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Journal of Biomechanics 37 (2004) 127134
Design and numerical implementation of a 3-D non-linear viscoelastic
constitutive model for brain tissue during impact
D.W.A. Brandsa,*, G.W.M. Petersb, P.H.M. Bovendeerdb
aDepartment of Mechanical Engineering, Division of Computational and Experimental Mechanics, Eindhoven University of Technology, P.O. Box 513,
5600 MB Eindhoven, The NetherlandsbDepartment of Biomedical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Abstract
Finite Element (FE) head models are often used to understand mechanical response of the head and its contents during impactloading in the head. Current FE models do not account for non-linear viscoelastic material behavior of brain tissue. We developed a
new non-linear viscoelastic material model for brain tissue and implemented it in an explicit FE code. To obtain sufficient numerical
accuracy for modeling the nearly incompressible brain tissue, deviatoric and volumetric stress contributions are separated.
Deviatoric stress is modeled in a non-linear viscoelastic differential form. Volumetric behavior is assumed linearly elastic. Linear
viscoelastic material parameters were derived from published data on oscillatory experiments, and from ultrasonic experiments.
Additionally, non-linear parameters were derived from stress relaxation (SR) experiments at shear strains up to 20%. The model was
tested by simulating the transient phase in the SR experiments not used in parameter determination (strains up to 20%, strain rates
up to 8 s1). Both time- and strain-dependent behavior were predicted accurately R2 >0:96 for strain and strain rates applied.However, the stress was overestimated systematically by approximately 31% independent of strain(rate) applied. This is probably
caused by limitations of the experimental data at hand.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Finite element modeling; Impact biomechanics; Non-linear viscoelastic constitutive model; Brain modelling
1. Introduction
Traumatic brain injury (TBI) caused by a mechanical
insult on the head, for example during traffic accidents,
sport accidents or falls, causes high mortality and
disability (Brooks et al., 1997; Viano et al., 1997;
Waxweiler et al., 1995). TBI occurs when the local
mechanical load, exerted on the brain tissue, exceeds
certain tolerance levels. Understanding how an external
mechanical load on a head is transferred to a local
mechanical load in the brain is needed to improve injury
protecting devices and diagnostic methods. To obtain
this understanding, finite element (FE) modeling is often
used (e.g. in Bandak and Eppinger, 1994; Claessens
et al., 1997; Turquier et al., 1996; Zhang et al., 2001).
Current FE head models contain a detailed geometrical
description of the intracranial contents but lack an
accurate description of brain material behavior.
At strain and strain rate levels associated with TBI
(approximately 20% (Bain and Meaney, 2000;Galbraith
et al., 1993;Schreiber et al., 1997) and typically 20 s1),
brain tissue behaves as a non-linear, viscoelastic material
(Bilston et al., 2001;Estes and McElhaney, 1970;Peters
et al., 1997). In stress relaxation (SR) experiments shear
softening occurs, i.e. the stiffness decreases as strain
increases (Arbogast et al., 1995; Bilston et al., 2001;
Brands et al., 2000; Prange et al., 2000, 2002) while
Darvish and Crandall (2001)found that shear hardening
occurred when increasing strains in oscillatory experi-
ments at frequencies exceeding 44 Hz;indicating full non-linear material behavior. The bulk modulus of brain
tissue is about 106 times higher than the shear modulus
(Etoh et al., 1994;Goldman and Hueter, 1956) indicating
nearly incompressible material behavior. This provides
the following requirements for a material model for use in
FE modeling brain tissue during impacts: accurate
replication of the non-linear viscoelastic behavior in
shear-like deformations for strain(rate)s up to 20%
20 s1 and special precautions for accurate modeling
of the nearly incompressible behavior.
ARTICLE IN PRESS
*Corresponding author. Tel.: +31-40-247-3135; fax: +31-40-244-
7355.
E-mail address: [email protected] (D.W.A. Brands).
0021-9290/$- see front matterr 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0021-9290(03)00243-4
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In this paper the development and FE implementation
of a new non-linear viscoelastic constitutive model for
modeling brain tissue is presented. The model is written
in a differential formulation as opposed to the QLV
formulations currently implemented in explicit FE
packages. Furthermore, the formulation is such that
the nearly incompressible material behavior is modeledwith sufficient numerical accuracy. Material parameters
are determined from two sets of shear measurements
with porcine brain tissue on a rotational viscometer and
from ultrasonic experiment data. The model and its
implementation are tested using a three-dimensional FE
simulation of transient shear experiments.
