Modeling of High Strain Rate Axial Response and Shear Failureof Thermoplastic Composites

Pierce D. 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

Dr. Scott Case, ChairDr. Romesh BatraDr. Michael Hyer

Dr. Sunghwan JungDr. Robert West

August 1, 2013Blacksburg, Virginia

Keywords: UHMWPE, time-temperature superposition, composite materials, shear lag,Monte Carlo, FEM, ABAQUS, punch shear

Copyright 2013 by Pierce D. Umberger

Contents

List of Figures iv

1 Background 1

1.1 Motivation & Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 High Strain Rate Tensile Properties and tTSP . . . . . . . . . . . . . . . . . 1

1.3 Axial Progressive Damage Modeling . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Through Thickness Shear of UHMWPE Laminates . . . . . . . . . . . . . . 6

1.4.1 High SPR Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.2 Low SPR Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Preliminary Results: Punch Shear 10

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Laboratory Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1.1 Punch Shear Test Apparatus . . . . . . . . . . . . . . . . . 12

2.2.1.2 Punch Configurations . . . . . . . . . . . . . . . . . . . . . 13

2.2.1.3 Punch Shear Test Procedure . . . . . . . . . . . . . . . . . . 14

2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

ii

3 Preliminary Results: Axial Model 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Proposed Work 28

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Punch Shear Modeling and Analysis . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Axial Modeling and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Project Time Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.5 Expected Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.6 Anticipated Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Bibliography 32

iii

List of Figures

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

1.2 Master curve developed from the data in Figure 1.1 with a reference temper-ature of 20◦C [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Weibull distribution of S3000 fiber strength at three thermorheologically "equiv-alent" strain rates [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Axial and shear stresses associated with 1 and 2 fiber breaks, respectively.Both stress components approach their far field value away from the fiberbreak. [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Schematic representation of the shear lag model of Okabe et al. [16] . . . . . 5

1.6 Node numbering convention for the shear lag model of Okabe et al. [16] . . . 6

1.7 Schematic of the fixture used in laboratory testing conducted by Xiao et al.[27] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.8 Finite element model of high-SPR punch testing by Xiao et al. [27] . . . . . 8

2.1 SS1214 laminate samples before and after testing. . . . . . . . . . . . . . . . 11

2.2 Punch shear test apparatus exploded assembly. . . . . . . . . . . . . . . . . 12

2.3 Side view of punch shear test apparatus exploded assembly. . . . . . . . . . . 13

2.4 Unsupported shear punch fabricated from A2 tool steel. . . . . . . . . . . . . 14

2.5 Supported shear punch fabricated from A2 tool steel. . . . . . . . . . . . . . 15

2.6 Assembled punch shear test apparatus in MTS hydraulic load frame. . . . . 16

2.7 Ultimate shear stress versus thickness. . . . . . . . . . . . . . . . . . . . . . 17

2.8 Load versus displacement for selected unsupported punch samples. . . . . . . 18

2.9 Load versus displacement for selected supported punch samples. . . . . . . . 18

2.10 Detail of 3D solid quarter model of punch shear test apparatus. . . . . . . . 20

iv

2.11 Boundary conditions of 3D solid quarter model of punch shear test apparatus. 21

3.1 Simulated stress-strain curve for SS3124 composite at 0.83s−1. . . . . . . . . 25

3.2 Simulated stress-strain curve for SS3124 composite at 8.3s−1. . . . . . . . . . 25

3.3 Simulated stress-strain curve for SS3124 composite at 6.4x105 s−1. . . . . . . 26

4.1 Expected project timetable. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

v

1 | Background

1.1 Motivation & Aims

Ultra-high molecular weight polyethylene (UHMWPE) composites are used in many appli-cations requiring light weight, high strength, and impact resistance, particularly armor ap-plications [1]. The Spectra/SpectraShield line, manufactured by Honeywell, is a UHMWPEmaterial 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 blendedpolyethylene matrix, formed by hot pressing. As is the case in viscoelastic polymers, mechan-ical response of UHMWPE fibers and composites is a function of temperature and strain rate[2, 3, 4, 5]. In particular, UHMWPE fibers and composites show an increase in stiffness andstrength with increased strain rate and/or decreased temperature [6, 7, 8, 9, 10]. Therefore,composite properties are of particular interest at strain rates equivalent to those occurringduring ballistic events, which are typically 105 s−1 to 106 s−1.

1.2 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 cannotcome close to reaching 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 notachieve all desired strain rates [5, 10]. Additionally, high strain rate tensile measurementspresent additional sample gripping and sensing concerns [11]. Viscoelastic modeling usingthe time-temperature superposition principle (tTSP) has been used to predict behavior ofviscoelastic materials at time scales or strain rates that are not physically achievable in alaboratory [3, 2].

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. Theauthors analyzed strain rate and temperature effects on tensile modulus and strength anddeveloped master curves for each. These master curves were used to predict the constitutivebehavior of the tapes and composites at various strain rates, including those that are diffi-

1

2

cult to achieve with physical testing. Figure 1.1 shows strength vs. strain rate for severaldifferent temperatures. Figure 1.2 shows the master curve developed from the data in Figure1.1, covering a much wider range of strain rate [2].

