective and efficient use in both existing and emerging applications. The materials

chosen for this investigation were two light weight non-ferrous metals, namely an aluminum alloy

and pure copper. The aluminum alloy chosen was 6061 in the T6 condition. There are a wide

variety of applications where 6061 is chosen for use in forging related applications, such as: (i) the

automobile industry, (ii) railroad-related applications, and (iii) structural and architectural

applications. Furthermore, this alloy offers good resistance to general corrosion and stress

corrosion cracking. A few other excellent features are (a) cold workability, and (b) capability to be

receptive to gas, arc, resistance and even spot weldability. AA 6061 is an ‘age- hardenable

aluminum, implying it can be strengthened using the techniques of solution heat treatment and

artificial aging. The T6 treatment for alloy 6061 involves solution heat treatment followed by

quenching in cold water and subsequent aging in an oil bath with the prime objective of increasing

strength of the alloy. The optical microstructure of alloy 6061 is shown in Figure 3.3 and reveals

a non-uniform distribution of both the coarse and intermediate-size second-phase particles

randomly through the microstructure [47]

21

Recent interests in a string of emerging engineering applications have led to the selection

and use of copper in the industries spanning the following: (i) automotive, (ii) electrical/ electronic,

and (iii) integrated circuits. Excellent high temperature properties are offered by the family of

dispersion strengthened copper alloys. The nominal chemical composition of the two chosen

nonferrous metals is provided in Table 3.2. The uniaxial tensile properties of the two chosen metals

are summarized in Table 3.4 [48]–[50]

Table 3.2: Nominal chemical composition of the two non-ferrous metals chosen for purpose of analysis.

Material Si Fe Mn Mg Cr Zn Ti Al Cu

6061 0.4- 0.8

0.7 0.15 0.8- 1.2

0.04-0.35 0.25 0.15 Balance 0.15-0.40

C-10200 - - - - - - - - 99.95

Table 3.4 Uniaxial tensile properties of the two non-ferrous metals chosen for this study.

Materials Used Density Elastic

Modulus Tensile Strength Yield Strength

Elongation in

50 mm (2 in.)

Poisson’s ratio

g/cm3 GPa Ksi MPa Ksi MPa Ksi (%)

C-10200 8.9 115 16679.3 221- 455 32-66 69-

365 10-53 5-55 0.33

6061-T6 2.7 69 10007.6 310 ~45 276 40 12-17 0.30

22

Figure 3.1: Optical micrographs showing microstructure of alloy steel 4140 at two different magnifications and the two key micro-constituents: pearlite and ferrite.

Figure 3.2: Optical micrographs showing microstructure of carbon steel 1018 at two different magnifications and the two key micro-constituents: cementite and ferrite.

(a) (b)

(a) (b)

100µm -------------

50µm -------------

_

50µm -------------

25µm -------------

23

Figure 3.3. Optical micrographs showing microstructure of aluminum alloy 6061 at two different magnifications showing grains of varying size and a non-uniform distribution of both the coarse and intermediate-sixe second phase particles through the microstructure.

50 µm

( ) a ( b )

25 µm

(a) (b)

24

CHAPTER IV

DESIGN OF TEST SPECIMEN

A square perforation pattern for the four metals was chosen for purpose of analysis. The

major idea for using a metal sheet having square-shape perforations is primarily because of its

ability to be versatile, offer adequate strength, be functionally acceptable, and importantly have an

overall acceptable visual appeal for use in products spanning a range of applications in the domain

of engineering. Depending upon size of the square perforation made the resultant perforated metal

sheet can now be considered to be a structure that contains a network of: (i) numerous fine links as

shown in Figure 4.1, or (ii) a network of coarse links as shown in Figure 4.2. The linked metal

structure has the ability to offer light weight, a high strength-to-weight ratio coupled with

acceptable mechanical strength when compared one-on-one with a solid piece of the same metal

having identical thickness. The perforated sheets of the chosen metal are essentially held together

by a network of links or link elements. The links are assumed to be uniform in thickness. A

perforated sheet of the chosen metal that is held together by a network of fine links does tend to

reveal differences in both strength and mechanical response depending upon the direction of

loading. For purpose of both selection and use in a spectrum of real-world applications both

strength and stiffness properties of the perforated metal sheet are of practical importance. In fact,

these two properties, i.e., strength and stiffness, are important for purpose of an analysis, numerical

in nature, of linked metal structures. In several real-world situations involving practical engineering

application, loading often involves a mixture of both bending and elongation.

25

The effective elastic constant (i.e., elastic modulus ‘E’ and Poisson’s ratio ‘υ’)are normally

for the plane stress condition and can be used during in-plane loading of a perforated metal sheet

that is held together by a network of fine links that are near uniform in thickness. By using

mechanical properties of the chosen metal, it is possible to determine the displacement and/or

deformation experienced by the link elements and the intersecting nodes in a perforated metal sheet

for the following two conditions:

(i) Any level of thickness, i.e., thick-link structure versus thin-link structure, and

(ii) Level of application of the load.

The dimensions of the square-shape perforations made in the chosen sheet of metal,

eventually resulting in a metal structure that is held together by a network of links, are summarized

in Table 5.1. The perforations made in the solid metal sheet were square in shape and uniformly

spaced through the metal plate from one end to the other end. The metal sheet now containing

perforations and essentially held together by a network of fine links was chosen for purpose of

analysis for various values of the applied load. The values of load taken in this study were fractions

of the yield load or yield stress. Two possible variations of the chosen metal sheet were considered

in this study, namely: (i) A metal structure containing thin links visualized from an analysis and (ii)

A perforated metal structure that is made up of thicker links, perspective to be the plane stress

condition,. A perforated metal plate that is held together by a network of thin links is shown in

Figure 4.1(b), while the perforated metal plate containing a network of thicker links is shown in

Figure 4.2(b). Dimensions of the metal plate chosen to incorporate perforations and thus form a

network of links are summarized in Table 4.1. The most interesting thing in the design is that we

have chosen the same shape for the perforation in the metal plate and to study how the perforated

metal plate responds based on the thickness of the network of links, or link elements, upon

application of load.

26

A 3-D view of the linked metal structure is shown in Figure 4.1(a) for a structure

held together by a network of thin links, and Figure 4.2 (a) for a structure held together by

a network of thicker link elements. A two-dimensional view of perforations in a perforated

metal plate or sheet is shown in Figure 4.3. Dimensions of the link elements are

summarized in Table 4.2.

Figure 4.1. The perforated metal plate comprising of a network of thin links or thick link elements: (a) Three-Dimensional view, (b) Two-dimensional view of the metal plate

27

Figure 4.2. The perforated metal plate comprising of a network of thick links or thick link elements (a) Three-Dimensional view, (b) Top-dimensional view of the metal plate

Table 4.1. Dimensions of the structure containing thin links and thick links. Thin Structure (mm) Thick Structure (mm)

Length 113.67 127.00 Breadth 76.84 86.36 Thickness 3.18 3.18

Table 4.2 Dimensions of the perforations chosen to form the linked metal structures used in this study.

Dimensions Thin Structure (mm) Thick Structure (mm)

External Length 21.59 25.40 External Breadth 21.59 25.40 Internal Length 15.24 15.24 Internal Breadth 15.24 15.24

28

Figure 4.3. Dimensions of the links in the perforated metal plate.

(a) Plate with thin links or link elements, and (b) Plate with thick links or link elements

( a ) ( b )

29

CHAPTER V

FORMULATION OF THE PROBLEM

5.1 Description

Consider the linked structures having links of varying thickness. The linked metal

structures were made using perforated metal sheets having a network of fine links. The structures

chosen for this study were the following: (i) a metal structure containing a network of thin links,

and (ii) a metal structure containing a network of thick links, as shown in Figure 5.1 and Figure

5.2. Dimensions of the structure containing thin links are L = 113.665mm, B (breadth) =

76.835mm, and t (thickness) = 3.175mm. The thin linked structure consisted of square perforation

of L (length) = 15.24 mm, and (B) breadth = 15.24mm. The perforations were made from one end

to the other of the chosen metal plate.

Dimension of the structure containing thick links are shown in Figure 4.2, with L (length)

=125mm, and B (breadth) = 86.36mm. The perforated metal sheet is considered to be uniform and

essentially isotropic in thickness [t = 3.175mm]. However, for purpose of tensile test simulation

performed on the linked metal structures they are considered to be in plane stress condition since

thickness of the chosen sheet of metal is small when compared to its length and breadth, and

minimal deformation is expected to occur through the thickness of the chosen metal plate.

The linked metal structures, now has specified locations at every intersection of two links.

This makes it easy for purpose of determining the location of a specific position of interest in our

analysis. The linked metal structure has been divided into rows and columns, which starts at Row

1 and Column 1 represented as [1,1], and ends as Row 5 and Column 7 represented as [5,7] , which

as referred to as “Nodes”. Each node is connected by a network of links, referred to as thin or thick

30

links with respect to size of the links. This is done with the purpose of observing the displacement

pattern experienced by the linked metal structure upon being subject to the influence of a load by

extracting the desired output from the finite element analysis. A pictorial representation of the

nodes for both the thin link structure and thick link structure is shown in Figure 5.3.

Figure 5.1: Iso metric view & XY plane view of the thin linked structure.

Figure 5.2: Iso metric view and XY plane view of the thick linked structure.

31

Figure 5.3 A schematic of the metal structure containing a network of links with identification of the different nodes.

32

5.2 Material Properties

The materials chosen in this study were two metals belonging to the family of ferrous

alloys, and two metals belonging to the family of non-ferrous alloys. The two metals belonging to

the ferrous family of alloys were: (i) alloy steel 4140, and (ii) carbon steel 1018. The two

nonferrous metals were (a) aluminum alloy 6061-T651, and (b) copper C-10200. Nominal

chemical composition of the metals chosen is given in Table 3.1 for the two ferrous alloys and

Table 3.2 for the two non-ferrous alloys. A summary of basic mechanical properties of: (i) the two

ferrous alloys are provided in Table 3.3, and (ii) the two non-ferrous alloys are provided in Table

3.4.

5.3 Deformation and Support.

For an analysis to be performed using the technique of the finite element method, the linked

metal structure had to be clamped at one end and allowed to move at the other end. It is then

subjected to tensile forces acting in opposite directions. This is done to ensure that when a tensile

force is applied on a perforated metal plate or linked structure, the structure is clamped in space

with the purpose of restricting its rigid body motion. Upon application of a tensile load, the linked

structure experiences deformation at both the ‘local’ level and ‘global’ level. Upon loading, the

problem becomes complex due to the fact the linked metal structure can experience bending

through the thickness. Also, care was taken while performing the analysis, since the linked metal

structure contains several locations of potential high ‘local’ stress concentration, which are

conducive for promoting failure of the structure upon application of a load.

33

CHAPTER VI

FINITE ELEMENT ANALYSIS: NUMERICAL PROCEDURE

6.1 Finite Element Formulation

Finite Element Analysis (FEA) is a powerful tool, which is being increasingly used to both

study and understand, following analysis, the basic aspects pertinent to modeling and numerical

analysis of elements having a wide range of sizes. Currently, the analysis technique has grown in

complexity for use by design engineers dispersed through a broad spectrum of industries. In

essence, this analysis technique is now being considered as a novel method for the purpose of

analysis of an unknown quantity, or a few quantities, by initially choosing a continuum, which is

discretely discretized into simple geometric shapes that are finite in size. Hence, the technique was

coined the name ‘finite element analysis’ and given the acronym FEA. Using material properties

and governing relationships, which are considered for the chosen elements, coupled with a known

set of loading conditions and boundary conditions results in a set of equations, which when solved

does provide a comprehensive overview pertinent to behavior of the chosen structure. It also adopts

the Newton technique for solving large deformation elastic-plastic problems. Since its initiation

and incorporation by way of use, the technique has gradually evolved to prove itself to be effective

in providing fairly accurate results once the model is properly formulated.

