Modeling the High Strain Rate Tensile Response and Shear ... · Modeling the High Strain Rate...

Modeling the High Strain Rate Tensile Response and ShearFailure of Thermoplastic Composites

Pierce David Umberger

Dissertation submitted to the Faculty ofVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophyin

Engineering Mechanics

Scott W. Case, ChairRomesh C. BatraMichael W. HyerSunghwan JungRobert L. West

September 6, 2013Blacksburg, Virginia

Keywords: UHMWPE, thermoplastic composites, time-temperature superposition, highstrain rate, composite materials, shear lag, Monte Carlo, punch shear

Copyright 2013 by Pierce D. Umberger

Modeling the High Strain Rate Tensile Response and Shear Failure ofThermoplastic Composites

Pierce David Umberger

ABSTRACT

The high strain rate fiber direction tensile response of Ultra High Molecular Weight Polyethy-lene (UHMWPE) composites is of interest in applications where impact damage may occur.This response varies substantially with strain rate. However, physical testing of these com-posites is difficult at strain rates above 10−1/s. A Monte Carlo simulation of compositetensile strength is constructed to estimate the tensile behavior of these composites. Loadredistribution in the vicinity of fiber breaks varies according to fiber and matrix properties,which are in turn strain rate dependent. The distribution of fiber strengths is obtained fromsingle fiber tests at strain rates ranging from 10−4/s to 10−1/s and shifted using the time-Temperature Superposition Principle (tTSP) to strain rates of 10−4/s to 106/s. Other fiberproperties are obtained from the same tests, but are assumed to be deterministic. Matrixproperties are also assumed to be deterministic and are obtained from mechanical testingof neat matrix material samples. Simulation results are compared to experimental data forunidirectional lamina at strain rates up to 10−1/s.

Above 10−1/s, simulation results are compared to experimental data shifted using tTSP.Similarly, through-thickness shear response of UHMWPE composites is of interest to supportcomputational modeling of impact damage. In this study, punch shear testing of UHMWPEcomposites is conducted to determine shear properties. Two test fixtures, one allowing, andone preventing backplane curvature are used in conjunction with finite element modeling toinvestigate the stress state under punch shear loading and the resulting shear strength of thecomposite.

Acknowledgments

Special thanks to the following people for helping make this work possible:

• Dr. Scott Case for the opportunity to study these problems, and for his advice, pa-tience, and support along the way.

• Drs. Romesh Batra, Michael Hyer, Sunghwan Jung, and Robert West for willinglysharing their knowledge and advice.

• US Army Research Lab, especially Michael Maher, James Wolbert, and Bryan Lovefor the offering of technical expertise and financial support of this work, along withassistance in sample fabrication.

• Lisa Smith and Beverly Williams for taking care of the work behind the scenes forwhich I am greatly in debt.

• David Simmons, and Darryl Link for fabrication of the jigs used in much of the exper-imental testing.

• My fellow MRG students for the conversations, advice, and friendship. It has been aprivilege working with you.

• My parents, David and Christina Umberger, and brother Dillon Umberger for theirtremendous support during the ongoing years of my education. Mom and Dad, youalways told me to "Stay in school." I don’t think you meant this long.

• My wife, Betsy Umberger, for her constant love, support, and – most importantly –patience. There is nothing more I could possibly wish for.

iii

Research was sponsored by the Army Research Laboratory and was accomplished under Coop-erative Agreement Number W911NF-06-2-0014. The views and conclusions contained in thisdocument are those of the authors and should not be interpreted as representing official poli-cies, either expressed or implied, of the Army Research Laboratory or the U.S. Government.The U.S. Government is authorized to reproduce and distribute reprints for Governmentpurposes notwithstanding any copyright notation heron.

iv

Contents

List of Figures viii

List of Tables xiii

1 Introduction 1

2 Literature Review 3

2.1 Single Fiber Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Weibull Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Bundle Strength Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Break Clusters and Cluster Sizes . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 High Strain Rate Tensile Properties and tTSP . . . . . . . . . . . . . . . . . 10

2.6 Axial Progressive Damage Modeling . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Through Thickness Shear of UHMWPE Laminates . . . . . . . . . . . . . . 16

2.7.1 High SPR Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7.2 Low SPR Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Objectives of this Study 20

4 UHMWPE Composites and Constituent Properties 22

v

4.1 UHMWPE Fiber Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Fiber Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.2 Fiber Tensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.3 Fiber Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 UHMWPE Composite Matrix Properties . . . . . . . . . . . . . . . . . . . . 30

4.3 UHMWPE Composite Properties . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.1 Composite Sample Preparation . . . . . . . . . . . . . . . . . . . . . 34

4.3.2 Composite Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Axial Modeling of UHMWPE Composites 36

5.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.1 Strength and Stress-Strain Predictions . . . . . . . . . . . . . . . . . 42

5.2.2 Break Cluster Results . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 Comparison to Experimental Data . . . . . . . . . . . . . . . . . . . . . . . 54

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Through-Thickness Shear Testing of UHMWPE Composites 58

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.3 Punch Shear Test Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.3.1 Punch Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.3.1.1 Unsupported Punch . . . . . . . . . . . . . . . . . . . . . . 64

6.3.1.2 Supported Punch . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4 Punch Shear Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

vi

6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7 Through-Thickness Shear Modeling of UHMWPE Composites 74

7.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

8 Discussion 89

8.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Bibliography 93

Appendices 99

A Stress-Strain Curves from Axial Simulation 100

B Content Licenses 107

vii

List of Figures

2.1 Bonding fibers to a fixture allows a gripping surface for testing without causingfiber failure at the grip location [9]. [Reprinted with permission] . . . . . . . 4

2.2 Capstan grip diagram and photograph of a quasi-static tensile test [5]. [Reprintedwith permission] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Critical cluster in a composite with fiber Weibull modulus m=5. [21]. [Reprintedwith permission] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Tensile strength vs. strain rate for several temperatures in a PP tape. Tensilestrength increases with increasing strain rate and decreasing temperature [2].[Reprinted with permission] . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Master curve developed from the data in Figure 2.4 with a reference temper-ature of 20 C [2]. [Reprinted with permission] . . . . . . . . . . . . . . . . . 12

2.6 Weibull distribution of S3000 fiber strength at three thermorheologically "equiv-alent" strain rates [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Axial and shear stresses associated with 1 and 2 fiber breaks, respectively.Both stress components approach their far field value away from the fiberbreak [28]. [Reprinted with permission] . . . . . . . . . . . . . . . . . . . . . 14

2.8 Schematic representation of the shear lag model of Okabe et al. [28]. [Reprintedwith permission] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9 Node numbering convention for the shear lag model of Okabe et al. [28].[Reprinted with permission] . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

viii

2.10 Schematic of the fixture used in laboratory testing conducted by Xiao et al.[39]. [Reprinted with permission] . . . . . . . . . . . . . . . . . . . . . . . . 18

2.11 Finite element mesh of high-SPR punch testing by Xiao et al. [39]. [Reprintedwith permission] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 An untested tabbed/bonded fiber sample, before mounting. . . . . . . . . . . 23

4.2 A tabbed and bonded fiber sample with the edges cut away. The sample isgripped by the card stock tabs. . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 A mandrel mounted fiber sample. The sample is gripped by the cardboardmandrels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 A tabbed and bonded fiber sample mounted in the Q800 DMA. . . . . . . . 25

4.5 380x magnification ESEM image of unprocessed UHMWPE fibers. Fibershape and cross section is non-constant. . . . . . . . . . . . . . . . . . . . . . 26

4.6 500x magnification ESEM image of unprocessed UHMWPE fibers. Fibershape and cross section is non-constant. . . . . . . . . . . . . . . . . . . . . . 27

4.7 Spectra S3000 fiber shift factors from creep compliance testing. . . . . . . . . 28

4.8 Spectra S3000 fiber creep compliance master curve from creep compliancetesting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.9 Unshifted Spectra S3000 fiber strength values at 8 temperatures and 3 ordersof magnitude of axial strain rate. . . . . . . . . . . . . . . . . . . . . . . . . 29

4.10 Spectra S3000 fiber strength values shifted using tTSP. . . . . . . . . . . . . 29

4.11 Unshifted creep compliance curves for Kraton D1161NS matrix material attemperatures from -70 C to 50 C. . . . . . . . . . . . . . . . . . . . . . . . 31

4.12 Shifted creep compliance curves for Kraton D1161NS matrix material. . . . . 32

4.13 Creep compliance shift factors for Kraton D1161NS matrix material. . . . . . 32

4.14 Completed unidirectional composite samples for tensile testing. . . . . . . . . 34

4.15 (a) MTS Servohydraulic load frame with thermal control chamber. (b) Uni-directional composite sample with mounted extensometer. . . . . . . . . . . 35

ix

5.1 Histogram of randomly generated fiber element strengths from a Weibull dis-tribution with σ0 = 5.2x109 Pa and m = 6.02. . . . . . . . . . . . . . . . . . 38

5.2 Simulated stress-strain curve for SS3124 composite at 1x10−4/s. . . . . . . . 42

5.3 Simulated stress-strain curve for SS3124 composite at 1x100/s. . . . . . . . . 43



5.6 Fiber elastic modulus and matrix shear modulus versus strain rate. . . . . . 45

5.7 Percent difference between Monte Carlo and bundle strength models. . . . . 46

5.8 Stress-strain curve predicted by the bundle strength model compared to theMonte Carlo model result at a strain rate of 10−4/s. . . . . . . . . . . . . . . 47

5.9 Stress-strain curve predicted by the bundle strength model compared to theMonte Carlo model result at a strain rate of 106/s. . . . . . . . . . . . . . . 48

5.10 Monte Carlo model composite strength predictions at strain rates of 10−4/sto 106/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.11 Weibull distribution plot of composite strength at a strain rate of 1x10−4/s. . 49

5.12 Weibull distribution plot of composite strength at a strain rate of 1x106/s. . 50

5.13 Critical cluster for UHMWPE composite simulation at a strain rate of 1x106/s. 51

5.14 Critical cluster size versus strain rate. . . . . . . . . . . . . . . . . . . . . . . 52

5.15 Values of Ω corresponding to strain rates used in this study . . . . . . . . . . 53

5.16 Shifted strength tTSP test data by Cook [3], bundle strength model predictedstrength, and the progressive damage axial model predicted strength. . . . . 54

5.17 Comparison of stress-strain behavior between Monte Carlo simulation andphysical testing at a strain rate of 0.17/s. . . . . . . . . . . . . . . . . . . . . 55

5.18 Ratio of fiber elastic modulus to matrix shear modulus versus strain rate, alongwith percent difference in ultimate strength distribution between Monte Carloand bundle strength models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

x

6.1 Schematic layout of punch shear samples. . . . . . . . . . . . . . . . . . . . . 61

6.2 SS1214 laminate samples before and after testing. . . . . . . . . . . . . . . . 61

6.3 Punch shear test apparatus exploded assembly. . . . . . . . . . . . . . . . . 63

6.4 Side view of punch shear test apparatus exploded assembly. . . . . . . . . . . 63

6.5 Unsupported shear punch fabricated from A2 tool steel. . . . . . . . . . . . . 65

6.6 Supported shear punch fabricated from A2 tool steel. . . . . . . . . . . . . . 65

6.7 Assembled punch shear test apparatus in MTS hydraulic load frame. . . . . 67

6.8 Ultimate shear stress versus thickness. . . . . . . . . . . . . . . . . . . . . . 68

6.9 Load versus displacement for selected unsupported punch samples. . . . . . . 69

6.10 Load versus displacement for selected supported punch samples. . . . . . . . 69

7.1 Detail of 3D quarter model of punch shear test apparatus. . . . . . . . . . . 76

7.2 Boundary conditions of 3D quarter model of punch shear test apparatus. . . 77

7.3 3D quarter model of punch shear test apparatus with supported punch. . . . 77