Figure 1.1: Tensile strength vs. strain rate for several temperatures in a PP tape. Tensilestrength increases with increasing strain rate and decreasing temperature [2].

Figure 1.2: Master curve developed from the data in Figure 1.1 with a reference temperatureof 20◦C [2].

Previous work with UHMWPE fibers attempted to predict fiber strength at high strain ratesusing tTSP and subambient temperature testing. While only directly applicable to fullyamorphous polymers, tTSP has been applied to failure mechanisms of composite systemsand has been successfully used to predict tensile modulus and strength at shifted time scales

3

[12]. Honeywell Spectra S3000 UHMWPE fibers were tested in a TA instruments Q800 DMAat a range of temperatures and strain rates in order to predict strength properties at ballisticstrain rates. Statistical tests of these strength distributions did not indicate problems withthese predictions – with a P-value of 0.374, it was not shown that the Weibull modulus ofUHMWPE fibers tested at several thermorheologically equivalent temperature/strain ratecombinations was different; Weibull modulus is the slope of fit lines shown in Figure 1.3[6]. This indicates that there is not a fundamental difference in the strength distribution ofUHMWPE fibers at thermorheologically equivalent temperature/strain rate combinations.However, while no problems were found with predicting raw fiber strengths, high strain ratepredictions using tTSP did not appear to work for manufactured composite laminates, asdiscovered by Cook in 2010 [7].

Figure 1.3: Weibull distribution of S3000 fiber strength at three thermorheologically "equiv-alent" strain rates [6].

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

4

is to conduct multiple batteries of ballistic tests on composite panels. Currently UHMWPEprices far exceed those of many other conventional armor systems, making these tests veryexpensive [1]. There is a need for UHMWPE composite strength predictions at a range ofelevated strain rates.

1.3 Axial Progressive Damage Modeling

The mechanical behavior of fiber reinforced composite materials is heavily dependent on thenature of the imperfect fibers that comprise the composite. The evolution of the damage inthe composite is critical to understanding and predicting ultimate composite strength andfailure.

As a unidirectional composite material has an axial tensile load applied, individual fiberfailures are generated due to localized imperfections [13]. Axial stress in a fiber is necessarilyzero at the location of a break, and stress concentrations form at this location in nearbyfibers [14]. Load is gradually transferred back to the broken fiber via the matrix material,until after some ineffective length, the fiber stress approaches its far-field value, as shownin Figure 1.4. It is generally assumed that axial stresses are carried entirely by the fiberstrands, while shear stresses are carried entirely in the matrix between broken and unbrokenfibers. However, in composite materials where the matrix stiffness is appreciable comparedto that of the fibers, the contribution of the axial stress in the matrix material has beenconsidered by Beyerlein and Landis [15].

Figure 1.4: Axial and shear stresses associated with 1 and 2 fiber breaks, respectively. Bothstress components approach their far field value away from the fiber break. [16]

5

Landis et al. developed a finite element shear lag model which determined the stress profilearound a broken fiber segment. This stress profile was then used to repeatedly simulate thedamage progression in the form of a Monte Carlo method [17]. Several models have beenproposed to describe the nature of composite damage evolution, in both 2 and 3 dimen-sions [18, 17, 19, 16, 13, 15, 20, 21, 22, 23, 24, 25]. For some sets of composite constituentproperties, fiber/matrix debonding failure modes can also occur [20, 19, 16]. Most recentwork involving fiber debonding and/or slipping has been in 2d, which cannot fully charac-terize the behavior and formation of fiber clusters that ultimately lead to composite failure[21, 25, 24, 22, 23], however, the most recent work on the topic by Okabe et al. considersfiber slipping and debonding in three dimensions [16].

Okabe et al. make the typical simplifying shear lag assumptions that fibers carry onlyaxial load, while the matrix carries only shear. Furthermore, they assume that in an NxMfiber array, fibers are round, of constant cross section, and evenly spaced [16]. The modelschematic is shown in Figure 1.5.

Figure 1.5: Schematic representation of the shear lag model of Okabe et al. [16]

Considering the composite fibers as axial springs and the matrix material as shear springs,the equilibrium equation for the arrangement shown in Figure 1.5 can be written as

Adσi,j,kdz

= −πr2

4∑l=1

τ li,j,k (1.1)

6

where A is the cross section area of the fiber, r is fiber radius, and τ is the interfacial shearstress, which can be written as

τ 1i,j,k = Gmui+1,j,k − ui,j,k

d(1.2a)

τ 2i,j,k = Gmui,j+1,k − ui,j,k

d(1.2b)

τ 3i,j,k = Gmui−1,j,k − ui,j,k

d(1.2c)

τ 4i,j,k = Gmui,j−1,k − ui,j,k

d(1.2d)

where d is the fiber spacing and the ui,j,k are the relative axial displacements of the individualfiber elements. The numbering convention used by Okabe et al. is shown in Figure 1.6

Figure 1.6: Node numbering convention for the shear lag model of Okabe et al. [16]

Equation 1.1 is solved numerically in combination with applied boundary conditions forcomposite ends and fiber breaks, resulting in the displacement field for a simulated compositewith an arbitrary number of broken fibers.