The finite element analysis software ABAQUS/ CAE 6.13.2 [13] was used in this research

study for the purpose of investigating the behavior of linked metal structures having a network of

links or link elements having two different thicknesses. The linked metal structure was obtained as

a consequence of incorporating square-shaped perforations in a solid metal plate. The perforated

metal plate, essentially held together by a network of links, is subject to uniaxial tension loading in

34

both the symmetric condition and asymmetric condition. The finite element analysis was carried

out using the general purpose finite element method. A static general non-linear elastic-plastic

analysis was performed on the linked structure having square-shaped perforations. Using the

VonMises yield criterion the equivalent stress was obtained for an isotropic material. This failure

criterion is often used for ductile metals, which tend to yield upon application of a load. For the

case of elastic-plastic analysis, when load is applied to the specimen, the stress is directly

proportional to strain until a steady state, such as the elastic limit or yield point is reached. The

linear region in which the stress is directly proportional to strain is governed by Hooke’s law. Once

the applied stress crosses the elastic limit or upper yield point, the region of non-linearity

commences, which necessitates the need for an elastic-plastic analysis.When the von mises stress

exceeds the yield stress of the chosen material, the metal plate of interest is no longer in the linear

region of deformation wherein the stress is directly proportional to strain.

A dependence of both elastic modulus (E) and Poisson’s ratio (υ) of a metal on direction

of loading is fairly well documented in the published literature. The presence of fairly high “local”

stress concentration at locations of intersection of the link elements, or links, in the linked metal

structure does make the overall structure complex. This makes it both interesting and necessary to

study the complicated model. Due to inherent complexity of the model upon application of a load,

it was both essential and desirable to observe the behavior of a linked metal structure in the elastic

regime, since presence of high level of ‘local’ stress concentration causes a non-uniform state of

stress to exist through bulk of the structure. This makes mechanical testing not only difficult but

also complex without causing failure of the linked metal structure by rupture.

The four chosen alloys [carbon steel 1018 and alloy steel 4140] and [aluminum alloy 6061T6 and

C-10200] for purpose of performing finite element analysis to help determine both the stress and

displacements experienced by the link elements in a perforated metal plate or linked structure. The

35

simulations were carried out using a commercial finite element code ABAQUS version 6.13.2 using

a laboratory-scale computer.

• Alloy steel 4140 had a yield strength [σ YS] in pure tension of 415 MPa, and an ultimate

tensile strength [σ UTS] of 655 MPa. The modulus of elasticity of this steel is 205 MPa.

• Carbon steel had a yield strength [σ YS] in pure tension of 220 MPa, and an ultimate tensile

strength [σ UTS] of 400 MPa. The modulus of elasticity (E) of the carbon steel is 200 MPa.

• Aluminum alloy 6061-T6 had a yield strength [σ YS] in pure tension of 276 MPa, and an

ultimate tensile strength [σ UTS] of 310 MPa. The modulus of elasticity (E) of aluminum

alloy is 69 GPa.

• Copper C-10200 had a yield strength [σ YS] in pure tension in the range 69-365 MPa, and

an ultimate tensile strength [σ UTS] in the range of 221-455 MPa. The modulus of elasticity

(E) of copper is 115 GPa.

In the non-linear region of the stress versus strain curve for the material chosen, the values

of plastic strain were determined with the aid of a web-plot digitizer [51] . The strain values

obtained from the stress versus strain curve comprises of both the elastic strain and the plastic

strain.

The finite element analysis involved the following steps:

(i) Creating and meshing both the 2-D geometry and 3-D geometry of the linked structure.

(ii) Specifying suitable material properties based on the metal chosen.

(iii) Applying the desired load/ displacement, and

(iv) Specifying the appropriate boundary conditions.

The analysis essentially deals with an understanding of the deformation kinetics that occurs

upon application of load, i.e., tensile load, using the finite element analysis software [ABAQUS].

Both the 2-D model and 3-D model were created for purpose of analysis using:

36

(a) ABAQUS 2D-CPS4 solid elements having 4 node brick elements for the 2-D structure,

(b) ABAQUS 3D solid elements C3D8 having 8 node brick elements for the 3-D structure.

Purpose of the current investigation was to simulate stress distribution in the linked structure of

the chosen metals, i.e., alloy steel 4140, carbon steel 1018, aluminum alloy 6061-T6 and copper C-

10200, upon application of a tensile load at two different intersections or nodal points in the linked

structure. In this simulation, five different load levels, spanning a varying percentage of the yield

strength of the chosen metal, i.e.[ 10 pct. σYS., 25 pct. σYS, 50 pct. σYS., 100 pct. σYS,, 102 pct. σYS (

for ferrous alloys ) , 110 σYS ( for non-ferrous alloys ) ]were chosen for purpose of: (i)

Characterizing stress distribution in the linked structure and thereby recording the variation of stress

with strain or stress-strain curve for the overall linked structure, and

(ii) Displacement experienced by each of the centroid nodes upon application of loading.

The numerical results include the following: (1) stresses and strains in each zone, (2) displacements

experienced by each ‘node’ for the different load levels applied to the structure comprising of thin

links and structure comprising of a network of thick links.

6.2 Finite Element Simulation

This section utilizes the data for the material’s chosen and taken from the Material’s Data

Handbook [44] for purpose of initiating simulation of the finite element method. The finite element

code that was used to conduct simulation, under conditions of tensile loading, was ABAQUS

[Version 6.13.2] [13]. The finite element method is useful primarily because it discretizes the

chosen model into relatively small elements, which are further divided into nodes for purpose of

simplifying a complex model. This discretization into small elements plays an important role in the

numerical analysis, such that displacement experienced by an element provides a measure of both

the extent and severity of distortion in shape of the element. The elastic-plastic analysis was

performed using the Newton Raphson iteration to determine both the stress and displacement with

the aid of the finite element method. Choice of the explicit solver arises from the fact that at

37

equivalent plastic strain the Von-Mises yield criterion should be applicable for structures, such as,

plates having a uniform thickness. The explicit solution technique does offer a few advantages

when compared to the implicit solution technique, which has been studied and documented in the

published literature by Prior [52]. Furthermore, for purpose of nonlinear analysis of the chosen

metal it requires an incremental load or displacement step such that after every increment the results

are easily extracted due to geometry of the initial structure having changed as a consequence of the

material having either yielded or deformed into the domain of the non-linear region, or plastic

region of the stress versus strain curve. This information is required for computing the stiffness

matrix for the subsequent increment in the analysis. Further, use of smaller elements, or finer

elements, for purpose of nonlinear analysis does make the solution both accurate and precise but

suffers from the drawback of being time consuming.

Numerical simulations of the tension test on perforated metal sheets were carried out using

the finite element commercial code to generate the required data. The criterion for yield at the

different nodal points or junctions in the perforated metal plate is expressed by the amount of load,

as a fraction of yield load of the chosen metal, which is applied at the nodes of interest, which

satisfies the criterion for loading at a point in uniaxial tension. The values of displacement

experienced by the nodes were recorded subsequent to the application of load. A node or nodal

point in the linked metal structure occurs at an intersection of two links. The displacement, or

deformation, experienced by the nodal points, or nodes, is graphically represented on a

threedimensional bar graph. This bar graph provides a visual representation of how displacement

occurs at a specific node occurs when load is applied at identical points on a hard material having

near similar value of elastic modulus (E) and Poisons ratio (υ), while a change in composition of

the chosen metal does bring about an observable difference in the values of displacement calculated

using the computational technique.

A two-dimensional [2-D] finite element analysis of the chosen mesh helps us to determine

the displacements experienced by the nodes in the linked metal structure. In addition to 2-D

38

modelling, calculations were performed for a full three-dimensional [3-D] representation of a

perforated metal sheet that was held together by links using the eight node brick elements. A careful

observation of the data resulting from numerical computation revealed the results provided by the

three-dimensional [3-D] model did not significantly differ from those of the 2-D plane stress model

in the domain of elastic deformation. Henceforth, all of the numerical results presented in this

thesis document will be for the 2-D model, or plane stress condition. The ultimate goal is to

compute the results using the correct input parameters, such as: (i) properties of the chosen metal,

(ii) dimensions of the linked metal structure, (iii) step time, (iv) magnitude of load used and/or

applied, as function of the yield load of the chosen metal, and (v) boundary conditions. Upon

performing the finite element analysis we obtained fairly consistent results. Before commencing

the numerical computation, it is essential to apply the boundary conditions carefully on the chosen

model. To fix, or stabilize, the chosen model in space for purpose of application of load we assumed

one end of the linked metal structure to be fixed, while the other end of the linked metal structure,

or perforated metal sheet, is subject to constraints in displacement.

The presence and/or occurrence of stress concentration, highly localized in nature,

occurring at the intersection of elements makes the starting structure to be complex, which makes

it interesting for purpose of study. In light of the complexity of the model upon application of a

load, it becomes essential to observe behavior of the structure in the elastic range, since due to

presence of stress concentrations, a non-uniform stress state exists at the ‘local’ level through the

structure of the perforated metal plate that is essentially held together by a network of links or link

elements. This makes mechanical testing both complex and challenging since loading of the linked

metal structure, held together delicately by a network of links, can result in early failure or rupture

of the different links.

39

CHAPTER VII

TWO DIMENSIONAL FINITE ELEMENT MODEL

7.1 Modelling

The perforated metal sheet or plate was initially considered to be in two-dimensional, or

plane stress problem, such that the approach for a solution was created using the finite element code

ABAQUS/CAE [13]. The model for a perforated metal plate containing a network of thin links

and a metal plate containing a network of thick links was modelled in 2D planar space. This helps

in studying a complex model using the assumption of 2-D, which is extended to the 3-D problem

after careful consideration of certain assumptions. The purpose of considering the 2-D problem

and resultant analysis is the thickness of the plate, in comparison to its length and breadth (or width)

is noticeably small. Herein, the plane stress is the state of stress for which the normal stress [σz]

and shear stresses [σxz and σ yz], are assumed to be non-existent or zero. Thus, an analysis of the

thin plates loaded in the plane of the plate was performed using the plane stress approximation. The

loads acting on both the X plane and Y plane are considered under conditions of plane stress. A

model for structure containing thin links and metal structure containing thick links was created

using dimensions provided in Table 5.

The structures chosen for this study are defined as the thin linked structure and a thick

linked structure as shown in Figure 5.1. Dimensions of the structure containing a network of thin

links are: L (length) = 113.665 mm, and B (breadth) = 76.835mm. The perforations used on a

solid metal plate were square in shape and measured L (length) = 15.24 mm, and (B) breadth =

15.24 mm. The square shaped perforations were made through the entire solid metal plate.

40

Dimensions of the thick linked structure shown in Figure 5.2 are L (length) = 125 mm and B

(breadth) = 86.36mm. The number of perforations made on the chosen metal plate was seven along

the ‘X’ axis along 5 along the ‘Y’ axis, which makes a total of 35 perforations on the chosen metal

plate for the structure containing thin links and the structure containing thick links. The plate

thickness was 3.175mm for both the thin link structure and the thick link structure.

7.2 Material Selection and Type of Analysis

The four chosen metals materials used in this study are considered to be isotropic,

homogeneous and linearly elastic to minimize overall complexity of the problem [2-5]. In the

region or domain of nonlinear behavior/ deformation of the metals chosen, the total strain comprises

of the elastic strain component and the plastic strain component. The values of the plastic strain

were obtained using a plot digitizer.

The type of analysis performed was static and the procedure used was general. For the

chosen metals, the non-linear large deformation formulation was used. A load step was created for

a time period of 1 second. The time chosen was 1 second since deformation experienced by the

metals chosen is not dependent on time. Hence, time was chosen to be 1 second. The non-linear

geometry was switched on. All the required outputs, such as stresses, strains and displacements

experienced by the links and nodal points, are defined in this module. In an attempt to obtain a

refined solution, a refinement of the number of iterations was attempted. The increment was chosen

to be automatic. The maximum number of increments, based on refinement was chosen to be

1,000,000. Minimum size of the increment chosen was 1E-007 while maximum size of the

increment chosen was 0.0001.

7.3 Size of Mesh and Configuration

The mesh that was chosen for use in finite element analysis of the metal plate that contained

a network of thin links was a refined 2-D mesh that in essence had 1869 nodes and 1292 elements.

For the same perforated metal plate that was held together by thicker links the chosen mesh was

refined and contained 1185 nodes and 836 elements. Size of the element for a thin structure was

41

1.59 * 1.59 mm2, and size of the element for a thick structure was 2.54*2.54 mm2. The meshing

was done subject to plane stress condition since the linked metal structure satisfies the plane stress

condition. The chosen metal sheet had one element through its thickness. For both cases, the

calculations were performed using two-dimensional, four node plane stress quadrilateral (CPS4)

elements [a four-node bilinear plane stress quadrilateral shape element].

