405

International Journal of Mechanical and Materials Engineering (IJMME), Vol.6 (2011), No.3, 405-413

STRESS WAVE METHOD FOR IDENTIFICATION OF VISCOELASTIC MATERIAL

PROPERTY BASED ON FINITE-ELEMENT INVERSE-ANALYSIS

F.E. Gunawan Jl KH Syahdan 9, Jakarta 11480, Indonesia

Industrial Engineering Department, Faculty of Engineering, Bina Nusantara University

Email: f.e.gunawan@gmail.com, fgunawan@binus.edu

Received 28 September 2011, Accepted 15 December 2011

ABSTRACT

This paper proposes a procedure to identify the viscoelastic

material constants using the inverse analysis and the finite

element analysis. The procedure requires two measured

data: the applied impact-force and the structural elastic

response at a point, and further assumes the material

viscoelasticity following the Prony series expansion. In this

procedure, we infer the viscoelastic material constants in the

following steps: we initialize the viscoelastic material

constants, and then calculate the stress state at the point, and

finally, we check the equilibrium of the constitute equation.

The procedure is repeated until the equilibrium of the

constitute equation is satisfied. In this repetition, the

viscoelastic material constants are updated following the

Gauss-Newton method. In addition, we evaluated the

method by using data obtained from a simulated impact on a

viscoelastic plate, and the results were rather promising.

Finally, we studied the convergence of the procedure using

various random material constants as the input data.

Keywords: Visco-elasticity, Stress-wave, Inverse-problem,

Finite element method

1. INTRODUCTION

The viscoelastic materials are playing an important role in

many engineering structures. Those materials such as

polymers (Drozdiv and Dorfmann, 2002) are being used, for

examples, to dissipate and to insulate vibration caused by

rotating or reciprocal movement. It also has potential

application in a new Hopkinson pressure bar testing

apparatus (Zhao and Gary, 1995; Zhao et al., 1997;

Salisbury, 2001; Benatar et al., 2003; Casem et al., 2003).

Therefore having the data of the material, particularly

related with their mechanical characteristics, are essential.

Unfortunately, identifying those data are often not easy due

to limitation enforced by the standard testing procedure.

American Society for Testing and Materials devised ASTM

E139-06 Standard Test Methods for Conducting Creep,

Creep-Rupture, and Stress-Rupture Tests of Metallic

Materials is, unfortunately, only suitable for an extremely

low loading rate. Another standard test, ASTM D6048-07

Standard Practice for Stress Relaxation Testing of Raw

Rubber, Unvulcanized Rubber Compounds, and

Thermoplastic Elastomer, is only suitable for a certain

specimen design and well-specified loading technique.

Neither standard can be used on medium to high strain rate

loading condition, or on a thin film polymer specimen (Ju

and Liu, 2002).

Despite of those standards, many scientists, such as Blanc

(1993); Hillstrom et al. (2000); Lemerle (2002); Soderstrom

(2002), have devised a number of testing procedures, and

those procedures can be classified into the vibratory method,

the wave propagation method, and the ultrasonic method. It

is quite clear that those divisions are according to the strain

rate involved in the test. The vibratory method is only

suitable for the frequency response up to some hundreds Hz

(for examples, see Pintelon et al. (2004), Araujo et al.