1.4 Through Thickness Shear of UHMWPE Laminates

In addition to axial properties, through-thickness shear response of UHMWPE compositesis of interest for modeling impact events. Particularly at high speeds where inertial effectsare relevant, the through-thickness shear properties play a significant role in overall laminatebehavior during an impact event [1, 26]. The most common means of characterization of

7

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 thediameter of the punch implement, is a controlling factor in shear versus bending deformationmodes [27, 28]. SPR values near 1 lead to shear dominated deformation and failure, whilelarge unsupported spans (SPR » 1) lead to bending dominated deformation and failure.

1.4.1 High SPR Investigations

Xiao et al. studied delamination damage in S-2 glass composites from punch-shear loading[29]. A schematic of the test fixture is shown in Figure 1.7. Sun and Potti compared quasi-static punch shear experiments with ballistic penetration tests on AS4 GFRP composites[28]. In both the work of Gama and Gillespie and Xiao et al., the SPR was much greaterthan 1, meaning that bending mechanics were significant. In the work of Sun and Potti,the SPR varied, but was between 3 and 12, indicating that bending behavior was likewisepresent [28]. Additionally, Xiao et al. conducted finite element modeling of the laboratoryexperiments. Their quarter-plate FE model is shown in Figure 1.8.

Figure 1.7: Schematic of the fixture used in laboratory testing conducted by Xiao et al. [27]

8

Figure 1.8: Finite element model of high-SPR punch testing by Xiao et al. [27]

1.4.2 Low SPR Investigations

ASTM D732 specifies a SPR ≈ 1 test of through thickness shear strength of plastics [30].It is noted in the standard that the nature of the stress concentration between the fixtureand sample as well as between the punch and sample can have an appreciable effect on theoutcome of the testing. Liu and Piggott used ASTM D732 on epoxy matrix materials andthermoplastic polymers. They noted that this method might not be well suited to testingof composites due to the close clearances between the punch and fixture. It was suggestedthat fiber pullout and friction between the punch, fixture, and sample would hamper results[29]. This standard was successfully employed by Crescenzi et al. on S-2 glass composites[31]. No problems were encountered with friction or fiber pullout/entanglement. To date,there is no published work on the characterization of the through-thickness shear strengthof thermoplastic composite laminates using a test similar to ASTM D732.

1.5 Summary

High strain rate constitutive properties of UHMWPE composite materials are not predictableusing low strain rate testing accelerated by tTSP [7], however prior work indicates that

9

constituent properties are predictable using low strain rate testing accelerated by tTSP[6, 12]. Failure models incorporating progressive damage are well established and validatedagainst laboratory data [18, 17, 19, 16, 13, 15, 20, 21, 22, 23, 24, 25].

To date, a progressive damage model that incorporates fiber debonding has not been applied topredict the high strain rate response of UHMWPE composites. Additional studies are neededto determine the efficacy of using such a model to predict composite strength at a rangeof strain rates approaching 106 s−1. Furthermore, validation of these predictions againstexisting experimental data needs to be performed. Finally, subject to model validation, astudy of composite failure mode, and other important parameters such as critical clustersizes, ineffective fiber lengths, and fiber/matrix debonding behavior needs to be conducted toprovide information that cannot be obtained through laboratory investigation.

Through-thickness shear properties of UHMWPE laminates have not been determined. Apunch test with a SPR ≈ 1 demonstrates primarily shear behavior. ASTM D732 has beenused successfully on neat matrix samples as well as rigid thermoplastic polymers and epoxymatrix composites. Finite element simulation has been performed and validated againstlaboratory 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. Characterization of the through-thickness behavior of UHMWPE laminatesneeds to be performed. Due to the nature of stress concentrations inherent in this test, finiteelement analysis needs to be conducted concurrent with the laboratory testing. Investigationof the stress concentrations and resulting implications for the physical testing needs to be com-pleted in order to estimate constitutive properties for the through-thickness shear strength ofthese laminates.

2 | Preliminary Results: Punch Shear

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

2.1 Introduction

Much work has been done to characterize the properties of laminates in terms of their energyabsorption via a punch type test. Gama and Gillespie and Sun and Potti studied the relationbetween quasi-static punch-shear behavior and ballistic penetration models for thick sectionS-2 glass composites [26, 28, 32]. They found that different span to punch ratios changed themechanics of failure. SPR close to 1 leads to shear dominated failure, while large supportspans lead to bending dominated failure. In 2005, Xiao et al. studied delamination damage inS-2 glass composites due to punch-shear loading [27]. In both of these cases, the supporteddiameter is substantially larger than the punch diameter, leading to a significant mix ofbending and shear.