7.4 Fiber direction tensile stress in layers 1-4 (top to bottom, respectively) of an8 layer composite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.5 Fiber direction tensile stress in layers 5-8 (top to bottom, respectively) of an8 layer composite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.6 Fiber direction tensile stress of an 8 layer composite, side view. . . . . . . . . 82

7.7 Through-thickness shear stress in an 8 layer composite. . . . . . . . . . . . . 82

7.8 Fiber direction stress components for 8 layer composite with unsupported punch. 83

7.9 Through thickness shear stress components for 8 layer composite with unsup-ported punch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.10 Fiber direction stress components for 16 layer composite with unsupportedpunch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xi

7.11 Through thickness shear stress components for 16 layer composite with un-supported punch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.12 Fiber direction stress components for 8 layer composite with supported punch. 85

7.13 Through thickness shear stress components for 8 layer composite with sup-ported punch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.14 Fiber direction stress components for 16 layer composite with supported punch. 86

7.15 Through thickness shear stress components for 16 layer composite with sup-ported punch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A.1 Simulated stress-strain curve for 8 composite samples at a strain rate of 10−4/s.100




A.5 Simulated stress-strain curve for 8 composite samples at a strain rate of 100/s. 102




A.9 Simulated stress-strain curve for 8 composite samples at a strain rate of3.16x103/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104


A.11 Simulated stress-strain curve for 8 composite samples at a strain rate of3.16x103/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105



xii

List of Tables

2.1 Critical cluster size versus Weibull modulus. [21]. [Reprinted with permission] 10

4.1 Predicted fiber strength and modulus values versus strain rate. . . . . . . . . 30

4.2 Predicted matrix shear modulus versus strain rate. . . . . . . . . . . . . . . 33

5.1 Comparison of Monte Carlo simulation average strength with bundle strengthestimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Weibull distribution parameters for predicted composite strength distributionsat strain rates of 10−4/s and 106/s. . . . . . . . . . . . . . . . . . . . . . . . 49

5.3 Summary of critical cluster size data . . . . . . . . . . . . . . . . . . . . . . 51

5.4 Values of Ω corresponding to strain rates used in this study . . . . . . . . . . 53

6.1 Average thickness and thickness coefficient of variation for SpectraShield 3124composite laminate samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2 Average ultimate shear strength and deviation for supported and unsupportedpunch types (MPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Failure load and apparent ultimate shear strength for 32 SpectraShield 3124test samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.4 Ultimate shear strength data for Aluminum 6061-T6 sheet samples tested inpunch-shear fixture with unsupported punch. . . . . . . . . . . . . . . . . . . 72

7.1 Key dimensions of punch shear assembly FEA model. . . . . . . . . . . . . . 76

xiii

7.2 Boundary conditions for punch shear assembly FEA model. . . . . . . . . . . 76

7.3 Composite material properties for punch shear assembly FEA model. . . . . 78

7.4 Summary of finite element modeling cases. . . . . . . . . . . . . . . . . . . . 79

7.5 Average ratio of fiber direction stress to through thickness shear stress inengaged layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xiv

Attribution

In Chapter 2, Figures 2.1, 2.2, 2.3, 2.1, 2.4, 2.5, 2.7, 2.8, 2.9, 2.10, and 2.11 are the work oftheir respective authors and are reprinted with permission of the Copyright holders. Copy-right releases can be found in Appendix B of this document.

The fabrication and testing described in Section 4.3 was conducted by F.P. Cook.

xv

Chapter 1

Introduction

Ultra-high molecular weight polyethylene (UHMWPE) composites are extensively used in

lightweight armor and other applications where impact damage may occur [1]. Their high

strength, stiffness, and light weight make them a natural choice for this role. UHMWPE

has been shown to exhibit increased stiffness and strength with increased strain rate [2]

[3]. During a ballistic impact, strain rates on the order of 105/s are typical [4]. These strain

rates are not readily achievable in a laboratory environment using conventional test methods.

Hydraulic load frames are typically limited to strain rates on the order of 10/s, while screw

driven frames are typically limited to strain rates on the order of 1/s. Wave propagation

techniques such as the Split Hopkinson bar have been employed to estimate high strain rate

composite properties, but strain rates above 104/s are unattainable, and there are many

sample gripping difficulties, primarily issues with securely gripping fibers to prevent slipping

without causing failures in the grip region [5].

Previous studies by this author have predicted UHMWPE constituent properties at high

strain rates using the time-temperature superposition principle (tTSP) [6]. A prediction of

high strain rate composite properties is desired as a function of predicted high strain rate

1

2

constituent properties. However, work by Cook suggested possible problems in applying

tTSP to the strength of composites [3]. A model is needed to estimate composite properties

at high strain rates. Chapter 5 describes a Monte Carlo model implementation to provide

these predicted composite properties at a range of strain rates from 10−4/s to 106/s.

In addition to tensile properties, it has been determined that through-thickness shear proper-

ties are of particular importance during ballistic and other impact events. A punch shear test

was devised and fabricated, consisting of a clamping die fixture with a cylindrical punch as-

sembly. Two versions of this punch, one allowing, and one constraining backplane curvature

are implemented. This experimental work is discussed further in Chapter 6.

In conjunction with the experimental evaluation in Chapter 6, finite element modeling was

conducted to provide insight into the stress state induced during punch testing. The model

design, implementation, and results are discussed in Chapter 7.

Chapter 8 summarizes the key findings of Chapters 4 - 7 and discusses recommended future

investigations to further expand on this body of work.

Chapter 2

Literature Review

2.1 Single Fiber Properties

Work by Peijs and others found that due to the high degree of anisotropy of UHMWPE fibers

and the method of composite fabrication, fiber properties largely dominate the behavior of

the composite as a whole [7]. Unfortunately both the size and viscoelastic nature of the

fibers complicates material characterization. Gripping individual fibers or even fiber bundles

is difficult without introducing stress concentrations and causing premature failure. In the

case of fiber bundles, care must be taken to ensure that load is distributed evenly among

all fibers [8]. Knotting or other similar methods of attachment result in considerable stress

concentrations, often threefold or larger, causing premature fiber failure [5].

Bonding has been used successfully in tensile test of various fibers [9, 10, 11]. Fiber bonding

must be carefully controlled, as any uneven bonding or any adhesive seepage between fibers

within the gage length will result in stress concentrations during testing. Testing of single

filaments rather than bundles is one solution to the inter-fiber issues associated with bonding

3

4

or other gripping methods. In 2005, Feih et al. performed single filament tests on sized

and unsized glass fibers using a cardboard sample fixture to which individual samples were

bonded. Figure 2.1 shows the fixture used. The fiber is oriented and bonded between the

holes. After curing, the center section is cut away, exposing the tabbed fiber for testing [9].

Figure 2.1: Bonding fibers to a fixture allows a gripping surface for testing without causingfiber failure at the grip location [9]. [Reprinted with permission]

Scharfeld et al. utilized a capstan method in which the fibers were wound on metal pegs

for quasi-static testing. The capstan grips were successful in testing high strength fibers to

failure without introducing stresses sufficient to cause premature fiber failure occurring at

the grips. Figure 2.2 shows a diagram and photograph of an Aramid fiber test using these

grips. However, the mass of the capstan grips can pose a problem for dynamic and high

strain rate testing [8].

Figure 2.2: Capstan grip diagram and photograph of a quasi-static tensile test [5]. [Reprintedwith permission]

5

2.2 Weibull Distributions

The presence of defects is commonly a controlling factor in the failure of composite fibers.

Typically, these defects are randomly located along the length of a fiber and the strength

of a fiber is dependent on the number and severity of defects present. Fiber strength is

often expressed as a distribution, expressing both the fiber strength as well as the amount of

variability in strength. The Weibull distribution is commonly used to express fiber strength

parameters [12, 13, 14].

A statistical distribution was developed by Weibull for the analysis of strength populations

of materials. The Weibull distribution is based upon the assumption that material failure

is defect driven, and that the number of defects present is proportional to the material

volume. As stress is applied to the material, failure will occur when the applied stress is

greater than the strength of the weakest defect area in the material. It then follows that

as the material volume increases, there will be a greater number of defects, and therefore

a higher probability of a defect with particularly low strength. As a result, as sample

volume is increased, the corresponding strength will be decreased. A two parameter Weibull

distribution for UHMWPE fiber strength was discussed in previous work by the author [6].

For strengths described by a two-parameter Weibull distribution, the probability of failure

of the material is given as a function of the material volume and applied stress

Pf = 1− exp[−V

(σ

σ0

)m](2.1)

where m is the Weibull modulus, V is the material volume, and σ and σ0 are the applied

axial stress and Weibull characteristic strength, respectively.

6

Pf = 1− exp[− L

L0

(σ

σ0

)m](2.2)

where L and L0 are the sample axial length and reference length, respectively. For a battery

of tests performed at a constant gage length, we let L = L0 and rewrite Equation 2.2 as

ln

(ln

[1

1− Pf

])= mln (σ)−mln (σ0) (2.3)

We estimate the probability of failure using Bernard’s approximation to median rank [15]

P(i)f =

i− 0.3

N + 0.4(2.4)

where N is the number of samples and i is the failure order rank. The values of m and σ0

can them be determined through linear regression.

2.3 Bundle Strength Models

Given a distribution of single fiber strengths, perhaps the simplest model of composite

strength is the consideration of a dry bundle of fibers. In this model, the contribution

of the matrix material is neglected entirely, meaning that there is no load transfer between

fibers. This assumption may be rationalized in cases where the fibers are much stiffer and

higher strength than the matrix material, meaning that fibers carry the majority of the load

in the composite.

Bundle strength models were first studied by Peirce, who developed a means of statistical

analysis of the strength of bundles of threads or fibers [16]. If the strength distribution of

individual fibers can be expressed as a two parameter Weibull distribution [17], then the

7

probability of failure of each individual fiber can be written as [18, 19]

Pf = 1− exp

(− L

L0

[σ

σ0

]m)(2.5)

The corresponding reliability would be

R = 1− Pf (2.6a)

R = exp

(− L

L0

[σ

σ0

]m)(2.6b)

The average axial stress in a fiber bundle at a given axial strain ε is

σ = EfεR(ε) (2.7)

where Ef is fiber modulus, and where R(ε) is defined as

R(ε) = exp

(− L

L0

[ε

ε0

]m)(2.8)

and

ε0 =σ0E

(2.9)

Solving Equation 2.7 for the maximum value of σ yields the expression

σmax = Efε0

(L0

mLe

)1m (2.10)

8

where e is the natural log base.

Finally, given a fiber volume fraction, Vf , the composite bundle strength estimate can be

written as

Xt = VfEfε0

(L0

mLe

)1m (2.11)

Given that this model neglects the contribution of the matrix to the overall composite be-

havior, this estimate is best suited to composites where fiber strength and stiffness are much

greater than matrix strength and stiffness. In the case of the UHMWPE composites in this

study, fiber strength and stiffness are many orders of magnitude greater than those of the

matrix.

2.4 Break Clusters and Cluster Sizes

In more complex models, the contribution of the matrix material is included in the overall

composite behavior. The matrix acts to transfer loads away from fiber breaks and results

in stress concentrations in fibers adjacent to a broken fiber. These stress concentrations can

lead to the development of several adjacent breaks (break clusters) in a composite.