To obtain a smooth curve for the variation of stress with strain, smaller increments were

chosen and used. Re-meshing was avoided in the solution, since interest was in determining the

response of the centroid nodes, i.e. at the intersection of any two links, with the primary objective

of extracting and understanding the results obtained.

Figure 7.1 Size of mesh for a structure containing a network of thin links.

42

Figure 7.2 Size of mesh for a structure containing a network of thick links

7.4 Boundary Conditions :

This part of the study is most essential since it introduces an efficient and systematic

method to predict or evaluate the behavior of linked metal structures. For a 2D geometry modelled

for the linked structure, Table 7.1 represents the boundary conditions applied on the structure

containing thin links and structure containing thick links. The boundary conditions were applied

to restrict rigid body rotation, since the linked metal structure, in finite element software, is free in

2-D planar space. The linked metal structures are meshed into nodes and elements, such that each

node in the structure comprises of 3 degrees of freedom [i.e. 2 displacements, and 1 rotation along

the X and Y axis]. For the linked metal structure to be stable in space, two sets were created for

the boundary conditions (BC) at the end nodes .These were named as BC1 and BC2. For boundary

condition BC1, displacement of the nodes was fixed along the coordinate axis such that

displacement was zero and rotation was restricted along the axis of rotation. For boundary

condition BC2 the vertical displacement of the nodes was restricted along the Y axis whereas

movement was permitted along the X axis.

43

Table 7.1 The boundary conditions for two-dimensional FEM for uniaxial tension.

BC Ux Uy UR3

BC1 Ux=0 Uy=0 UR3 =0

.BC2 - Uy=0 UR3 =0

Figure 7.3 Pictorial view for the boundary conditions applied on the linked structure.

7.5 Loading

The load was applied in tension and the metal structure was assumed to be symmetric along

the X axis and Y axis. The loading on the linked-metal structure was different at the other positions

causing us to analyze the complete linked metal structure. To study the nature of loading on the

BC1

BC2

44

linked metal structure, the loading process was studied for both the symmetric and asymmetric

conditions. The chart below provides an overview of the work done for the purpose of analyzing

both the thin linked structure and the thick linked structure.

Figure 7.4. Structural Chart depicting the work performed for purpose of analyzing the thin linked and thick linked metal structures of the four chosen metals.

7.5.1 Symmetric Loading

In this type of loading, the load was applied on Node [2, 6] and Node [4, 2], such that they

follow the same line of action, thereby creating a tensile pull on the structure containing thin links

and the structure containing thick links.

Elastic Plastic Analysis

Thin Linked Structure

Displacement Control

Symmetric Loading

Asymmteric Loading

Thick Linked Structure

Pure Displacement

Control

Symmetric Loading

Asymmteric Loading

45

7.5.2 Asymmetric Loading

In this type of loading the load was applied on Node [2, 6] and Node [4, 1], such that the

tensile pull is slightly offset when compared to case of symmetric loading.

The loading approach was chosen to be displacement type using the boundary conditions

and the maximum displacement on load step was allowed to be 10mm in x and y direction, The

method of loading was chosen to be ramp type, which means the load increased uniformly at a

certain rate, which helps us to determine the behavior of the linked metal structures when subjected

to loading by way of tensile pull or tensile force. Further, for purpose of analysis it is assumed that

the applied load is distributed equally among the two nodes on which it is applied.

The two loading conditions are chosen and Case I- refers to symmetric loading for loading

applied at the two chosen nodes and the chosen linked metal structure is symmetric about both the

X axis and Y axis. For the case of asymmetric loading obtained by a slight shift / offset in the point

of load application to the adjacent node. This causes a change in the distribution of the local stress,

strain and displacement experienced by the network of links as a direct consequence of a shift in

the point of load application from one node to another.

46

Figure 7.5 Pictorial representation of metal structure containing a network of thin links and subject to “Symmetric” Loading

47

Figure 7.6 Pictorial representation of metal structure containing a network of thin links and subject to “Asymmetric” Loading

48

TABLE 7.2: Node locations chosen for application of load for both symmetric and asymmetric loading for both thin link and thick link metal structure of the four chosen metals belonging to the ferrous alloy family and non-ferrous family.

Yield stress [PCT.]

Symmetric loading for

4140 ALLOY STEEL

Asymmetric loading for

4140 ALLOY STEEL


1018 CARBON STEEL


1018 CARBON STEEL


ALUMINUM ALLOY 6061T6


ALUMINUM ALLOY 6061-T6

Symmetric Loading for COPPER

C-10200


COPPER C-10200

THIN LINK and THICK LINK STRUCTURE

10 [2,6] and [4,2] [2,6] and [4,2] [2,6] and [4,2] [2,6] and [4,1] [2,6] and [4,2] [2,6] and [4,1] [2,6] and [4,2] [2,6] and [4,1]




102 [2,6] and [4,2] [2,6] and [4,2] [2,6] and [4,2] [2,6] and [4,1] - - - -

110 - - - - [2,6] and [4,2] [2,6] and [4,1] [2,6] and [4,2] [2,6] and [4,1]

49

Figure 7.7. Methodology used for location of the Nodes in the linked metal structure (Thin link structure and Thick link structure).

50

The eight simulation models that were designed for the chosen metal plate having square

perforations, under conditions of plane stress, are compared. The simulation models are the

following:

(i) The first model is a perforated metal plate comprising of thin links in alloy steel 4140.

(ii) The second model is a perforated metal plate of carbon steel 1018 containing thin links.

(iii) The third model is a perforated metal plate of aluminum alloy 6061 containing thin links.

(iv) The fourth model is perforated metal plate of copper [C-10200] containing thin links.

(v) The fifth model represents the perforated plate of alloy steel 4140 containing thick links.

(vi) The sixth model represents the perforated plate of carbon steel 1018 containing thick links.

(vii) The seventh model represents the perforated plate of aluminum alloy 6061-T6 containing

thick links.

(viii) The eighth model represents a perforated plate of copper [C-10200] containing thick links.

Five different levels of load were chosen for this study. The first four levels of load chosen

and used were well within the elastic domain of the chosen metal plate, i.e., 10 pct. σYS, 25 pct. σYS,

50 pct. σYS, and 100 pct. σYS, while the fifth load chosen was above the yield load i.e. well into the

plastic region of the stress versus strain curve of the chosen metal. This load levels or parameters

were used to both obtain and analyze the distribution of stress, strain and resultant displacements

experienced by the different links, or link elements, in the chosen linked metal structure.

51

CHAPTER VIII

RESULTS AND DISCUSSIONS

A 3D bar graph was used to plot the results and a pattern was observed after plotting the

displacements experienced by the nodal points of the linked metal structure. The outputs were

extracted for the five chosen levels of load that was applied to the linked structure containing a

network of thin links, and linked metal structure containing a network of thick links. The 3-D bar

graph shown in Figure 8.1 provides a pattern for the displacement experienced by the nodal points

and is compared with the pattern observed using electrical mesh topology. For the case of

symmetric loading and with the requirement of mutually compatible load conditions at the common

boundary of adjacent squares of the element. For the case of symmetric loading, it was sufficient to

consider a 45° segment and the boundary conditions were applied.

The primary focus is to determine the displacements occurring at the nodal points, or

experienced by the nodal points, of the chosen linked metal structure. This will help establish the

displacement pattern for the perforated metal plate that is held together by a network of links when

subjected to loading in the tensile direction for the case of both symmetric loading and asymmetric

loading. A pattern, or profile, was observed for the displacements by way of contours, and the

numerical values were obtained using finite elements in synergism with numerical analysis. The

patterns that was obtained is carefully analyzed by a comparison of the results obtained for the two

chosen steel structures, i.e., alloy steel 4140 and carbon steel 1018, for varying levels of applied

load, using the finite element analysis. The 2-D approximation of the linked structure of the chosen

52

steel plate was analyzed assuming conditions of plane stress for purpose of finite element analysis.

For both symmetric loading and asymmetric loading the contours obtained were quite similar to the

results obtained for the 3-D model. Hence, the results discussed in this section will be focused on

the plane stress condition. For the case of symmetric loading, the load was applied at Node (2, 6)

and Node (4, 2); while for asymmetric loading the load was applied at Node (2, 6) and Node (4, 1).

8.1 A Comparison between Thin Structures of Alloy Steel 4140 And Carbon Steel 1018

The linked metal structure was found to deform from its original shape and the following

observations, with specific reference to displacements experienced by the nodes and links, are

recorded for the five chosen load levels, as a function of yield stress of the chosen metal.

8.1.1 Symmetric Loading of Alloy Steel 4140 and Carbon Steel 1018

The linked metal structure of alloy steel 4140 was found to deform from its original shape

and the following observations, with specific reference to displacements experienced by the nodes

and links, are recorded for the five chosen load levels, as a function of yield stress of the chosen

metal. A few of the key observations are highlighted with respect to the method chosen. Figure

8.1 shows the behavior of deformation of linked structure for symmetric loading of alloy steel 4140

at the yield stress, when the structure containing a network of thin links is perfectly elastic. Upon

close examination of the results reveals the presence of plastic strain on the lower elements, which

initiated at Node [2, 6]. This node experiences a higher level of more stress, which is shown in

Figure 8.1.

The stresses are randomly distributed in the metal structure of alloy steel 4140 containing

a network of thin links, which can be observed by examining Figure 8.1. The structure does contain

few points of high local stress concentration due to the presence of square shaped perforations.

Hence, upon being subject to loading, the region both at and near the perforation experiences high

level of stress, whereas the region between the links contains a negligible value of stress. The

53

magnitude of stress both at and around the nodes is noticeably high in comparison with the stress

at the middle of the links.

The linked metal structure of alloy steel containing a network of thin links, the maximum

displacement was found to occur at the nodal points situated towards the upper half of the perforated

metal plate. The magnitude of displacement at the middle of the structure containing a network of

thin links was noticeably less.

The three dimensional bar graph shown in Figure 8.2 to Figure 8.6 reveals the pattern of

displacement experienced by the centroidal nodes in the metal structure of alloy steel 4140

containing a network of thin links. The values of displacement were obtained from results of the

finite element analysis that was performed on a laboratory-scale computer. The outputs were

extracted in the visualization module of ABAQUS/CAE-6.13-2. The point of attention is the

centroid identified by its location at the intersection of two links. Deformation experienced by the

linked metal structure initiates once a higher stress occurs at the center of the links in a given mesh.

The area of interest is the region contained within the linked metal structure comprising of a

network of thin links, which upon being subject to loading, the magnitude of displacement

experienced by the centroidal nodes are represented by a 3D bar graph. The graphs for the linked

metal structure of alloy steel 4140 that was subject to symmetric loading is shown in Figure 8.2-

Figure 8.6. The displacements were recorded for σYS the five levels of load applied as a function of

yield stress of the chosen metal, i.e., 10 pct. σYS, 25 pct., 50 pct. σYS, 100 pct. σYS and 102 pct. σYS.

of the yield stress. The percentages of the yield stress are chosen with respect to the material used

for the analysis. Below the yield stress the metal structure containing a network of thin links and

subjected to symmetric loading, the maximum deflection occurred at the point of application of

load, i.e. Node [2, 6] and Node [4, 2], whereas Node [3,4] experiences a minimal amount of

deformation, since the structure is assumed to be symmetric along X and Y axis. Hence, we

observed a minimal effect on the centroidal node of the structure containing a network of thin links.

54

The pattern of the deformation of the linked structure is shown in Figure 8.3 -8.7 for the percentages

of the yield stress chosen for 4140 alloy steel. The upper region of the linked structure experiences

more deformation on Row 4 and Row 5, when compared to the lower region i.e. Row 2 and Row

1. The pattern upon close observation reveals a minimal effect on the centroidal node.

The linked metal structure of carbon steel 1018 containing a network of thin links is shown

in Figure 8.2. The Von Mises stress in carbon steel 1018 containing a network of thin links is

relatively high when compared one-on-one with alloy steel 4140 at 100 pct. of yield stress [σYS].

This is based on an observation of the contours. Figure 8.1 and Figure 8.2 provide a similar contour,

which reveals the occurrence of stress centration at and near the intersection of two links. The Von

Mises stress and resultant plastic strain was found to be noticeably more in carbon steel 1018 when

compared one to one with alloy steel 4140. This is rationalized on the basis of strength of the two

chosen steels. The lower strength and resultant higher ductility of carbon streel makes it receptive

to experience degradation by way of displacement of both the links and nodes upon loading. The

deformation was evident from the values recorded for displacement of the links, or link elements,

and the nodes.