In order to support modeling efforts, it is desired to understand the through-thickness shearbehavior of UHMWPE composites in terms of material properties. While any mechanicaltest is always a measurement of a mechanical system property rather than a true materialproperty, 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 [29].Liu and Piggott used ASTM standard D732-85 as a basis for evaluating shear propertiesof thermoplastic polymers as well as several epoxy matrix materials. They concluded thatthe punch shear test may have problems associated with friction between the punch anddie, 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 fiberreinforced composites in a similar manner and found that the primary damage modes weredelamination and plugging [33, 34]. Nemes et al. performed quasi-static as well as high strainrate punch shear characterization of graphite/epoxy laminates using a SHPB apparatus.They observed that the overall penetration behavior of the laminate was relatively insensitiveto construction parameters other than thickness [35]. Liu and Piggott also tested samplesusing an Iosipescu type test geometry. They found that it provided unreliable results forpolymers in the rubbery state [29]. To date, literature has not been published to establishconstitutive property estimates for the shear behavior of UHMWPE or similar laminates.

10

11

2.2 Laboratory Testing

Samples tested were commercially available Honeywell SpectraShield 1214 product. Either4 or 8 layers of SpectraShield 1214, each having a lay-up of [0,90]2 were combined to formlaminates of lay-up [0,90]4 or [0,90]8. Sheets of the SpectraShield product were cut intoapproximately 30.5 cm x 30.5 cm panels using a CNC cutting table. Sheets were stacked inthe aforementioned configuration. Multiple sheets were pressed in one cycle, with a sheetof silicone coated paper between each. The stack of panels was placed between two 0.635cm thick smooth aluminum sheets and placed in a hot press. Panels were pressed at atemperature of 118◦C and pressure of 19 MPa for 15 minutes. The pressure was removedand the panel was allowed to cool in the press for approximately 15 minutes before beingremoved. These pressed plates were clamped between rigid fiberglass plates and cut into50 mm x 50 mm samples using a wet saw. Samples were sufficiently large that cut edgeswere far away from the punch area. Figure 2.1 shows SpectraShield samples before and aftertesting. Typical sample thicknesses were 0.53 mm and 1.04 mm, for 8 and 16 layer samples,respectively. Typical fiber volume fractions range from 70 to 75%.

Figure 2.1: SS1214 laminate samples before and after testing.

12

2.2.1 Testing Procedure

Specimens were tested using an in-house punch shear assembly. The test setup is similar toASTM D732-85, but with smaller dimensions to achieve failure within a desired load rangebased on load frame specifications and predicted strength.

2.2.1.1 Punch Shear Test Apparatus

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 clampedin between the two die halves. The punch to die fit is 0.025 mm nominal radius, effectivelyminimizing the unsupported span of the sample during testing. Laminate thickness is 25 to50 times this clearance. The goal is to create a region of intense through-thickness shear inthe test specimen at the punch/die interface, while minimizing other mechanics phenomenasuch as bending. Figures 2.2 and 2.3 show the assembly. Up to eight 1

2-20 inch bolts are

used to provide clamping pressure to the specimen. The entire assembly was manufacturedout of A2 tool steel and then air-hardened to approximately Rockwell C50 to minimize wearon the edge of the hole at the punch-die-sample interface.

Figure 2.2: Punch shear test apparatus exploded assembly.

13

Figure 2.3: Side view of punch shear test apparatus exploded assembly.

2.2.1.2 Punch Configurations

Two main punch configurations were considered; one allowing and one preventing curvatureof the sample during testing. It was hypothesized that disallowing curvature of the sampleduring testing would create a stress state closer to pure shear, and thereby more effectivelydecouple tensile and shear modes of failure.

Unsupported Punch The unsupported punch was simply a right circular cylinder ofdiameter 19.00 mm and a length of 50 mm. The punch was intended to create a region ofintense shear in the sample at the punch-die interface. The punch was manufactured fromA2 tool steel and then air-hardened to approximately Rockwell C50 to minimize wear of thepunch edge. Additionally, the face of the punch was surface ground to make the edge assquare as possible to reduce bending at the punch-die interface. Figure 2.4 shows the punchused.

Supported Punch The supported punch was designed to prevent curvature of the sampleduring testing. The main body of the punch was a right circular cylinder of diameter 19.00mm and a length of 50 mm. One end of the punch was drilled and tapped with a 1

4-20

14

Figure 2.4: Unsupported shear punch fabricated from A2 tool steel.

inch hole and surface ground to create a square edge. A second right circular cylinder wasmachined with a 19.00 mm diameter and a 10 mm length with a 6.35 mm diameter holethrough the center to form the clamp. Both pieces were fabricated from A2 tool steel andair-hardened to Rockwell C50, as before. Figure 2.5 shows the punch assembly.

2.2.1.3 Punch Shear Test Procedure

Samples of size 50 mm x 50 mm were centered on the bottom portion of the die assembly. A 5mm diameter hole was cut using a razor blade for samples tested with the supported punch.The top die section was installed. Proper alignment of the holes in the upper and lowersection of the die was achieved via pressed dowel pins. Four 1

2-20 inch bolts were inserted

in the bolt holes located at the corners of the die halves and tightened to a torque of 68N-m. The die assembly was designed with eight bolt holes due to anticipated problems withsatisfactorily gripping the samples, however four bolts proved to be sufficient to achieve thenecessary 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 MTSForce Transducer 661.20 load cell was used to measure load. Displacement measurementswere taken from the load frame stroke transducer. An aluminum block was clamped in the

15

Figure 2.5: Supported shear punch fabricated from A2 tool steel.

upper MTS 647 Hydraulic Wedge Grip and used to transfer load to the punch head. Controland data collection was provided using an in-house National Instruments (NI) LabVIEWprogram. The punch shear test assembly was rested on the lower grip frame, which wasverified to be parallel to the aluminum block in the upper grip to ensure load was appliedvertically and did not cause the punch to jam in the die assembly. Figure 2.6 shows theassembled test apparatus in the load frame prior to a typical test.