Ibnabdeljalil and Curtin, as well as Zhou and Curtin investigated the behavior of break

clusters in the failure of unidirectional fiber reinforced composites [20, 21]. They studied an

idealized model of a composite, subjected to a tensile load, with different load sharing rules,

characterized by a non-dimensional load sharing parameter Ω, defined as [20]

Ω =

√µeffδ2

2Efa0r(2.12)

9

where µeff is the effective matrix shear modulus, δ is fiber element length, r is fiber radius,

a0 is fiber spacing, and Ef is fiber elastic modulus. As Ω approaches zero, load sharing

becomes more local, while large values of Ω indicate more global load sharing relationships.

They investigated the role of this load sharing parameter Ω on cluster size and load sharing

relationships. Typical values of Ω chosen are 0.001, 1, and 10 (spanning 4 orders of magni-

tude). Figure 2.3 shows a typical critical cluster resulting from a simulated unidirectional

composite with a fiber Weibull modulus of 5. It was also determined that composite failures

tended to occur at lower stress levels with more localized load sharing. Table 2.1 shows the

results obtained in [21] for critical cluster size versus Weibull modulus.

Figure 2.3: Critical cluster in a composite with fiber Weibull modulus m=5. [21]. [Reprintedwith permission]

10

Table 2.1: Critical cluster size versus Weibull modulus. [21]. [Reprinted with permission]Weibull Modulus, m Critical Cluster Size

2.0 1663.0 994.0 684.5 595.0 515.5 456.0 416.5 377.0 337.5 318.0 288.5 269.0 249.5 2310.0 21

2.5 High Strain Rate Tensile Properties and tTSP

Laboratory testing of composites at strain rates relevant to ballistic events is difficult. Con-

ventional tensile test methods such as servohydraulic and electromagnetic load frames cannot

come close to achieving the high strain rates needed. Some high strain rate methods exist,

such as Split Hopkinson Pressure Bar (SHPB) testing, but even these test methods can

not achieve all desired strain rates [5, 22]. Additionally, gripping of fiber and composite

samples presents the same stress concentration and grip slipping concerns as are present in

quasi-static testing [23].

These difficulties lead us to investigate other approaches for determining high strain rate

composite response. Viscoelastic modeling using the time-temperature superposition prin-

ciple (tTSP) has been used to predict behavior of viscoelastic materials at time scales or

strain rates that are not physically achievable in a laboratory [24, 2].

11

In 2007, Alcock et al. investigated the effects of temperature and strain rate on the me-

chanical properties of highly oriented polypropylene (PP) tapes and all-PP composites. The

authors analyzed strain rate and temperature effects on tensile modulus and strength and

developed master curves for each. These master curves were used to predict the behavior of

the tapes and composites at various strain rates, including those that are difficult to achieve

experimentally. Figure 2.4 shows strength vs. strain rate for several different temperatures.

Figure 2.5 shows the master curve developed from the data in Figure 2.4, covering a much

wider range of strain rate [2].

Figure 2.4: Tensile strength vs. strain rate for several temperatures in a PP tape. Tensilestrength increases with increasing strain rate and decreasing temperature [2]. [Reprintedwith permission]

Previous work with UHMWPE fibers attempted to predict fiber strength at high strain rates

using tTSP and subambient temperature testing [6]. While only directly applicable to fully

amorphous polymers, tTSP has been applied to failure mechanisms of composite systems

and has been successfully used to predict tensile modulus and strength at shifted time scales

[25]. Honeywell Spectra S3000 UHMWPE fibers were tested in a TA instruments Q800 DMA

at a range of temperatures and strain rates in order to predict strength properties at ballistic

strain rates. Statistical tests of these strength distributions did not indicate problems with

12

Figure 2.5: Master curve developed from the data in Figure 2.4 with a reference temperatureof 20 C [2]. [Reprinted with permission]

these predictions – with a P-value of 0.374, it was not shown that the Weibull modulus of

UHMWPE fibers tested at several thermorheologically equivalent temperature/strain rate

combinations was different; Weibull modulus is the slope of fit lines shown in Figure 2.6

[6]. This indicates that there is not a fundamental difference in the strength distribution of

UHMWPE fibers at thermorheologically equivalent temperature/strain rate combinations.

However, while no problems were found with predicting raw fiber strengths, high strain rate

predictions using tTSP did not appear to work for manufactured composite laminates due

to differing strength distributions at equivalent thermorheologically equivalent temperature

strain rate combinations, as discovered by Cook in 2010 [3].

The current alternative to a viable model for high strain rate material property prediction

is to conduct multiple batteries of ballistic tests on composite panels. Currently UHMWPE

prices far exceed those of many other conventional armor systems, making these tests very

expensive [4]. There is a need for UHMWPE composite strength predictions at a range of

elevated strain rates.

13

Figure 2.6: Weibull distribution of S3000 fiber strength at three thermorheologically "equiv-alent" strain rates [6].

2.6 Axial Progressive Damage Modeling

The mechanical behavior of fiber reinforced composite materials is heavily dependent on the

nature of the imperfect fibers that comprise the composite. The evolution of the damage in

the composite is critical to understanding and predicting ultimate composite strength and

failure.

As an axial tensile load is applied to a unidirectional composite material, individual fiber

failures are generated due to localized imperfections [26]. The axial stress in a fiber is

necessarily zero at the location of a break, and stress concentrations form at this location

14

in nearby fibers [27]. Load is gradually transferred back to the broken fiber via shear in the

matrix material, until after some ineffective length, the fiber stress approaches its far-field

value, as shown in Figure 2.7. It is commonly assumed that axial stresses are carried entirely

by the fiber strands, while shear stresses are carried entirely in the matrix between broken

and unbroken fibers [26, 27, 28]. However, in composite materials where the matrix stiffness

is appreciable compared to that of the fibers, the contribution of the axial stress in the matrix

material has been considered by Beyerlein and Landis [29].

Figure 2.7: Axial and shear stresses associated with 1 and 2 fiber breaks, respectively. Bothstress components approach their far field value away from the fiber break [28]. [Reprintedwith permission]

Landis et al. developed a finite element shear lag model which determined the stress profile

around a broken fiber segment. This stress profile was then used to repeatedly simulate the

damage progression in the form of a Monte Carlo method [30]. Several models have been

proposed to describe the nature of composite damage evolution, in both 2 and 3 dimensions

[31, 30, 32, 28, 26, 29, 33, 34, 35, 36, 21, 37, 20]. For some sets of composite constituent

properties, fiber/matrix debonding failure modes can also occur [33, 32, 28]. Most recent

work involving fiber debonding and/or slipping has been in two dimensions, which cannot

fully characterize the behavior and formation of fiber clusters that ultimately lead to com-

posite failure [34, 37, 21, 20, 35, 36], however, work on the topic by Okabe et al. considers

15

fiber slipping and debonding in three dimensions [28].

Okabe et al. make the typical simplifying shear lag assumptions that fibers carry only axial

load, while the matrix carries only shear load. Furthermore, they assume that in an NxM

fiber array, fibers are round, of constant cross section, and evenly spaced [28]. The model

schematic is shown in Figure 2.8.

Figure 2.8: Schematic representation of the shear lag model of Okabe et al. [28]. [Reprintedwith permission]

Considering the composite fibers as axial springs and the matrix material as shear springs,

the equilibrium equation for the arrangement shown in Figure 2.8 can be written as

Adσi,j,kdz

= −πr2

4∑l=1

τ li,j,k (2.13)

where A is the cross section area of the fiber, r is fiber radius, and τ is the interfacial shear

stress, which can be written as

τ 1i,j,k = Gmui+1,j,k − ui,j,k

d(2.14a)

τ 2i,j,k = Gmui,j+1,k − ui,j,k

d(2.14b)

16

τ 3i,j,k = Gmui−1,j,k − ui,j,k

d(2.14c)

τ 4i,j,k = Gmui,j−1,k − ui,j,k

d(2.14d)

where d is the fiber spacing, Gm is the matrix shear modulus, and the ui,j,k are the relative

axial displacements of the individual fiber elements. The numbering convention used by

Okabe et al. is shown in Figure 2.9

Figure 2.9: Node numbering convention for the shear lag model of Okabe et al. [28].[Reprinted with permission]

Equation 2.13 is solved numerically in combination with applied boundary conditions for

composite ends and fiber breaks, resulting in the displacement field for a simulated composite

with an arbitrary number of broken fibers.

2.7 Through Thickness Shear of UHMWPE Laminates

In addition to axial properties, through-thickness shear response of UHMWPE composites

is of interest for modeling impact events. Particularly at high speeds where inertial effects

are relevant, the through-thickness shear properties play a significant role in overall laminate

behavior during an impact event [4, 38]. The most common means of characterization of

17

the through-thickness behavior of composite laminates is a punch test. In punch testing,

the span-to-punch ratio (SPR), that is, the diameter of the composite fixture relative to the

diameter of the punch, is a controlling factor in shear versus bending deformation modes

[39, 40]. SPR values near 1 lead to shear dominated deformation and failure, while large

unsupported spans (SPR » 1) lead to bending dominated deformation and failure.

2.7.1 High SPR Investigations

Xiao et al. studied delamination damage in S-2 glass composites from punch-shear loading

[41]. A schematic of the test fixture is shown in Figure 2.10. Sun and Potti compared quasi-

static punch shear experimental results with ballistic penetration test results on AS4 GFRP

composites [40]. In both the work of Gama and Gillespie and Xiao et al., the SPR was much

greater than 1, meaning that bending mechanics were significant [38, 39]. In the work of

Sun and Potti, the SPR varied, but was between 3 and 12, indicating that bending behavior

was likewise present [40]. Additionally, Xiao et al. conducted finite element modeling of the

laboratory experiments. Their quarter-plate FE model is shown in Figure 2.11.

18

Figure 2.10: Schematic of the fixture used in laboratory testing conducted by Xiao et al.[39]. [Reprinted with permission]

Figure 2.11: Finite element mesh of high-SPR punch testing by Xiao et al. [39]. [Reprintedwith permission]

19

2.7.2 Low SPR Investigations

ASTM D732 specifies a SPR ≈ 1 test of through thickness shear strength of plastics [42].

It is noted in the standard that the nature of the stress concentration between the fixture

and sample as well as between the punch and sample can have an appreciable effect on the

outcome of the testing. Liu and Piggott used ASTM D732 on epoxy matrix materials and

thermoplastic polymers. They noted that this method might not be well suited to testing

of composites due to the close clearances between the punch and fixture. It was suggested

that fiber pullout and friction between the punch, fixture, and sample would hamper results

[41]. ASTM D732 was successfully employed by Crescenzi et al. on S-2 glass composites

[43]. No problems were encountered with friction or fiber pullout/entanglement. To date,

the author is not aware of published work on the characterization of the through-thickness

shear strength of thermoplastic composite laminates using a test similar to ASTM D732.

Chapter 3

Objectives of this Study

Ultra-high molecular weight polyethylene (UHMWPE) composites are used in many applica-

tions requiring light weight, high strength, and impact resistance, particularly armor appli-

cations [4, 1]. The Spectra/SpectraShield line, manufactured by Honeywell, is a UHMWPE

material system commonly used in ballistic armors. SpectraShield consists of a cross-ply lam-

inate of high molecular weight polyethylene fibers in a weak low molecular weight blended

polyethylene matrix, formed by hot pressing. As is the case in viscoelastic polymers, me-

chanical response of UHMWPE fibers and composites is a function of temperature and strain

rate [2, 24, 44, 5]. In particular, UHMWPE fibers and composites show an increase in stiff-

ness and strength with increased strain rate and/or decreased temperature [6, 3, 7, 45, 22].