From Table 8.1 we can see the difference in the values obtained at the centroidal nodes,

which brings to light the significance of the results obtained using finite element analysis (FEA).

To facilitate ease in understanding of the displacements, all the nodal points are shown through

three dimensional bar graphs to provide an overview of the pattern obtained upon subjecting the

structure containing a network of thin links to loading. From the bar graphs, we observe a much

higher displacement to occur in carbon steel 1018 when compared one to one with alloy steel 4140.

55

Table 8.1 A comparison of the values of displacements occurring at the internal nodes of the linked metal structure containing a network of thin links upon being subject to 100 pct. of the yield stress for symmetric loading.

Node Percentage of Yield Stress (%)

Displacement (mm)

Row Column Alloy steel 4140

Carbon Steel 1018

2 2 100 0.1225 0.1641 2 3 0.0985 0.1313 2 4 0.0785 0.1013 2 5 0.1072 0.1373 2 6 0.1577 0.2049 3 2 0.1145 0.1561 3 3 0.0731 0.1016 3 4 0.0169 0.028 3 5 0.0571 0.0739 3 6 0.105 0.1384 4 2 0.1807 0.245 4 3 0.1423 0.1941 4 4 0.1166 0.1591 4 5 0.1248 0.1682 4 6 0.1457 0.1956

56

Figure 8.1: Profile showing contours of the Von Mises stress for alloy steel 4140 that was

subjected to symmetric loading at Node (2,6) and Node (4,2) for the metal structure containing a network of thin links at the yield stress ( σYS ) of the material.

Figure 8.2: Profile showing the contours of the Von Mises stress for carbon steel 1018 that was subjected to symmetric loading at Node (2,6) and Node (4,2) for the structure containing a network of thin links at a load equal to yield stress( σYS ) of the material.

57

Figure 8.3 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of alloy steel 4140 upon being subject to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Figure 8.4 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Series1

Series3

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns

4140 Alloy Steel - Thin Structure - 10 % Yeild Stress -

Symmetric Loading

Series1 Series3

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns

4140 Alloy Steel - Thin Structure - 25 % Yield Stress -

Symmetric Loading

58



.

Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


Symmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Column


Symmetric Loading

59

Figure 8.7 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140 containing a network of thin links, when subjected to symmetric loading, that is 102 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Figure 8.8 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140, containing a network of thin links, when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


Symmetric Loading

0.0000

0.1000

0.2000

0.3000

1 2 3 4 5 6 7 Rows

Columns

1018 Carbon Steel - Thin Structure - 10 % Yield Stress - Symmetric Loading

60


Figure 8.10 Profile showing the displacement experienced by the different nodes of the “thin” linked structure of alloy steel 4140 containing a network of thin links when subjected to symmetric loading, that is 50 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Series1 Series2

Series3 Series4

Series5

0.0000

0.1000

0.2000

0.3000

1 2 3 4 5 6 7 Columns

1018 Carbon Steel - Thin Structure - 25 % Yield Stress -

Symmetric Loading

Series1 Series3

Series5

0.0000

0.1000

0.2000

0.3000

1 2 3 4 5 6 7 Columns


Symmetric Loading

61

Figure 8.11 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140, containing a network of think links, when subjected to symmetric loading, that is 100 percent of the yield stress, applied at Node (2,6) and Node (4,2).


Series1 Series3

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000

1 2 3 4 5 6 7 Columns


Series1 Series3

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000

1 2 3 4 5 6 7 Columns


62

Figure 8.13 A contour profile showing the magnitude of displacements experienced by

linked structure of alloy steel 4140, containing a network of thin link elements, when subjected to symmetric loading, 102 pct. of the yield stress.


linked structure of carbon steel 1018, containing a network of thin links, when subjected to symmetric loading, 102 pct. of the yield stress.

63

8.1.2 Asymmetric Loading of Alloy Steel 4140 and Carbon Steel 1018:

For the case of asymmetric loading, i.e. when the point of application of load is changed,

or offset, and a similar analysis is performed, we observe that the displacement pattern does not

change appreciably. The magnitude of displacement is fairly high and observable up to the nodes

situated on Row 4. The magnitude of displacement experienced by nodes located on Row 5

decreases, which is the opposite of what was observed for the case of symmetric loading. Also, the

overall pattern of displacement recorded for the case of asymmetric loading was quite similar to the

pattern obtained for symmetric loading. This is an interesting observation for the two chosen steels,

and the pattern obtained takes the shape of an “S” when the values are plotted and represented

graphically.

Upon application of an asymmetric load to alloy steel 4140 containing a network of thin

links at the chosen load levels of 10 pct. σYS, 25 pct. σYS, 50 pct. σYS, 100 pct. σYS, and 102 pct. σYS,

the maximum displacement was evident at the nodal points, or nodes, located towards upper half

of the perforated metal plate. The displacement experienced by the different links, or link elements,

in alloy steel 4140 were found to be quite similar when compared one-on-one with the

displacements experienced by the network of thin links, or link elements, in carbon steel 1018.

Also, the displacement pattern when represented graphically on a 3-D bar graph reveals a kind of

symmetry between the two chosen steels upon application of a load. While the magnitude of

displacement experienced by the links is different; the two chosen steels reveal a near-similar

behavior when compared one-on-one with each other. The pattern represents low displacements

occurring at the nodes on approximately 30 percent of the loaded metal plate while maximum

displacement was observed to be occurring at the nodes located towards the upper half of the plate.

64

Figure 8.15: Profile showing the contours of the von mises stress for alloy steel 4140 that was

subjected to asymmetric loading at Node (2,6) and Node (4,1) for the structure containing a network of thin links and at load corresponding to yield stress of the metal.

Figure 8.16: Profile showing the contours of the Von Mises stress for carbon steel 1018 that was subjected to asymmetric loading at Node (2,6) and Node (4,1) for the structure containing a network of thin links at a load corresponding to yield stress of the metal.

65

From Figure 8.16 and Figure 8.17 it is observed that when the loading is asymmetric, the

maximum stress occurs at the upper node of the steel metal structure. The nodes in carbon steel

1018 provide a higher value of stress on the linked metal structure when compared one-to-one with

alloy steel 4140. This is a key factor that either governs or dictates the viability of this material for

purpose of machining and use in products. The contours also provide a globally ductile behavior

for carbon steel 1018 when compared one-on-one with alloy steel 4140.

66

Figure 8.17 Profile showing the displacement experienced by the different nodes of the linked structure of alloy steel 4140, containing a network of thin links, when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1).


Series1 Series3

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


Asymmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns

4140 Alloy Steel - Thin Structure - 25 % Yield Stress - Asymmetric Loading

a

67

Figure 8.19 Profile showing the displacement experienced by the different nodes of the

linked structure of alloy steel 4140, containing a network of thin links, when subjected to asymmetric loading, that is 50 percent of the yield stress, applied at Node (2,6) and Node (4,1).


Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Column


Series1 Series3

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


68




Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500

1 2 3 4 5 6 7 Columns

1018 Carbon Steel - Thin Structure - 10 % Yield Stress - Asymmetric Loading

69





Series1 Series2

Series3 Series4

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500

1 2 3 4 5 6 7 Columns


Asymmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500

1 2 3 4 5 6 7 Columns


Asymmetric Loading

70





Series1 Series2

Series3 Series4

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500

1 2 3 4 5 6 7 Columns


71

Table 8.2 A comparison of the displacements obtained by different internal nodes of the thin linked structure, when subjected to 100 pct. of the yield stress for asymmetric loading.


Displacement (mm)

Row Column Alloy Steel 4140

Carbon Steel 1018

2 2 100 0.0756 0.0973 2 3 0.0884 0.1159 2 4 0.1163 0.1532 2 5 0.1494 0.1966 2 6 0.1822 0.2395 3 2 0.0108 0.0104 3 3 0.0348 0.0491 3 4 0.0719 0.0972 3 5 0.1056 0.1408 3 6 0.1233 0.1636 4 2 0.1144 0.1492 4 3 0.1083 0.143 4 4 0.1186 0.1573 4 5 0.138 0.1829 4 6 0.1441 0.1906

From table 8.2, is seen a noticeable difference in the values of displacement experienced

by the centroidal nodes, which reiterates both the accuracy and importance of the results obtained

using finite element analysis. To provide a better appreciation, all of the nodal points detailed in

Figure 5.3 are shown in the three dimensional bar graphs to facilitate an understanding of the

displacement pattern obtained when the structure containing a network of thin links is subject to

loading. From the bar graphs we infer a much higher value of displacement experienced by the

nodes in carbon steel 1018 when compared one-on-one with the nodes in alloy steel 4140.

72

Figure 8.27 A contour profile showing the magnitude of displacements experienced by linked structure of alloy steel 4140, containing a network of thin links, when subjected to asymmetric loading, at 102 pct. of the yield stress.

Figure 8.28 A contour profile showing the magnitude of displacements experienced by linked structure of carbon steel 1018, containing a network of thin links, when subjected to symmetric loading at 102 pct. of the yield stress.

73

From Figure 8.1 to Figure 8.28, it is clear that when the structure containing a network of

thin links is subjected to symmetric loading, the center of the linked structure experiences minimal

amount of deformation. However, when the loading is offset to an adjacent node, the least

deformation occurs at Node [3, 2], which is an interesting observation. From the results obtained

it is clear that when the loading is offset to another point, the overall contour of deformation

experienced by both the nodes and links does reveal a noticeable change.

For alloy steel 4140, the lower node [2, 6] experiences a higher amount of deformation

when compared to carbon steel 1018. This is observed from the tabular data presented in Table 8.2

and the bar graphs depicting the displacement pattern. Upon close examination is observed that the

displacement for C.S. 1018 and A.S. 4140 and provides information pertinent to the displacement

experienced by the nodes in the structure containing a network of thin links. An interesting

observation is the intensity of deformation experienced by Node [5, 1], which is adjacent to Node

[4, 1] where the actual load is applied, the displacement experienced is noticeably more or higher.

Yielding of the metal structure containing a network of links initiates at Node [4, 1] for

both carbon steel 1018 and Alloy steel 4140, for the case of asymmetric loading, and the

displacements were observed to be more towards the lower right half of the linked metal structure.

The displacements are linearly increasing towards the right side of the chosen metal plate and rather

irregularly distributed towards the left half of the linked metal structure.

74

8.2 A Comparison between Thin Structures of Aluminum Alloy 6061 and Copper C 10-200.

For the purpose of this study, commercially available two nonferrous metals were chosen.

These two metals, namely aluminum alloy 6061 and Copper are a preferred candidate for use in a

spectrum of industry-relevant applications, with specific emphasis on light weight, were chosen to

establish their behavior or response in the role of linked metal structure. Few studies have been

reported in the “open” literature on the behavior of perforated metal plates and sheets. To study the

tensile behavior of linked metal structures the classic deformation theories can be used to study a

solid plate element. The presence of one or more perforations in a metal plate or sheet resulting

essentially in a structure containing a network of links develops the shearing stresses, which does

influence the overall stress induced and resultant deformation behavior of the structure.

The stresses and displacement distributions computed for a linked metal structure upon

being subject to a tensile force or load was studied for two different designs of the structure for the

materials chosen, i.e. AA 6061-T6 and copper C-10200. The distribution pattern was studied when

the loading was applied uniformly at two different locations in the linked metal structure. The five

levels of load chosen were fractions of the yield stress of the chosen metal with the prime objective

of studying the behavior of links in a linked structure.

8.2.1 Symmetric Loading of Aluminum Alloy 6061-T6 and Copper C 10-200

The deformation experienced by the centroidal nodes was extracted and shown in 3D bar

graphs for the case of symmetric loading upon being subject to a tensile load. Figure 7.4 represents

the case of symmetric loading, which occurs upon application of the tensile load at Node (2, 6) and

Node (4, 2) of the two chosen metals, i.e., aluminum alloy 6061-T6 and copper C-10200. The

values of deformation, represented on 3-D bar graphs, indicates that a comparison can be made

between the values of displacement experienced by the nodal points in aluminum alloy, i.e.,

AA6061, which is relatively higher when compared one-on-one with the value of displacement

experienced by the nodal points in copper C-10200. For the chosen levels of the load, i.e., 10 pct.

