409_ftp chaboche with stochastic methods

GAMM-Mitt. 30, No. 2, 409 429 (2007)

Identification of Material Parameters for InelasticConstitutive Models Using Stochastic MethodsTobias Harth1 and Jurgen Lehn11 Fachbereich Mathematik, Technische Universitat Darmstadt, Schlogartenstrae 7,

64289 Darmstadt, Germany

Received 19 February 2007

Key words Viscoplasticity, material parameters, stochastic simulation, optimization.

The parameters of a constitutive model are usually identified by minimization of the dis-tance between model response and experimental data. However, measurement failures anddifferences in the specimens lead to deviations in the determined parameters. In this arti-cle we present our results of a study of these uncertainties for two constitutive models ofChaboche-type. The models differ only by a kinematic hardening variable. It turns out,that the kinematic hardening variable proposed by Haupt, Kamlah, and Tsakmakis yieldsa better description quality than the one of Armstrong and Frederick. For the parameteroptimization as well as for the study of the deviations of the fitted parameters we applystochastic methods. The available test data result from creep tests, tension-relaxation testsand cyclic tests performed on AINSI SS316 stainless steel at 600OC. Since the amountof test data is too small for a proper statistical analysis we apply a stochastic simulationtechnique to generate artificial data which exhibit the same stochastic behaviour as theexperimental data.

c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Constitutive models are applied to describe the response of a material which is subjected toa given loading history. The models of Bodner and Partom [5, 6], Chaboche [10], Kremplet al. [27], and Steck [40] are typical examples of so called unified inelastic constitutivemodels for isotropic materials.

Inelastic constitutive models usually consist of first order differential equations, whichdescribe the time-dependent evolution of stress or strain respectively. These equationscontain parameters which have to be determined for a given material by fitting the modelequations to experimental data. Normally, uniaxial experiments like creep tests or tensiontests with or without intermediate relaxation periods are performed. However, measure-ment failures or differences in the specimens have a great affect on the obtained parameterfits. Therefore, the estimation of material parameters becomes a problem which has to beanalyzed by statistical methods. A precise statistical investigation requires a large data baseof experimental data. In [3, 7] statistical investigations of the deviations in the experimen-tal data and the resulting uncertainties in the identified parameters are presented. However,these studies are restricted by a rather small amount of available identification experiments.

To overcome the problem that the amount of test data is not sufficient for a statisticalanalysis, we apply a method of stochastic simulation in order to generate artificial data on acomputer which exhibit the same stochastic properties as the measured experimental data.Thus, a data base of artificial data can be build up which provides a large enough amountof data. The stochastic simulation method is described and analyzed in detail in [21, 37].

Corresponding author: e-mail: [email protected], Phone: +49 6151 16 2188, Fax:+49 6151 16 6822


410 T. Harth and J. Lehn: Identification using Stochastic Methods

The material on which experiments have been performed is AINSI SS316 stainless steel.It is applied for manufacturing acid resistant parts for the textile industry and chemicalindustries and which is used for medical and pharmaceutical applications.

The applied constitutive models and the identification experiments are described in Sec-tion 2. We investigate a viscoplastic model of Chaboche [11] and an extended version ofthis model where the kinematic hardening variable proposed by Haupt, Kamlah, and Tsak-makis [25] is applied instead of the Armstrong and Frederick equation [2]. The modelscontain 15 and 17 material parameters, respectively, which have to be determined fromexperimental data. Three different types of experiments have been performed: Creep ex-periments at two different hold stresses, tension-relaxation tests measured at three differ-ent constant strain rates and cyclic tension-compression tests performed at three differentstrain rates and two different strain amplitudes respectively. Each test has been performedat 600OC with 12 specimens.

In Section 3 the methods applied for the integration of the model equations and for fittingthe optimal material parameters are briefly introduced. Integration methods for viscoplasticmodels have been thoroughly studied e. g. in [23, 38]. For the identification of the modelparameters we apply stochastic methods. The studies performed in [38, 39] have shown thatcombined optimization methods yield the best results. We present the algorithm PRINOwhich consists of the global cluster-oriented method of Price [35] in combination with thelocal optimization strategy of Muller, Nollau, and Polovinkin [33]. More research aboutparameter identification can be found e. g. in [4, 9, 26, 28, 30, 31].

In Section 4 the results of the parameter identification from the measured data for thetwo considered viscoplastic models are presented. The parameter vector determined simul-taneously from the entire test data is called optimal parameter vector, since all availableinformation about the mechanical properties of the steel SS316 is applied for its identifi-cation. However, parameter fits to single experiments are also presented and the resultsobtained from the two models are compared.

Usually, stochastic simulations are applied to produce a large amount of parameters asinput of the investigated model in order to investigate the resulting model responses. Wedecided to generate artificial data as described in [21, 37], since it is the scatter in thesedata, which produces the uncertainties in the parameter fits. A validation and robustnessstudy of the stochastic simulation method is given in [20, 21]. In [16, 32, 36] similarapplications of stochastic simulations in engineering are given. The identification resultsobtained from the artificial data are studied thoroughly in Section 5. A large amount of datais generated and material parameters are repeatedly determined. The identified parametersas well as the corresponding model responses are analyzed. The predictions of the modelsare then compared with the measured data. A similar study is performed in [22, 37] for theconstitutive model of Chan, Bodner, and Lindholm [14].

