An Augmented Hybrid Constitutive Model for Simulation of ...An Augmented Hybrid Constitutive Model for Simulation of Unloading and Cyclic Loading Behavior of Conventional and Highly

An Augmented Hybrid Constitutive Model for Simulation of Unloading and Cyclic Loading Behavior of Conventional

and Highly Crosslinked UHMWPE

J.S. Bergström1*

, C.M. Rimnac2, S.M. Kurtz

3

1Veryst Engineering, 47A Kearney Road, Needham, MA

2Musculoskeletal Mechanics and Materials Laboratories,

Departments of Mechanical and Aerospace Engineering and Orthopaedics,

Case Western Reserve University, Cleveland, OH

3Implant Research Center, School of Biomedical Engineering, Science

and Health Systems, Drexel University, 3141 Chestnut St., Philadelphia PA

*

Corresponding Author:

Jörgen S. Bergström

Veryst Engineering

47A Kearney Road

Needham, MA 02494

Tel: 781-433-0433

Email: [email protected]

Submitted to Biomaterials, June 2003

Revised August 2003

1

Abstract

Ultra-high molecular weight polyethylene (UHMWPE) is extensively used in total joint

replacements. Wear, fatigue, and fracture have limited the longevity of UHMWPE components.

For this reason, significant effort has been directed towards understanding the failure and wear

mechanisms of UHMWPE, both at a micro-scale, and at a macro-scale, within the context of

joint replacements. We have previously developed, calibrated, and validated a constitutive model

for predicting the loading response of conventional and highly crosslinked UHMWPE under

multiaxial loading conditions (Biomaterials 24 (2003) 1365). However, to simulate in vivo

changes to orthopedic components, accurate simulation of unloading behavior is of equal

importance to the loading phase of the duty cycle. Consequently, in this study we have focused

on understanding and predicting the mechanical response of UHMWPE during uniaxial

unloading. Specifically, we have augmented our previously developed constitutive model to

allow also for accurate predictions of the unloading behavior of conventional and highly

crosslinked UHMWPE during cyclic loading. It is shown that our augmented hybrid model

accurately captures the experimentally observed characteristics, including uniaxial cyclic

loading, large strain tension, rate-effects, and multiaxial deformation histories. The augmented

hybrid constitutive model will be used as a critical building block in future studies of fatigue,

failure, and wear of UHMWPE.

Key Words—constitutive modeling, ultra-high molecular weight polyethylene, UHMWPE,

Hybrid model, FEM, radiation crosslinking, multiaxial mechanical behavior, small punch test

2

1. Introduction

Wear of the articulating surface of ultra-high molecular weight polyethylene (UHMWPE)

implant components is an important problem that can significantly limit the life expectancy of

total joint replacements. Wear of UHMWPE components is multifactorial and is influenced by

the functional loading environment, joint kinematics, component geometry, and material

properties. Recently, efforts to reduce UHMWPE wear have involved changes to the resin type,

sterilization method, radiation crosslinking, and thermal treatments [1,2]. However, there is, at

present, an incomplete understanding of the wear characteristics and mechanisms of damage

evolution, complicating and impeding rapid progression and improvements in performance of

UHMWPE joint replacement components.

There are two complementary approaches for improving the general understanding of the

wear behavior of UHMWPE components used in total joint replacements: macroscopic

experimental testing and microstructural material characterization. Wear simulators and other

mechanical testing techniques can provide information related to wear rates of different

tribological systems and can rank the performance of materials subjected to different

environments and thermomechanical histories [3]. Although useful, systematic empirical testing

has thus far not enabled a priori predictions of the mechanisms causing the actual wear of

UHMWPE. To predict the evolution in microscopic and macroscopic damage for new

UHMWPE materials from fundamental polymer physics principles requires an understanding of

the performance and response of the material on the microstructural level.