Tests were conducted in load control at a load rate of 667 Ns. Load and displacement

channels were recorded 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 thefailure surface.

16

Figure 2.6: Assembled punch shear test apparatus in MTS hydraulic load frame.

2.2.2 Results

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

τ =F

Ashear

(2.1)

where Ashear is defined as

Ashear = πDpt (2.2)

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

The results for ultimate shear stress versus sample thickness are shown in Figure 2.7. Averageultimate shear stress for each thickness and punch type are given in Table 2.1. Additionally,standard deviation and coefficient of variation (COV) for each data set is shown in Table2.2.

17

Figure 2.7: Ultimate shear stress versus thickness.

Table 2.1: Average ultimate shear strength and deviation for supported and unsupportedpunch types (MPa).

Unsupported Supported

8-layer 266 19616-layer 242 179

Absolute Difference 24 17Percent Difference 9.0 8.7

Typical load-displacement curves for unsupported punch samples are given in Figure 2.8.We note that there is a distinct slope change for both 8 and 16 layer samples. Similarly,Figure 2.9 shows typical load-displacement curves for samples tested using the supportedpunch.

2.2.3 Discussion

There is a trend of increased apparent shear strength with decreasing thickness. If the stateof stress was truly pure shear through the thickness of the sample, apparent shear strength

18

Figure 2.8: Load versus displacement for selected unsupported punch samples.

Figure 2.9: Load versus displacement for selected supported punch samples.

19

Table 2.2: Standard deviation and COV for supported and unsupported punch types.

Unsupported Supported

8-layer Std. Dev. (MPa) 12.1 15.1COV (%) 4.6 7.7

16-layer Std. Dev. (MPa) 17.1 12.2COV (%) 7.1 6.8

would not be expected to vary with specimen thickness. Initially, when only unsupportedpunch tests had been performed, it was hypothesized that this increase in apparent strengthwas due to a membrane type behavior. Thinner 8-layer samples have less bending stiffnessand more ability to rotate when compared to 16-layer samples, given the constant clearancein the punch and die geometry. It stands to reason that apparent shear strength would thenbe higher for samples that are experiencing more of a membrane type state of stress, due tothe extreme anisotropy of UHMWPE fibers that comprise the SpectraShield laminates.

This notion is further supported by the distinct change in slope of the load-displacementcurves shown in Figures 2.8 and 2.9. It is possible that the increase in slope during thecourse of the punch shear test is due to a nonlinear geometric effect that is governed by amembrane type behavior. The same phenomenon that governs this increased mechanicalstiffness could be governing the increase in observed shear strength.

It was initially expected that the supported punch that prevented curvature of the samplewould reduce this hypothesized membrane behavior and would result in apparent shearstrengths that were more consistent versus sample thickness. However, given the results inFigure 2.7 and Tables 2.1 and 2.2, this does not appear to be the case. For both supportedand unsupported punch types, an 8 to 9 percent decrease in apparent shear strength isobserved with a two fold increase in sample thickness. Finite element modeling is in progressto investigate these phenomena.

20

2.3 Finite Element Modeling

2.3.1 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 2.10. The modelcurrently treats the punch and fixture assembly as rigid, but has the ability to model themas elastic structures. Composite elements are 3D C3D20 quadratic 20-node elements withfull integration. The bottom edge of the test fixture is fixed in space, and a load is applied tothe top half of the fixture to simulate clamping pressure. A second load is applied to the topof the punch to simulate an applied load. A schematic of the boundary conditions is shownin Figure 2.11. Interaction between the punch, fixture, and composite sample is modeled ascontact and friction loads.

Figure 2.10: Detail of 3D solid quarter model of punch shear test apparatus.

It is assumed that composite layers are perfectly bonded and that no slipping of the compositeoccurs in the grip region of the fixture.

21

Figure 2.11: Boundary conditions of 3D solid quarter model of punch shear test apparatus.

2.3.2 Preliminary Results

The development of the aforementioned finite element model is complete, but substantialresults have not yet been produced and analyzed. Further work is proposed in Section 4.2.

3 | Preliminary Results: Axial Model

3.1 Introduction

3.2 Model Formulation

A shear-lag type model is being used to characterize the load transfer away from fiber breaks.This load transfer is governed by several parameters, most important in this context are theelastic modulus of the fiber and the shear modulus of the matrix. Both of these propertieschange with strain rate, but the rate of change is profoundly different between the two.Fiber modulus changes are small compared to the several orders of magnitude change thatthe matrix material exhibits over 5 orders of magnitude of strain rate. Due to these changes,the nature of the load transfer away from fiber breaks is expected to very with strain rate.

The notation used is the same as that of Okabe et al. [16]. The arrangement of elements inthe model is shown in Figure 1.5. The i and j indices indicate fiber number, and the k indexis the element number along the length of a given (i,j) fiber.