2 Model Equations

The equations of a constitutive model describe the relation between stress and strain for agiven material. Since the response of a material to a given loading is strongly dependenton its mechanical properties, the model contains parameters, which have to be fitted fromexperimental data.

2.1 Model of Chaboche

It is observed from experiments that if the absolute value of stress in a material stays be-low a certain limit Y , then the relation between stress and strain is not rate-dependent.This behaviour is called elastic and the elastic strain is denoted by e. Usually, the stressis assumed to be proportional to the strain, which is expressed in Hookes law of linear


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elasticity

= E e, (1)

where the modulus of elasticity E is a temperature-dependent parameter for the stiffness ofthe material.

When the stress in the material approaches the limit Y or Y respectively, materialslike steel begin to yield, i. e. when the stress is reduced to zero a certain amount of strainremains. This permanent strain is called plastic strain denoted by p and the limit Y iscalled yield stress. It is assumed that the total strain can be additively decomposed into anelastic and plastic part, such that

= e + p (2)

holds. If the evolution of strain is time-dependent and if the stress may exceed the yieldstress, then this behaviour is called viscoplastic.

If only overstress is responsible for the evolution of plastic strain, a simple model forviscoplasticity can be given by a friction element with constant Y connected in parallelwith a nonlinear dashpot which obeys the equation

= K 1n . (3)

The constants K and n are again free parameters. Hence, the resulting model for plasticstrain, a so called flow rule, is given by

g() = p =

{ (f(,Y )

K

)n sign() if || > Y ,

0 otherwise,(4)

with f(, Y ) = || Y . Thus, in viscoplasticity overstress determines the plastic flow,i. e. plastic strain occurs if the condition f > 0 holds. The function f is called yieldfunction.

However, from experiments with steel it is known that the yield stress is not a constantvalue but dependent on the loading history. When a specimen of a material is subjectedto a tension test an increase of the yield stress is observed, which is called hardening. If atension period is followed by compression, then the material has hardened in the tensionperiod, but it is observed that it has softened in compression. The lower stress limit forelastic behaviour after a tension-compression experiment has been carried out lies above theyield stress which would have been observed in a compression test only. This phenomenonis called Bauschinger effect. Thus, the elastic domain does not only change in size, but itcan also be translated. The translation of the elastic region is called kinematic hardeningand is dependent on the plastic strain, whereas the increase of the size is called isotropichardening and is dependent on the accumulated plastic strain. For a virgin material thecenter of the elastic domain is initially zero with the limits Y in tension and Y incompression. Let X(t) and R(t) with X(0) = 0 and R(0) = 0 be variables which describekinematic and isotropic hardening respectively, then the function

f(,X,R) = | X | R Y , (5)

defines the onset of plastic flow by f > 0 under consideration of hardening effects.Another phenomenon which has to be taken into account when modeling the behaviour

of steel at high temperatures is a time-dependent recovery process. When a specimen issubjected to a relaxation test, then a softening of the material is observed in time. This effectmust be considered in the evolution equations for the kinematic and isotropic hardeningvariable, respectively. The influence of recovery on the isotropic hardening is a decreasein size of the elastic region in time, whereas the influence on the kinematic hardening is



a time-dependent translation of the center of the elastic region towards the origin. Therecovery effect is usually negligible at room temperature.

The flow rule of a Chaboche model [11, 29] is defined by equation (4) with hardeningvariables as in equation (5). Thus, the flow rule of this model is given by

p =

| X | R YK

n sign( X), (6)

with = max(0, x).In [12] the isotropic hardening variable R is defined by the differential equation

R = b (q R) p r Rmr , (7)where the last term is applied for the description of recovery effects at high temperaturesas proposed in [12, 26], since the experimental data were observed at 600 OC. The variablep denotes the accumulated plastic strain, i. e. p = |p|. The parameter b indicates the speedof stabilization, whereas the value of the parameter q is an asymptotic value according tothe evolution of the isotropic hardening. In the case of a low plastic strain rate recoverytakes place which is controlled in its intensity by the parameters r and mr.

In order to obtain a more accurate modelling of kinematic hardening we follow theadvice in [11] to consider more than only one variable. We apply a sum of two non-linearkinematic hardening variables X = X1 + X2. Their evolution is given by Armstrong andFrederick type equations [2]

Xi = ci p ai Xi p i |Xi|mi sign(Xi) (i = 1, 2), (8)where the last terms as in [12, 26] are considered for recovery effects. The values of theparameters a1 and a2 denote the speed of saturation and according to these values theparameters c1 and c2 are asymptotic values of the kinematic hardening variables. Thecomplete model is stated in Table 1.

Table 1 The constitutive model of Chaboche.

Strain: (t) = e(t) + p(t)Hookes Law: (t) = E e(t)Flow Rule: p(t) =

|(t)X(t)|R(t)Y

K

n sign((t) X(t))

Hardening: X(t) = X1(t) + X2(t)Xi(t) = ci p(t) ai Xi(t) |p(t)| i |Xi(t)|mi sign(Xi(t)) (i = 1, 2)R(t) = b (q R(t)) |p(t)| r R(t)mr

Initial Conditions: p(0) = 0, X1(0) = 0, X2(0) = 0, R(0) = 0Parameters: Y (Yield Stress)

K,n (Flow Rule)a1, a2, c1, c2, b, q (Hardening)1, 2, r,m1,m2,mr (Recovery)

A superposition of two kinematic hardening variables leads to a better description, sinceeach variable covers its own strain range, where it yields a suitable modelling. Hence,one variable describes the hardening for rather large strains, where the other one modelsparticularly the transition from the elastic to the plastic domain. Hence, one variable isstabilizing rapidly.