Therefore, composite properties are of particular interest at strain rates equivalent to those

occurring during ballistic events, which are typically 105 s−1 to 106 s−1.

High strain rate constitutive properties of UHMWPE composite materials are not predictable

using low strain rate testing accelerated by tTSP [3], however prior work indicates that

constituent properties are predictable using low strain rate testing accelerated by tTSP

[6, 25]. Failure models incorporating progressive damage are well established and validated

20

21

against laboratory data at low strain rates [31, 30, 32, 28, 26, 29, 33, 34, 35, 36, 21, 20, 37].

To date, a progressive damage model has not been applied to predict the high strain rate

response of UHMWPE composites. This study uses such a model to predict composite

strength at a range of strain rates approaching 106 s−1. Furthermore, validation of these

predictions against existing experimental data needs to be performed. Finally, a study of

composite failure mode, and other important parameters such as critical cluster sizes and,

ineffective fiber lengths needs to be performed.

Through-thickness shear properties of UHMWPE laminates have not been determined. A

punch test with a SPR ≈ 1 demonstrates primarily shear behavior. ASTM D732 has been

used successfully on neat matrix samples as well as rigid thermoplastic polymers and epoxy

matrix composites [42]. Finite element simulation has been performed and validated against

laboratory testing at a variety of SPR values substantially greater than 1.

A punch-shear test similar to ASTM D732 is a viable means of characterizing the through-

thickness shear strength of UHMWPE laminates. However, such testing has not been pub-

lished to date. This study examines the characterization of the through-thickness behavior

of UHMWPE laminates. Due to the non-uniform stress state inherent in this test, finite

element analysis is conducted concurrent with the laboratory testing. Investigation of the

stress state and resulting implications for the physical testing is completed in order to gain

insight into the behavior of these composites under through-thickness shear loading.

Chapter 4

UHMWPE Composites and Constituent

Properties

4.1 UHMWPE Fiber Properties

Ultra-high molecular weight polyethylene fibers are manufactured by gel-spinning bulk high

molecular weight polyethylene material dissolved in a solvent at high temperature. The

solvent is then removed and the fiber cooled under tension resulting in a UHMWPE fiber

with a highly oriented molecular structure [46]. The contents of this section are intended to

provide a brief review of the fiber characterization process. For a more thorough presentation

of the fiber characterization work, see the author’s Masters’ Thesis [6].

4.1.1 Fiber Sample Preparation

For the purposes of this study, Spectra 3000 fibers, manufactured by Honeywell are used.

These fibers comprise commercially available SpectraShield 3124 product. Fiber tensile prop-

22

23

erties were measured using a TA Instruments Q800 DMA. Individual fibers were separated

from spooled fiber tows using a razor blade and forceps, taking care not to kink or otherwise

damage the fiber.

Fiber samples were either bonded to card stock tabs as shown in Figures 4.1 and 4.2, or

wrapped on adhesive coated cardboard mandrels as shown in Figure 4.3 and mounted in the

Q800 DMA.

Figure 4.1: An untested tabbed/bonded fiber sample, before mounting.

24

Figure 4.2: A tabbed and bonded fiber sample with the edges cut away. The sample isgripped by the card stock tabs.

Figure 4.3: A mandrel mounted fiber sample. The sample is gripped by the cardboardmandrels.

25

4.1.2 Fiber Tensile Testing

Creep compliance testing was conducted on the UHMWPE fibers in order to develop master

curves and shift factors for the material. Fibers were gripped in the DMA film/fiber clamp

and subjected to a creep compliance test. The DMA thermal control chamber, a liquid

nitrogen cooling chamber, is capable of temperatures below -140 C. The creep compliance

test consisted of a temperature sweep from -70 C to 40 C in 10 C increments. At each

temperature, the chamber was allowed to equilibrate for 10 minutes, then a tensile stress

of 375 MPa was applied for 10 minutes, while displacement was measured. The test setup,

including gripped fiber and thermal chamber, is shown in Figure 4.4. The resulting shift

factors were used for shifting tensile strength data in order to predict high strain rate tensile

strength [6].

Figure 4.4: A tabbed and bonded fiber sample mounted in the Q800 DMA.

Fiber tensile testing was conducted in the Q800 DMA with attached thermal control cham-

ber. For all tests, fibers were mounted in the DMA film/fiber grip and allowed to equilibrate

26

in the thermal chamber for 10 minutes before being tested to failure at the desired temper-

ature/strain rate combination. Strain rates ranged from 5x10−4/s to 5x10−1/s.

For purposes of computing fiber strength, average fiber cross section area was determined

via imaging of raw fibers. As can be seen in the ESEM image in Figures 4.5 and 4.6, the

fibers are non-round, and are of non-constant cross section shape, and non-constant cross

section area. A nominal average cross section area of 2.375x10−9 m2 was calculated from

end-on SEM images. This corresponds to a nominal fiber diameter of 27.5 µm.

Figure 4.5: 380x magnification ESEM image of unprocessed UHMWPE fibers. Fiber shapeand cross section is non-constant.

27

Figure 4.6: 500x magnification ESEM image of unprocessed UHMWPE fibers. Fiber shapeand cross section is non-constant.

4.1.3 Fiber Properties

The master curve corresponding to the shift factors given in Figure 4.7 is shown in Figure

4.8. Using the shift factors obtained from creep compliance testing, shown in Figure 4.7 the

fiber strength results shown in Figure 4.9 were shifted using tTSP. The resulting master plot

of shifted fiber strength values is shown in Figure 4.10. The corresponding fiber strength

and modulus values are summarized in Table 4.1

28

Figure 4.7: Spectra S3000 fiber shift factors from creep compliance testing.

Figure 4.8: Spectra S3000 fiber creep compliance master curve from creep compliance testing.

29

Figure 4.9: Unshifted Spectra S3000 fiber strength values at 8 temperatures and 3 orders ofmagnitude of axial strain rate.

Figure 4.10: Spectra S3000 fiber strength values shifted using tTSP.

30

Table 4.1: Predicted fiber strength and modulus values versus strain rate.Strain Rate (1/s) Fiber Modulus (Pa) Fiber Strength (Pa)

1.00E-04 7.44E+10 3.05E+091.00E-03 7.96E+10 3.26E+091.00E-02 8.48E+10 3.48E+091.00E-01 9.01E+10 3.69E+091.00E+00 9.53E+10 3.91E+091.00E+01 1.01E+11 4.12E+091.00E+02 1.06E+11 4.34E+091.00E+03 1.11E+11 4.55E+093.16E+03 1.17E+11 4.66E+091.00E+04 1.19E+11 4.77E+093.16E+04 1.22E+11 4.88E+091.00E+05 1.25E+11 4.98E+091.00E+06 1.3E+11 5.2E+09

4.2 UHMWPE Composite Matrix Properties

The matrix material for the composites used in this study is polyethylene blended with

styrene and other unknown polymers. The exact composition of the matrix material is a

trade secret and was not disclosed. As a result, matrix properties are based upon laboratory

testing of Kraton D1161NS matrix, which is believed to be similar to the matrix material

used in the laminates tested by Cook [3].

Bulk cast Kraton D1161NS material of 2 mm thickness was cut into 10 mm x 30 mm rect-

angular samples. These samples were mounted in the film/fiber clamp of a TA Instruments

Q800 DMA and allowed to equilibrate at -70 C for 10 minutes. Samples were then sub-

jected to a 0.25 MPa applied stress, while displacement was logged at 1 Hz for a period

of 10 minutes. This process was repeated at temperatures from -70 C to 50 C in 10 C

increments, with a 10 minute isothermal period prior to each test. Figure 4.11 shows the

unshifted creep compliance curves at these temperature ranges.

31

By shifting the curves in Figure 4.11 to form a continuous curve, a set of shift factors for

the material can be developed. Figure 4.12 shows the data from Figure 4.11 shifted. The

resulting shift factors from this translation are shown in Figure 4.13. The corresponding

matrix shear modulus values are summarized in Table 4.2

Figure 4.11: Unshifted creep compliance curves for Kraton D1161NS matrix material attemperatures from -70 C to 50 C.

These shifted matrix material properties will be used to provide model inputs to the com-

posite tensile model discussed in Chapter 5.

32

Figure 4.12: Shifted creep compliance curves for Kraton D1161NS matrix material.

Figure 4.13: Creep compliance shift factors for Kraton D1161NS matrix material.

33

Table 4.2: Predicted matrix shear modulus versus strain rate.Strain Rate (1/s) Matrix Shear Modulus (Pa)

1.00E-04 3.88E+041.00E-03 4.88E+041.00E-02 6.02E+041.00E-01 8.16E+041.00E+00 1.14E+051.00E+01 1.38E+051.00E+02 1.91E+051.00E+03 3.35E+053.16E+03 6.05E+051.00E+04 7.20E+063.16E+04 2.61E+071.00E+05 5.27E+071.00E+06 8.99E+07

4.3 UHMWPE Composite Properties

This section provides a brief overview of the construction and testing process for UHMWPE

composite laminates for axial tensile testing. Cook’s Masters’ Thesis provides a greater level

of detail of the physical construction and testing of the composites that will be simulated in

Chapter 5. The work described in this section was conducted by FP Cook [3].

The composites studied in this work consist of UHMWPE fibers in a blended polyethylene-

based matrix. Fibers are nominally aligned unidirectionally, however commercially-available

Honeywell SpectraShield consists of a cross-ply layup of four unidirectional layers. In order to

procure unidirectional composites for testing, unidirectional filament wound prepreg samples

were obtained directly from Honeywell.

34

4.3.1 Composite Sample Preparation

Eight 30 cm x 30 cm prepreg layers were arranged on silicone sheets with their fibers aligned

in a unidirectional manner. The stack was consolidated in a hot press at a temperature of

65 C and a pressure of 2.75 MPa for 10 minutes. The resulting panels were allowed to cool

under a pressure of 2.75 MPa [3].

After cooling, panels were cut into 20 mm wide strips parallel with the fiber direction,

creating 300 mm x 20 mm axial test samples. Figure 4.14 shows the completed samples.

Figure 4.14: Completed unidirectional composite samples for tensile testing.

4.3.2 Composite Testing

Axial testing of composite samples was carried out on an MTS servohydraulic load frame

with an attached thermal control chamber. Samples were gripped with a gage length of

approximately 75 mm, and a 25 mm gage length extensometer was attached for strain mea-

35

surement. Figure 4.15 shows the load frame with attached temperature chamber as well as

a mounted sample with attached extensometer ready for testing.

Figure 4.15: (a) MTS Servohydraulic load frame with thermal control chamber. (b) Unidi-rectional composite sample with mounted extensometer.

Composite samples were tested in displacement control mode at temperatures from -60 C

to 23 C and strain rates ranging from 10−4/s to 5/s. Results and composite strength

predictions are shown in Figure 5.16.

Chapter 5

Axial Modeling of UHMWPE

Composites

5.1 Model Formulation

A shear-lag type Monte Carlo model is implemented in MATLAB to predict the behavior of

axially loaded unidirectional composites. This behavior is governed by several parameters;

the most important in this context are the elastic modulus and strength distribution of the

fibers and the shear modulus of the matrix. All of these properties change with strain rate,

as discussed in Chapter 4, but the rate of change is profoundly different between the fiber

and matrix. Fiber modulus and strength changes are small compared to the several orders of

magnitude that the matrix modulus changes with strain rates ranging from 10−4/s to 106/s.