75

σYS, 25 pct. σYS, 50 pct. σYS, 100 pct., σYS and 110 pct. σYS, and for the case of symmetric loading,

maximum displacement was observed to occur at Node [4, 2] and in the region in the immediate

vicinity. The forces of reaction were highest at the lower node where actual load was applied.

Hence, from the 3D bar graphs plotted, it is observed that the maximum displacement experienced

by the centroidal nodes is both at and near the upper region of the linked metal structure, i.e. node

[4, 2] and its immediate surroundings. Upon careful observation of the upper region of the linked

metal structure experiences a noticeably larger displacement in comparison with the lower region,

such that the pattern of displacement experienced by the nodes is gradually increasing in

comparison with displacements experienced by the nodes located towards the lower row, i.e., Row

1. Nodes at the center of the plate experience minimum deformation, which was evident from the

displacement contour for the thin-link metal structure of aluminum alloy 6061.

Also, for the case of pure copper C-10200 the deformation pattern experienced by the

overall linked-metal structure was quite similar to the trend shown by aluminum alloy 6061. The

concentration of Von Mises stress initiates at Node [2, 6] upon being subject to loading. The copper

begins to yield at the same section, since the overall stress concentration at this region is higher

when compared to the other nodes in the structure containing a network of thin links. The internal

nodes, which are connected by surrounding links, experience a higher degree of stress concentration

when compared to the outer links, shown in Figure 8.30. This causes the reaction force to be greater

than the other node, where actual load is applied. The displacement contours, upon careful

observation, provides a completely different pattern when compared with the stress contours of the

same linked structure at similar values of the applied load. The deformation experienced by the

links and nodes in the upper region was noticeably more when compared to the links and nodes in

the lower region of the linked metal structure. Further, a distinct increase in the values of

displacement experienced by both the nodes and links was observed in the upper regime of the

linked metal structure. The nodes and links in copper C- 10200 experienced a lower displacement

when compared one-on-one with the nodes in the linked metal structure of AA 6061-T6.

76

Table 8.3 provides a comparison of displacement for AA 6061 and copper C-10200 for the

internal nodes of the structure containing a network of thin links. The table also provides an

overview of the deformation experienced by metal plate containing a network of thin links upon

being subject to symmetric loading that is applied at Node [2,6] and Node [4,2].

77

Figure 8.29: Profile showing the contours of the von mises stress for aluminum alloy 6061 that was subjected to symmetric loading for Node (2,6) and Node (4,2) of the metal plate containing a network of thin links and at a load corresponding to yield stress of the metal.

Figure 8.30: Profile showing the contours of the Von Mises stress for pure copper C-10200

that was subjected to symmetric loading at Node (2,6) and Node (4,2) of the structure containing a network of thin links and corresponding to a load that is equal to yield stress of the chosen metal.

78

Table 8.3 A comparison of the displacements obtained by different internal nodes of the thin linked metal structure, when subjected to 100 pct. of the yield stress and for the case of symmetric loading.


Displacement (mm)

Row Column Aluminum

Alloy 6061

C-10200

2 2 100 0.2251 0.0287 2 3 0.1823 0.0235 2 4 0.1533 0.0205 2 5 0.2113 0.0287 2 6 0.3039 0.0403 3 2 0.2038 0.0253 3 3 0.1248 0.0152 3 4 0.0166 0.0008 3 5 0.1132 0.0158 3 6 0.1987 0.0259 4 2 0.3233 0.0406 4 3 0.2512 0.0313 4 4 0.2063 0.0256 4 5 0.2278 0.0291 4 6 0.2669 0.0338

79


“thin” linked structure of aluminum alloy 6061 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2).



Series1 Series2

Series3 Series4

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

1 2 3 4 5 6 7 Columns

6061 - T6 Aluminum Alloy - Thin Structure - 10 % -

Yield Stress - Symmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000 0.0100 0.0200 0.0300

0.0400 0.0500 0.0600

1 2 3 4 5 6 7 Columns

6061 - T6 Aluminum Alloy - Thin Structure -

% 25 - Yield Stress - Symmetric Loading

80





Series1 Series2

Series3 Series4

Series5

0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200

1 2 3 4 5 6 7 Columns

6061 - T6 Aluminum Alloy - Thin Structure - % 50 -


Series1 Series2

Series3 Series4

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500

1 2 3 4 5 6 7 Columns


100 % - Yield Stress - Symmetric Loading

81




“thin” linked structure of copper C-10200 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Series1 Series2

Series3 Series4

Series5

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

1 2 3 4 5 6 7 Columns



Series1 Series2

Series3 Series4

Series5

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

1 2 3 4 5 6 7 Columns

C - 10200 - Thin Link-Symmetric Loading - 10 % Yield Stress

82





Series1 Series2

Series3 Series4

Series5

0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

1 2 3 4 5 6 7 Columns


83





Series1 Series2

Series3 Series4

Series5

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

1 2 3 4 5 6 7 Columns

C - 10200 - Thin Link_Symmetric Loading - 110 % Yield Stress

84


“thin” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, 110 pct. of the yield stress.


“thin” linked structure of copper C 10200 when subjected to asymmetric loading, 110 pct. of the yield stress.

85

8.2.2 Asymmetric loading of Aluminum Alloy 6061 and Copper C 10-200.

The deformation experienced by the centroidal nodes are extracted and shown in 3D bar

graphs for the case of asymmetric loading applied in tension. In Figure 7.5 is shown symmetric

loading obtained by applying the load applied at Node (2, 6) and Node (4, 1) of the two chosen

metals, i.e., aluminum alloy 6061-T6 and copper C-10200. The Von Mises stress experienced at

Node [4, 1] was maximum primarily because of actual load application at this node in tension. The

presence of reaction forces at this same node causes the structure to initiate yielding in this region.

Further, the Von Mises stress was observed to be higher at the links in the immediate vicinity, which

is shown in Figure 8.43 for AA 6061-T6. The stresses were found to be high at the intersection of

the links, particularly in an area where four links are connected when compared to the outer

boundary or periphery of the linked metal structure that is connected by two links. Node [2, 6] and

links in the immediate surrounding also experience a higher value of the Von Mises stress as shown

in Figure 8.43. Pattern shown by the displacement contour was noticeably different from the pattern

shown by the Von Mises stress. Prior to yielding the maximum displacement occurred in the region

where load was applied but once the structure begins to deform at the yield load, Node [4, 1]

experiences a higher level of deformation when compared to the other nodes upon loading in

tension. Also, center of the linked metal structure for the case of symmetric loading experiences

very little deformation. However, for asymmetric loading the overall structure is stable both at and

around the region surrounding Node [3, 2]. This observation is favored to occur due to a change in

the position of actual load application on the upper node of the thin linked metal structure.

For the case of copper alloy C-10200, when subjected to asymmetric loading, the maximum

Von Mises stress was experienced by Node [2,6], which is similar to the behavior shown by the

thin linked structure of AA 6061. Also, the reaction forces are more in this region when compared

to remaining portion of the structure. This causes the initiation of maximum strain in the structure,

which is conducive for the initiation of localized yielding, as shown in Figure 8.44. The stress

86

concentration occurring at the intersection of the links was quite similar to the trend shown or

observed in AA 6061-T6 for the case of asymmetric loading.

The magnitude of displacement experienced was more upon asymmetric loading in tension.

This was similar to the behavior shown by AA 6061. The displacements were comparatively low

for copper C-10200, which had noticeably higher yield strength when compared to the aluminum

alloy chosen, which tended to easily deform upon application of a load when compared to copper.

The displacement patterns are shown for the five chosen load levels in Figure 8.50 to Figure 8.54.

In Table 8.4 is provided a detailed comparison of the displacement experienced by the centroidal

nodes for the metal structure containing a network of thin links, for the two non-ferrous metals

chosen In Figure 8.55 and Figure 8.56 is shown a view of the contour for the metal structure

containing a network of thin links in the non-linear region for the two metals chosen, and a

noticeable difference in deformation behavior can be observed.

87

Figure 8.43: Profile showing the contours of the Von Mises stress for aluminum alloy 6061

that was subject to asymmetric loading at Node (2,6) and Node (4,1) of the thin linked structure at a load corresponding to yield stress of the metal.

Figure 8.44: Profile showing the contours of the von mises stress for copper C 10200 that

was subjected to asymmetric loading at Node (2,6) and Node (4,1) of the thin linked structure at a load corresponding to yield stress of the metal.

88


“thin” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1).



Series1 Series2

Series3 Series4

Series5

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300

1 2 3 4 5 6 7 Columns

6061 - T6 Aluminum Alloy - Thin Structure - % 10 -

Yield Stress - Asymmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 0.0600 0.0700

1 2 3 4 5 6 7 Columns



89





Series1 Series2

Series3 Series4

Series5

0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 0.1400

1 2 3 4 5 6 7 Columns



Series1 Series2

Series3 Series4

Series5

0.0000

0.1000

0.2000

0.3000

0.4000

1 2 3 4 5 6 7 Columns


% 100 - Yield Stress - Asymmetric Loading

90




“thin” linked structure of copper C-10200 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1).

Series1 Series2

Series3 Series4

Series5

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

1 2 3 4 5 6 7 Columns


110 % - Yield Stress - Asymmetric Loading

Series1

Series3

Series5

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

1 2 3 4 5 6 7 Columns

C - 10200 - Thin Link_Asymmetric Loading - 10 % Yield Stress

91





Series1 Series2

Series3 Series4

Series5

0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120

1 2 3 4 5 6 7 Columns


Series1

Series3

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

1 2 3 4 5 6 7 Columns


92





Series1 Series2

Series3 Series4

Series5

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

1 2 3 4 5 6 7 Columns


93

Table 8.4 A comparison of the vale of displacements experienced by different internal nodes of the thin linked structure, when subjected to a load that was 100 pct. of the yield stress for the case of asymmetric loading.


Displacement (mm)

Row Column Aluminum Alloy 6061 C-10200

2 2 0.1498 0.0205 2 3 0.1724 0.0211 2 4 0.2261 0.0278 2 5 0.2907 0.0370 2 6 0.3546 0.0459 3 2 0.0271 0.0080 3 3 0.0644 0.0038 3 4 0.1368 0.0148 3 5 0.203 0.0249 3 6 0.2372 0.0300 4 2 0.2167 0.0310 4 3 0.2036 0.0272 4 4 0.2238 0.0289 4 5 0.2624 0.0341 4 6 0.2744 0.0360

94


“thin” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, 110 pct. of the yield stress.


“thin” linked structure of copper C 10200 when subjected to asymmetric loading, 110 pct. of the yield stress.

95

8.3 A Comparison between thick structures of Alloy Steel 4140 and Carbon Steel

1018

The 4140 alloy steel link structure initiated deforming from its original shape and

the following observations, with specific reference to displacement, are recorded for the

five chosen loading conditions, as fraction of the yield stress obtained from the stress versus

strain curve following a tensile test simulation performed on the structure containing a

network of thick links. Upon symmetric loading in tension of alloy steel 4140 containing

a network of thick links, the reaction forces were noticeably more towards the lower end,

i.e. node [2, 6 ], resulting in a higher magnitude of the Von Mises stress in this region

culminating in the early initiation of yielding as shown in Figure 8.56. The contour of

stress concentration occurring at the nodes obtained at the intersection of four links was

noticeably high and minimal at the center of the links. The trend shown by the displacement

provides a completely different contour, when compared with the stress contour at an

identical value of the applied load.

8.3.1 Symmetric Loading Alloy Steel 4140 and Carbon Steel 1018

Upon application of load to the linked metal structure of alloy steel 4140 containing

a network of thick links, the maximum displacement was observed to occur at the

intersecting nodes situated both at and near the upper regime of the perforated metal plate.