Theoretical and experimental research supports the notion that the wear observed in vivo

and in vitro is the result of localized high stresses and strains in the surface region of the

UHMWPE component [4,5]. To better understand and predict these stresses it is necessary to

have a well-calibrated and accurate constitutive model of UHMWPE. In the orthopaedic

research community, the J2-plasticity model has been the most widely used approach for

simulating the behavior of UHMWPE. It has been shown [6] however, that the J2-plasticity

model is not an accurate general tool for predicting the large-deformation-to-failure behavior of

UHMWPE. In addition, the J2-plasticity model does not accurately predict cyclic loading of

3

UHMWPE. These are serious limitations since UHMWPE joint components undergo large

deformations locally at the articulating surface and are also subject to cyclically applied loads.

To address these limitations, a new constitutive model was recently developed for

conventional and highly crosslinked UHMWPEs [6]. This new model, which is inspired by the

physical micromechanisms governing the deformation resistance of polymeric materials, is an

extension of specialized constitutive theories for glassy polymers that have been developed

during the last 10 years. The new model, named the Hybrid Model (HM), has been shown to

accurately predict the mechanical response of both conventional and highly crosslinked

UHMWPE materials in uniaxial tension, compression and multiaxial loading. However, to

simulate in vivo changes to orthopedic components, accurate simulation of unloading behavior is

also of importance. Consequently, the objective of this study was to better understand and predict

the mechanical response of UHMWPE during uniaxial unloading. In this regard, we have

developed an augmented hybrid constitutive model that is capable of accurately predicting the

experimentally observed stress-strain response in cyclic loading for conventional and well as

highly crosslinked UHMWPEs.

2. Augmented Hybrid Constitutive Model for Predictions of UHMWPE

The new augmented Hybrid Model (HM) is a modification of our earlier constitutive

models [6,7] aimed at predicting the large strain time-dependent behavior of both crosslinked

and uncrosslinked UHMWPE. The modification in the augmented HM specifically addresses

the unloading behavior during cyclic loading. The kinematic framework used in the augmented

HM is based on a decomposition of the applied deformation gradient into elastic and viscoplastic

components: F = Fe F

p (Figure 1). The spring and dashpot representation shown in Figure 1a is a

one-dimensional embodiment of the model framework used to capture the viscoplastic flow

characteristics. With the exception of the top spring (E), all spring and dashpot elements are

highly nonlinear (described in detail, below). Figure 1b depicts a map of the decomposition of a

given material deformation state. This decomposition specifies how the three-dimensionality of

the deformation gradients and stress tensors are connected and evolve during an applied

deformation history.

4

As in our previous modeling approaches [6,7], the deformation state is decomposed into

elastic, backstress, and viscoplastic components. Compared with our previous models, we have

now incorporated time-dependent viscoplasticity to the backstress network to improve the

predictive capabilities of the model with respect to unloading. The rationale for this approach is

as follows: the interaction between the amorphous and crystalline domains in UHMWPE is

complicated by entanglements due to its very high molecular weight and also due to chemical

crosslinks (when present). At large deformations, however, the underlying molecular

deformation resistance, the “backstress” network of molecular chains, has the ability to undergo

viscoplastic flow. This flow behavior is caused by the absence of an isotropic crosslinked

microstate in the material, which creates both regions with highly stretched molecular chains and

regions that are less stretched. The flow behavior is a function of the highly deformed material

state and the interaction between the amorphous and crystalline domains, and can be accurately

captured using an energy activation representation. The kinematics of the viscoplastic flow of

the backstress network is captured by decomposing the deformation gradient acting on part B of

the backstress network (Figure 1a) into elastic and viscoelastic components: Fp = F

eB F

vB.

The Cauchy stress in the system is given by the isotropic linear elastic relationship:

( )1 2 tre ee eeJ

µ ! " #= + $ %T E E 1 , (1)

where µe and !e are Lame’s constants which can be obtained from the Young’s modulus and

Poisson’s ratio by µe = Ee / (2(1+!e)) and !e = Ee!e / ((1+!e)(1-2!e)), Je = det[F

e] is the relative

volume change of the elastic deformation, Fe is the deformation gradient, E

e = ln[V

e] is the

logarithmic true strain, and Ve is the left stretch tensor [8] which can be obtained from the polar

decomposition of Fe.