Fiber properties are randomly assigned based on the Weibull distribution. Previous work onthis project has demonstrated the Weibull distribution to be a suitable fit for this materialsystem [6]. Each fiber element is assigned an ultimate fiber tensile strength as shown inEquation 3.1. Matrix properties and all other fiber properties are assumed to be determin-istic. This is partly because the composite strength is largely governed by the fiber strengthdistribution, and partly because an accurate distribution for other material properties isdifficult to obtain.

σuti,j,k =

(ln

(1

1 − ηi,j,k

)L0

∆z

)1mσ0 (3.1)

Following the aforementioned method of Okabe et al., we note that the equilibrium equationfor axial stress on a fiber element is

Adσi,j,kdz

= −πr2

4∑l=1

τ li,j,k (3.2)

22

23

And we assume that the shear stress at the fiber-matrix interface is a function of the dis-placements of the surrounding fibers.

τ 1i,j,k = Gmui+1,j,k − ui,j,k

d(3.3a)

τ 2i,j,k = Gmui,j+1,k − ui,j,k

d(3.3b)

τ 3i,j,k = Gmui−1,j,k − ui,j,k

d(3.3c)

τ 4i,j,k = Gmui,j−1,k − ui,j,k

d(3.3d)

We can then write a finite difference method equation to solve for the displacements of thefiber nodes, using the approach of Oh [20].

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

](3.4)

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

24

The boundary conditions are

ui,j,1 = 0 (3.5a)

ui,j,K = εcL (3.5b)

We solve the above system of equations. From the computed displacements, the fiber elementstresses are computed, and fiber breaks are calculated. The above steps are then repeatedfor the new composite configuration with fiber breaks. This continues until the compositehas failed.

3.3 Preliminary Results

Simulations have been completed for ultra-high molecular weight polyethylene (UHMWPE)composites at a variety of strain rates, using fiber and matrix constitutive properties previ-ously measured. As expected due to the very low matrix stiffness of these materials relativeto fiber stiffness, the low strain rate simulation results are very close to the bundle strengthresult for the UHMWPE fibers, despite the presence of load transfer between fibers thatthe model allows. Figure 3.1 shows the simulation result for 32 simulations of a SS3124composite at a strain rate of 0.83s−1, while Figure 3.2 shows result of 32 simulations at astrain rate 10 times faster, 8.3s−1.

At low strain rates results are within 2% of the bundle strength result. Monte Carlo simula-tion results have a relatively low amount of scatter, with a coefficient of variation of <2.5%.High strain rate simulations exhibit a strength substantially higher than the bundle strengthresult, as the matrix becomes relatively stiffer compared to the fibers with changes in strainrate. The ultimate tensile strengths shown in Figures 7 and 8 indicate an increase of 7% withone order of magnitude change in strain rate. When the strain rate is increased further to6.4x105 s−1, the average ultimate tensile strength increases to 2.9 GPa, as shown in Figure3.3.

25

Figure 3.1: Simulated stress-strain curve for SS3124 composite at 0.83s−1.

Figure 3.2: Simulated stress-strain curve for SS3124 composite at 8.3s−1.

26

Figure 3.3: Simulated stress-strain curve for SS3124 composite at 6.4x105 s−1.

3.4 Discussion

At high strain rate simulated strength tends to overestimate the experimentally measuredcomposite strength. The current model does not incorporate fiber-matrix debonding ormatrix yielding, although it has the ability to do so. Current efforts are focusing on obtainingfiber-matrix bond strength data so as to enable the model to simulate fiber-matrix debonding.Since the current model does not use any fit parameters, efforts are being made to obtainan estimate of the bond strength, rather than treat the value as a fit parameter.

The results of the axial strength model agree well with both bundle strength estimates aswell as measured composite strength at low strain rates. However, at high strain rates, themodel predicts strengths substantially higher than measured values, while bundle strengthestimates are substantially lower. It was hypothesized that this was due to the fact that theaxial simulation model did not implement debonding at the fiber/matrix interface. This isa planned feature of the model that has now been implemented. However, the interfacialstrength of the fiber/matrix bond is being estimated. Several techniques, including micro-bead debond, and fiber pushout were deemed to be impractical to perform on this composite.Currently, work is in progress to determine an estimate of this property from two sources.The first source is tensile tests of [(+/-45)N ]S composites that were previously conducted.This method incorporates other modes of deformation than those of interest, but the test

27

data already exists. The second method involves estimating the debond strength from atransverse tension test of a unidirectional specimen. This is a much simpler test, but thedebonding mode is not correct. In addition to obtaining estimates for the fiber/matrixdebonding strength, a sensitivity study is in progress to determine how much the modelstrength estimates are affected by the debonding stress. The necessary precision of theestimate is dependent on how sensitive the model is to the variable.

Further work is proposed in Section 4.3.

4 | Proposed Work

4.1 Introduction

The objective of the proposed studies is to develop models capable of producing reliableestimates for constitutive properties of thermoplastic composites to enable a robust modelingeffort.