The displacement experienced by the different links in the thick linked structure of alloy

steel 4140 was found to be noticeably lower when compared one-on-one with the structure

containing a network of thin links. Also, the pattern of displacement when plotted on a 3D

bar graph reveals a symmetry to exist between the two chosen metals, when subjected to

loading at different values. While the values of displacement are different they do show a

similar behavior when compared one-on-one with respect to each other. At the upper left

96

end of the perforated metal plate the displacement experienced by both the links and nodes

was relatively low. This is ascribed to the fact that in this location the actual effect of

loading is reduced. Also, the load was applied equally at the two chosen nodes of the linked

metal structure and the displacement was noticeably high at the upper half of the metal

structure, when compared to the lower half of the same structure for the five chosen levels

of load. For CS 1018, the structure containing thick links when subjected to loading, i.e.,

Node [2,6], high values of the “local” stress initiate at the lower nodes of the linked

structure due to which yielding is favored to initiate in this region. The upper half the

metal structure experiences noticeably lower stresses when compared to rest of the metal

structure. The displacements experienced by the nodes, were plotted on a 3D bar graph

(Figure 8.58 - Figure 8.67). The displacements experienced by the nodes in carbon steel

1018 were observably higher towards the right half of the metal plate and gradually

decreased towards the left side of the linked metal structure. The deformation or

displacement experienced by the following nodes provides an insight that the nodes in

carbon steel 1018 experience higher value of deformation when compared one-on-one with

the nodes in alloy steel 4140. This is attributed to the higher strength of the chosen alloy

steel 4140 and resultant lower ductility when compared to carbon steel 1018. A similarity

between the linked structures of the two chosen steels is that the upper regions near the

point of application of load experiences a higher level of deformation experienced by the

nodes, which is observed from the pattern shown by the bar graphs. In Table 8.5 is provided

a comparison for the centroidal nodes for alloy steel 4140 and carbon steel 1018 containing

a network of thick links and subject to symmetric loading.

97

Figure 8.57: Profile showing the contours of the Von Mises stress for alloy steel 4140 that was subjected to symmetric loading for Node (2,6) and Node (4,2) of the structure containing a network of thick links at a loads equal to the yield stress of the metal. .

Figure 8.58: Profile showing the contours of the von mises stress for carbon steel 1018 that

was subjected to symmetric loading for node (2, 6) and node (4, 2) of the thick linked structure at the elastic limit.

98

Figure 8.59 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Figure 8.60 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Series1 Series3

Series5

0.0000

0.0020

0.0040

0.0060

0.0080

1 2 3 4 5 6 7 Column

4140 Alloy Steel - Thick Structure - 10 % Yield Stress -

Symmetric Loading

Series1

Series3

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

1 2 3 4 5 6 7 Columns


Symmetric Loading

99


“thick” linked structure of alloy steel 4140 when subjected to symmetric loading, that is 50 percent of the yield stress, applied at Node (2,6) and Node (4,2).



Series1 Series2

Series3 Series4

Series5

0.0000

0.0100

0.0200

0.0300

0.0400

1 2 3 4 5 6 7 Columns


Symmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

1 2 3 4 5 6 7 Columns


Symmetric Loading

100




“thick” linked structure of carbon steel 1018 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

1 2 3 4 5 6 7 Columns


Symmetric Loading

Series1 Series3

Series5

0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120

1 2 3 4 5 6 7 Columns

1018 Carbon Steel - Thick Structure - 10 % Yield Stress - Symmetric Loading

101





Series1 Series3

Series5

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250

1 2 3 4 5 6 7 Columns


Series1 Series3

Series5

0.0000 0.0100 0.0200 0.0300 0.0400 0.0500

1 2 3 4 5 6 7 Columns


102





Series1 Series3

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


Series1 Series3

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


103

Table 8.5 A comparison of the displacements experienced by the internal nodes of a thick linked metal structure, when subjected to 100 pct. of the yield stress under

symmetric conditions


Node Displacement (mm)


Carbon Steel 1018

2 2 840 0.0668 0.091 2 3 885 0.0518 0.070 2 4 1042 0.0362 0.045 2 5 1132 0.0551 0.067 2 6 1105 0.0943 0.120 3 2 818 0.0758 0.106 3 3 811 0.0494 0.071 3 4 892 0.0195 0.032 3 5 1049 0.0310 0.041 3 6 1180 0.0603 0.079 4 2 930 0.1252 0.174 4 3 905 0.0931 0.131 4 4 913 0.0732 0.104 4 5 1065 0.0726 0.101 4 6 1148 0.0817 0.112

104

Figure 8.69 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of alloy steel 4140 when subjected to symmetric loading, 102 pct. of the yield stress.

Figure 8.70 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of carbon steel 1018 when subjected to symmetric loading, 102 pct. of the yield stress.

105

8.3.2 Asymmetric loading: Alloy Steel 4140 and Carbon Steel 1018

For alloy steel 4140 and asymmetric nature of loading, i.e. when application of the

load is changed to an adjacent nodal point, we observe the contour pattern to change

appreciably at the point of application of the tensile load. The upper region, i.e., the region

near node [4, 1] experiences a higher level of the Von Mises stress, whereas the area

towards the lower right half of the perforated metal plate reveals a similar profile, when

compared one-on-one with symmetric loading. The stresses are noticeably more at the

intersection of four links, or link elements, i.e. at the internal nodes, and comparatively low

both at and near the outer region of the metal structure. The magnitude of displacement

was noticeably more on the nodes situated in Row 4. The magnitude of displacement

decreases for the nodes on Row 5, which is vice versa of the observation for the case of

symmetric loading. Thus, for the two chosen metals, an ‘S’ shaped pattern was observed

when the data is plotted on bar graphs.

Upon application of an asymmetric load to alloy steel 4140 containing a network

of thick links, the maximum displacement at the five chosen load levels occurs at the

intersecting nodes located towards the upper half of the perforated metal plate. It is

interesting to note that the lower half of the perforated metal plate, or linked metal structure,

does not experience appreciable deformation when compared to the link elements and nodal

points located towards upper half of the plate. When the load is applied gradually, the

displacement contour reveals maximum deformation to occur in the region near Node [4,

1] and the surrounding links. Upon increasing the load, the minimal deformation

experienced by Node [3, 4] under symmetric loading, starts to shift towards the right of the

perforated metal plate. From the 3D bar graphs, it can be concluded that alloy steel 4140

106

provides a more uniform distribution of the centroidal nodes when compared one-on-one

with carbon steel 1018 containing a network of thick links.

For carbon steel 1018, a similar trend for the Von Mises stress having a higher

concentration at the upper end of the metal structure was observed. This facilitates in the

structure to begin yielding from and around the region of Node [4, 1] and the surrounding

links. The Von Mises stress shown in the contour were much higher in numerical value

when compared with alloy steel 4140 at identical values of the applied load. The

deformation experienced at the center of the structure containing a network of thick links

was interesting since the stresses were initially low at Node [3, 4] and upon gradual increase

in the load, the centroidal nodes of the complete linked metal structure towards the right

half experiences minimal deformation, which is different from the contours shown by alloy

steel 4140 containing a network of thick links and subjected to asymmetric loading. The

bar graphs (Figure 8.77- Figure 8.81) provide a detailed view of the deformation

experienced by the nodes in carbon steel 1018. From the stress versus strain curve, when

subjected to asymmetric loading, the following observation was made for carbon steel in

the occurrence of strain hardening following yielding. This was followed by softening to

failure. An observation that was quite dissimilar, when compared one-on-one, with alloy

steel 4140 at the same value of applied load. Overall, both the link elements and the nodal

points experience low displacement on approximately 30 pct., of the plate, with maximum

impact occurring at the upper half of the perforated metal plate where pull by a tensile load

is applied.

Figure 8.82 and Figure 8.83 provide a clearer view of the behavior of the metal

structure containing a network of thick links and subjected to deformation as a consequence

of application of load in the non-linear regime. Table 8.6 provides a comparison of the

107

amplitude of the displacements for the nodes connected by four links inside the linked metal

structure containing a network of thick links.

108

Figure 8.71: Profile showing the contours of the von mises stress for alloy steel 4140 that was subjected to asymmetric loading for Node (2, 6) and Node (4, 1) of the thick linked structure at load corresponding to yield stress of the chosen metal.

Figure 8.72: Profile showing the contours of the von mises stress for carbon steel 1018 that

was subjected to asymmetric loading for Node (2, 6) and Node (4, 1) of the thick linked structure at a load corresponding to the yield stress.

109

Figure 8.73 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of alloy steel 4140 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1).


Series1 Series2

Series3 Series4

Series5

0.0000

0.0020

0.0040

0.0060

0.0080

1 2 3 4 5 6 7 Columns


Asymmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

1 2 3 4 5 6 7 Columns


Asymmetric Loading

110



Series1

Series3

Series5

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

1 2 3 4 5 6 7 Columns


Asymmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200

1 2 3 4 5 6 7 Columns

4140 Alloy Steel - Thick Structure - 100 % Yield Stress - Asymmetric Loading

111


Figure 8.78 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of carbon steel 1018 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1).

Series1 Series2

Series3 Series4

Series5

0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200

1 2 3 4 5 6 7 Columns


Asymmetric Loading

Series1 Series3

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

1 2 3 4 5 6 7 Columns

1018 Carbon Steel - Thick Structure - 10 % Yield Stress - Asymmetric Loading

112



Series1

Series3

Series5

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

1 2 3 4 5 6 7 Columns


Series1

Series3

Series5

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

1 2 3 4 5 6 7 Columns

1018 Carbon Steel - Thick Structure - 50 % Yield Stress -

Asymmetric Loading

113



Series1

Series3

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

1 2 3 4 5 6 7 Columns


114

Table 8.6 A comparison of the displacements obtained by different internal nodes of the thick linked structure, when subjected to 100 pct. of the yield stress for asymmetric loading.

Node

Percentage of Yield Stress (%)

Displacement (mm)


Carbon Steel 1018

2 2 0.0312 0.064 2 3 0.0376 0.064 2 4 0.0581 0.089 2 5 0.0827 0.125 2 6 0.1136 0.168 3 2 0.0142 0.047 3 3 0.0141 0.034 3 4 0.0374 0.060 3 5 0.0586 0.093 3 6 0.0736 0.115 4 2 0.0699 0.049 4 3 0.0577 0.019 4 4 0.0607 0.049 4 5 0.0721 0.083 4 6 0.0761 0.095

115


“thick” linked structure of alloy steel 4140 when subjected to asymmetric loading, 102 pct. of the yield stress.

Figure 8.84 A contour profile showing the magnitude of displacements experienced by “thick” linked structure of carbon steel 1018 when subjected to symmetric loading, 102 pct. of the yield stress.

116

8.4 A Comparison between Thick Structures of Aluminum Alloy 6061-T6 and Copper

C 10-200.

When a linked metal structure containing a network of thick links was subjected to loading,

it does tend to experience deformation at every node.

8.4.1 Symmetric loading of Aluminum Alloy 6061 and Copper 10200.

For aluminum alloy 6061-T6, the stresses were higher at the lower node [4, 2] of the

structure containing a network of thick links, which is similar upon comparison one-on-one with a

thin linked structure of the same material. The stresses increase uniformly with the magnitude of

applied load eventually resulting in the initiation of yielding at a node. This behavior is attributed

essentially to a concentration of the reaction forces occurring on a node. The values of deformation

or displacement behavior were obtained from the analysis and plotted using a three dimensional bar

graph, as shown in Figure 8.86 to Figure 8.95. The deformation was noticeable at an upper node (4,

2), which experienced a maximum amount of deformation as inferred by the value of displacement.

The displacement was noticeably uniform in the region of elastic range. However, when the

material started to yield, the values of displacement revealed a non-linear trend. Upon examining

the displacement data, represented by way of 3D bar graphs, the upper half of the linked metal

structure experiences a greater degree of deformation, quantified by displacement, while the lower

half of the linked metal structure experiences observably less deformation quantified by

displacement of the links and the nodes. The deformation at the center of the linked metal structure

containing a network of thick links for aluminum alloy 6061 starts reducing and gradually shifts to

the lower nodes and links. This was quite dissimilar to what was observed for structure of the same

metal containing a network of thin links. The deformation pattern observed from the graphs shows

a trend for the displacement experienced by the centroidal nodes towards the lower region to be

less when compared with the centroid nodes at the upper half of the chosen linked metal structure.

The magnitude of displacement experienced by the links and nodal points of aluminum was

117

noticeably higher when compared to the linked metal structure of copper C-10200. In fact, the value

of deformation, or displacement, experienced by the links in AA6061-T6 was found to be

noticeably higher than the deformation experienced by the links in copper.