The stress acting on the equilibrium portion of the backstress network is given by the

same expression as used in our earlier work [6]:

( ) ( )28

1; , , ;

1

p lock p

A chain A A A A I A

A

qq

µ ! " µ# $= +% &+T T F T F , (2)

5

( )*

2* * *2

2 1

2

3

pp p p

I A

IIµ! "

= # #$ %& '

T B 1 B , (3)

where TA is a tensor-valued function of the viscoplastic deformation gradient Fp and the material

parameters {µA, "Alock

, #A, qA}, where µA is the shear modulus, "Alock

is the locking stretch, #A is

the bulk modulus, and qA is a material parameter specifying the relative magnitudes of T8chain and

TI2, and Bp*

is the left Cauchy-Green deformation tensor. This hyperelastic stress representation

is based on the 8-chain model [9], and a term containing I2-dependence of the strain energy

density. The I2-dependence is introduced by the crystalline domains and is manifested by the

asymmetry in the response between tension and compression [6].

The stress driving the viscoplastic flow of the backstress network is obtained from the

same hyperelastic representation that was used to calculate the backstress, and has a similar

framework as used in the Bergström-Boyce representation of crosslinked polymers at high

temperatures [10,11]:

()eBBABs=!TTF, (4)

where sB is a dimensionless material parameter specifying the relative stiffness of the backstress

network. At small deformations, the stiffness of the backstress network is constant and the

material response is linear elastic. At larger applied deformations, viscoplastic flow caused by

molecular chain sliding is initiated. With increasing viscoplastic flow, the crystalline domains

become distorted and provide additional molecular material to the backstress network. This is

manifested by an initial reduction in the effective stiffness of the backstress network with

imposed viscoplastic deformation and is captured in the model by allowing the parameter sB to

evolve with the plastic deformation. The parameter sB evolves with imposed plastic deformation

to capture the distributed yielding:

( )B B B Bf Cs p s s != " # " #& & , (5)

6

where pB is a material parameter specifying the transition rate of the distributed yielding event,

sBf is the final value of sB reached at fully developed plastic flow, and C!& is the magnitude of the

viscoplastic flow rate (Eq. (9)):

0BmvBBbaseB!""!#$=%&'(&&

. (6)

The velocity gradient of the viscoelastic flow of the backstress network is given by

[ ]1 dev Bv v e e

B B B B

B

!"

#=T

L F F& , (7)

where v

B!& is the rate of viscoplastic flow of the time-dependent network B, [ ]devB B

F! = T ,

base

B! and mB are material parameters, and 0!& is a constant coefficient with a value of 1/s.

The yielding and plastic flow of the material is captured in the same way as in our earlier

work [6,7]:

dev[]peTeCCC!"#$=%&'(TLRR&

, (8)

where

1ppp!=LFF&,

[()]/eeTeCABJ=!+TTFTTF is the stress acting on the relaxed

configuration convected to the current configuration, dev[]CCF!=T

is the effective shear

stress (calculated using the Frobenius norm) driving the viscoplastic flow,

0(/)CmbaseCCC!!""=#&&, (9)

is the magnitude of the viscoplastic flow, baseC! and mC are material parameters, and

0!& a

constant coefficient with a value of 1/s.

7

In total, the augmented HM contains 13 material parameters: 2 small strain elastic

constants (Ee, !e); 4 hyperelastic constants for the back stress network (µA, "Alock

, #A, qA); 5 flow

constants of the backstress network (sBi, sBf, pB, $Bbase

, mB); and 2 yield and viscoplastic flow

parameters ($Cbase

, mC). These parameters can readily be determined from a few select

experiments, as will be discussed in the next section.