(2010), and Barkanov et al. (2009)). Therefore, this

particular method is designed for materials with high

damping property. For an extremely high strain-rate, within

the frequency spectrum of 0.5 MHz to 5 MHz, one would

need an ultrasonic method (Lemerle, 2002). Recently,

Casimir and Vinh (2012) enhanced Le Rolland-Sorin’s

double pendulum proposed in 1934 by replacing the original

cantilever specimen with a simple supported bending

specimen. This particular method is interesting in its

simplicity and it can be used to obtain the complex Young

modulus at a low frequency of 0.1–5 Hz. Another proposal

in the group of the vibratory-based method is given by Kim

and Lee (2009), which proposed an identification scheme by

minimizing data obtained from an experiment and those

from a finite element analysis. This proposal is similar to the

present proposal in this respect. However, the two are

diametrically different in the aspects of the viscoelastic

material representation and range of the frequency of

interest. Kim and Lee (2009) proposal relied on the

fractional-derivative model of the material; our proposal is

based on the Prony series expansion. We interested in

application at higher frequency range. Using a similar

viscoelastic representation, Martinez-Agirre and Maria Jesus

Elejabarrrieta (2011) proposed another vibratory based

method that was designed for a high damping material or

material having mechanical properties that strongly

depending on the frequency. In this approach, instead of

minimizing discrepancy with numerical data, the

discrepancy was measured with respect to a theoretical

prediction. For the strain-rate range in the between of the

406

two, one should rely on a wave propagation method (Blanc,

1993). Many existing wave propagation methods exploit the

benefit of the simplicity of the wave propagation theory in

one dimension (see, for examples, Kolsky (1963); Graff

(1975); Doyle (1989)). This phenomenon can be observed in

a slender bar specimen subjected to a pulse impact-force.

Using the bar specimen, Blanck (Blanc, 1993), Lemerle

(Lemerle, 2002), and Lundber and Blanc (Lundberg and

Blanc, 1988) demonstrated that viscoelastic property could

be inferred from some mechanical responses such

displacement or strain at two different locations. The

inference is on the basis of the following formula:

),(),()(

12

12

xx

xxc

(1)

where 1x and

2x are the measurement locations at the bar,

is the phase angle of the quantity (the displacement of the

strain), is the angular frequency, and )(c is the phase

velocity. And then, the wave speed is used to compute the

complex Young's modulus. Hilton et al. (2004) argued that

the viscoelastic material constants could be identified by

inversely solving the convolution integral for viscoelastic

material derived by Christensen (1981) using the elastic-

viscoelastic analogy. However, until this time, it is still very

difficult to convert the complex Young's into the Young's

modulus in the time domain. Expressing the property in the

time domain is necessary because the existing mechanics for

viscoelastic materials is much well-established in the time

domain than that in the frequency domain (see, for

examples, (Zienkiewics and Taylor, 2000, Section 3.2

Viscoelasticity) and (Mesquite and Code, 2003)).

Furthermor existing commercial finite element packages

such as ANSYS (Kohnke, 1999) and ABAQUS

(Abaqu2008) require a user to supply the material data in

the time domain. Those reasons are major limitation of the

existing identification method in the group of the wave

propagation. In addition, the method is only applicable when

the involved wave length is much longer than the largest

dimension of the specimen cross section. Soderstrom (2002)

shows that if the wave length is longer than d10 , where

d is the specimen diameter, then the valid frequency range

is smaller than )10/( dc , where c is the longitudinal wave

speed. However, this limitation is due to the underlying

theory where Eq. (1) was derived, and it seems to us, this

limitation can be avoided when a higher order

approximation, see for an example Anderson (2006), is used

in deriving the governing dynamics.

This work aims to develop a new technique to identify the

viscoelastic material property within the wave propagation

method. It is designed to overcome limitations mentioned

above. In addition, the technique allows one to use a more

complex specimen geometry such as a plate specimen; On

some circumstances, such a specimen geometry is more

preferable. Therefore, the method does not limit the valid

frequency range of the stress-wave involved during the test.

We should note that the underlying theory used in deriving

the equation of motion often strictly limits the use of the

theory such as those in the slender bar case.

In the present work, we combine the finite element method

and the nonlinear least-squares method to infer the

viscoelastic material property. In general, the use of such

approach is not completely new; Aoki et al. (1997), for an

example, has adopted the approach to infer the Gurson

material constants because those data are difficult to be

measured directly from an experiment. In addition, Mahata

et al. (2004); Mahata and Soderstrom (2004) employed the

nonlinear least-squares method to establish a non-parametric

wave propagation function in the frequency domain.

Matzenmiller and Gerlach (2001) utilized the

correspondence principle of viscoelastic materials and a

numerical technique to inverse the relaxation function in the

Laplace domain. They also assumed that the material bulk

modulus does not depend on time; the same assumption is

used in this work. A more common approach is to

parametrize the material property function such as using a

Prony series (Tschoegl, 1989), and the parameters are

determined accordingly. Emri and Tschoegl (1993)

presented a recursive computer algorithm technique to

determine the relaxation time from relaxation modulus data.