The model of Chaboche in the form as presented in Table 1 has 15 material parameters,where Youngs modulus is considered to be a fixed value. Its inelastic part consists of asystem of four ordinary differential equations.


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2.2 Extended Model of ChabocheIn order to improve the description of cyclic behaviour more complex variables for kine-matic hardening have to be applied. In [25] an extension of the Armstrong and Frederickequation is introduced which considers an enhanced dependence on the strain range by anappropriate positive function g dependent on an additional internal variable S. The kine-matic hardening equation becomes

X2 = c2 pg(S) X2 p2 |X2|m2 sign(X2), with g(S) = a21 + v S (9)

where the last term in the equation for X2 is again considered for recovery effects. Sinceone hardening variable in the previously described Chaboche model stabilizes rapidly andserves more or less only for modelling purposes we exchange only one of the two hardeningvariables to obtain a new constitutive model. The flow rule (6) and the other equations ofthe hardening variables stated in (7) and in (8) for i = 1 are not changed. The evolutionequation of the variable S is given by

S =p

p0 (|X2| S) (10)

with initial condition S(0) = 0. In the case v = 0 in equation (9) equation (8) from thefirst model is obtained. The full constitutive model is presented in Table 2.

Table 2 The extended constitutive model of Chaboche.

Strain: (t) = e(t) + p(t)Hookes Law: (t) = E e(t)Flow Rule: p(t) =

|(t)X(t)|R(t)Y

K

n sign((t) X(t))

Hardening: X(t) = X1(t) + X2(t)X1(t) = c1 p(t) a1 X1(t) |p(t)| 1 |X1(t)|m1 sign(X1(t))X2(t) = c2 p(t) a21+vS(t) X2 |p(t)| 2 |X2(t)|m2 sign(X2(t))S(t) = |

p(t)|p0

(|X2(t)| S(t))R(t) = b (q R(t)) |p(t)| r R(t)mr

Initial Conditions: p(0) = 0, X1(0) = 0, X2(0) = 0,S(0) = 0, R(0) = 0

Parameters: Y (Yield Stress)K,n (Flow Rule)a1, a2, c1, c2, v, p0, b, q (Hardening)1, 2, r,m1,m2,mr (Recovery)

This extension of the model of Chaboche has 17 material parameters (without Youngsmodulus) and its inelastic part consists of a system of five differential equations.

It should be noted that both systems of differential equations, the system for the Chabochemodel as well as the system for the extended Chaboche model, have unique solutions forthe loading histories considered in this paper. This was derived by Hans-Dieter Alber froma result in [1]. More about existence and uniqueness results for constitutive models can befound in [8, 13].

2.3 Experimental DataThe applied material is the austhenitic steel SS316, which belongs to the group of stainlesssteel. The short notation is 1.4404 X2 CrNiMo 18 10.



The available test data consist of measurements from 132 experiments at a constanttemperature of 600OC. Three different types of experiments have been performed, whichare creep tests, tension-relaxation tests, and cyclic tension-compression tests:

Tension-tests with intermediate relaxation. The tension tests are displacement-con-trolled tests, which have been performed at constant strain rates of 103s1, 104s1and 105s1, respectively. After an increase of strain up to 1% a relaxation period of15 000 s is initiated. This process is repeated four times such that a maximum strainof 4% is reached. Each test has been performed with 12 specimens. In Figure 1 theloading history (idealized) and the resulting measurements of the tension-relaxationtests at strain rate 103s1 are presented.

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Time [s] b)Fig. 1 Loading history a) and measured data b) of the tension-relaxation tests at strain rate 103s1.

Tension-compression tests. The tension-compression tests are displacement-controlledtests, which have been performed at constant strain rates of 103s1, 104s1 and105s1 with a maximum strain of 0.25% and 0.5% respectively. Every experimentconsists of 5 tension-compression cycles and has been performed with 12 specimens.The loading history and the data of the 12 performed tension-compression tests atstrain rate 105s1 and a maximum strain of 0.5% are presented in Figure 2.

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Time [s] b)Fig. 2 Loading history a) and measured data b) of the cyclic tests at strain rate 105s1 and maxi-mum strain of 0.5%.

Creep tests. Creep tests are stress controlled experiments, where the specimen isloaded with increasing stress. After 10 seconds the hold stress is reached and theresulting strain is measured for a duration of 100 and 1000 hours, respectively. Thecreep tests with a duration of 100 hours have a stress level of 230 MPa and the creeptests with a duration of 1000 hours have a stress level of 160 MPa. Both kinds of creeptests are performed with 12 specimens. In Figure 3 the loading history and the data ofthe 12 measured curves of the creep tests with a hold stress of 230 MPa and a durationof 100 hours are presented.