3. Materials and Methods

Section 3.1 first describes the different types of UHMWPE that were examined in this

study and the experimental techniques that were used to characterize the material behavior. The

methods and procedures that were used to calibrate and validate the predictions from the Hybrid

Model (HM) are then described in Section 3.2.

3.1. Experimental

In this work, we have focused on one radiation sterilized, and two highly crosslinked

GUR 1050 materials. These materials have been characterized and tested in previous studies

using uniaxial tension, uniaxial compression, uniaxial cyclic loading, and small punch testing.

The details of the material preparations and the experimental data can found elsewhere [6,12];

however, a short summary is provided herein for clarity.

3.1.1. Previous Materials and Testing

Ram-extruded GUR 1050 was used as the base material. All test samples were cut such

that the loading direction coincided with the extrusion direction. Three groups of specimens

were created. The first group was gamma radiation sterilized in nitrogen with a dose of 30 kGy

(“30 kGy %-N2”), the second group was gamma irradiated with a dose of 100 kGy and then heat

treated at 110°C for 2 hours (“100 kGy (110°C)”), and the third group was gamma irradiated

with a dose of 100 kGy and then heat treated at 150°C for 2 hours (“100 kGy (150°C)”). After

all material preparations, all specimens were stored in a –20°C freezer to minimize aging and

oxidation effects. The microstructure of the materials studied in this work has been extensively

8

examined elsewhere [6,12]; e.g., the degree of crystallinity of the three materials has been

determined to be 0.51 for the sterilized materials (30 kGy %-N2), 0.61 for the crosslinked material

that was heat treated at 110°C, and 0.46 for the crosslinked material heat treated at 150°C.

Data from three different types of room-temperature experiments was analyzed in this

study. The first test type was uniaxial tension to failure at three different deformation rates

(approximately corresponding to true strain rates of 0.007/s, 0.018/s and 0.035/s). The second

test type was cyclic uniaxial fully-reversed tension-compression experiments. In these

experiments, cylindrical specimens were cyclically loaded and unloaded to a maximum true

strain of 0.12, and a minimum true strain of –0.12. The first two load-unload cycles were

analyzed. The last type of experimental data that was analyzed was from a multiaxial small

punch test. In these multiaxial tests, miniaturized disc specimens with a diameter of 6.4 mm and

a thickness of 0.5 mm were tested by indentation with a hemispherical head punch at a constant

punch displacement rate of 0.5 mm/min. The experimental test setup recorded the punch force

as a function of punch displacement.

3.2. Analytical

The capability of the augmented Hybrid Model (HM) to predict the response of

UHMWPE was evaluated by comparing the model predictions with the aforementioned

experimental data for the three materials. The first step in this effort was to calibrate the HM to

the uniaxial tensile and cyclic experimental data, for each of the materials. For this purpose, the

same procedure that was described in our previous work [6] was followed and is briefly

summarized. The first step, the bootstrapping step, is to find an initial estimation of the material

parameters. In this study, we used material parameters determined from our earlier work [6].

Then, a specialized computer program based on the Nelder-Mead simplex minimization

algorithm was used to iteratively improve the correlation between the predicted data sets and the

experimental data. The quality of a theoretical prediction, and therefore of the chosen material

parameters, was evaluated by calculating the coefficient of determination (r2). The reported

material parameters for each material are from the set having the highest r2-value.

9

After the optimal set of material parameters was found, the same parameters were then

used to simulate the small punch test. This validation simulation was performed to check the

capability of the augmented HM to predict a multiaxial deformation history. It is well known

that many constitutive models can predict uniaxial deformation histories relatively well, but that

it is significantly more difficult to accurately predict multiaxial deformation states. It has been

shown, for example, that the J2-plasticity model can accurately predict monotonic uniaxial

tension or compression data for UHMWPE, but is very poor at predicting cyclic or multiaxial

deformation states [6].