For pure copper C-10200 containing a network of thick links, the linked metal structure

under conditions of symmetric loading experiences a higher concentration of stress, when subjected

to loading at the lower node [4, 2]. This results in the early initiation of yielding at the intersection

of thick links. The non-uniformity of the stresses was more in the central area of the linked structure,

while the outer region experiences a lower stress when compared to the internal region of the

structure containing a network of thick links. . The displacement patterns revealed a higher

magnitude of displacement at the point of application of the load, which provides non-uniformity

in the value or magnitude of displacement experienced by both the links and nodes towards the

upper right corner of the linked metal structure. This is shown in Figure 8.97. The displacements

experienced at the mid-region of the linked metal structure are low and gradually increases to the

sides. The contour of deformation when the structure containing a network of thick links begins to

yield and reaches a non-linear profile for the linked structures of both AA 6061 and pure copper

C10200 is shown in Figure 8.96 and Figure 8.97. The bar graphs provide an overview of the

magnitude of deformation experienced by the links in AA 6061 and pure copper C-10200 when

subject to loading. The value of displacement or deformation was higher for AA 6061 when

compared with the magnitude of displacement experienced by pure copper C-10200. In Table 8.7

is provided a summary of the difference in magnitude of deformation experienced by the links,

when compared one-to-one, of AA 6061 and pure copper C-10200.

118

Figure 8.85: Profile showing the contours of the Von Mises stress for aluminum alloy 6061 that was subjected to symmetric loading for Node (2,6) and Node (4,2) of the thick linked structure at load corresponding to yield stress.

Figure 8.86: Profile showing the contours of the von mises stress for pure copper C-10200 that was subjected to symmetric loading for Node (2, 6) and Node (4, 2) of the thick linked structure at a load corresponding to yield stress.

119

Table 8.7 A comparison of the displacements obtained by different internal nodes of the thick linked structure, when subjected to 100 pct. of the yield stress for symmetric loading.


Displacement (mm)

Row Column Aluminum

Alloy 6061

C-10200

2 2 0.1211 0.0147 2 3 0.0950 0.0117 2 4 0.0766 0.0105 2 5 0.1187 0.0164 2 6 0.1931 0.0257 3 2 0.1310 0.0152 3 3 0.0799 0.0088 3 4 0.0213 0.0013 3 5 0.0630 0.0089 3 6 0.1181 0.0153 4 2 0.2203 0.0261 4 3 0.1595 0.0184 4 4 0.1244 0.0143 4 5 0.1294 0.0157 4 6 0.1484 0.0180

120

Figure 8.87 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of aluminum alloy 6061 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2).


“thick” linked structure of aluminum alloy 6061 when subjected to symmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Series1 Series2

Series3 Series4

Series5

0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140 0.0160

1 2 3 4 5 6 7 Columns

AA 6061 - T6 - 10 % Yield Stress - Symmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400

1 2 3 4 5 6 7 Columns


121





Series1 Series2

Series3 Series4

Series5

0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 0.0600 0.0700 0.0800

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

1 2 3 4 5 6 7 Columns


122

Figure 8.91 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of aluminum alloy 6061 when subjected to symmetric loading, that is 110 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Figure 8.92 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of copper C-10200 when subjected to symmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,2).

Series1 Series2

Series3 Series4

Series5

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

1 2 3 4 5 6 7 Columns

C - 10200 - 10 % Yield Stress - Symmetric Loading

123


“thick” linked structure of copper C-10200 when subjected to symmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,2).



Series1 Series2

Series3 Series4

Series5

0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140

1 2 3 4 5 6 7 Columns


124





Series1 Series2

Series3 Series4

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

1 2 3 4 5 6 7 Columns


125

Figure 8.97 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of aluminum alloy 6061 when subjected to symmetric loading, 110 pct. of the elastic limit.

Figure 8.98 A contour profile showing the magnitude of displacements experienced by “thin” linked structure of copper C 10200 when subjected to symmetric loading, 110 pct. of the elastic limit.

126

8.4.2 Asymmetric loading of Aluminum Alloy 6061-T6’ and Copper 10200. Upon changing the load from symmetric to asymmetric, the deformation behavior of the

links in the linked metal structure does reveal a noticeable change. For aluminum alloy 6061-T6

the only node, which experiences a higher level of reaction was node [4, 2].This is kind of unique

and different observation in this case, since for thin linked structures when subjected to asymmetric

loading, the intensity of load and stresses experienced during far field loading was Node [4, 1],

which is opposite of what was observed for the structure containing a network of thick links. This

provides a glimpse that when thickness of the links are increased, the yielding and overall behavior

of the structure does differ even for the same material being used. The stresses were found to be

non-uniform in the region between the two points of application of the load. Even when the loading

points are away from each other, the profile when observed reveals low values of stress at the center

of the links. The deformation contours reveal a high magnitude of the displacement upon

application of the load. The upper right portion of the metal plate containing a network of thick

links in AA 6061 does experience greater deformation when compared to the lower left corner of

the structure. The 3D bar graphs help us to quantify the magnitude of displacements and an

irregular shaped contour, forming a ‘U’ shape pattern, can be observed. The amplitude of the

displacement was high at Node [4, 1] when compared with the equal and opposite loading point for

the metal structure containing a network of thick links. Further, the overall deformation, by way of

displacement, experienced by the links was more concentrated towards the right half of the metal

structure made up of thick links. The area that experiences a minimum amount of deformation, by

way of displacement, was at and around the vicinity of Node [3, 4] for the case of symmetric

loading.

For the case of copper C-10200, a similar trend of maximum Von Mises stress being

induced towards the upper regime of the linked metal structure i.e. node [4,1], which also represents

the maximum reaction induced on a similar node that assists the thick linked structure to initiate

127

yielding at a specific area when subjected to loading. The stresses provide an observable pattern at

the intersection of any four links between the two loading points for the case of asymmetric loading.

The centroidal nodes in Row 3 experience a low magnitude of stresses, which is shown in Figure

8.99. The magnitude of displacements was low for the case of pure copper C-10200 when compared

one-on-one with AA 6061. Also, the deflection contours up until the point of yielding was quite

similar for the two non-ferrous metals chosen. Subsequent to yielding of the two chosen metals

there was a remarkable difference in the contours of the metal structure containing a network of

thick links, as shown in Figure 8.110 and Figure 8.111. The deflection observed and recorded was

more at the point of application of load, while the nearby nodes and adjacent links experienced

lower deformation quantified by displacement.

Table 8.8 provides the maximum values of displacement recorded for elastic behavior and

location of its occurrence. From the values of displacement recorded, the following key

observations are made:

(i) For the case of asymmetric loading aluminum alloy 6061-T6 experiences an observable

amount of displacement of the metal links, and

(ii) For asymmetric loading of the linked metal structure of copper experiences the least.

128

Results obtained from the finite element simulations also provide the following highlights.

(a) The displacement experienced by the links and centroid nodes was noticeably high

for the case of asymmetric loading of aluminum alloy 6061-T6 and observed to be

concentrated at the lower right half of the structure. For symmetric loading, the

linked metal structure of copper experienced the least amount of displacement of

the links and the centroid nodes.

(b) Mechanical response of the linked metal structure to symmetric loading shows the

same response, when the structure begins to yield and the pattern of displacement

by both the links and nodal points reveals an uneven trend, as can be inferred from

the 3D bar-graph. When the point of loading is shifted to ensure asymmetric nature

of loading, the deformation pattern does reveal an observable shift, or inclination,

to the left half of the linked metal structure resulting in ‘U’ trend, which is inferred

from the 3D bar graphs.

129

Figure 8.99: Profile showing the contours of the von mises stress for aluminum alloy 6061 that was subjected to asymmetric loading for Node (2,6) and Node (4,1) of the thick linked structure at load commensurate with yield stress of the metal .

Figure 8.100: Profile showing the contours of the von mises stress for copper C 10200 that was

subjected to asymmetric loading at Node (2, 6) and Node (4, 1) of the thick linked structure at load commensurate with yield stress of the metal

130

Figure 8.101 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6), and Node (4,1).

Figure 8.102 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, that is 25 percent of the yield stress, applied at Node (2,6) and Node (4,1).

Series1 Series2

Series3 Series4

Series5

0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140 0.0160 0.0180

1 2 3 4 5 6 7 Columns

AA 6061 - T6 - 10 % Yield Stress - Asymmetric Loading

Series1 Series2

Series3 Series4

Series5

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450

1 2 3 4 5 6 7 Columns


131



Series1 Series2

Series3 Series4

Series5

0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 0.0600 0.0700 0.0800 0.0900

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

1 2 3 4 5 6 7 Columns


132


Figure 8.106 Profile showing the displacement experienced by the different nodes of the “thick” linked structure of copper C-10200 when subjected to asymmetric loading, that is 10 percent of the yield stress, applied at Node (2,6) and Node (4,1).

Series1 Series2

Series3 Series4

Series5

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

0.3000

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

1 2 3 4 5 6 7 Columns

C - 10200 - 10 % Yield Stress - Asymmetric Loading

133



Series1 Series2

Series3 Series4

Series5

0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 0.0080

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140 0.0160

1 2 3 4 5 6 7 Columns


134



Series1 Series2

Series3 Series4

Series5

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

1 2 3 4 5 6 7 Columns


Series1 Series2

Series3 Series4

Series5

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350

1 2 3 4 5 6 7 Columns


135

Table 8.8 A comparison of the displacements obtained by different internal nodes of the thick linked structure, when subjected to 100 pct. of the yield stress for asymmetric loading.


Displacement ( mm)

Row Column Aluminum

Alloy 6061

Copper C-10200

2 2 100 0.0659 0.0093 2 3 0.0767 0.0098 2 4 0.1164 0.0149 2 5 0.1652 0.0217 2 6 0.2265 0.0299 3 2 0.0299 0.0057 3 3 0.0244 0.002 3 4 0.0717 0.0086 3 5 0.1142 0.0146 3 6 0.1436 0.0184 4 2 0.134 0.0181 4 3 0.1085 0.0141 4 4 0.1146 0.0146 4 5 0.138 0.0178 4 6 0.1459 0.0188

136


“thick” linked structure of aluminum alloy 6061 when subjected to asymmetric loading, 110 pct. of the yield stress of the chosen metal.


“thick” linked structure of copper C 10200 when subjected to asymmetric loading, 110 pct. of the yield stress of the chosen metal.

137

CHAPTER IX

CONCLUSIONS

Based on a study on the effective use of finite element analysis for purpose of assessing the

mechanical response of linked metal structures, having a network of links that can be classified as

being thin and thick, and upon being subject to both symmetric loading and asymmetric loading,

following are the key findings:

(1) Nature of loading, that is symmetric versus asymmetric, was observed to have minimal

influence on stress versus strain behavior of a metal plate having an array of square

perforations, that resulted in a structure that was essentially held together by a network of

links, of two varying thicknesses, categorized as thin links and thick links.

(2) The Von-Mises failure criterion was used for purpose of analysis. A detailed analysis

revealed plane stress conditions to prevail over a significant portion of the chosen

perforated metal plate structure.

(3) With the use of numerical computation it was possible to obtain a substantial amount of

information pertinent to: (i) displacement and stresses developed in the links or link

elements, and (ii) displacements occurring or experienced by the nodal points, or nodes, in

a perforated metal structure that is essentially held together by a network of link elements

of varying thickness and at the intersection of these link elements are the nodes or nodal

points.

(4) Under the influence of an applied load, the values of both stress and displacement

experienced by the links, or link elements, and the nodal points in a perforated metal plate

138

were determined. A two-dimensional [2-D] analysis was chosen and performed for the

case of both symmetric loading and asymmetric loading. The intrinsic advantages in

choosing a 2-D formulation was it involved fewer number of calculations with a

concomitant decrease in time required for numerical computation. An analysis of the 3-D

model was not considered since thickness of the chosen metal plate is comparatively low

when compared to the other two dimensions of the chosen metal plate.

(5) Upon application of a given magnitude of load using the displacement approach, as a

fraction of yield stress of the chosen metal, the values of displacement experienced by the

links or link elements, and the nodes dispersed through the linked metal structure were

obtained. The value of displacement, or deformation experienced by both the nodes and

links was observed to be noticeably different or non-uniform for both thick links and thin

links but did provide a meaningful shape when represented graphically on a 3-D bar graph.

(6) The pattern shown by displacement of the link elements of the two chosen steels was

different depending on: (i) the nature of loading, symmetric versus asymmetric, and (ii) for

a specific magnitude of the load, as a function of the yield load, used.

(7) The finite element method was successfully used to study the mechanical response of the

linked metal structure, quantified in this study by deformation or displacement experienced

by the links and centroid node upon application of a load. This was made possible for the

perforated metal structure containing a network of thick links and perforated metal structure

containing a network of thin links and the materials chosen for the overall linked structure

belonging to the families of both ferrous alloys and non-ferrous alloys.