3 Methods for Parameter IdentificationIn [38, 39] several methods for numerical integration of the Chan, Bodner, and Lindholmmodel [14] are analyzed. The study consists of a comparison of an explicit Euler scheme,


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Time [s] b)Fig. 3 Loading history a) and measured data b) of the creep tests with a hold stress of 230 MPa.

an implicit Euler scheme, a Runge-Kutta-Fehlberg method and a generalized Runge-Kuttamethod. All methods were implemented with an adaptive step size control. It turned outthat the explicit Euler scheme is most efficient for integrating the constitutive model witha loading history of tension experiments and that the generalized Runge-Kutta method ismost efficient for the integration with a loading history of a tension-compression or a creepexperiment. We apply exactly the same methods for the Chaboche models. The coefficientsfor the generalized Runge-Kutta method are the ones of Shampine given in [34] and theapplied step size control is described in [19].

3.1 Target FunctionIn [38] and [39] different distance functions are studied and it is shown that the identifica-tion results are hardly affected by the choice of such functions. The L2 distance is givenby

dL2() =

ki=1

(yi yi())2 (11)

where is a parameter vector which consists of the parameters of a constitutive model, yi isthe measured value at discrete time ti, yi() is the model response obtained from at timeti and k is the number of all discrete time points. However, for simultaneous parameterfits from different types of experiments it is important to apply a distance function whichyields comparable values for the fits from the different experiments. This is the reason whythe relative deviation of the model response to the data

d() = (k

i=1

y2i ) 1

2 dL2() (12)

is applied. In order to determine simultaneously the distance between l different exper-iments and the corresponding model responses dependent on one parameter vector wedefine the function G given by

G() =

lj=1

(j) d(j)(), (13)

where d(j) is the distance according to (12) between the data of experiment j and the fittedmodel response, calculated with weights (j) > 0. A minimization of the function G yieldsone parameter vector such that the distances between the data of the l experiments andthe model responses according to are minimized with respect to the weights (j). Foran appropriate choice of these weights the number of experiments belonging to differenttypes of experiments should be considered, thus a choice of (1) = . . . = (l) is notalways suitable. In this work the sum of weights of all creep experiments, all tension



experiments, and all cyclic experiments, respectively, sum up to 13 . Each single experimentof one kind of experiments gets an equal weight. However, depending on the focus ofa study, another choice of weights could be reasonable, as e. g. higher weights on longtime data would increase the prediction quality for these experiments. It is also possible toinclude weights in equation (11) in order to put more emphasis on long term behaviour. Thedistance function in (13) is the target function of the optimization process for the parameteridentification.

3.2 Controlled Random SearchIn order to minimize the target function (13), we apply the controlled random search al-gorithm of Price [35] in order to find an initial parameter vector for a local optimizationstrategy. In this section the algorithm of this global search is briefly introduced.

Let m be the number of material parameters of the constitutive model and let S Rmbe a convex set, which specifies limits for each parameter. In order to find a start clusterof N > m parameter vectors, a stochastic search is performed, which randomly generatesM > N trial vectors in S. The start cluster C will consist of the N vectors with the lowestfunction values of G. Since we apply a search area of the form

S = [u1, v1] [um, vm], ui, vi R, i = 1, . . . ,m, (14)a trial vector can be generated component-wise by m independent random numbers uni-formly distributed in [ui, vi]. For every trial vector the target function G defined in (13) isevaluated.

The optimization step of the algorithm is as follows. At first, m + 1 vectors(1), . . . , (m+1) are randomly chosen from the cluster C and the center of gravity

gr =1

m

mi=1

(i) (15)

of the first m vectors is computed. The reflection of (m+1) at gr

r = 2 gr (m+1) (16)leads to a new trial vector if r S. If this is not the case the step is repeated. IfG((m+1)) < maxG(C), i. e. if the step was successful, then the new trial vector is ex-changed by the worst vector C with G() = maxG(C), which leads to a modifiedcluster C. The algorithm continues by a repetition of this optimization step with C insteadof C.

If the step of the algorithm described above was not successful, but the number of suc-cesses in relation to the overall number of steps so far is greater than 50%, the clusterremains unchanged and the step of the algorithm is repeated. If the success rate is lowerthan 50% a secondary trial point

s =1

2 (gr + (m+1)) (17)

is computed and the optimization step of the algorithm is repeated.If a total number n0 of steps is reached the algorithm is halted and the vector in the last

modified cluster with the lowest function value of G is stored and is applied as initial vectorfor the local optimization strategy. We choose M = 2500, N = 2 m and n0 = 9000 forthe parameters of this global search algorithm.

3.3 Local OptimizationThe method for local optimization applied in this work is the stochastic method of Muller,Nollau, and Polovinkin [33]. This algorithm receives an initial parameter vector start


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which should be close to the location of the minimum of the target function. Initially, thisvector presents the center c of the search domain, where the optimal parameter vector, i. e.the vector which minimizes the target function, is expected. The vector c is applied as thefirst trial vector tr. Thus, initially start = c = tr holds.

The optimization step of the algorithm consists of the generation of a new trial vectorby realizations of normally distributed random variables. Let G be the target function asdefined in equation (13) and let

i = 0 G(tr)

G(start)(vi ui), (18)

where i = 1, . . . ,m and 0 denotes a compression factor for the calculation of the valuesi. Then the next trial vector

tr is generated component-wise by independent realizationsof normally N(ci , 2i )-distributed random variables.