2. Methods
2.1. Kinematics
The stress in an arbitrary solid material is determined
by changes of volume and shape, described by the
deformation gradient tensor F: Deformation gradienttensor Fis split multiplicatively into an elastic part, Fe;and an inelastic part, Fp:
d~xx F d~xx0; F Fe Fp; 1
where d~xx0 and d~xx represent a material line element inundeformed stateC0and deformed state Ct;respectively(see Fig. 1). The inelastic contribution refers to the
deformation (with respect to the undeformed state), of
the relaxed stress-free configurationCp;which is defined
as a fictitious state that would be recovered instanta-
neously when all loads were removed from the material
element.
For later use in the constitutive model, we introduce
the, recoverable, elastic Finger tensor Be; and itsinvariantsI1;2;3:
Be Fe Fce;
I1traceBe;
I2 12traceBe
2
traceB2
e ;I3detBe: 2
To separate changes in volume from changes in shape,
we introduce the isochoric elastic Finger tensor %Be and
its associated invariants as
%Be I1=33 Be; %I1;2 I1;2 %Be; %I3 1: 3
The description of the rate of deformation is based on
the velocity gradient tensor, L which, is decomposed
additively into an elastic part, Le and an inelastic part,
Lp;
L F F1; L Le L
p;
Le Fe F1e ; Lp Fe Fp F
1p F
1e : 4
Both parts, Le and Lp; are decomposed additively as
L D W; D 12L LT; W 1
2L LT 5
with D; the symmetric rate of deformation tensor (LT
denotes the transpose of tensor L) and W; the skew-symmetric spin tensor. To obtain a unique relaxed
stress-free state, Cp; it is assumed that the inelasticdeformation occurs spin-free,
We W and Wp 0: 6
2.2. The constitutive model
For accurate modeling of the nearly incompressible
brain tissue, the Cauchy stress, r; is written as the sumof a volumetric part, rv; which depends on volumetricchanges only, and a deviatoric part, rd;which depend onchange of shape only:
r rv rd: 7
The volumetric part of the stress, rv; is assumed linearelastic, implying:
r
v
Kffiffiffiffi
I3p
1I 8with unitary tensor Iand bulk modulus, K:
The deviatoric part of the stress is modeled non-linear
viscoelastic. It is decomposed in a number of viscoelastic
modes, rdi:
rd
XNi1
rdi: 9
The number of modes used, N; is determined by thefrequency range for which the model has to be valid. To
derive rdi we apply Eq. (1) for each mode separately
obtaining F Fe;i Fp;i: We then define the strain rate
ARTICLE IN PRESS
C0C t
CP
F
Fp Fe
Fig. 1. Graphical representation of the multiplicative decomposition
of the deformation gradient tensor F: The inelastic part Fp of Ftransforms the undeformed stateC0 to a relaxed stress-free configura-
tion, Cp; which is a fictitious state that would be recoveredinstantaneously when all loads were removed from the material
element. The elastic part FeofFtransforms the stress free state, Cp;tothe deformed state Ct: To obtain a unique fictitious stress free state,Cp; it is assumed that the inelastic deformation occurs spin free i.e. allrotations must be accounted for in the constitutive model governing
the elastic part.
D.W.A. Brands et al. / Journal of Biomechanics 37 (2004) 127134128
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dependent behavior, governed by viscosity, Z i; as
Dp;i r
di
2Zi: 10
The elastic behavior is modeled using a strain energy
density function (SEDF), Wi:
Wi C10;i %I1;i 3 C01;i %I2;i 3 C20;i %I1;i 32
C02;i %I2;i 32; 11
where %I1;i and %I2;irepresent invariants of the tensor %Be;i(Eq. (3)).
The deviatoric part of the Cauchy stress tensor,
follows fromW:
rdi
2ffiffiffiffiffiffiI3;i
p fC10;i 2C20;i %I1;i 3g %Bde;i
2
ffiffiffiffiffiffiI3;ip fC01;i 2C02;i %I2;i 3g %B
1e;i
d: 12
2.3. Numerical implementation
The time evolution of stress and strain in each mode,
is described by an evolution law, based on kinematics
only. For each mode, the inelastic right CauchyGreen
tensor, Cp; is defined as
Cp FTp Fp F
T B1e F: 13
Taking the time derivative of Cp and using the
requirement of spin free inelastic deformation, Eq. (6),
provides a evolution equation insensitive for large rigid-body rotation and translation
Cp 2Cp F1 Dp F: 14
This evolution equation is used for updating Cpnumerically in the time integration procedure. At the
new time increment, the updated elastic Finger tensor
then follows from Be F C1p F
T: Application ofBein constitutive equations (8) and (12) yields updated rdiand rv: Finally, the inelastic rate of deformation Dp isdetermined from Eq. (10) and serves as basis for
proceeding to the next time step using Eq. (14) again.