The small punch validation simulations were performed using the ABAQUS (HKS Inc.,

RI) finite element package. The simulations used an axisymmetric representation with 360

quadratic triangular elements (CAX8H) to represent the small punch geometry (see inset in

Figure 8). In the simulations the friction coefficient between the specimen and the punch, and

between the specimen and the die was taken as 0.1 [6]. The quality of the validation simulation

was evaluated by plotting the predicted and experimental force-displacement data and by

calculating the r2-value of the predictions.

4. Results

The material parameters for the three UHMWPE materials for the augmented Hybrid

Model (HM) are given in Table 1. As with our previous constitutive theory, nine of the

parameters of the augmented HM were found to be the same for the conventional and the two

highly crosslinked UHMWPEs; that is, only four material parameters are dependent on

crosslinking density and thermal treatment. These four material parameters are: elastic

(Young’s) modulus (E); yield strength (base

B! ); the effective stiffness after yield (µA); and the

limiting chain stretch ( lockA! ), which controls the large strain behavior.

A direct comparison between the experimental and the predicted data used for the

calibration is shown Figures 2 to 7. Figures 2 and 3 show the results for the sterilized GUR 1050

10

(30 kGy, %-N2) in monotonic large strain tension to failure, and cyclic loading with a strain

amplitude of 0.12, respectively. Figures 4 and 5 show the results for highly crosslinked GUR

1050 (100 kGy, 110°C), and Figures 6 and 7 show the results for the highly crosslinked GUR

1050 (100 kGy, 150°C) that was heat treated at 150°C for 2 hours. For all materials, the HM

does a very good job of predicting both the large strain tensile data and the small strain cyclic

data. The r2-values were 0.98 or higher for all cases, except the prediction of the tensile behavior

of the sterilized conventional material (30 kGy, %-N2). In this case the r2-value was slightly

lower (0.973), mainly due to variability in the experimental data (the stress-strain curves at

different rates crossed each other at high strain levels). A summary of the predictive

performance of the HM is given in Table 2.

The performance of the old HM [6] is illustrated in Figure 8. The figure compares cyclic

experimental data for GUR 1050 (30 kGy, %-N2) with predictions from the old HM. The

material parameters that are used in this simulation are the same as was used in the original work

[6]. The figure shows that the old model representation, which has been shown to work very

well [6] for large strain tension, compression, and small punch loading, is not accurate at

predicting cyclic loading. The new model specifically addresses this issue and enables accurate

simulations of both monotonic and cyclic loading conditions using one set of material

parameters.

The calibrated material models were then used in a finite element model to predict the

behavior in the small punch tests. The results from these validation simulations are summarized

in Table 2 and in Figures 9 to 11. The figures show that for all three materials the new HM does

a good job of predicting also the multiaxial deformation in the small punch test, including the

initial elastic slope, small strain yielding, large scale yielding, and strain localization during the

biaxial stretching. The r2-values for the small punch predictions are between 0.937 and 0.960 for

the three materials.

11

5. Discussion

The mechanical response of UHMWPE at large deformations is very complex,

considering the nonlinear behavior during both loading and unloading. Initially, at small strains,

the response is linear elastic. With increasing deformation, localized yielding is initiated at sites

where the flow resistance is the lowest. The flow resistance then evolves and becomes more

homogeneous in both the crystalline and the amorphous domains. Finally, at large deformations

the imposed molecular chain stretching and alignment causes a stiffening in the response which

continues to increase until final failure. To model these events is challenging, but necessary for

developing a better understanding of the fatigue, fracture, and wear response.

Despite the complexity inherent in the constitutive framework of our augmented Hybrid

model, we found that only four independent material properties were needed to define the overall

mechanical behavior of the conventional and highly crosslinked UHMWPE investigated in the

present study for the loading and unloading histories that were considered. As in our previous

constitutive model, the majority of the material parameters associated with the elastic, plastic,

and backstress (recovery) behavior of UHMWPE appear to be unaffected by radiation

crosslinking and thermal treatment. Although the constitutive equations used to describe our

augmented HM have increased somewhat in complexity, as compared with our previous hybrid

theory [6], the number of independent material properties necessary to characterize conventional

and highly crosslinked UHMWPE has remained unchanged in both our previous and current

theoretical frameworks. Consequently, the augmented hybrid model outlined in our present

study is proposed to be a unified constitutive theory for conventional and highly crosslinked

UHMWPE materials, in the sense that it is consistent with our previous constitutive modeling

approach, as well as in the sense that it appears equally applicable to conventional and highly

crosslinked UHMWPE.