If G(tr) < G(tr), then the new center of the search domain is moved to

c=

tr+ c G(

tr)

G(start) (tr tr), (19)

where initially c = 1. If the vector c

is not contained in S as given in (14), a new vectoris computed by (19) with the factor c2 instead of c. This process is repeated as long as thenew center of the search domain is outside of S, otherwise the optimization step of thealgorithm is repeated with the new center c of the search domain instead of c and tr

instead of tr.On the other hand, in the the case that G(tr) > G(tr), then a new center of the search

domain is determined by

c= tr +

(1 G(

tr)

G(start)

) (c tr), (20)

and the optimization step is repeated with c instead of c.If a total number n0 of steps is reached the local optimization method is halted. The

last trial vector with the lowest value of the target function is called the optimal parametervector. In this work we choose n0 = 4000 and 0 = 1600 .

The method PRINO is a combined method which consists of the cluster-oriented methodof Price and the local optimzation strategy of Muller, Nollau, and Polovinkin, where thebest parameter vector resulting from the method of Price is applied as initial parametervector for the local optimzation strategy. In [38, 39] this method turned out to be muchmore efficient than evolution strategies.

4 Identification ResultsIn this section we investigate the constitutive models by comparison of their model re-sponses obtained by fits to the different kinds of experimental data. We also perform pa-rameter fits simultaneously to all 132 experiments. The obtained parameter vector from theentire data is called optimal parameter vector, because it yields the best description of thedata. A computation of an optimal parameter vector with the applied numerical methodstakes between 9 and 12 hours on a PC, whereas a parameter fit to only one experiment takesbetween 2 and 5 minutes depending on the applied model and experiment.

In Table 3 the optimal parameter vector of the Chaboche model is given. Youngs mod-ulus is fixed to a value of 145 000 MPa, which was computed by linear regression in theelastic regions of the tension experiments. For this parameter fit to all experimental datawith the Chaboche model, we obtain the value of 7.72% of the target function G defined



Table 3 Optimal parameter vector of the Chaboche model.

a1 c1 a2 c2 K n h q b

3547.78 43006.94 314.07 15655.34 17.03 6.38 62.64 265.70 9.69

1 m1 2 m2 r mr

6.11e-5 2.59 8.34e-9 1.47 3.36e-9 1.79

in equation (13). The evolution of stress and the evolution of the hardening equations for atension test at strain rate 104s1 are stated in Figure 4.

The identified parameters from all 132 experiments for the extended Chaboche modelare given Table 4. The distance value for the fit with this model is 7.03%, which is clearlybetter than the result from the fit with the Chaboche model. This fact can only be due to thekinematic hardening variable of Haupt, Kamlah, and Tsakmakis. The evolution of stressand the evolution of the hardening equations are given in Figure 4 for a tension experimentat strain rate 104s1.

Table 4 Optimal parameter vector of the extended Chaboche model.

a1 c1 a2 c2 v p0 K n h

3776.96 36920.24 674.75 24979.44 11.36 27.24 17.69 3.26 67.44

q b 1 m1 2 m2 r mr

93.99 32.52 3.82e-4 1.57 1.33e-8 1.08 3.01e-9 1.93

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b)Fig. 4 Response and evolution of inner variables of Chaboche a) and extended Chaboche model b)for a tension experiment at strain rate 104s1 computed with optimal parameter vector.

The improvement from the Chaboche model to the extended Chaboche model resultsmainly from the better description of cyclic experiments, what is shown in Table 5, wherethe accuracy values of the parameter fits of the two models to all available cyclic experi-ments, tension-relaxation, and creep experiments are stated. It can be seen, that the kine-matic hardening variable of Haupt, Kamlah, and Tsakmakis yields a better description thanthe Armstrong and Frederick variables in the Chaboche model.

Table 5 Accuracy values of simultaneous fits to the different types of experiments.

constitutive model cyclic tension-relaxation creep all experiments

Chaboche 7.73% 3.98% 4.38% 7.72%

extended Chaboche 6.94% 3.63% 4.18% 7.03%


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Figure 5 shows that the fit of the extended Chaboche model to a cyclic experiment atstrain rate 103s1 and strain amplitude of 0.25% lies very close to the data. The sameholds for the fit to a creep experiment with a duration of 100 hours. The Chaboche modeland the extended Chaboche model yield very similar model responses for fits to singlecreep experiments.

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Time [s] b)Fig. 5 Fit of extended Chaboche model to a cyclic experiment at strain rate 103s1 and strainamplitude of 0.25% a) and fit of Chaboche model to a creep experiment with a duration of 100 hoursb).

A similar good result as for the fits to cyclic and creep experiments can be obtained fromfits to tension-relaxation experiments. The problem for these tests is, that the model has todescribe not only the tension periods correctly, but also the relaxation periods. In Figure 6the result of a fit to a tension-relaxation experiment at strain rate 103s1 is presented.

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It can be concluded that the applied numerical methods are appropriate for the parameteridentification and that both models describe single experiments of a high accuracy.

5 Analysis of the Parameter EstimatorSince material differences in the test specimens and measurement inaccuracies lead to dif-ferent results for the identification of parameters, we interpret the process of parameteridentification as a statistical parameter estimation. We are interested in the mean valuesand standard deviations of the fitted parameters. In order to perform many parameter fitsand to compute the desired values we apply a stochastic simulation method.