Subsequently, Tschoegl and Emri (1993) also utilized the

theoretical storage and loss functions, and demonstrated

using experimental data (Emri and Tschoegl, 1994, 1995).

2. MECHANICAL BEHAVIOR OF VISCOELASTIC

MATERIALS

2.1 General

The state of stress in an elemental volume of a loaded body

is defined in terms of six components of stress, and

expressed in a vector form as

Tzxyzxyzyx ttttttt )()()()()()()( (2)

where x , y

, and z are the normal components of stress,

and xy ,

yz , and zx are the shear components of stress.

Corresponding to the six stress components in Eq. (2) is the

state of strain, which can also be written in a vector:

Tzxyzxyzyx ttttttt )()()()()()()( (3)

Using the elastic-viscoelastic analogy Christensen (1981),

the theory of elasticity has been extended to the viscoelastic

material. For the viscoelastic material, the material

properties are a function of time and the past response

affecting the present stress state. Both the phenomena can be

expressed by a hereditary integral:

t

tt

tttEt

0

'd'

)'()'()(

(4)

where )(tE is a matrix of relaxation function and 't is a

dummy variable. For the isotropic viscoelastic material, the

relaxation function is expressed as:

407

)(00000

0)(0000

00)(000

000)())(1()()()()(

000)()()())(1()()(

000)()()()()())(1(

)(

tG

tG

tG

tctvtctvtctv

tctvtctvtctv

tctvtctvtctv

tE

(5)

where v is Poisson's ratio and is defined as

)21)(1( vv

Ec

(6)

Young's modulus E can be expressed as a function of the

shear modulus G :

)1(2 vGE (7)

For general viscoelastic material in small deformation

regime, the dilatational modulus and the shear modulus

should be assumed to be a function of time. However, there

are many engineering material where the dilatational

deformation does not depend on time (Flugge, 1975).

Hence, the dilatation deformation is completely elastic

Kes 3 (8)

where s is the hydrostatic pressure, K is the bulk modulus,

and e is the volumetric change. Hence, a simple uniaxial

tensile test or a compression test is sufficient to obtain the

bulk modulus K . For this reason, in this study, K is

assumed to be a known data and G is the only unknown to

be sought. The viscoelastic stress-strain relationship can also

be written in form of compliance function:

t

tt

tttCt

0

'd'

)'()'()(

(9)

where )(tC is a matrix of the compliance retardation

function. We should note that in Eq. (4) and Eq. (9), we

assume the initial stress 0)0( and the initial strain

0)0( , which are generaly applicable.

2.2 Parameterization by Prony Series Expansion

When the data of the stress (Eq. (2)) and the strain (Eq. (3))

at a point exist, Eq. (4) or Eq. (9) can be used to obtain the

material data )(tE or )(tG . However, solving the both

equations inversely are not easy due to the numerical

stability. Such as a process is called deconvolution, and its

results are often sensitive to the data error depending on the

condition number of the problem. Parameterization of the

unknown variable in a deconvolution often leads to much

accurate solution as demonstrated in Gunawan et al. (2006).

Fortunately, parameterization of the viscoelastic material,

such as using Prony series expansion, has been widely used

in representing the viscoelastic characteristics. The Prony

series expansion of the shear relaxation )(tG is given by a

series of the exponential function:

N

i

t

i GeGtG i

1

/)(

(10)

Where )( 0 GGCG ii,

0G is the initial shear modulus, G

is the final shear modulus, i is the relaxation time, N is the

number of Maxwell elements, and iC is a constant. For the

number of Maxwell unit 1N , a standard linear model for a

solid viscoelastic material is obtained. The model has three

unknown parameters, and it is capable to express the stress

relaxation or the strain retardation phenomena of many solid

viscoelastic materials particularly within a short range of

time. For the standard linear solid model, the shear

relaxation function can be reduced to

GeGGtG t /0 )()( (11)

The number of Maxwell unit N is related to the range of

time duration of interest. Emri and Tschoegl (1997) noted

following regarding N and difficulty in finding unique

value for the parameter: two spectrum lines (or two

Maxwell units) per logarithmic decades (i.e., a spacing of

0.5) appears to be most convenient. Larger spacings would

be less accurate although Baumgaertel and Winter (1989)

obtained good results with a spacing of 0.7. Smaller

spacings, on the other hand, are likely to be difficult to

handle (Emri and Tschoegl, 1993). This difficulty also

appears in the present proposal. Therefore, we limit our

discussion, following Emri and Tschoegl (1997)'s advise,

that for the time duration of interest of 100 s , only one

Maxwell unit is relevant, and the related constants need to

be determined.