This procedure is implemented in an explicit FE Code
typically used for crash impact simulations (madymo,
TNO-Automotive, 1999).
2.4. Determination of material parameters
The multi-mode model contains one material para-
meter for the volumetric behavior K; and fiveparameters for each viscoelastic mode C10;i; C01;i; C20;i;C02;i and Zi).
The bulk modulus, K;is determined from the velocityof dilatational waves,cp;measured in brain tissue (Etohet al., 1994; Goldman and Hueter, 1956) and mass
density r; using:
Kc2pr: 15
Linear parameters C10;i; C01;i and Zi were determinedfrom data of oscillatory shear experiments or dynamic
frequency sweeps (DFS) with porcine brain tissue on a
rotational plate-plate viscometer (ARES, RheometricScientific, 1993) published previously (Brands et al.,
2000). A strain amplitude of 0.01 was applied in a
frequency range of 1:6216 Hz at various temperaturesand a master curve, valid for frequencies ranging from
1.6 to 684 Hz at 37C; was constructed using the timetemperature superposition principle (TTS) first applied
to brain tissue in (Peters et al., 1997). The results of these
experiments were presented in terms of the storage
modulus , G0 and the loss modulus, G00;as a function offrequency o: For small shear strains, the non-linearviscoelastic constitutive equations (10) and (12) reduce
to a linear multimode Maxwell model for which, G0 and,
G00; are found as:
G0 Xni0
Gil2io
2
1l2io2; 16
G00 Xni0
Gilio
1l2io2
with Gi2C10;i C01;i the shear modulus and li
Zi=Githe relaxation time of mode i:Note that, only the sumC10;i C01;ican be obtained
from the shear experiments. We uniquely determined
C10;iand C01;iby assuming them to be equal as inMiller
and Chinzei (1997).
Next, the parameters describing non-linear behavior,
C20;i and C02;i; were derived from large strain stressrelaxation experiments(SR), using the linear parameters
determined before. These were performed, with the same
sample, within minutes after finishing the DFS at shear
strains of 0.05, 0.1 and 0.2. Experiments were finished
within 4 h after sacrifice.
In the SR experiments it was found that, once the
strain had obtained a constant value, (within 0:1 s), thestress could be written as a function of the shear strain
at the plate edge, gR; a strain-dependent normalized
stiffness (damping function), hagR; and the linearrelaxation modulus, Gt:
tat; gR GthagRgRt; 17
where
hagR
tagRgR
limgR-0tagRgR
: 18
Before using these data in the present work, they were
corrected for the radial inhomogeneity in the strain field,
according to the procedure described in the appendix, to
obtain the true damping function htruegR:
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Evaluation of Eq. (12) for simple shear strain provides
the following expression for normalized stiffness for
each mode:
hig 1 2g2fnls;i; fnls;i
C02;i C20;i
C01;i C10;i: 19
The non-linear shear parameter, fnls;i describes thenonlinearity in mode i: Since htruegR is independentof time, we assume equal nonlinearity for each mode, i.e.
hig htruegR providing fnls;ifnls:As for the linear parameters, also for the nonlinear
parameters we uniquely determined C20;i and C02;i by
assuming them to be equal.
2.5. Test of the constitutive model
The model is tested by a three-dimensional FE
simulation of the SR experiments on the rotational
rheometer including the transient strain onset that has
not been used for material parameter determination. In
this manner we test the constitutive model and its
numerical implementation as well as the validity of the
strain correction method applied to the experimental
data.
The geometry of the brain sample is modeled by a
quarter cylinder with diameter 25 mm and thickness
2 mm corresponding to the typical sample size used in
the experiments. Spatial discretisation is obtained using
1288 brick elements with linear interpolation functions
(Fig. 2). A preliminary convergence study revealed a
maximum deviation of 2%. To prevent mesh locking,
likely to occur due to the nearly incompressible materialbehavior, reduced spatial integration is used. Symmetry
boundary conditions are applied on the cross-sectional
planes. The lower plane is rotated according to the
experimental data. The upper-plane is rigidly supported.