5.1 Stochastic SimulationThe identification experiments consist of scattered data and thus cause uncertainties in theidentified parameters. Since not enough test data is available for a statistical analysis, weapply a method of stochastic simulation, where artificial data are generated which exhibitthe same properties as the experimental data. By a detailed time series analysis of the test



data it turns out that the behaviour of the scattered data can be modeled by appropriatestochastic processes. We refer to [20, 37] for a detailed description. The stochastic modelsof the data are based on first order autoregressive processes (AR(1)-processes). A valida-tion and robustness study of the simulated data can be found in [20, 21]. In Figure 7 themeasured data of creep experiments with a duration of 100 hours and 12 simulated tests areshown. It can be seen that the applied simulation method yields reasonable and appropriateresults.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 100000 200000 300000 400000

Str

ain

[ ]

Time [s] a)0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 100000 200000 300000 400000

Str

ain

[ ]

Time [s] b)Fig. 7 Measured a) and generated data b) of a creep test with a duration of 100 hours.

5.2 AnalysisIn order to analyze the identified parameters artificial data of one experiment of each ofthe eleven different kinds of experiments is generated and the material parameters are de-termined by a simultaneous parameter fit. This process is repeated 250 times, which takesbetween 9 and 12 days computation time on a PC. We make the reasonable assumption thatrepeated experiments with specimens of the same material are (stochastically) independent.Thus, for generating the random numbers applied for the stochastic simulations as wellas for the stochastic optimization methods, we apply an inversive congruential generator,which has better independence properties than the widely used linear congruential genera-tors and is suitable for parallel processing. This type of generator has been introduced byEichenauer and Lehn [17]. In this work an explicit inversive congruential generator [18] isapplied, which generates pseudorandom numbers xi, i = 1, 2, . . . , by

xi a i + b (mod p) (21)with p = 231 1, a = 50000, b = 1, and where for any x {1, . . . , p 1} the numberx {1, . . . , p 1} is uniquely defined by x x 1 (mod p) and 0 = 0. For a parallelcomputation of parameter fits on k computers it is necessary to generate k parallel streamsof random numbers x(j)i , j = 1, . . . , k, which can be achieved by applying the generators

x(j)i a(j) i + b(j) (mod p), (22)

where a(j) = 50000 and b(j) = j.The vector of mean values and the covariance matrix of the parameter fits (l) for

l = 1, . . . , 250 are computed by

=1

250

250l=1

(l) and = 1249

250i=1

((l) ) ((l) )T, (23)

where T denotes transposition. The components of denote the mean values of the esti-mated parameters whereas the components of the matrix denote the empirical variancesof the fitted parameters.

In the following analysis the mean values i and the standard deviations

ii (ii di-agonal entries of) of the estimated parameters for each constitutive model are presented.Furthermore, the mean values are compared with the optimal parameter vector from


GAMM-Mitt. 30, No. 2 (2007) 421

Section 4 by relating the distances |i i | to the standard deviations

ii. Additionally,the model responses to the parameter estimates are analyzed and compared with the corre-sponding stress and strain values of the measured data. As in [22, 37] the results are statedexemplary for three reference experiments, namely the creep tests with a duration of 100hours, tension-relaxation tests at strain rate 103s1, and cyclic tests with strain amplitude0.5% and strain rate 105s1. The correlation coefficients corresponding to the parameterestimates for each model have also been computed and a discussion of the results can befound at the end of this section.

5.3 Analysis of the Chaboche ModelIn Table 6 the optimal parameter vector from Section 4 and the mean values and standarddeviations of the 250 performed parameter fits are stated.

Table 6 Optimal parameter vector, mean values, standard deviations, and the relations|ii |/

ii of the estimated parameters of the Chaboche model computed component-

wise.

i parameter i i

ii |i i |/

ii1 a1 3547.8 3483.9 740.159 0.0862 c1 43007 38095 5094.5 0.9643 1 6.11 105 4.77 105 2.422 105 0.5534 m1 2.59 3.38 0.737 1.0725 a2 314.1 300.5 48.357 0.2816 c2 15655 14894 2471.7 0.3087 2 8.34 109 1.69 108 5.86 108 0.1468 m2 1.47 1.41 0.210 0.2869 K 17.03 18.45 6.898 0.206

10 n 6.38 7.13 1.456 0.51511 q 265.7 245.1 39.135 0.52612 b 9.69 11.45 3.115 0.56513 Y 62.64 62.70 6.078 0.0114 r 3.36 109 1.79 108 1.666 108 0.87315 mr 1.79 1.56 0.230 1

Although the optimal parameter vector was determined from all 132 experiments simul-taneously and the parameter estimates in this section are obtained by identifications fromartificial data of only 11 experiments, there is a good coincidence between the componentsof the optimal parameter vector i and the mean values i.

The model responses to the tension-relaxation experiments at strain rate 103s1 andthe model response to the optimal parameter vector are evaluated in Table 7. It turns out,that the model response obtained from and the mean values of stresses at the end ofeach tension and relaxation period of the model responses obtained from the 250 parameterestimates (1), (2), . . . , (250) coincide well. The comparison with the mean values ofthe measured data shows that the model of Chaboche overestimates the stress at all timeinstants. However, the model responses at the end of the tension periods are very close tothe values of the measured data. The greatest difference between the stress of the modelresponse obtained from and the corresponding mean values of stresses of the measuredcurves is only 8.91 MPa. The accuracy of the relaxation periods is significantly worse. Thegreatest difference of stresses is here even 37.7 MPa. It can be concluded, that the modelof Chaboche describes the relaxation periods not precisely.