3. INVERSE ANALYSIS FOR MATERIAL

PROPERTY IDENTIFICATION

To simplify derivation of the present approach, we utilize a

simple experimental setup as shown in Fig. 1; however, the

technique is also applicable for more complex specimen

geometry as demonstrated in Section 4.

Figure 1 A uniaxial impacted bar.

Consider a linear viscoelastic bar subjected to an impact

load as shown in Fig. 1, and we express the material

constants to be estimated in a vector

TGGx 0 (12)

To obtain x , Eq. (4) or Eq. (9) should be solved for )(tG ,

and finally, Eq. (11) is used to calculate the values of 0G ,

408

G , and so that the differences between )(tG obtained by

Eq. (4) and by Eq. (11) are minimum. It is clear that the

necessary data for the calculation are the stress and the

strain at the same location. Although, the strain can be easily

measured by a strain-gage, but the stress is not easily

evaluated, especially for a rather complex structure.

To circumvent such difficulty, the finite element analysis is

employed. The experimental setup shown in Fig. 1 is quite

simple and the applied load )(tP and )(t can be measured

in the experiment. The stress at the measurement point can

be computed by a finite element method for the presumed

viscoelastic material constants as the first approximation.

The measured strain )(t is taken as the reference strain

)(tr ; therefore, an error vector can be established as:

''

)'()'()()(~

0

dtt

tttEtt r

t

r

(13)

In the subsequent iterations, we minimize the error function,

in the least-squares sense, by adjusting the material

constants using Gauss-Newton method and adjusting the

stress using the finite element method. The process is

repeated until 2

)(~ tr is smaller than a certain limit . The

detail of the technique is presented chronologicaly in

Algorithm 1.

A few notes needs to be outlined regarding Algorithm 1.

The data involved in a high impact test often are quite large

in size. Therefore, the convolution equation of Eq. (13) must

be solved as quickly as possible. By applying the

convolution theorem, the equation can quickly be solved in

the frequency domain. And, the data can be transformed into

the frequency domain using the fast Fourier transformation

(Brigham, 1974). However, prior the transformation, zeros

as long as the data length should be padded to the tail of

data r and to improve the accuracy of the data spectrum

in the frequency domain. In addition, because the magnitude

of strain is significantly smaller than the stress, those data

have to be normalized prior the optimization process,

otherwise the Gauss-Newton algorithm may fail to converge.

And as a final note, in the present implementation, the

central difference approach is used to compute the time

derivative of the stress.

A similar procedure can also be used to minimize the error

function on the basis of the compliance function of Eq. (14);

the both will be evaluated numerically in this work.

)(''

)'()'()(~

0

tdtt

tttCt r

t

r

(14)

3.1 Experimental Apparatus and Procedures

The experimental apparatus depicted in Fig. 2 is one of

many possible arrangements to perform the experiment for

this purpose. In this arrangement, a short-stress pulse is

generated by striking the load transfer rod with a ball. The

generated stress pulse can be measured using a pair of gages

attached to the load transfer road. The gages should be

connected to the Wheatstone bridge in a half-bridge

configuration; hence, the strains due to the bending wave, if

any, could be canceled out.

The data from the bridge will be recorded in a computer via

a transient converted and a signal conditioner. The transient

converter allows us to temporarily store the data; it becomes

necessary particularly when the data need to be sampled at

an extremely high rate. Modern signal conditioner usually

has filter feature, which is usefull if the data containing

excessive noise.

Figure 2 The schematic diagram of the air gun impact

system

409

4. NUMERICAL EVALUATIONS

We have verified the above procedure using a number of

specimen types such as a bar specimen, a one-point bend

specimen, and a plate specimen. However, only the results

for the case of the plate specimen are presented due to

limitation of the space. This particular case is also used to

illustrate the proposed procedure.