The augmented HM has the same foundation as our previous modeling efforts [6]. The

main difference is that the augmented model now also incorporates relative sliding (reptation) of

the molecular chains of the backstress network that carries the main load at moderate to large

deformations. The results from this study implicitly show that the relative sliding of the

molecular chains in the back stress network is a unified feature of the UHMWPE, both

12

uncrosslinked and crosslinked, mechanical behavior. Figures 2 to 7 show that the HM accurately

captures large strain tension and small strain cyclic loading of conventional and highly

crosslinked UHMWPEs. These tests are straightforward to perform and sufficient for calibrating

the model.

In this study, we have focused on creating a mathematical representation of the

deformation resistance and flow characteristics for conventional and highly crosslinked

UHMWPE at the molecular level. This effort has focused on the physics of the deformation

mechanisms by establishing the framework and equations necessary to model the behavior on the

macroscale. As already mentioned, to use the constitutive model for a given material requires a

calibration step where material specific parameters are determined. A variety of numerical

methods may be used to determine the material specific parameters for a constitutive theory. In

our study, we chose to employ numerical optimization techniques to identify the material

parameters for our constitutive theory, as opposed to graphical techniques or simple curve fitting.

Of greater importance is how well the physics-inspired model framework represents the

governing micromechanisms, and ultimately, how well the model can predict the behavior of a

given material under different loading conditions than that for which the model was originally

calibrated. The simulations of the small punch test performed in this study demonstrate that our

modeling approach provides satisfactory and valid predictions of large-deformation multiaxial

behavior of conventional and highly crosslinked UHMWPEs. Thus, our augmented hybrid model

yields similar consistent and valid results under large-deformation multiaxial behavior as were

observed with our earlier constitutive theory [6]. However, we have now introduced a key new

feature to our augmented constitutive theory, which was not incorporated in the previous hybrid

model; namely, the new ability to accurately capture the nonlinear unloading behavior of

conventional and highly crosslinked UHMWPEs.

In summary, the augmented HM is an accurate, validated and unified material model for

simulating the loading as well as the unloading behavior of conventional and highly crosslinked

UHMWPE used in joint replacements. In the present work, we have restricted our attention to

cyclic uniaxial mechanical behavior at room temperature. Based on earlier testing [12], some

adjustment of properties is expected for body temperature due to thermal softening.

13

Consequently, research is ongoing to evaluate the performance of the augmented HM at body

temperature during cyclic multiaxial loading. In addition, fatigue, fracture, and ultimately wear

are targeted to be studied using the augmented HM as an essential tool.

Acknowledgement

This work was supported by NIH Grant 1 R01 AR 47192. Special thanks for M.

Villarraga and L. Ciccarelli for assistance with the uniaxial and small punch testing.

14

References

[1] Kurtz SM, Muratoglu OK, Evans M, Edidin AA. Advances in the processing,

sterilization, and crosslinking of ultra- high molecular weight polyethylene for total joint

arthroplasty. Biomaterials 1999;20: 1659-1688.

[2] Muratoglu OK, Kurtz SM. Alternative bearing surfaces in hip replacement. In: R. Sinha,

editor Hip Replacement: Current Trends and Controversies. New York: Marcel Dekker,

2002.

[3] Wang A, Essner A, Polineni VK, Stark C, Dumbleton JH. Lubrication and wear of ultra-

high molecular weight polyethylene in total joint replacements. Tribology International

1998;31: 17-33.