To compare numerical and experimental results, the
reaction torque at the upper plate,T; is determined andthe apparent stress, tat; gR; is calculated, as in theexperiments, using
tat; gR 2T
pR3 20
with, R the radius of the sample. Brain tissue material
parameters shown inTable 1are applied. The time step
used is limited by the conditional stability of the explicit
central difference time integration scheme used in the
code and is set to 1:66107 s:
3. Results
3.1. Determination of material parameters
With cp 155972 m=s (average7range in literaturevalues) and r 1040 kg=m3 Eq. (15) provides a bulkmodulus of 2:3 GPa:
The small strain DFS experimental data could be
fitted well over the complete frequency range using four
viscoelastic modes with first-order material parameters
shown in Table 1 (see Fig. 3). The error in G0 equals
2:079:6% (average7standard deviation) and reachesmaximum absolute values of approximately 20% at
upper and lower end of the frequency spectrum (684 and
1:6 Hz;respectively) . The loss modulus is fitted with anaverage error of 3:574:6%:The error values remain lessthan 10% and are approximately distributed at random.
Application of more modes yielded no improvement to
the results while less modes deteriorated the accuracy of
the solution. The damping function obtained from the
SR data shows that shear softening is more prominent in
the normalized stiffness data which has been corrected
for strain inhomogeneity than in the raw experimental
data (Fig. 4). The shear softening equals 37% at 20%
strain. Fitting Eq. (19) to this data with fnls 4:49provides a relative error less than 2%.
ARTICLE IN PRESS
Fig. 2. Graphical representation of the three-dimensional mesh (Top
view and side view of cross-section shown). A quarter cylinder is
modeled using 1288 brick elements with 1672 nodes. Symmetry
boundary conditions are applied on the cross-sectional planes.
Table 1
Material parameters obtained from fitting the 4-mode non-linear
viscoelastic constitutive model on brain tissue DFS and SR shear data
(sample G2 inBrands et al. (2000))
Mode MooneyRivlin parameters Viscosity Bulk modulus
i C10;i C01;i(Pa) C20;i C02;i(Pa) Zi(Pas) K(GPa)
0 85.96 386:0 N 2.31 67.27 302:0 18.9 2 80.66 362:2 2.46 3 106.8 479:5 0.606 4 824.9 3704 0.0403
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3.2. Test of the constitutive model
The three-dimensional FE simulation of the rota-
tional viscometer experiments, with material parameters
in Table 1, provides predicted stress values which are
systematically higher than experimental results (Fig. 5).
A linear regression analysis (Statgraphics 5.1), shows
good correlation between numerical and experimental
results R2 >96%: The ratio of predicted over experi-mental stress equals 1:3370:28; 1:3470:25; 1:2670:18for 5%, 10% and 20% strain, respectively (average7
standard deviation) but does not statistically depend on
strain at the 95% confidence level (Students-T-test).
The model thus provides good prediction of time-
dependent behavior but an average over estimation of
the stress by 31% which does not depend on strain.
Scaling the simulation results by a single factor of 0.7
indeed provides much better simulation results (strain
averaged maximum relative error equals 7% at 0:2 s(Fig. 6)).
4. Discussion
A non-linear viscoelastic material law for brain tissue
is developed and implemented in a FE code, to improve
capabilities of FE head models to predict TBI in traffic
accidents, sport accidents or falls.
To prevent that errors in the computation of the
hydrostatic stress interfere with deviatoric stress, we
decoupled hydrostatic and deviatoric behavior by using
isochoric strain measures for the deviatoric behavior.
This is important since, due to the nearly incompressible
material behavior, deviatoric stresses are typically six
orders of magnitude smaller than hydrostatic stresses. A
drawback of this approach is that it introduces the
additional assumption that shear behavior is indepen-
dent of volumetric compression. However, to our
knowledge, evidence of such dependency is not reported
in literature.
The hydrostatic stress is modeled linearly elastic.