Due to these changes, the nature of the load transfer away from fiber breaks is expected to

vary with strain rate.

The notation used in this investigation is the same as that of Okabe et al. [28]. The

36

37

arrangement of elements in the model is shown in Figure 2.8. The i and j indices indicate

fiber number, and the k index is the element number along the length of a given (i,j) fiber.

Fiber properties are randomly assigned based on the Weibull distribution. Previous work on

this project has demonstrated the Weibull distribution to be a suitable fit for fibers used in

this material system [6]. Each fiber element is assigned an ultimate fiber tensile strength as

shown in Equation 5.1, where ηi,j,k is randomly generated.

σuti,j,k =

(ln

(1

1− ηi,j,k

)L0

∆z

)1mσ0 (5.1)

The fiber element strength distribution for a sample composite with σ0 = 5.2x109 Pa and m =

6.02 is shown in Figure 5.1. Generated values are shown in red, with the ideal distribution in

blue. Matrix properties and all other fiber properties are assumed to be deterministic. This

is both because the composite strength is largely governed by the fiber strength distribution

[4], and because an accurate distribution for other material properties is difficult to obtain.

38

Figure 5.1: Histogram of randomly generated fiber element strengths from a Weibull distri-bution with σ0 = 5.2x109 Pa and m = 6.02.

Following the aforementioned method of Okabe et al., we note that the equilibrium equation

for axial stress on a fiber element is

Adσi,j,kdz

= −πr2

4∑l=1

τ li,j,k (5.2)

where A is the fiber cross section area, σi,j,k is the axial stress in a fiber element, r is the

fiber radius, and τi,j,k is the shear stress at the fiber-matrix interface, which is a function of

the displacements of the surrounding fibers.

τ 1i,j,k = Gmui+1,j,k − ui,j,k

d(5.3a)

τ 2i,j,k = Gmui,j+1,k − ui,j,k

d(5.3b)

39

τ 3i,j,k = Gmui−1,j,k − ui,j,k

d(5.3c)

τ 4i,j,k = Gmui,j−1,k − ui,j,k

d(5.3d)

We can then write a finite difference method equation to solve for the displacements of the

fiber nodes, using the approach of Oh [33].

4EfA (γi,j,k (ui,j,k+1 − ui,j,k)− γi,j,k−1 (ui,j,k − ui,j,k−1)

(2 + γi,j,k−1 + γi,j,k) ∆z2

+h[Gm

ui+1,j,k − ui,j,kd

(1− P 1

i,j,k

) (1−D1

i,j,k

)+ζ1P

1i,j,k

(1−D1

i,j,k

)τy + ζ1D

1i,j,kτs

]+h[Gm

ui,j+1,k − ui,j,kd

(1− P 2

i,j,k

) (1−D2

i,j,k

)+ζ2P

2i,j,k

(1−D2

i,j,k

)τy + ζ2D

2i,j,kτs

](5.4)

+h[Gm

ui−1,j,k − ui,j,kd

(1− P 3

i,j,k

) (1−D3

i,j,k

)+ζ3P

3i,j,k

(1−D3

i,j,k

)τy + ζ3D

3i,j,kτs

]+h[Gm

ui,j−1,k − ui,j,kd

(1− P 4

i,j,k

) (1−D4

i,j,k

)+ζ4P

4i,j,k

(1−D4

i,j,k

)τy + ζ4D

4i,j,kτs

]= 0

where P and D are equal to 1 if matrix yielding or debonding has occurred, respectively, and

are zero otherwise. γ = 1 only if the indexed fiber element has broken and is zero otherwise.

The boundary conditions are

ui,j,1 = 0 (5.5a)

40

ui,j,K = εcL (5.5b)

that is to say that at ui,j,1, the bottom edge of the composite is fixed with zero displacement.

At ui,j,K , the top edge of the composite is subjected to a uniform displacement based upon

the globally prescribed composite strain, εc.

The above M x N x K system of equations in Equation 5.4 is flattened and written to a

box-banded MATLAB sparse matrix which has had memory preallocated. It was found that

MATLAB sparse matrix inversion took substantially less time than the original solution

method of successive over relaxation (SOR). The primary downside of the sparse matrix

solution method is that assigning values to indexed matrix locations is slow, even with

memory preallocation. For this reason, formulation of the stiffness matrix accounts for

approximately 25% of the solution time.

An incremental value of εc is applied to the composite, and ui,j,K is computed at the top

composite face. The boundary conditions in Equation 5.5 are enforced using the penalty

method. The load vector entries corresponding to the prescribed displacements are replaced

by βuprescribed where β is a "large" value. Similarly, in the stiffness matrix, Ki,i is replaced

by Ki,i + β. By this method, the essential boundary conditions are approximately satisfied

at the composite ends. Since (Ki,i + β) >> |Ki,j|, the corresponding equation is effectively

reduced to

u =uprescribed

1 + Kii

β

≈ uprescribed (5.6)

The matrix is inverted and the fiber element displacements ui,j,k are computed.

From the resulting fiber displacements, the fiber element stresses are computed in post-

processing based upon the differential displacement between adjacent fiber elements along

41

the length of a fiber, as shown in Equation 5.7.

σi,j,k = Efui,j,k+1 − ui,j,k

∆z(5.7)

Fiber breaks are calculated based on the individual fiber element strengths. For broken fiber

elements, γi,j,k is set to 1. The same process is carried out with matrix debonding, with Di,j,k

being set to 1 if the shear stress in an element is greater than the debonding strength value.

The global composite stress is defined in Equation 5.8 and is post processed based upon the

results computed in Equation 5.7, neglecting normal stress carried in the matrix.

σapp = Vf

(1

NM

N∑i=1

M∑i=1

σi,j,k

)(5.8)

The system of equations in Equation 5.4 is recomputed with these changes, and a stiffness

matrix for the next increment is reformulated. The above steps are then repeated for the

new composite configuration with fiber breaks. This continues until the composite has failed.

Failure is defined as when the global composite stress from Equation 5.8 has dropped below

some fraction of its peak value, indicating that as strain is increased, the load carried by the

composite is decreasing. A maximum composite strain condition may also be imposed.

This method as described produces stress-strain information for a single simulated test.

This is a Monte Carlo method in that fiber properties are randomly assigned based upon a

(Weibull) strength distribution. The entire simulation is repeated a minimum of 32 times,

resulting in an estimate of the distribution of composite properties. Given that the strengths

and defect locations in the fibers that comprise a composite are randomly distributed in

reality, this method is intended to capture this distribution.

42

5.2 Results

5.2.1 Strength and Stress-Strain Predictions

Simulations have been completed for ultra-high molecular weight polyethylene (UHMWPE)

composites at strain rates ranging from 10−4/s to 106/s, using fiber and matrix constitutive

properties previously measured and discussed in Chapter 4, and a fiber volume fraction of

70

Figure 5.2 shows 8 simulated stress-strain curves for a composite at 1x10−4/s, while Figures

5.3, 5.4, and 5.5 show stress-strain curves at 1x100/s, 1x103/s, and 1x106/s, respectively.

Stress-strain curves at the other simulated strain rates can be found in Appendix A.

Figure 5.2: Simulated stress-strain curve for SS3124 composite at 1x10−4/s.

At low strain rates these simulation results are within 3% of the bundle strength result. This

is expected, given the very low matrix stiffness relative to fiber stiffness. The Monte Carlo

43

Figure 5.3: Simulated stress-strain curve for SS3124 composite at 1x100/s.


44


simulation results have a relatively low amount of scatter, with a coefficient of variation

of <2.5%. High strain rate simulations exhibit a strength higher than the bundle strength

result by approximately 7%, as the matrix becomes relatively stiffer compared to the fibers

with changes in strain rate. This result is still relatively close to the bundle strength result.

Even at elevated strain rates, the contribution of the matrix to overall composite properties

is relatively small. Simulation results, bundle strenghth estimates, and the percent difference

between the two are given in Table 5.1

It is apparent from Table 5.1 that at higher strain rates, there is an increase in the disparity

between the Monte Carlo axial model and the bundle strength model. In order to gain insight

into the nature of this difference, it is helpful to consult the important constitutive properties

and examine their variation with strain rate. Figure 5.6 shows fiber elastic modulus and

matrix shear modulus versus strain rate.

It is noted that fiber modulus changes very little relative to the matrix modulus, which

45

Table 5.1: Comparison of Monte Carlo simulation average strength with bundle strengthestimate.

Strain Rate(1/s) M.C. Strength (Pa) Bundle Strength (Pa) Pct. Diff.

1.00E-04 9.89E+08 9.62E+08 2.71.00E-03 1.06E+09 1.03E+09 2.61.00E-02 1.13E+09 1.10E+09 2.71.00E-01 1.20E+09 1.17E+09 2.61.00E+00 1.27E+09 1.23E+09 2.71.00E+01 1.34E+09 1.30E+09 2.61.00E+02 1.41E+09 1.37E+09 2.71.00E+03 1.47E+09 1.43E+09 3.03.16E+03 1.51E+09 1.46E+09 3.61.00E+04 1.55E+09 1.48E+09 4.43.16E+04 1.60E+09 1.52E+09 5.11.00E+05 1.64E+09 1.55E+09 5.71.00E+06 1.73E+09 1.62E+09 6.4

Figure 5.6: Fiber elastic modulus and matrix shear modulus versus strain rate.

46

changes modulus by over 3 orders of magnitude. This transition occurs in the strain rate

range around 103/s - 105/s. Figure 5.7 shows the percent difference between the Monte Carlo

and bundle strength results. From this figure, appear that the contribution of the matrix

material to composite strength is relatively consistent below strain rates of 103/s. At strain

rates above 103/s, the increasing modulus of the matrix material relative to the fibers leads

to strengths greater than those predicted by the bundle strength model.

Figure 5.7: Percent difference between Monte Carlo and bundle strength models.

To illustrate the practical differences in the predictions of the two models, Figure 5.8 shows

the stress strain curve predicted by the bundle strength model and the Monte Carlo model

at a strain rate of 10−4/s. It is apparent that the predictions of the two models agree within

3 percent, both in the region before the onset of damage, and in the region where damage

is occurring. In contrast, figure 5.9 shows the stress strain curve predicted by the bundle

strength and Monte Carlo models at a strain rate of 106/s. At this strain rate, the matrix

modulus is much higher relative to the fiber modulus. The models still agree well at the early

pre-damage region, however, once fiber breaks begin to occur, the Monte Carlo model allows

47

stress to be transferred back into broken fibers, allowing them to contribute to the overall

strength of the composite. The bundle strength model does not take this stress transfer into

account and thus predicts a lower strength.

Figure 5.8: Stress-strain curve predicted by the bundle strength model compared to theMonte Carlo model result at a strain rate of 10−4/s.

Figure 5.10 shows the model predicted strength as a function of strain rate.

48

Figure 5.9: Stress-strain curve predicted by the bundle strength model compared to theMonte Carlo model result at a strain rate of 106/s.

Figure 5.10: Monte Carlo model composite strength predictions at strain rates of 10−4/s to106/s.

49

It also appears that the spread of the distribution of predicted composite strength does

not appreciably vary with strain rate. The Weibull distribution parameters for strength

distributions at 10−4/s and 106/s are shown in Table 5.2. Figure 5.11 shows the Weibull

distribution plot at a strain rate of 10−4/s, and Figure 5.12 shows the Weibull distribution

plot at a strain rate of 106/s. However, from Figures 5.11 and 5.12, it is clear that the lower

tails of the distribution do not agree well with the Weibull fit. This has been previously

observed in flaw-driven failure by Todinof [47] and Harlow et al. [48]. Other methods exist

for predicting low-percentile strength values that better fit the lower tails, however such

predictions are outside the scope of this work.