Finite element modeling of through-thickness punch shear testing in conjunction with anal-ysis of stress concentrations will be used in conjunction with test data to predict values ofthrough-thickness shear strength and modulus. Additionally, the axial progressive damagemodel will be used in conjunction with measured and predicted constituent properties topredict composite response at strain rates not attainable in a laboratory setting.

4.2 Punch Shear Modeling and Analysis

Objectives: Validate and refine FE model to examine the mechanics governing the failureof composite laminates as tested in Section 2.2.

Methodology: The finite element model is being refined to include a finer mesh with severalelements through the thickness of each composite layer.

Additional investigations will be performed as to the stress state at the interface of the punchand sample:

• Refine FE model to >1 element through the thickness of each composite layer.

• Determine the ratio of in plane to out of plane stresses from the FE model as a functionof applied load, sample thickness, and punch configuration.

• Determine the magnitude of stress concentration at punch-composite interface and theimplications of this stress concentration on measured strength.

• Develop analytical half-space contact solution for punch-composite interface for com-parison to FE model results.

• Estimate through-thickness shear strength of UHMWPE composite laminates.

28

29

4.3 Axial Modeling and Analysis

Objectives: Expand the progressive damage model to include fiber/matrix debonding andvalidate the model. Use the model to predict composite properties at high strain rates thatare not attainable in laboratory testing.

Methodology: The progressive damage model is currently being expanded to include ma-trix plasticity and fiber/matrix debonding. After determining interfacial bond strength,implement debonding and simulate composite failure at a range of strain rates.

• Estimate fiber/matrix bond strength from off-axis tensile test data

• Implement fiber/matrix debonding and simulate composite failure at strain rates rang-ing from quasi-static to 106 s−1

• Examine and explain any changes in failure mode, fiber ineffective length, criticalcluster size, or other relevant failure characteristics

• Estimate composite tensile performance at strain rates ranging from quasi-static to 106

s−1

4.4 Project Time Table

Work on all major areas of proposed study is underway. An expected timetable of work isshown in Figure 4.1.

Completion is expected in July 2013.

30

Figure 4.1: Expected project timetable.

4.5 Expected Contributions

To date, a prediction of through-thickness shear properties of UHMWPE composites hasnot been realized. A method for their prediction is outlined in Section 2.3. This researchconsisting of combined mechanical testing and simulation will produce an estimate of shearmodulus and strength along with a discussion of the confidence in this prediction and anyassociated issues.

To date, a progressive damage tensile strength model that incorporates fiber debonding hasnot been applied to predict the high strain rate response of UHMWPE composites. Thisresearch will produce a model framework and a prediction of composite strength at a rangeof strain rates from quasi-static to 106 s−1. These predictions will be validated againstexisting experimental data.

To date, a study of UHMWPE composite failure including critical break cluster sizes, inef-fective fiber lengths, and fiber/matrix debonding behavior has not been conducted. Using theframework developed in this research, a study of these parameters will be performed.

The above outlined contributions constitute a significant addition to the body of knowledgeregarding thermoplastic composite laminates and their mechanical response.

31

4.6 Anticipated Publications

Umberger, P.D., Case, S.W., Tensile strength and modeling of UHMWPE composites athigh strain rates, Journal TBD

Umberger, P.D., Cook, F.P. Case, S.W., Characterization and response of UHMWPE re-inforced thermoplastic composites and constituents using time-temperature superposition,Journal of Thermoplastic Composite Materials

Umberger, P.D., Case, S.W., Through-thickness shear behavior of UHMWPE composites,Applied Composite Materials

Bibliography

[1] Bhatnagar, A. Lightweight ballistic composites: military and law-enforcement applica-tions. Woodhead, 2006.

[2] Alcock B., Cabrera N.O., Barkoula N.M., Reynolds C.T., Govaert L.E., Peijs T. Theeffect of temperature and strain rate on the mechanical properties of highly orientedpolypropylene tapes and all-polypropylene composites. Composites Science and Tech-nology, 67:2061–2070, 2007.

[3] Brinson H., Brinson L. Polymer engineering science and viscoelasticity. Springer, 2008.

[4] Ariyama T., Mori Y., Kaneko K. Tensile properties and stress relaxation of polypropy-lene at elevated temperatures. Polymer Engineering and Science, 37 Iss. 1:81–90, 2004.

[5] Tan V.B.C., Zeng X.S. Characterization and constitutive modeling of aramid fibers athigh strain rates. International Journal of Impact Engineering, 32:1737–1751, 2006.

[6] Umberger, P.D. Characterization and response of thermoplastic composites and con-stituents. Virginia Polytechnic Institute and State University, 2010.

[7] Cook, F.P. Characterization of uhmwpe laminates for high strain rate applications.Virginia Polytechnic Institute and State University, 2010.

[8] Peijs, T., et al. Role of interface and fibre anisotropy in controlling the performance ofpolyethylene-fibre-reinforced composites. Composites Science and Technology, 52:449–466, 1994.

[9] Tan, V.B.C., Ching T.W. Computational simulation of fabric armour subjected toballistic impacts. International Journal of Impact Engineering, 32:1737–1751, 2006.

[10] Koh C.P., Shim V.P.W., Tan V.B.C. Response of a high-strength flexible laminate todynamic tension. International Journal of Impact Engineering, 35:559–568, 2007.