Linearity seems valid as volumetric strains are small at
pressures expected during traffic impacts (typically
105 Pa (Nahum et al., 1977)). Viscous effects are
neglected, since ultrasonic experiments indicate signifi-
cant damping due to hydrostatic deformation only at
ARTICLE IN PRESS
0 5 10 15 200.4
0.6
0.8
1
[%]
h()
Raw dataCorrected dataFit result
Fig. 4. Normalized stiffness,hg;of sample G2 inBrands et al. (2000)obtained from the constant strain part of SR experiments on a plate
plate viscometer. The raw experimental data is compared with data
corrected for the non-homogeneous radial strain field by applying
second-order MooneyRivlin (MR2) model in Eq. (A.1). To obtain
realistic fits of the normalized stiffness, (i.e. limg-0 hg 1), the
normalized stiffness value at 5% strain is assumed to be valid at 1%
strain also. Shear softening is more prominent in corrected data whilefitted result shows that the MR2 model provides excellent fit.
0 0.1 0.2 0.3 0.4
0
10
20
30
Time [s]
Strain[%]
0 0.1 0.2 0.3 0.4
0
20
40
60
80
100
Time [s]
Stress[Pa]
0=0.05
0=0.1
0=0.2
NumExp
Fig. 5. Experimental apparent shear stresses (sample G2 in Brands
et al. (2000)), at 5%, 10% and 20% maximum edge strain, together
with the FE simulation results. Left: sample edge shear strain histories,
Right: apparent shear stresses, Num result of three-dimensional FE
model, Exp experiment: The numerical predictions are system-atically higher than the experimental results regardless of strain and
time.
0 0.1 0.2 0.3 0.40
20
40
60
80
Time [s]
Stress[Pa]
0=0.05
0=0.1
0=0.2
NumExp
0 0.05 0.10
20
40
60
80
0=0.05
0=0.1
0=0.2
Time [s]
Stress[Pa]
Fig. 6. Investigation of error source of simulation results versus
experimental data. Left: simulation results multiplied by a constant
factorX0:7 showing much better simulation results. Right: closeup of the stress history during transient part of the deformation (scaled
result).
100
101
102
10310
2
103
104
Frequency [Hz]
G [Pa]
ExpFit
100
101
102
10310
2
103
104
Frequency [Hz]
G [Pa]
Fig. 3. Master curves of storage and loss modulus G0 and G00 from
sample G2 obtained from DFS with TTS applied, at 1% shear strain in
Brands et al. (2000) together with four mode fit showing good
resemblance of small strain behavior.
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frequencies above 50 kHz (Etoh et al., 1994), well above
1000 Hz typical for impacts of interest in this study.
The deviatoric behavior is described using a differ-
ential formulation. Differential formulations have been
proposed before (Bilston et al., 2001; Pamidi and
Advani, 1978). However, these predict infinite stress at
instantaneous loading (Pamidi and Advani, 1978) orwhere not implemented in a FE package (Bilston et al.,
2001; Pamidi and Advani, 1978). Based on the experi-
mental finding of strain and time separability in stress
relaxation (SR) experiments, we could have opted
for a quasi-linear viscoelastic (QLV) model (Mendis
et al., 1995; Miller and Chinzei, 1997; Prange et al.,
2002).
However, at frequencies above 44 Hz fully non-linear
material behavior might be present (Darvish and
Crandall, 2001) which cannot be described with QLV
theory but can be included in the present formulation.
The elastic behavior was modelled by a hyperelastic
SEDF written in polynomial form with integer powers
of first and second invariants of the isochoric Finger
tensor. We took a second-order model, the simplest
model which predicts non-linear shear behavior. For
correct prediction of shear softening observed (Fig. 4)
negative second-order parameters were required. If the
model is used outside its range of validity (i.e. strains
more than 20%) this might result in negative stiffness
when shear strains exceed 27%.
Negative stiffness in simulations in a head model, in
which it is unknown beforehand which strain levels will
occur, can be avoided by extending the model with a
third order MooneyRivlin term with suitable para-meter settings as done inBrands (2002)andBrands et al.
(2002). However, for realistic parameter settings, experi-
ments at higher strains must be performed. Negative
stiffness can also be avoided by using an Ogden SEDF
with fractional powers of stretches Ogden (1972),
applied to brain tissue in Miller and Chinzei (2002)
andPrange et al. (2002, 2000).However, we found that
this SEDF cannot predict the amount of shear softening
observed in our experiments.
The inelastic behavior is modeled by a simple linear
Newtonian law. To describe the viscoelastic behavior in
a broad frequency range, we used a multi-mode
approach with discrete time constants as opposed to a
relaxation spectrum, i.e. some arbitrarily continuous
function of relaxation time (Macosko, 1994). A draw-
back of this method that each additional mode
introduces five new material parameters in the model.