Table 5.2: Weibull distribution parameters for predicted composite strength distributions atstrain rates of 10−4/s and 106/s.

Strain Rate Weibull Modulus Characteristic Strength

1.0E-04 30.44 1.02E+091.0E+06 30.16 1.77E+09

Figure 5.11: Weibull distribution plot of composite strength at a strain rate of 1x10−4/s.

50

Figure 5.12: Weibull distribution plot of composite strength at a strain rate of 1x106/s.

5.2.2 Break Cluster Results

As studied by Zhou, Ibnandeljalil and Curtin, some composite materials develop clusters of

adjacent fiber breaks due to the stress concentrations on adjacent fibers [21, 20]. Figure 5.13

shows the critical cluster, consisting of 8 adjacent broken fibers, for a UHMWPE composite

at a strain rate of 1x106/s.

Critical cluster size was found to be highly variable from simulation to simulation. Table 5.3

summarizes the critical cluster size results. Figure 5.14 shows the average critical cluster size.

The standard deviation of critical cluster size is very large, and there is not a discernible trend

of cluster size versus strain rate. Likewise, there is no discernible trend in the variability of

cluster size with strain rate.

While Zhou, Ibnandeljalil, and Curtin investigated the nature of load sharing at different

values of Ω, they performed these investigations at values of 0.001, 1, and 10. However,

51

Figure 5.13: Critical cluster for UHMWPE composite simulation at a strain rate of 1x106/s.

Table 5.3: Summary of critical cluster size dataStrain Rate Critical Cluster Size (Avg) Std. Dev. COV %

1.00E-04 8.6 2.5 29.01.00E-03 8.6 2.5 29.01.00E-02 12.4 4.1 33.11.00E-01 9.1 2.9 31.81.00E+00 12.8 4.1 32.21.00E+01 5.5 1.8 32.71.00E+02 12.3 3.7 30.21.00E+03 8.5 2.5 29.43.16E+03 13.0 4.2 32.31.00E+04 8.0 3 37.53.16E+04 16.5 3.9 23.61.00E+05 4.3 1.5 35.31.00E+06 9.4 3.2 34.1

52

Figure 5.14: Critical cluster size versus strain rate.

the approximate values of Ω for UHMWPE composites corresponding to the range of strain

rates in this study do not come close to spanning this range. Recalling that Ω is defined as

Ω =

√µeffδ2

2Efa0r(5.9)

Table 5.4 and Figure 5.15 show the relationship between Ω and strain rate for the UHMWPE

material in this study.

It is clear from Figure 5.15 that the range of Ω resulting from the constituent properties used

in this study are far lower than the idealized values chosen by Curtin et al. in their studies,

so no comparison can readily be made between break cluster sizes.

53

Table 5.4: Values of Ω corresponding to strain rates used in this studyStrain Rate Omega

1.00E-04 0.00018051.00E-03 0.00019561.00E-02 0.00021041.00E-01 0.00023781.00E+00 0.00027271.00E+01 0.00029281.00E+02 0.00033541.00E+03 0.00043383.16E+03 0.00056941.00E+04 0.00194193.16E+04 0.00365551.00E+05 0.00513621.00E+06 0.0065687

Figure 5.15: Values of Ω corresponding to strain rates used in this study

54

5.3 Comparison to Experimental Data

Cook studied the strength response of unidirectional UHMWPE composite laminates at a

variety of strain rates [3]. Using a servohydraulic load frame, actual strain rates were limited

to approximately 5/s. However, using tTSP, composite properties were predicted for strain

rates approaching 107/s. Although there is much scatter in the experimental data, the

simulation results appear to be generally in agreement with the shifted composite strengths.

Figure 5.16: Shifted strength tTSP test data by Cook [3], bundle strength model predictedstrength, and the progressive damage axial model predicted strength.

In addition, the predicted stress-strain behavior agrees well with the composite test data

published by Cook. Figure 5.17 shows stress strain curves for both a UHMWPE composite

test and a Monte Carlo simulation result at a strain rate of 0.17/s.

55

Figure 5.17: Comparison of stress-strain behavior between Monte Carlo simulation andphysical testing at a strain rate of 0.17/s.

5.4 Discussion

The results of the Monte Carlo axial strength model described in this chapter agree well with

bundle strength estimates at low strain rates. As strain rate is increased past approximately

104/s, matrix shear modulus increases by a factor of roughly 1000. This results in better

stress redistribution and increased strength as compared to a bundle strength model. The

performance of the Monte Carlo model relative to bundle strength is relatively constant below

the matrix modulus transition phase occurring at strain rates around 104/s. During and after

this transition phase, the Monte Carlo result predicts increasingly higher strengths compared

to the bundle strength model. Figure 5.18 shows the ratio of fiber elastic modulus to matrix

shear modulus with changes in strain rate, overlaid with the difference between the ultimate

strength predictions between the Monte Carlo and the bundle strength models. It is clear

56

that the predicted strength difference increases by a factor of 2.5 after the matrix modulus

transition region. However, compared to many other composites the matrix material in the

UHMWPE composites is several orders of magnitude more compliant relative to the fiber

stiffness. Typical epoxy matrix materials have shear moduli in the range of 1-2GPa, and

even typical polypropylene thermoplastic resins have shear moduli in the region of 250-500

MPa. By contrast, at a low strain rate of 10−4/s, the matrix in the composites studied has a

shear modulus of only 38 kPa, and at a high strain rate of 106/s, a shear modulus of 89MPa,

several orders of magnitude below properties of other typically used matrix materials.

Figure 5.18: Ratio of fiber elastic modulus to matrix shear modulus versus strain rate, alongwith percent difference in ultimate strength distribution between Monte Carlo and bundlestrength models.

The stress-strain behavior predicted by the Monte Carlo axial model agrees reasonably well

with stress-strain data obtained experimentally by Cook [3], in that initial modulus and

strain at failure are nearly identical and ultimate strength is within 10 percent. In Figure

5.17, it is worth noting that during Cook’s experimental testing, the extensometer becomes

57

unfastened after the composite begins to fail, resulting in the apparent decrease in strain

after the stress maximum.

On the topic of break clusters and their size and influence on composite performance, there

is nothing to suggest that there is a measurable change in break cluster size as a function

of applied strain rate. As shown in Figure 5.14, break cluster size is highly variable and

does not show a correlation as strain rate increases. Given that the composite matrix is

so compliant relative to other commonly studied composite matrices, stress transfer lengths

are sufficiently long relative to composite length that overall composite strength remains to

be mainly dependent on fiber properties, regardless of the strain rate dependent changes in

matrix properties at the strain rate ranges investigated.

Chapter 6

Through-Thickness Shear Testing of

UHMWPE Composites

A partial form of this investigation was presented at the SAMPE 2010 Technical Conference.

6.1 Introduction

Much work has been done to characterize the properties of laminates in terms of their energy

absorption via a punch type test. Gama and Gillespie and Sun and Potti studied the relation

between quasi-static punch-shear behavior and ballistic penetration models for thick section

S-2 glass composites [38, 40, 49]. They found that different span to punch ratios changed the

mechanics of failure. SPR close to 1 leads to shear dominated failure, while large support

spans lead to bending dominated failure. In 2005, Xiao et al. studied delamination damage in

S-2 glass composites due to punch-shear loading [39]. In both of these cases, the supported

diameter is substantially larger than the punch diameter, leading to a significant mix of

bending and shear.

58

59

In order to support modeling efforts, it is desired to understand the through-thickness shear

behavior of UHMWPE composites in terms of material properties. While any mechanical

test is always a measurement of a mechanical system property rather than a true material

property, it is desired to isolate the through-thickness shear behavior inasmuch as is possible.

In 1995, Liu and Piggott studied the shear properties of polymers and fiber composites [41].

Liu and Piggott used ASTM standard D732-85 as a basis for evaluating shear properties

of thermoplastic polymers as well as several epoxy matrix materials. They concluded that

the punch shear test may have problems associated with friction between the punch and

die, as well as problems developing a stress state close to pure shear, rather than a tension-

dominated state. Lee and Sun and Potti and Sun performed punch testing on graphite fiber

reinforced composites in a similar manner and found that the primary damage modes were

delamination and through-thickness shear [50, 51]. Nemes et al. performed quasi-static as

well as high strain rate punch shear characterization of graphite/epoxy laminates using a

SHPB apparatus. They observed that the overall penetration behavior of the laminate was

relatively insensitive to construction parameters other than thickness, and that peak load

varied linearly with thickness [52]. Liu and Piggott also tested samples using an Iosipescu

type test geometry. They found that it provided unreliable results for polymers in the

rubbery state [41]. At room temperature and quasi-static strain rates, the Kraton matrix

material of the UHMWPE composites used in this study is rubbery. As a result, Iosipescu

test geometry, as well as similar methods such as rail shear testing are not practical for this

class of UHMWPE composite. For this reason, tight tolerance punch testing was chosen to

evaluate the shear properties of this material system.

60

6.2 Sample Preparation

Samples tested were commercially available Honeywell SpectraShield 3124 product. Either

1, 2, 4, or 8 layers of SpectraShield 3124, each having a lay-up of [0,90]S were combined to

form laminates of lay-up [0,90]2S, [0,90]4S, [0,90]8S, [0,90]16S. Sheets of the SpectraShield

product were cut into approximately 50 cm x 50 cm panels using a CNC cutting table.

Sheets were stacked in the aforementioned configuration. Multiple sheets were pressed in

one cycle, with a sheet of silicone coated paper between each. The stack of panels was placed

between two 0.635 cm thick smooth aluminum sheets and placed in a hot press. Panels were

pressed at a temperature of 118 C and pressure of 19 MPa for 15 minutes. The pressure

was removed and the panel was allowed to cool in the press for approximately 15 minutes

before being removed. These pressed plates were clamped between rigid fiberglass plates and

cut into 50 mm x 50 mm samples using a water jet cutting table. Samples were sufficiently

large that cut edges were far away from the punch area. Post-test inspection showed no

interaction between composite edges and the damage zone from punch testing. Figure 6.1

shows a schematic layout of the sample panel configuration. Figure 6.2 shows SpectraShield

samples before and after testing. Typical sample thicknesses are shown in Table 6.1. Typical

fiber volume fractions range from 70 to 75%.

Table 6.1: Average thickness and thickness coefficient of variation for SpectraShield 3124composite laminate samples.

N Layers Thickness (mm) Thickness COV (%)

4 0.322 1.98 0.551 2.616 1.100 1.332 2.145 1.1

61

Figure 6.1: Schematic layout of punch shear samples.

Figure 6.2: SS1214 laminate samples before and after testing.

62

6.3 Punch Shear Test Apparatus

Specimens were tested using an in-house fabricated punch shear assembly. This setup is

similar to ASTM D732-85, but with smaller dimensions to achieve failure within a desired

load range based on load frame specifications and predicted strength.