4.1 Data for Validation

In this section, we present the forward analysis to establish

data for verification the proposed procedure and to select the

objective function among the existing options of Eq. (13)

and Eq. (14). The analysis was performed numerically using

the finite element method so that the necessary data could be

obtained in a well-controlled environment. However, further

verifications on the basis of experimental data are certainly

necessary.

The specimen geometric data, material properties, and

loading conditions are following: the specimen shape is a

square plate with the edge length of 500 mm, and a

thickness of 10 mm. The specimen is assumed to be hanged

in the air, and is made of a viscoelastic material with

following properties: 0G = 161.54 GPa, G = 16.154 GPa,

and = 20 s , and K = 175.0 GPa. A concentrated load is

applied to the middle of the specimen edge where the load

varies with time following a half-sine function with a

loading duration of 50 s . A strain gage is assumed to be

attached at a point located 50 mm ahead of the impact-site.

This experimental design leads to specimen response where

the incident stress-wave can clearly be separated from the

reflected stress-wave. This issue is crucial because when the

reflected stress-wave superimposing the incident stress-

wave, the structural response become too complex, and

convergence is hard to be attained.

The above impact event is performed numerically using

finite element analysis where the finite element mesh of the

model is shown in Fig. 3. Only a half of the specimen is

modeled due to symmetry. The model mesh has 5000 linear

solid elements and 5151 nodes. Although the event is

performed numerically, but it is certainly easy to establish

an experimental apparatus for such a case where a short

stress-pulse can be produced using a small sphere projectile

and the plate specimen can be hanged on the air. The finite

element analysis is conducted for various time-steps and

various element sizes, and their effects on the stress and the

strain at selected measurement point in the analyzed

structure are studied. The largest time-step and element-size

that do not substantially affect the computed stress and

strain are selected for the next analysis. This selection of the

time-step and element-size must be carefully made to save

the computation time, because the present approach requires

that the finite element analysis have to be performed for a

few times. From the analysis, we obtain the stress (see Fig.

5) and the strain (see Fig. 4) as a function of time at the

measurement location. In addition, to accommodate the

error that inherently exists in the measurement process,

small pseudo-random noises are superimposed to the data.

Figure 3 Finite element mesh of a square plate.

Figure 4 The strain time history in x -direction obtained by

the finite element analysis (solid line) and those by solving

Eq. (9) (dotted line).

Figure 5 The stress time history in x -direction obtained

by the finite element analysis (solid line) and those by

solving Eq. (4) (dotted line).

410

It is important to select the best objective function between

Eqs. (13) or (14). For a given strain shown in Fig. 4, the

hereditary integral of Eq. (4) is solved to obtain the stress

presented in Fig. 5 with the dotted line. In the similar

manner, the dotted line in Fig. 4 is obtained by solving Eq.

(9). Figures 4 and 5 show that the strain computed (dotted

line in Fig. 4) from the stress data (solid line in Fig. 5) is

less accurate than the stress computed (dotted line in Fig. 5)

from the strain data (solid line in Fig. 4). The phenomenon

seems to be clear by considering the fact that the

displacement-based finite element analysis produces the

more accurate strain than the stress. Because of this fact, Eq.

(13) is selected as the objective function.

4.2 Inverse Analysis for Identification of Material

Constants

In the inverse analysis, the required data are the reference

strain )(tr , which are depicted in Fig. 4 with a solid line.

To create realistic `measured data', a small amount of the

normally distributed pseudo-random noise is superimposed

to the data. The ratio of noise to the data is taken as 1.0%.

Another required data are initial assumptions of the

viscoelastic material constants. Because this data will be

used in the finite element analysis, the initial data should

reasonably represent the exact viscoelastic constants of a

given material. Too small initial data of 0G and G could

lead to an un-compressible material and ``hour glassing''

may occur during deformation, while too large initial data

may cause the ``locking''. One best way is to employ the

data obtained from a static test for the initialization.