To reduce the number of independent parameters we
assumed the ratio between first- and second-order
MooneyRivlin parameters governing the elastic
strain-dependent behavior in each mode to be constant,
based on the observation of (approximately) time-strain
separability in SR data (Bilston et al., 2001; Brands et al.,
2000;Prange et al., 2002).
We presented a new approach for determining
material parameters of brain tissue on a rotational
viscometer. Often all material parameters are deter-
mined from of SR experiments only while assuming
perfect instantaneous strain application (Bilston et al.,
2001;Prange et al., 2002). A drawback of this approach
is that time constants valid during the rise time of thestrain cannot be determined. Instead, we used two data
sets from a single sample (Brands et al., 2000). Linear
viscoelastic material parameters were fitted to DFS
results valid for small strains (0.01) but high frequencies
(up to 684 Hz) while parameters C20 C02;describingthe strain-dependent decrease of stiffness, were deter-
mined from the constant strain part of SR experiments.
This method implies that time constants, determined for
small strains and high frequencies, are also valid at large
strains. Evaluating the predicted material behavior
during transient strain application in the SR serves as
a test of this assumption.
During simulation of the SR experiments, stress
values were overestimated by 31%, independent of the
strain level applied, also during transient onset of the
strain. A potential explanation is that we assumed no
shear softening for strains between 1% (at which the
DFS is performed) and 5%, the lowest strain value in
the SR experiments, resulting in a normalized stiffness
of 1.0 (Fig. 4). As a result, the fitted model predicts only
2% shear softening at 5% strain. This softening is low
compared to the 3050% softening reported in litera-
ture, when increasing shear strains from 1% to 5%
(Brands et al., 1999;Bilston et al., 2001).
Correct strain-dependent behavior shows that thecorrection method for the radially inhomogeneous
strain field is indeed valid. Correct time-dependent
behavior indicates that time constants obtained from
the DFS results are valid for frequencies over 20 Hz
also. The DFS results were composed from isothermal
DFS results up to 16 Hz; using the time temperaturesuperpositioning (TTS) principle. This provides evi-
dence on the validity of TTS for determining time
constants of brain tissue for given model, as well as the
validity of these time constants determined at small
strains, for large strains and high frequencies. However,
strain rates applied range up to 8 s 1 which is less than
the rate values expected during injurious impacts (15
21 s1 Brands (2002)). The behavior at higher strain
rates remains to be investigated.
It is impossible to fully characterize the three-
dimensional non-linear viscoelastic behavior of a
material using simple shear experiments from a rota-
tional viscometer. Effects of anisotropic material beha-
vior (Prange et al., 2002) cannot be determined. Also,
MooneyRivlin parameters C10;i; C01;i; C20;i and C02;icould not be defined uniquely. The ratio ratios C10=C01in a first-order MooneyRivlin model did have a
significant effect on stress response in free compression
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experiments byMendis et al. (1995). Adjusting the ratios
Cj0;i=C0j;i while keeping Cj0;i C0j;iconstant provides apowerful tool to tune free compression results without
influencing shear behavior. The performance of the
model for other deformation modes remains to be
investigated.
In this paper, we developed a non-linear viscoelasticmaterial law for brain tissue and implemented it in a FE
code, to improve capabilities of FE head models to
predict TBI in traffic accidents, falls or sports. We
provided an approach to determine material parameters
for the model and found that the model is capable of
predicting realistic shear material behavior of brain
tissue observed in SR experiments including the
transient application of the strain. However, full
characterization of the material properties is not
possible using experimental shear data at hand and
requires extra experiments.
Acknowledgements
The authors would like to acknowledge TNO Prins
Maurits Laboratory and Ford Motor Company for
their financial support of this research and P. Nauta of
TNO Automotive for his assistance during the numer-
ical implementation of the model.
Appendix. A
Brands et al. (2000) determined the stress assuminglinear viscoelastic theory, thus neglecting the effect of
the radial inhomogeneous strain field between the plates
of the rotational viscometer. For any isotropic material,
the true experimental normalized stiffness htruegR can
be obtained when the strain rate is zero, using Soskey
and Winter (1984),
htruegR hagR 1q ln hagR
4q ln gR
: A:1
When experimental data at sufficient strain levels is
present, q ln hagR=4q ln gR can be estimated without
making assumptions on material behavior. As we havedata at three strain values only, we fitted Eq. (19) to
experimentally found hag and applied Eq. (21) to
obtain htrueg which is then fitted to Eq. (19) again
to obtain the correct material parameters.
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