[4] Pooley CM, Tabor D. Friction and molecular structure: the behavior of some

thermoplastics. Proc. R. Soc. Lond. 1972;329: 251-274.

[5] Edidin AA, Pruitt L, Jewett CW, Crane DJ, Roberts D, Kurtz SM. Plasticity-induced

damage layer is a precursor to wear in radiation- cross-linked UHMWPE acetabular

components for total hip replacement. Ultra-high-molecular-weight polyethylene. J

Arthroplasty 1999;14: 616-627.

[6] Bergström JS, Rimnac CM, Kurtz SM. Prediction of multiaxial behavior for conventional

and highly crosslinked UHMWPE using a hybrid constitutive model. Biomaterials

2003;24: 1365-1380.

[7] Bergström JS, Kurtz SM, Rimnac CM, Edidin AA. Constitutive modeling of ultra-high

molecular weight polyethylene under large-deformation and cyclic loading conditions.

Biomaterials 2002;23: 2329-2343.

15

[8] Gurtin ME. An introduction to continuum mechanics (Academic Press, Inc., 1981).

[9] Arruda EM, Boyce MC. A Three-Dimensional Constitutive Model for the Large Stretch

Behavior of Rubber Elastic Materials. J. Mech. Phys. Solids 1993;41: 389-412.

[10] Bergström JS, Boyce MC. Large strain time-dependent behavior of filled elastomers.

Mech. Mater. 2000;32: 627-644.

[11] Bergström JS, Boyce MC. Constitutive Modelling of the Large Strain Time-Dependent

Behavior of Elastomers. J. Mech. Phys. Solids 1998;46: 931-954.

[12] Kurtz SM, Villarraga ML, Herr MP, Bergström JS, Rimnac CM, Edidin AA.

Thermomechanical behavior of virgin and highly crosslinked ultra-high molecular weight

polyethylene used in total joint replacements. Biomaterials 2002;23: 3681-3697.

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List of Tables

Table 1. Hybrid Model (HM) material parameters for the three different types of GUR 1050. Ee

is the Young’s modulus, e

! is the Poisson’s ratio, µA is the shear modulus of network A,

lock

A! is the locking stretch of network A, "A is the bulk modulus of network A, qA is a

parameter specifying the asymmetry between tension and compression, sBi is a parameter

that controls the initial flow resistance, sBf is a parameter that controls the final flow

resistance, pB is a parameter that controls the distributed yielding, base

B! is a parameter that

control the yield strength of network B, mB is a parameter controlling the rate-dependence of

network B, base

C! is a parameter that controls the yield strength of network C, and mC is a

parameter that controls the rate-dependence of network B. Parameters that are unique for

each material are written in bold text................................................................................. 19

Table 2. Summary of the performance of the HM to predict the response of GUR 1050........... 20

17

List of Figures

Figure 1. (a) Rheological representation of the augmented HM. (b) Deformation map showing

the kinematics and stress tensors used in the augmented HM. These figures illustrate how

the model represents the viscoplastic flow, and how the deformation state is generalized into

three dimensions. .............................................................................................................. 21

Figure 2. Comparison between experimental uniaxial compression data and predictions from the

HM for GUR 1050 (30 kGy, %-N2). The three data sets are for true strain rates of 0.007/s,

0.018/s and 0.035/s. .......................................................................................................... 22

Figure 3. Comparison between experimental uniaxial cyclic tension and compression data and

predictions from the HM for GUR 1050 (30 kGy, %-N2). The experimental data correspond

to a true strain rate of 0.05/s. ............................................................................................. 23


HM for GUR 1050 (100 kGy, 110°C). The three data sets are for true strain rates of

0.007/s, 0.018/s and 0.035/s. ............................................................................................. 24


predictions from the HM for GUR 1050 (100 kGy, 110°C). The experimental data

correspond to a true strain rate of 0.05/s. ........................................................................... 25



0.007/s, 0.018/s and 0.035/s. ............................................................................................. 26



correspond to a true strain rate of 0.05/s. ........................................................................... 27

18

Figure 8. Comparison between experimental cyclic tension and compression data predictions

from the original HM [6] for GUR 1050 (30 kGy, %-N2). The experimental data correspond

to a true strain rate of 0.05/s. ............................................................................................. 28

Figure 9. Comparison between experimental small punch data and predictions from the HM for

GUR 1050 (30 kGy, %-N2). The experimental data correspond to a punch rate of 0.5

mm/min. The figure also shows the FE mesh that was used in the small punch simulations.