Theoretically, long-term response of a viscoelastic solid will

approach to its elastic response. Therefore, the G can be

initialized with the static shear modulus, in the present case

8.80G GPa. The 0G must be larger than G , so then

we assumed 0.880 G GPa, or 10.0% higher than the final

shear modulus. The initial relaxation time is taken as 1.0 s ,

which is the same as the sampling-time in the finite element

analysis. The time is also a logical choice by considering the

duration of the impact-force. It is certainly impossible to

identify the relaxation time longer than the loading duration

even of the data actually exists. Therefore, the initial design

variable is 0.10.800.880 x . The strain data presented

in Fig. 4 are utilized, and Algorithm 1 is evaluated for 10

iterations. Figures 6 to 8 show the evolution of the estimated

viscoelastic constants along the iterations. On each iteration,

the finite element analysis is performed to update the stress,

and several sub-iterations are performed to update the

viscoelastic material constants. In Fig. 9, the estimated shear

modulus at the 10-th iteration is compared to the exact shear

modulus. Although the shear relaxation function shows a

relatively good agreement, the estimated viscoelastic

material constants converge to a certain value slightly

deviated from the exact data.

Figure 6 Estimated instantaneous shear modulus along

iteration

Figure 7 Estimated final shear modulus along iteration

Figure 8 Estimated relaxation time along iteration

411

Figure 9 Comparison of estimated and exact shear

relaxation function

4.2.1 Convergence of the Finite-Element Inverse-Analysis

Numerical results in Figs. 6 to 8 indicate that the combined

finite-element inverse-analysis method has a fast rate of

convergence. Following, we study the robustness of the

method particularly with respect to the initial data. The

robustness is examined with respect to the ability of the

procedure to consistently converge for given any initial data

in the feasible domain. For this purpose, wide ranges of

initial data of the viscoelastic material constants are

provided. The initial data of the initial shear modulus is

selected from normally distributed pseudo-random numbers

in the range of the elastic shear modulus to five times larger

than the elastic shear modulus. The initial data of the final

shear modulus is set as the same with the exact final shear

modulus. The initial relaxation time is randomly selected

from uniformly distributed random numbers in the range of

the sampling time to the whole analysis time. For each set of

initial data, the analysis of Algorithm 1 is performed for 5

iterations. Ten data sets are used for the evaluation. The

results are presented in Fig. 10, and it is seen that the

convergence of the Algorithm 1 is consistent and falls

within a reasonable degree of accuracy.

Figure 10 Convergence map of 0G and

4.2.2 Effect of the Loading Rate

The coupling method of finite element and inverse analysis

can identify the viscoelastic material constants using the

data of viscoelastic response of the structure. If the structure

is statically loaded, the response data will not contain the

viscoelastic property, i.e., the stress relaxation and the strain

retardation. Consequently, such data cannot be used to

identify the viscoelastic material constants. This fact

suggests that the loading rate affects the accuracy in the

estimation of viscoelastic material constants. Different

loading rate can be obtained by varying the loading

duration. The shorter loading duration provides the higher

loading rate. In this section, the effect of loading

duration, load , on accuracy of the estimated viscoelastic

material constants is examined. The loading profile is

assumed to be similar to the profile of a half-sine function.

The loading duration is varied to 40, 50, 75 and 100 s , and

for each case, the analysis is performed for 100 s . It is

easy to produce such variation of the loading duration, for

an example, by changing the length of the impact.The

results, summarized in Table 1, suggest that the estimated

initial shear modulus, final shear modulus and the relaxation

time are slightly affected by the loading duration.

Table 1 Effect of the loading duration to the estimated

viscoelastic material constants

s)(load Exact

00 / GG Exact/ GG Exact/

0 0.94 0.93 1.15

50 0.93 0.93 1.13

75 0.92 0.93 1.12

100 0.91 0.90 1.11

5 CONCLUSIONS

A new method is proposed to identify the parameters of a

simple viscoelastic model in form of the Prony series

expansion. The method is basically based on a coupling of

the finite element analysis and inverse analysis. The method

estimates the parameters by minimizing discrepancy of the

stress-time histories—utilizing the Gauss-Newton method—

at a reference point selected by the user. It takes advantages

of the finite element analysis; hence, the proposal, unlike the

existing methods, is also suitable for complex specimen

geometry. However, verification on the basis of

experimental data for various specimen geometries is

necessary to further assess the accuracy and robustness of

the proposal.

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