(8) From the results obtained it is clear that when the nature of loading is changed from

symmetric to asymmetric, the linked structure does experience a noticeable increase in the

139

extent of deformation, or displacement, experienced by both the links and the centroid

nodes.

(9) Higher the magnitude of applied load, then (i) greater and distinctly evident was the extent

of deformation, or displacement, experienced by the different links, (ii) greater was the

displacement occurring at the centroid nodes, and (iii) greater was the displacement

experienced at the node where actual load was applied.

(10) The numerical analysis used in this research study helps in predicting the variation of stress

with strain for the chosen metal, which is conventionally obtained using a tensile test, in

the configuration of a linked metal structure. The approach used through the entire research

study was simple and based on static general analysis using the displacement approach

(11) The finite element method in conjunction with the numerical technique was used to

compare perforated metal structures made from four metals, each perforated metal structure

containing a network of thin links and a network of thick links. Since the linked metal

structure does contain a number of junctions that experience fairly high ‘local’ stress

concentration, the corresponding value of displacement calculated for both the links and

the nodes is noticeably lower. This necessitates the need for changes in design of the

starting structure, i.e., perforated metal plate, followed by a careful analysis using the finite

element method and numerical technique.

140

REFERENCES

[1] F.-K. Chen and Y.-C. Lee, “Plastic Deformation of a Perforated Sheet With Non-Uniform Circular Holes Along the Thickness Direction,” J. Eng. Mater. Technol., vol. 125, no. 1, p. 75, 2003.

[2] I. P. Association, Designers, Specifiers and Buyers Handbook for Perforated Metals.

Industrial Perforators Association, 1978. [3] W. J. O’Donnell and B. F. Langer, “Design of perforated plates,” J. Eng. Ind., vol. 84, no.

3, pp. 307–319, 1962. [4] J. E. Goldberg, “I. introduction,” Nucl. Struct. Eng. 2, vol. 2, no. 4, pp. 360–381, 1965. [5] P. L. E. Fort, “WITH SQUARE PENETRATION PATTERNS,” vol. 12, pp. 122–134, 1970. [6] J. S. Sirkis and T. J. Lim, “Displacement and strain measurement with automated grid

methods,” Exp. Mech., vol. 31, no. 4, pp. 382–388, 1991. [7] A. Litewka, Poznan, and A. Sawczuk, “A yield criterion for perforated sheets,”

IngenieurArchiv, vol. 50, no. 6, pp. 393–400, 1981. [8] I. Arrayago, E. Real, and L. Gardner, “Description of stress – strain curves for stainless steel

alloys,” vol. 87, pp. 540–552, 2015. [9] C. Pellegrino, E. Maiorana, and C. Modena, “Linear and non-linear behaviour of steel plates

with circular and rectangular holes under shear loading,” Thin-Walled Struct., vol. 47, no. 6–7, pp. 607–616, 2009.

[10] F.-K. Chen, “Analysis of plastic deformation for sheet metals with circular perforations,” J.

Mater. Process. Technol., vol. 37, no. 1–4, pp. 175–188, 1993. [11] Z. Lopez and A. Fatemi, “A method of predicting cyclic stress-strain curve from tensile

properties for steels,” Mater. Sci. Eng. A, vol. 556, pp. 540–550, 2012. [12] E. Mirambell and E. Real, “On the calculation of deflections in structural stainless steel

beams: an experimental and numerical investigation,” J. Constr. Steel Res., vol. 54, no. 1, pp. 109–133, 2000.

141

[13] D. S. Simulia, “ABAQUS 6.13 User’s Manual,” Dassault Syst. Provid. RI, 2013. [14] N. Senthilnathan, G. Venkatachalam, and N. N. Satonkar, “A Two Stage Finite Element

Analysis of Electromagnetic Forming of Perforated Aluminium Sheet Metals,” Procedia

Eng., vol. 97, pp. 1135–1144, 2014. [15] S. K. Park, J. Kim, Y. C. Chang, and B. S. Kang, “Analysis of the deformation of a

perforated sheet under thermal and tension load using finite element method,” J. Mater.

Process. Technol., vol. 113, no. 1–3, pp. 761–765, 2001. [16] G. Venkatachalam, P. N. S, P. M. Neil, P. Agarwal, and S. Narayanan, “INFLUENCE OF

PERFORATIONS ON FLEXURAL STRENGTH OF ALUMINIUM 8090 ALLOY SHEETS,” ARPN J. Eng. Appl. Sci., vol. 8, no. 4, pp. 277–279, 2013.

[17] T. Slot and J. P. Yalch, “STRESS ANALYSIS OF PLANE PERFORATED

STRUCTURES BY POINT-WISE MATCHING OF BOUNDARY CONDITIONS *,” Nucl. Eng. Des., vol. 4, pp. 163–176, 1966.

[18] A. L. Yettram and E. S. Awadalla, “A direct matrix method for the elastic stability analysis

of plates,” Int. J. Mech. Sci., vol. 10, no. 11, pp. 887–901, 1968.

[19] A. L. Yettram and C. J. Brown, “The elastic stability of square perforated plates,” Comput.

Struct., vol. 21, no. 6, pp. 1267–1272, 1985. [20] S. C. Baik, K. H. Oh, and D. N. Lee, “Forming limit diagram of perforated sheet,” Scr.

Metall. Mater., vol. 33, no. 8, pp. 1201–1207, 1995. [21] C. García-Garino, F. Gabaldón, and J. M. Goicolea, “Finite element simulation of the simple

tension test in metals,” Finite Elem. Anal. Des., vol. 42, no. 13, pp. 1187–1197, 2006. [22] D. M. Norris Jr, B. Moran, J. K. Scudder, and D. F. Quiñones, “A computer simulation of

the tension test,” J. Mech. Phys. Solids, vol. 26, no. 1, pp. 1–19, 1978. [23] B. R. Dewey, “Finite-element analysis of creep and plasticity tensile-test specimens,” Exp.

Mech., vol. 16, no. 1, pp. 16–20, 1976. [24] Y. A.L. and B. C.J., “PLATES UNDER BI-AXIAL LOADING,” Comput. Struct., vol. 22,

no. 4, pp. 589–594, 1986. [25] C. J. Brown, “Elastic buckling of perforated plates subjected to concentrated loads,”

Comput. Struct., vol. 36, no. 6, pp. 1103–1109, 1990. [26] A. L. Yettram and C. J. Brown, “Improving the Elastic Stability of Square Perforated

Plates.,” J. Constr. Steel Res., vol. 7, no. 5, pp. 371–383, 1987. [27] S. Huybrechts and S. W. Tsai, “Analysis and behavior of grid structures,” Compos. Sci.

Technol., vol. 56, no. 9, pp. 1001–1015, 1996.

142

[28] Seung Chul Baik, Kyu Hwan Oh, and Dong Nyung Lee, “Analysis of the deformation of a perforated sheet under uniaxial tension,” J. Mater. Process. Technol., vol. 58, no. 2–3, pp. 139–144, 1996.

[29] S. C. Baik, H. N. Han, S. H. Lee, K. H. Oh, and D. N. Lee, “Plastic Behaviour of Perforated

Sheets under Biaxial Stress State,” Int. J. Mech. Sci., vol. 39, no. 7, pp. 781– 793, 1997. [30] Y. C. Lee and F. K. Chen, “Yield criterion for a perforated sheet with a uniform triangular

pattern of round holes and a low ligament ratio,” J. Mater. Process. Technol., vol. 103, no. 3, pp. 353–361, 2000.

[31] F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, F. Pourboghrat, S. H.

Choi, and E. Chu, “Plane stress yield function for aluminum alloy sheets - Part 1: Theory,” Int. J. Plast., vol. 19, no. 9, pp. 1297–1319, 2003.

[32] J.-W. Yoon, F. Barlat, R. E. Dick, K. Chung, and T. J. Kang, “Plane stress yield function

for aluminum alloy sheets—part II: FE formulation and its implementation,” Int. J. Plast., vol. 20, no. 3, pp. 495–522, 2004.

[33] A. Cirello, F. Furgiuele, C. Maletta, and A. Pasta, “Numerical simulations and experimental

measurements of the stress intensity factor in perforated plates,” Eng. Fract. Mech., vol. 75, no. 15, pp. 4383–4393, 2008.

[34] D. C. Webb, K. Kormi, and S. T. S. Al-Hassani, “Use of FEM in performance assessment

of perforated plates subject to general loading conditions,” Int. J. Press. Vessel. Pip., vol. 64, no. 2, pp. 137–152, 1995.

[35] L. A. Dobrzański, A. Pusz, A. J. Nowak, and M. Górniak, “Application of FEM for solving

various issues in material engineering,” J. Achiev. Mater. Manuf. Eng., vol. 42, no. 1–2, pp. 134–141, 2010.

[36] K. Zhao, L. Wang, Y. Chang, and J. Yan, “Identification of post-necking stress-strain curve

for sheet metals by inverse method,” Mech. Mater., vol. 92, pp. 107–118, 2016. [37] J. A. Benito, R. Cobo, W. Lei, J. Calvo, and J. M. Cabrera, “Stress-strain response and

microstructural evolution of a FeMnCAl TWIP steel during tension-compression tests,” Mater. Sci. Eng. A, vol. 655, pp. 310–320, 2016.

[38] E. E. Cabezas and D. J. Celentano, “Experimental and numerical analysis of the tensile test

using sheet specimens,” Finite Elem. Anal. Des., vol. 40, no. 5–6, pp. 555–575, 2004. [39] A. Litewka, “EXPERIMENTAL STUDY OF THE EFFECTIVE YIELD SURFACE OF

PERFORATED MATERIALS *,” Nucl. Eng. Des. © North-holl. Publ. Co., vol. 57, pp. 417–425, 1980.

[40] V. M. N, P. T, V. P, and R.Subramanian, “Design and Structural Analysis of a Dual

Compression Rotor,” Int. J. Adv. Sci. Tech. Res., vol. 4, no. 3, pp. 1, 417–421, 2013.

143

[41] Seung Chul Baik, Kyu Hwan Oh, and Dong Nyung Lee, “Analysis of the deformation of a perforated sheet under uniaxial tension,” J. Mater. Process. Technol., vol. 58, no. 2–3, pp. 139–144, 1996.

[42] H. Khatam, L. Chen, and M.-J. Pindera, “Elastic and Plastic Response of Perforated Metal

Sheets With Different Porosity Architectures,” J. Eng. Mater. Technol., vol. 131, no. 3, pp. 31014–31015, 2009.

[43] A. L. Yettram and C. J. Brown, “The Elastic Stability of Square Perforated Plates Under Bi-

Axial Loading.,” Comput. Struct., vol. 22, no. 4, pp. 589–594, 1986. [44] Handbook, ASM Handbook, Volume 01 - Properties and Selection: Irons, Steels, and High-

Performance Alloys. 1990. [45] Kuhn, Howard, and Dana Medlin, Mechanical Testing and Evaluation. ASM International,

2000. [46] T. S. Srivatsan, H. Mike, T. S. Sudarshan, and K. O. Legg, “Influence of nitrogen ion

implantation on tensile behavior of 1018 carbon steel,” vol. 213, pp. 27–33, 1992. [47] S. H. Lee, Y. Saito, T. Sakai, and H. Utsunomiya, “Microstructures and mechanical

properties of 6061 aluminum alloy processed by accumulative roll-bonding,” Mater. Sci.

Eng. A, vol. 325, no. 1–2, pp. 228–235, 2002. [48] A. S. M. I. H. Committee, “ASM Handbook, Volume 02 - Properties and Selection:

Nonferrous Alloys and Special-Purpose Materials.” ASM International. [49] J. R. Davis, “ASM Specialty Handbook - Copper and Copper Alloys.” ASM International. [50] J. R. Davis, “ASM Specialty Handbook - Aluminum and Aluminum Alloys.” ASM

International. [51] A. Rohatgi, “Web Plot Digitizer. ht tp,” arohatgi. info/WebPlotDigitizer/app/(accessed June

2, 2014). [52] A. M. Prior, “Applications of Implicit and Explicit Finite Element Techniques to Metal

Forming,” J. Mater. Process. Technol., vol. 45, no. X, pp. 649–656, 1994.

investigating and understanding the mechanical response of linked structures of hard and soft

Documents

Transcript of investigating and understanding the mechanical response of linked structures of hard and soft