-
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
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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
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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
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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.
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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# $= +% &+T T F T F , (2)
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( )*
2* * *2
2 1
2
3
pp p p
I A
IIµ! "
= # #$ %& '
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)
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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.
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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
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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.
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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
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(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.
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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
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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.
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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.
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References
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processing,
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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
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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
Figure 4. Comparison between experimental uniaxial compression
data and predictions from the
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
Figure 5. Comparison between experimental uniaxial cyclic
tension and compression data and
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
Figure 6. Comparison between experimental uniaxial compression
data and predictions from the
HM for GUR 1050 (100 kGy, 150°C). The three data sets are for
true strain rates of
0.007/s, 0.018/s and 0.035/s.
.............................................................................................
26
Figure 7. Comparison between experimental uniaxial cyclic
tension and compression data and
predictions from the HM for GUR 1050 (100 kGy, 150°C). The
experimental data
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
Figure 10. Comparison between experimental small punch data and
predictions from the HM for
GUR 1050 (100 kGy, 110°C). The experimental data correspond to a
punch rate of 0.5
mm/min.
...........................................................................................................................
30
Figure 11. Comparison between experimental small punch data and
predictions from the HM for
GUR 1050 (100 kGy, 150°C). The experimental data correspond to a
punch rate of 0.5
mm/min.
...........................................................................................................................
31
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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
uniaxial tension 0.987
100 kGy % 110°C uniaxial cyclic loading 0.988
small punch 0.960
uniaxial tension 0.980
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
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.
-
23
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.
-
24
Figure 4. Comparison between experimental uniaxial compression
data and predictions from the
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.
-
25
Figure 5. Comparison between experimental uniaxial cyclic
tension and compression data and
predictions from the HM for GUR 1050 (100 kGy, 110°C). The
experimental data
correspond to a true strain rate of 0.05/s.
-
26
Figure 6. Comparison between experimental uniaxial compression
data and predictions from the
HM for GUR 1050 (100 kGy, 150°C). The three data sets are for
true strain rates of
0.007/s, 0.018/s and 0.035/s.
-
27
Figure 7. Comparison between experimental uniaxial cyclic
tension and compression data and
predictions from the HM for GUR 1050 (100 kGy, 150°C). The
experimental data
correspond to a true strain rate of 0.05/s.
-
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
correspond to a true strain rate of 0.05/s.
-
29
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.
-
30
Figure 10. Comparison between experimental small punch data and
predictions from the HM for
GUR 1050 (100 kGy, 110°C). The experimental data correspond to a
punch rate of
0.5 mm/min.
-
31
Figure 11. Comparison between experimental small punch data and
predictions from the HM for
GUR 1050 (100 kGy, 150°C). The experimental data correspond to a
punch rate of
0.5 mm/min.