In Table 8 the model responses to the creep tests with a duration of 100 hours andthe measured data are presented. The mean values of strains of the experimental data areunderestimated at all time instants by the model responses obtained from the 250 parameterestimates, except at the first time instant. The same holds for the model response from the



Table 7 Comparison of model responses and experimental data of the Chaboche modelfor the tension-relaxation tests at strain rate 103s1.

time [s] 10 15 010 15 020 30 020mean value [MPa] 149.39 134.07 175.42 159.94

model maximum [MPa] 161.20 145.12 188.41 170.96minimum [MPa] 139.78 124.46 164.20 153.09

responses std dev [MPa] 4.408 3.877 3.977 3.147opt vector [MPa] 149.48 133.41 174.44 158.22mean value [MPa] 141.81 117.18 168.63 136.88

measured maximum [MPa] 153.13 122.50 177.91 143.18minimum [MPa] 135.37 110.58 162.15 131.24

data std dev [MPa] 5.707 3.843 5.056 3.608time [s] 30 030 45 030 45 040 60 040mean value [MPa] 195.87 180.32 213.92 198.31




data std dev [MPa] 6.214 3.354 7.247 3.921

optimal parameter vector. However, the differences between the predicted strains of themodel responses and the measured ones are small. The model of Chaboche describes creeptests with a duration of 100 hours accurately.

Table 8 Comparison of model responses and experimental data of the Chaboche modelfor the creep tests with a duration of 100 hours.

time [s] 50 000 100 000 150 000 200 000 250 000 300 000 350 000mean value [%] 6.007 6.100 6.175 6.244 6.311 6.376 6.441

model maximum [%] 7.401 7.525 7.623 7.714 7.802 7.890 7.977minimum [%] 4.906 4.967 5.018 5.067 5.116 5.165 5.213

responses std dev [%] 0.4263 0.4362 0.4451 0.4542 0.4639 0.4742 0.4851opt vector [%] 5.976 6.073 6.145 6.210 6.272 6.332 6.392mean value [%] 5.999 6.187 6.327 6.441 6.545 6.655 6.769

measured maximum [%] 6.706 6.863 7.004 7.138 7.259 7.437 7.646minimum [%] 5.233 5.385 5.494 5.590 5.665 5.751 5.817

data std dev [%] 0.4827 0.5020 0.5178 0.5271 0.5405 0.5594 0.5967

In Table 9 the evaluations of the model responses and the measured data are presentedfor the cyclic tension-compression experiments with stain amplitude 0.5% and strain rate105s1. Again the differences of the mean values for the stresses predicted by the modelfrom the 250 parameter estimates and predicted from the response to the optimal parametervector are very small. The greatest difference is only 3.74 MPa at the last time instant.However, the absolute mean values of the measured data are overestimated by the Chabochemodel for the last two cycles, although the predicted stress (except for the first time instantat 500 s) is always between the minimum and maximum values of the measured data.

It can be concluded that the model of Chaboche describes all three types of experimentswith high accuracy except relaxation. The model responses from the parameter vectors(1), (2), . . . , (250) coincide well with the ones obtained from .

5.4 Analysis of the Extended Chaboche ModelThe mean values and standard deviations of the 250 parameter estimates obtained fromsimultaneous fits to the artificial data of one experiment from each of the eleven kinds of


GAMM-Mitt. 30, No. 2 (2007) 423

Table 9 Comparison of the model responses and experimental data of the Chabochemodel for the cyclic tests at strain rate 105s1 and maximum strain of 0.5%.

time [s] 500 1 500 2 500 3 500 4 500mean value [MPa] 120.96 -147.72 163.94 -179.16 192.61

model maximum [MPa] 131.61 -138.24 169.80 -167.90 203.46minimum [MPa] 110.37 -154.20 152.94 -187.79 179.73

responses std dev [MPa] 4.035 2.637 2.506 3.111 3.960opt vector [MPa] 123.38 -149.53 165.54 -180.65 194.24mean value [MPa] 126.36 -159.47 167.06 -190.76 191.02

measured maximum [MPa] 132.41 -143.73 172.39 -166.89 199.81minimum [MPa] 122.71 -165.79 151.04 -201.04 168.70

data std dev [MPa] 2.482 6.685 5.863 9.782 8.459time [s] 5 500 6 500 7 500 8 500 9 500mean value [MPa] -204.68 215.53 -225.35 234.24 -242.34

model maximum [MPa] -190.05 232.83 -207.40 257.75 -221.62minimum [MPa] -218.52 199.41 -245.85 214.78 -268.65

responses std dev [MPa] 4.777 5.559 6.316 7.064 7.813opt vector [MPa] -206.57 217.80 -228.06 237.45 -246.08mean value [MPa] -210.41 207.03 -224.34 218.92 -234.61

mean value maximum [MPa] -183.32 216.82 -196.72 230.18 -208.31minimum [MPa] -225.32 185.17 -242.33 195.50 -252.65

data std dev [MPa] 11.669 8.514 11.715 8.667 11.231

experiments together with the optimal parameter vector determined in Section 4 arestated in Table 10.