[11] Chen, W.W., Lim, J., Zheng, J.Q. Dynamic small strain measurements of kevlar 129single fibers with a miniaturized tension kolsky bar. Polymer Testing, 29:701–705, 2010.

[12] Miyano Y., Nakada M., Naoyuki S. Accelerated testing for long-term durability of gfrplaminates for marine use. Composites: Part B, 35:497–502, 2004.

32

33

[13] Hedgepeth, J.M., vanDyke, P. Local stress concentrations in imperfect filamentarycomposite materials. Journal of Composite Materials, pages 294–309, 1967.

[14] Hedgepeth, J.M. Stress concentrations in filamentary structures. NASA TN D-882,1961.

[15] Beyerlein, I.J., Landis, C.M. Shear-lag model for failure simulations of unidirectionalfiber composites including matrix stiÂĂness. Mechanics of Materials, 31:331–350, 1999.

[16] Okabe, T., Takeda, N., Kamoshida, Y., Shimizu, M., Curtin, W.A. A 3d shear-lagmodel considering micro-damage and statistical strength prediction of unidirectionalfiber-reinforced composites. Composites Science and Technology, 61:1773–1787, 2001.

[17] Landis, C.M., Beyerlein, I.J., McMeeking, R.M. Micromechanical simulation of thefailure of fiber reinforced composites. Journal of the Mechanics and Physics of Solids,48:621–648, 2000.

[18] Landis, C.M., McGlockton, M.A., McMeeking, R.M. An improved shear lag model forbroken fibers in composites. Journal of Composite Materials, 33:667–680, 1999.

[19] Landis, C.M., McMeeking, R.M. Stress concentrations in composites with interfacesliding, matrix stiÂĂness, and uneven fiber spacing using shear lag theory. InternationalJournal of Solids and Structures, 36:4333–4361, 1999.

[20] Oh K.P. A monte carlo study of the strength of unidirectional fiber-reinforced compos-ites. Journal of Composite Materials, 13:311–328, 1979.

[21] Beyerlein I.J., Phoenix S.L. Stress concentrations around multiple fiber breaks in anelastic matrix with local yielding or debonding using quadratic influence superposition.Journal of Mechanics and Physics of Solids, 44:1997–2039, 1996.

[22] Beyerlein, I.J., Phoenix, S.L. Statistics of fracture for an elastic notched compositelamina containing weibull fibers -part i. features from monte-carlo simulation. Eng.Frac. Mech., 57:241–265, 1997.

[23] Beyerlein, I.J., Phoenix, S.L. Statistics of fracture for an elastic notched compositelamina containing weibull fibers - part ii. probability models of crack growth. Eng.Frac. Mech., 57:267–299, 1997.

34

[24] Ibnabdeljalil, M., Curtin, W.A. Strength and reliability of fiber-reinforced composites:localized load-sharing and associated size effects. Int. J. Solids Structures, 34:2649–2668,1997.

[25] Sastry, A.M., Phoenix, S.L. Load redistribution near non-aligned fiber breaks in a twodimensional unidirectional composite using break influence superposition. J. Mater.Sci. Lett., 12:1596–1599, 1993.

[26] Gama, B.A., Gillespie, J.W. Punch shear based penetration model of ballistic impactof thick-section composites. Composite Structures, 86:356–369, 2008.

[27] Xiao, J., Gama, B., Gillespie Jr., J. Progressive damage and delamination in plainweave s-2 glass/sc-15 composites under quasi-static punch-shear loading. CompositeStructures, 78:182–196, 2007.

[28] Sun C.T., Potti S.V. A simple model to predict residual velocities of thick compositelaminates subjected to high velocity impact. International Journal of Impact Engineer-ing, 18:339–353, 1996.

[29] Liu, K, Piggott, M. Shear strength of polymers and fibre composites: 1. thermoplasticand thermoset polymers. Composites, 26(12):829–840, 1995.

[30] ASTM D732. Standard test method for shear strength of plastics by punch tool. ASTM,2009.

[31] Crescenzi, F., Marini, F., Nardi, C., Pizzuto, A., Rossi, P., Verdini, L., Rajainmaki, H.,Knaster, J., Bettinali, L. Mechanical characterization of glass fibreâĂŞepoxy compositematerial for iter pre-compression rings. Proceedings of the 26th Symposium of FusionTechnology, 86:2553–2556, 2011.

[32] BA Gama, M Rahman, and JW Gillespie Jr. Energy absorbing damage mechanisms inquasi-static punch shear of thick-section composites. CD Proceedings, SAMPE, pages909–21, 2004.

[33] S-W. R. Lee and C.T. Sun. A Quasi-Static Penetration Model for Composite Laminates.Journal of Composite Materials, 27(3):251–271, January 1993.

[34] S.V. Potti and C.T. Sun. Prediction of impact induced penetration and delamination inthick composite laminates. International Journal of Impact Engineering, 19(1):31–48,January 1997.

35

[35] JA Nemes, H Eskandari, and L Rakitch. Effect of laminate parameters on penetrationof graphite/epoxy composites. International journal of impact engineering, (97):97–112,1998.