The in-house punch-shear assembly consists of a two piece rigid steel die with a 19.05 mm di-

ameter hole through which a 19.00 mm diameter punch is pressed. Thin samples are clamped

in between the two die halves. The punch to die fit is 0.025 mm nominal radius, effectively

reducing the unsupported span of the sample during testing. The laminate thickness is 13 to

86 times this clearance. The goal is to create a region of intense through-thickness shear in

the test specimen at the punch/die interface, while minimizing other mechanics phenomena

such as bending. Figures 6.3 and 6.4 show the assembly. Up to eight 12-20 inch bolts are

used to provide clamping pressure to the specimen. The entire assembly was manufactured

out of A2 tool steel and then air-hardened to approximately Rockwell C50 to minimize wear

on the edge of the hole at the punch-die-sample interface.

63

Figure 6.3: Punch shear test apparatus exploded assembly.

Figure 6.4: Side view of punch shear test apparatus exploded assembly.

64

6.3.1 Punch Configurations

Two punch configurations were considered; one allowing and one preventing curvature of the

sample during testing. It was hypothesized that disallowing curvature of the sample during

testing would create a stress state closer to pure shear, and thereby more effectively decouple

tensile and shear modes of failure.

6.3.1.1 Unsupported Punch

The unsupported punch was simply a right circular cylinder of diameter 19.00 mm and a

length of 50 mm. The punch was intended to create a region of intense shear in the sample

at the punch-die interface. The punch was manufactured from A2 tool steel and then air-

hardened to approximately Rockwell C50 to minimize wear of the punch edge. Additionally,

the face of the punch was surface ground to make the edge as square as possible to reduce

bending at the punch-die interface. Figure 6.5 shows the punch used.

6.3.1.2 Supported Punch

The supported punch was designed to prevent curvature of the sample during testing. The

main body of the punch was a right circular cylinder of diameter 19.00 mm and a length

of 50 mm. One end of the punch was drilled and tapped with a 14-20 inch hole and surface

ground to create a square edge. A second right circular cylinder was machined with a 19.00

mm diameter and a 10 mm length with a 6.35 mm diameter hole through the center to form

the clamp. Both pieces were fabricated from A2 tool steel and air-hardened to Rockwell

C50, as before. Figure 6.6 shows the punch assembly.

65

Figure 6.5: Unsupported shear punch fabricated from A2 tool steel.

Figure 6.6: Supported shear punch fabricated from A2 tool steel.

66

6.4 Punch Shear Test Procedure

Samples of size 50 mm x 50 mm were centered on the bottom portion of the die assembly.

A 5 mm diameter hole was punched for samples tested with the supported punch. The top

die section was installed. Proper alignment of the holes in the upper and lower section of

the die was achieved via pressed dowel pins. Four 12-20 inch bolts were inserted in the bolt

holes located at the corners of the die halves and tightened to a torque of 68 N-m. The die

assembly was designed with eight bolt holes due to anticipated problems with satisfactorily

gripping the samples, however four bolts proved to be sufficient to achieve the necessary grip

pressure on the sample to prevent slipping, based upon post-test inspection.

Punch shear testing was conducted in an MTS hydraulic load frame. An 88 kN capacity MTS

Force Transducer 661.20 load cell was used to measure load. Displacement measurements

were recorded from the load frame stroke transducer. An aluminum block was clamped in the

upper MTS 647 Hydraulic Wedge Grip and used to transfer load to the punch head. Control

and data collection was provided using an in-house National Instruments (NI) LabVIEW

program. The punch shear test assembly was rested on the lower grip frame, which was

verified to be parallel to the aluminum block in the upper grip to ensure load was applied

normal to the face of the composite and die assembly. Figure 6.7 shows the assembled test

apparatus in the load frame prior to a typical test.

Tests were conducted in stroke control at a rate of 0.127 mms. Load and displacement channels

were logged at 50 Hz using the in-house NI LabVIEW program. After each test, the punch

was removed, die halves unbolted, and the sample removed for inspection of the failure

surface.

67

Figure 6.7: Assembled punch shear test apparatus in MTS hydraulic load frame.

6.5 Results

For this test geometry, average through-thickness shear stress is ideally represented as

τ =F

Ashear(6.1)

where Ashear is defined as

Ashear = πDpt (6.2)

where Dp is the average of the punch and die diameters and t is the sample thickness.

The results for average ultimate shear stress versus sample thickness are shown in Figure 6.8

68

and Table 6.3. Average ultimate shear stress for each thickness and punch type are given in

Table 6.2. It was determined that 4 layer samples were too thin to test effectively. With these

thin samples, fiber pullout occurred resulting in parts of the composite being pulled between

the punch and die. For the 4 layer laminates, sample thickness was only approximately 12

times the punch-to-die clearance, resulting in problems obtaining a shear failure as opposed

to fiber pullout. As a result, 4 layer composite properties are not considered henceforth.

Figure 6.8: Ultimate shear stress versus thickness.

Typical load-displacement curves for unsupported punch samples are given in Figure 6.9.

We note that there is a distinct stiffness increase as load increases for both 8 and 16 layer

samples. Similarly, Figure 6.10 shows typical load-displacement curves for samples tested

using the supported punch.

69

Figure 6.9: Load versus displacement for selected unsupported punch samples.

Figure 6.10: Load versus displacement for selected supported punch samples.

70

Table 6.2: Average ultimate shear strength and deviation for supported and unsupportedpunch types (MPa).

Unsupported Punch Average Strength (MPa) Std Dev COV (%)

4 157.55 9.19 5.838 192.02 2.58 1.3416 190.91 4.33 2.2732 168.01 7.76 4.62

Supported Punch4 123.25 8.03 6.528 138.64 7.10 5.1216 130.52 7.12 5.4532 125.15 2.58 2.06

71

Table 6.3: Failure load and apparent ultimate shear strength for 32 SpectraShield 3124 testsamples.Sample Number N Layers Thickness (mm) Max Load (N) Shear Strength (MPa)

PS-1-2-31 4 0.368 3181 144.31PS-1-2-32 4 0.362 3481 160.70PS-1-2-29 4 0.343 3397 165.55PS-1-2-30 4 0.356 3397 159.64PS-2-A-2 8 0.572 6471 189.18PS-2-A-3 8 0.572 6525 190.78PS-2-A-4 8 0.572 6673 195.09PS-2-A-5 8 0.578 6675 193.03PS-3-A-1 16 1.022 11835 193.43PS-3-A-2 16 1.092 12135 185.65PS-3-A-3 16 1.105 12513 189.23PS-3-A-4 16 1.092 12767 195.32PS-4-A-1 32 2.083 21436 171.97PS-4-A-2 32 2.108 21138 167.54PS-4-A-3 32 2.127 22301 175.17PS-4-A-4 32 2.172 20452 157.36PS-1-2-33 4 0.318 2353 123.82PS-1-2-34 4 0.324 2598 134.04PS-1-2-35 4 0.330 2275 115.11PS-1-2-36 4 0.318 2281 120.02PS-2-A-6 8 0.533 4485 140.50PS-2-A-7 8 0.559 4889 146.20PS-2-A-8 8 0.546 4220 129.11PS-2-A-9 8 0.565 4693 138.75PS-3-A-5 16 1.118 8231 123.06PS-3-A-6 16 1.092 8349 127.73PS-3-A-7 16 1.105 8690 131.41PS-3-A-8 16 1.086 9091 139.89PS-4-A-5 32 2.172 16764 128.98PS-4-A-6 32 2.153 16002 124.21PS-4-A-7 32 2.140 15800 123.37PS-4-A-8 32 2.115 15696 124.03

72

Additionally, in order to provide data for validation purposes, seven 50mm x 50mm Alu-

minum 6061-T6 samples were also punch tested. The thickness of all seven samples was 1.55

mm. The aluminum test results are summarized in Table 6.4. Generally accepted values for

shear strength of 6061-T6 are 190-230 MPa, indicating that the punch shear test apparatus

produces reasonable results for aluminum samples.

Table 6.4: Ultimate shear strength data for Aluminum 6061-T6 sheet samples tested inpunch-shear fixture with unsupported punch.

Sample Number Failure Load (N) Shear Stress (MPa)

1 21897.3 236.152 21747.2 234.533 21900.2 236.184 22177.2 239.175 22185.9 239.266 22116.6 238.517 21732.8 234.37

Average 21965.3 236.9COV (%) 0.9 0.9

6.6 Discussion

There is a trend of increased apparent shear strength with decreasing thickness. If the state

of stress was truly pure shear through the thickness of the sample, apparent shear strength

would not be expected to vary with specimen thickness. Initially, when only unsupported

punch tests had been performed, it was hypothesized that this increase in apparent strength

was due to a membrane type behavior. Thinner 8-layer samples have less bending stiffness,

a span-to-thickness ratio of 33:1, and more ability to rotate when compared to 32-layer

samples with a span-to-thickness ratio of 9, given the constant clearance in the punch and

die geometry. It stands to reason that apparent shear strength would then be higher for

73

samples that are experiencing more of a membrane type state of stress, due to the extreme

anisotropy of UHMWPE fibers that comprise the SpectraShield laminates.

This notion is further supported by the distinct change in slope of the load-displacement

curves shown in Figures 6.9 and 6.10. It is possible that the increase in slope during the

course of the punch shear test is due to a nonlinear geometric effect that is governed by a

membrane type behavior. The same phenomenon that governs this increased mechanical

stiffness could be governing the increase in observed shear strength.

It was initially expected that the supported punch that prevented curvature of the sample

would reduce this hypothesized membrane behavior and would result in apparent shear

strengths that were more consistent versus sample thickness. Given the results in Figure

6.8 and Table 6.2, it is possible that this is the case. The unsupported punch configuration

results in a 12.5% decrease in apparent shear strength between the 8 and 32 layer composites,

while the apparent strength decrease in the case of the supported punch is 9.7%.

Chapter 7

Through-Thickness Shear Modeling of

UHMWPE Composites

7.1 Model Formulation

To better understand the results of the through thickness punch shear testing, a finite el-

ement model of the test apparatus was constructed, as shown in Figure 7.1. The model

currently treats the punch and fixture assembly as rigid, but has the ability to model them

as elastic structures. Composite elements are 3D C3D20 quadratic 20-node elements with

full integration.

The bottom edge of the test fixture is fixed in space, and a pressure load is applied to the

top half of the die to apply clamping pressure. Clamping pressure was estimated based upon

bolt clamping force and sample area

σc =PcAc

(7.1)

74

75

where σc is the composite clamping pressure, Pc is the total clamping load, and Ac is the

clamped sample area. The total bolt clamping load, Pc is

Pc =T

KdNb (7.2)

where T is bolt torque, d is bolt diameter, and K is an empirical correction factor. For the

12in-20 bolts used in testing, the accepted value of K is 0.2. For all tests, bolt torque was 68

N-m, resulting in an approximate clamping pressure of 5.5 MPa.

A distributed load is applied to the top of the punch to simulate an applied load. A schematic

of the boundary conditions is shown in Figure 7.2. Interaction between the punch, fixture,

and composite sample is modeled as contact and friction loads.

In the unsupported punch cases, the bottom of the sample test section was unconstrained.

However, in the cases of the supported punch discussed in Chapter 6, an additional punch

section constrains the bottom of the sample punch section, with an applied pressure to

simulate the punch clamping pressure, as shown in Figure 7.3

Model dimensions match those of the test fixture used in Chapter 6. Important dimensions

are given in Table 7.1, boundary conditions in Table 7.2, and material properties in Table 7.3.

The model is meshed with C3D20 20-node quadratic brick elements with full integration. A

structured mesh is created with 2 elements through the thickness of each composite layer,

for a total of 16 to 64 elements through the total thickness. Convergence of the model

was checked in both displacement at an applied load, as well as stress at the center of the

composite at r = 19 mm.

