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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:

ARTICLE IN PRESS

D.W.A. Brands et al. / Journal of Biomechanics 37 (2004) 127134 129


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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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