......................................................................................................................................... 29


GUR 1050 (100 kGy, 110°C). The experimental data correspond to a punch rate of 0.5

mm/min. ........................................................................................................................... 30


GUR 1050 (100 kGy, 150°C). The experimental data correspond to a punch rate of 0.5

mm/min. ........................................................................................................................... 31

19

Table 1. Hybrid Model (HM) material parameters for the three different types of GUR 1050. Ee

is the Young’s modulus, e

! is the Poisson’s ratio, µA is the shear modulus of network

A, lock

A! is the locking stretch of network A, "A is the bulk modulus of network A, qA is

a parameter specifying the asymmetry between tension and compression, sBi is a

parameter that controls the initial flow resistance, sBf is a parameter that controls the

final flow resistance, pB is a parameter that controls the distributed yielding, base

B! is a

parameter that control the yield strength of network B, mB is a parameter controlling

the rate-dependence of network B, base

C! is a parameter that controls the yield strength

of network C, and mC is a parameter that controls the rate-dependence of network B.

Parameters that are unique for each material are written in bold text.

Material

Parameter

30 kGy %-N2 100 kGy %

110°C

100 kGy %

150°C

Ee (MPa) 2020 2009 1270

#e 0.46 0.46 0.46

µA (MPa) 8.22 10.15 8.14

lock

Aë 4.40 2.80 2.52

"A (MPa) 2000 2000 2000

qA 0.20 0.20 0.20

sBi 40.0 40.0 40.0

sBf 10.0 10.0 10.0

pB 27.0 27.0 27.0

!Bbase

(MPa) 25.0 26.2 20.7

mB 9.50 9.50 9.50

$Cbase

(MPa) 8.00 8.00 8.00

mC 3.30 3.30 3.30

20

Table 2. Summary of the performance of the HM to predict the response of GUR 1050.

GUR1050

Material

Test Mode r2-value

uniaxial tension 0.978

30 kGy %-N2 uniaxial cyclic loading 0.984

small punch 0.937


100 kGy % 110°C uniaxial cyclic loading 0.988

small punch 0.960


100 kGy % 150°C uniaxial cyclic loading 0.990

small punch 0.948

21

(a)

(b)

Figure 1. (a) Rheological representation of the augmented HM. (b) Deformation map showing

the kinematics and stress tensors used in the augmented HM. These figures illustrate

how the model represents the viscoplastic flow, and how the deformation state is

generalized into three dimensions.

22


HM for GUR 1050 (30 kGy, %-N2). The three data sets are for true strain rates of

0.007/s, 0.018/s and 0.035/s.

23


predictions from the HM for GUR 1050 (30 kGy, %-N2). The experimental data

correspond to a true strain rate of 0.05/s.

24



0.007/s, 0.018/s and 0.035/s.

25




26



0.007/s, 0.018/s and 0.035/s.

27




28

Figure 8. Comparison between experimental cyclic tension and compression data predictions

from the original HM [6] for GUR 1050 (30 kGy, %-N2). The experimental data


29


GUR 1050 (30 kGy, %-N2). The experimental data correspond to a punch rate of 0.5

mm/min. The figure also shows the FE mesh that was used in the small punch

simulations.

30


GUR 1050 (100 kGy, 110°C). The experimental data correspond to a punch rate of

0.5 mm/min.

31


GUR 1050 (100 kGy, 150°C). The experimental data correspond to a punch rate of

0.5 mm/min.

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