It is assumed that composite layers are perfectly bonded and that no slipping of the com-

posite occurs in the grip region of the fixture. Interlaminar properties are not known for

76

Figure 7.1: Detail of 3D quarter model of punch shear test apparatus.

Table 7.1: Key dimensions of punch shear assembly FEA model.Punch Diameter 19.00 mm

Die Diameter 19.05 mmPunch-Die Radial Clearance 0.025 mmComposite Layer Thickness 0.068 mm

Table 7.2: Boundary conditions for punch shear assembly FEA model.Location Boundary Condition

Lower Die Outer Face EncastreUpper Die Outer Face U_x = 0, U_y = 0Quarter Cut Edges SymmetryUpper Die Top Face 5.5 MPa Uniform PressureUpper Punch Face Increasing Uniform Pressure

Lower Punch Bottom Face 5.5 MPa Uniform PressureSample Outer Face U_x = 0, U_y = 0

77

Figure 7.2: Boundary conditions of 3D quarter model of punch shear test apparatus.

Figure 7.3: 3D quarter model of punch shear test apparatus with supported punch.

78

Table 7.3: Composite material properties for punch shear assembly FEA model.Property Value

Element Type C3D20E1 63 GPaE2 688 kPaE3 688 kPav12 0.43v13 0.43v23 0.35G12 220 kPaG13 1.8 GPaG23 1.8 GPa

this composite. During tensile testing by Cook, there was substantial evidence of inter- and

intralaminar shear separation and debonding. Cook attempted to investigate the apparent

interlaminar shear strength of UHMWPE composites, but material gripping and testing dif-

ficulties led to the conclusion that determining interlaminar shear strength was not practical

[3]. Furthermore, the assumption of perfect bonding is supported in this particular test

geometry by the lack of evidence of any slipping or interlaminar deformation in the post-

test inspection of UHMWPE composite samples tested in Chapter 6. The exception to this

observation was that 4 layer test samples (the thinnest samples tested) showed visible fiber

pullout and did not remain stationary in the fixture. The punch-die gap was large enough

to allow the thin samples to rotate and pass between the punch and die. For this reason, 4

layer composites are not investigated here.

7.2 Results

The FE model was first compared to the Aluminum 6061-T6 test data discussed in Chapter

6. The model predicted a shear stress of 227 MPa at the failure load from experimental

79

testing. This result falls within the generally accepted range for shear strength of Aluminum

6061.

Of particular interest in this investigation is the proportion of fiber direction tensile stresses

to through-thickness shear stresses in various situations. Figure 7.4 shows the fiber-direction

tensile stress in the first 4 layers of an 8 layer composite, while Figure 7.5 shows the stress in

layers 5-8. Similarly, Figure 7.7 shows the through-thickness shear stress in the composite.

Table 7.4 shows the model cases that were implemented.

Table 7.4: Summary of finite element modeling cases.Case Number # Layers Punch Type

1 8 Supported2 16 Supported3 32 Supported4 8 Unsupported5 16 Unsupported6 32 Unsupported

80

Figure 7.4: Fiber direction tensile stress in layers 1-4 (top to bottom, respectively) of an 8layer composite.

81

Figure 7.5: Fiber direction tensile stress in layers 5-8 (top to bottom, respectively) of an 8layer composite.

82

Figure 7.6: Fiber direction tensile stress of an 8 layer composite, side view.

Figure 7.7: Through-thickness shear stress in an 8 layer composite.

Figure 7.8 shows the elemental average fiber direction stress through the thickness of an

8-layer composite with the unsupported punch at r=19.025 mm. Similarly, Figure 7.9 shows

the elemental average through thickness shear stress in each layer at the same location.

Figures 7.10 and 7.11, respectively, show the same two quantities for a 16-layer composite

with an unsupported punch.

83

Figure 7.8: Fiber direction stress components for 8 layer composite with unsupported punch.

Figure 7.9: Through thickness shear stress components for 8 layer composite with unsup-ported punch.

84

Figure 7.10: Fiber direction stress components for 16 layer composite with unsupportedpunch.

Figure 7.11: Through thickness shear stress components for 16 layer composite with unsup-ported punch.

85

In the same manner as Figures 7.8 - 7.11 above, Figures 7.12, 7.13, 7.14, and 7.15 show

fiber direction and through thickness shear in 8 and 16 layer composites, respectively, with

supported punches.

Figure 7.12: Fiber direction stress components for 8 layer composite with supported punch.

86

Figure 7.13: Through thickness shear stress components for 8 layer composite with supportedpunch.

Figure 7.14: Fiber direction stress components for 16 layer composite with supported punch.

87

Figure 7.15: Through thickness shear stress components for 16 layer composite with sup-ported punch.

7.3 Discussion

From Figures 7.8 - 7.15 above, it can be seen that the ratio of stress in the fiber direction

to through thickness shear stress varies with both composite thickness and punch type. The

ratios of the average stresses through the thickness of the composite are summarized in Table

7.5.

Table 7.5: Average ratio of fiber direction stress to through thickness shear stress in engagedlayers

# Layers Punch Type S11:S13 Ratio

8 Unsupported 2.4016 Unsupported 1.8532 Unsupported 1.648 Supported 1.8616 Supported 1.8032 Supported 1.75

88

From Table 7.5 it is clear that for thinner composite sections, a larger fraction of fiber direc-

tion stress is present compared to shear stress. This effect is most pronounced in the unsup-

ported punch configuration, but is present to a much lesser degree in the supported/clamped

punch. Unfortunately, single fiber shear strength is unknown, but given its process of man-

ufacture, [46], it is likely that its axial strength is much higher than its shear strength. In

that case, given a larger proportion of the composite stress in the fiber direction, a higher

apparent strength would be predicted. This would account for the trend of increased appar-

ent shear strength that was measured in thinner composites. It would likewise account for

the apparent shear strength difference as observed between the unsupported and the sup-

ported/clamping punch. By constraining the out of plane rotation of the composite layer,

the thickness dependence of apparent shear strength is greatly diminished. With this infor-

mation in mind, based upon the data in Figure 6.8 and Table 6.2, the estimated composite

through-thickness shear strength in this configuration is approximately 125 MPa.

Chapter 8

Discussion

The objective of this work was to develop predictions of the properties and response of

UHMWPE composites. The work in Chapter 5 produced estimates of tensile properties

at a range of strain rates from 10−4/s to 106/s. These estimated properties were based

entirely upon measured and shifted constituent properties and did not involve the use of fit

parameters. It is not currently possible to conclusively evaluate the quality of these estimates,

as there has not been a successful effort to measure composite tensile properties at high strain

rates. However, the results agree reasonably well with composite data generated and shifted

using tTSP by Cook [3].

In addition to a a prediction of composite tensile properties as a function of strain rate, an

investigation was performed into the relationship of this data to the simple bundle strength

model. The results of the two models are similar at low strain rates where the matrix

material is extremely compliant relative to the fiber modulus. At higher strain rates above

the matrix modulus transition, the model of Chapter 5 predicts strength values higher than

those predicted by the bundle strength model. This is due to the more significant contribution

of the higher modulus matrix to the overall composite behavior, a behavior which is neglected

89

90

by the bundle strength model. However, even at higher strain rates, the matrix material is

much less stiff than other common matrices, meaning that the predicted composite strength

does not increase by a large amount above the bundle strength result. It is likely that the

matrix contribution to composite behavior would continue to increase with progressively

higher strain rate, however neither composite nor constituent tTSP estimates exist at these

strain rates due to thermal equipment limitations.

Other investigations have probed the effect of break cluster size on composite failure as a

function of the load transfer rules in the composite. In the case of the UHMWPE composites

in this study, the load transfer rules are primarily governed by the effective matrix stiffness,

which is a function of strain rate. However, at the range of strain rates of interest, there is

not an appreciable change in the load sharing parameter Ω, and concordantly, there was not

a detectable change in the critical break cluster size observed in axial composite simulations

as a function of strain rate.

The next portion of this study involved an investigation into the through-thickness shear

strength of UHMWPE composites. Owing to the extremely low matrix modulus, many

typical shear tests such as Iosipescu and rail shear are not viable options for this material

system. However, a punch shear test, similar in nature to ASTM D732 [42] was designed,

fabricated, and employed. The results of this shear test showed an apparent dependence of

shear strength on sample thickness and punch configuration. As a result, the final study was

undertaken to provide insight into the reason for this result.

The final portion of this work involved the creation of a finite element model of the punch

shear test, in order to study the nature of the stress state with respect to punch configuration

and composite thickness. As a result of this study, it was concluded that the reason for

lower apparent strength in thicker composites and with the supported punch was that these

composites and punch configurations had a lower proportion of in-plane stresses, that is to say

91

that they are closer to a state of pure shear. Due to the nature of the fiber manufacture, the

fiber direction strength is much higher than its shear strength, resulting in higher apparent

composite strengths when fibers are loaded in tension. The result of these two investigations

into the shear strength of UHMWPE composites was an estimated shear strength of 125

MPa, which was a previously unknown quantity.

8.1 Future Work

Much of the future work required to develop a better understanding of the behavior and

response of UHMWPE composites lies within the realm of material characterization. The

work in Chapters 6 and 7 involves a combined representation of fiber and matrix properties,

meaning that the results are only valid for this particular composite. Particularly given

the fact that this material system is commonly used in situations where impact loading

occurs normal to the composite surface, a better understanding of through-thickness material

properties - particularly at the constituent level - is desired. This is no trivial task, given

the difficulties in material testing outlined in this work, as well as previous work by Cook

and Umberger [3, 6]. Ideally, the above characterizations of transverse properties will be

expanded to include strain rate dependence of these properties. It is possible that this

may be undertaken with the use of tTSP, although problems with the use of tTSP at the

composite level have been noted [3].

Future efforts to measure the high strain rate response of UHMWPE composites and other

similar thermoplastic composite systems would also be useful. Methods such as Split Hop-

kinson bar testing may be used to measure composite and constituent tensile response at

some of the higher strain rate ranges simulated in the study in Chapter 5. The success of

these efforts will be largely dependent on the ability to successfully grip composite or con-

92

stituent samples, as grip slipping and grip failures have been a nearly constant problem for

all tensile testing undertaken on this material system, and a solution mitigating this problem

would be welcomed. Such a development could be used to validate the predictions made in

Chapter 5.

Overall, this work provides some insight into the response of UHMWPE composites in tension

and through-thickness shear. There is much about this class of composites that is poorly

understood, and future work - particularly focusing on material characterization and strain

rate response - is needed.

In closing, the work contained herein provides insight and previously unknown estimates of

the properties and behavior of UHMWPE composites. Tensile properties at a wide range

of strain rates are predicted. Through-thickness shear properties are measured and inves-

tigated. Together, these tensile and shear property estimates provide a starting point for

more effective modeling of UHMWPE composites at high strain rates, whether for ballistic

armor applications or other purposes. Extreme difficulty has been present in all experimen-

tal testing efforts for this material. In the future, improvement of gripping techniques and

high strain rate testing may enable further material characterization may enable improved

composite modeling capabilities for a variety of applications and conditions.

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Appendices

99

Appendix A

Stress-Strain Curves from Axial

Simulation

Figure A.1: Simulated stress-strain curve for 8 composite samples at a strain rate of 10−4/s.

100

101



102


Figure A.5: Simulated stress-strain curve for 8 composite samples at a strain rate of 100/s.

103



104


Figure A.9: Simulated stress-strain curve for 8 composite samples at a strain rate of3.16x103/s.

105


Figure A.11: Simulated stress-strain curve for 8 composite samples at a strain rate of3.16x103/s.

106



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