Table 10 Optimal parameter vector, mean values, standard deviations, and relations|i i |/

ii of the estimated parameters of the extended Chaboche model computed

component-wise.

i parameter i i

ii |i i |/

ii1 a1 3777.0 3121.7 405.493 1.6162 c1 36920 29887 6152.073 1.1433 1 3.82 104 4.55 104 2.444 104 0.2994 m1 1.57 3.68 1.712 1.2325 a2 674.8 548.1 68.061 1.8626 c2 24979 22590 2626.602 0.917 v 11.36 8.55 3.340 0.8418 p0 27.24 28.26 5.574 0.1839 2 1.33 108 4.12 108 3.314 108 0.842

10 m2 1.08 1.37 0.366 0.79211 K 17.69 25.72 10.504 0.76412 n 3.26 4.75 1.656 0.913 q 93.99 112.50 14.86 1.24614 b 32.52 25.52 6.073 1.15315 Y 67.44 69.09 4.806 0.34316 r 3.01 109 6.10 109 1.258 108 0.24617 mr 1.93 1.84 0.206 0.437

For a few parameters as e. g. a1 and a2 there is a considerable difference between themean values of the 250 estimated parameters and the corresponding component of theoptimal parameter vector. However, it must be considered that the parameter estimates(1), (2), . . . , (250) are identified from only one experiment of each kind simultaneously,whereas the is determined from all twelve available experiments from each kind. Thus,extreme data have a smaller weight in the computation of the optimal parameter vector.



Moreover, there is a highly non-linear dependence of the model responses on the materialparameters.

In Table 11 the evaluations of the model responses to the tension-relaxation experimentsat strain rate 103s1 are presented. The values for the stresses obtained from the optimalparameter vector, except at the ends of the last two relaxation periods, are closer to themean values of the measured data than the ones of the Chaboche model investigated in theprevious section. At the ends of the tension periods the mean values of the model responsesfrom the 250 parameter estimates also yield better results than the ones of the Chabochemodel. However, the extended model of Chaboche also overestimates the stresses at theends of the relaxation periods. The greatest difference between the model response from and the mean values of the model responses from (1), (2), . . . , (250) is 4.76 MPa.

Table 11 Comparison of model responses and experimental data of the extendedChaboche model for the tension-relaxation tests at strain rate 103s1.

time [s] 10 15 010 15 020 30 020mean value [MPa] 148.36 136.06 173.14 160.74




data std deviation [MPa] 5.707 3.843 5.056 3.608time [s] 30 030 45 030 45 040 60 040mean value [MPa] 193.96 181.48 212.18 199.63




data std dev [MPa] 6.214 3.354 7.247 3.921

The description of the creep experiments with a duration of 100 hours is not exactly asgood as the description of these data by the Chaboche model. The evaluations of the strainvalues for this kind of test is shown in Table 12. The extended model of Chaboche alsounderestimates the mean values of the strains obtained from the measured data. However,a comparison with Table 8 shows that the values for the strains of the two Chaboche modelsat the last time instant of the responses from the optimal parameter vectors differ by only0.144. The mean values of the model responses from the 250 parameter estimates differby only 0.106. Thus, it can be concluded that the creep experiments with a duration of 100hours are described accurately by the extended Chaboche model.

In Table 13 the model responses to cyclic tension-compression experiments with strainamplitude 0.5% at strain rate 105s1 are stated. The values for the stresses of the modelresponse obtained from and the mean values of the model responses from the 250 pa-rameter estimates coincide very well. At the ends of the tension periods these values fitalmost exactly to the mean values of the measured data. In compression however, the ex-tended model of Chaboche underestimates the mean values for the stresses computed fromthe measured data. The greatest difference between the mean values from the model re-sponses corresponding to the estimates (1), (2), . . . , (250) and the mean values of themeasured data is 11.63 MPa. Thus, the extended Chaboche model describes these tension-compression experiments with a high accuracy.


GAMM-Mitt. 30, No. 2 (2007) 425

Table 12 Comparison of model responses and experimental data of the extendedChaboche model for the creep tests with a duration of 100 hours.

time [s] 50 000 100 000 150 000 200 000 250 000 300 000 350 000mean value [%] 5.949 6.031 6.100 6.164 6.223 6.280 6.335

model maximum [%] 7.402 7.546 7.659 7.763 7.860 7.953 8.043minimum [%] 4.637 4.723 4.802 4.876 4.946 4.978 5.008

responses std dev [%] 0.4484 0.4570 0.4646 0.4724 0.4806 0.4893 0.4984opt vector [%] 5.956 6.027 6.077 6.123 6.166 6.208 6.248mean value [%] 5.999 6.187 6.327 6.441 6.545 6.655 6.769

measured maximum [%] 6.706 6.863 7.004 7.138 7.259 7.437 7.646minimum [%] 5.233 5.385 5.494 5.590 5.665 5.751 5.817

data std dev [%] 0.4827 0.5020 0.5178 0.5271 0.5405 0.5594 0.5967

Table 13 Comparison of the model responses and experimental data of the extendedChaboche model for the cyclic tests at strain rate 105s1 and maximum strain of 0.5%.

time [s] 500 1 500 2 500 3 500 4 500mean value [MPa] 125.21 -149.40 165.76 -179.13 190.11

model maximum [MPa] 132.69 -144.15 171.99 -174.12 198.04minimum [MPa] 117.76 -155.87 160.88 -186.24 185.28

responses std dev [MPa] 2.885 2.196 1.944 1.856 1.893opt vector [MPa] 124.77 -149.10 166.35 -179.98 190.88mean value [MPa] 126.36 -159.47 167.06 -190.76 191.02

measured maximum [MPa] 132.41 -143.73 172.39 -166.89 199.81minimum [MPa] 122.71 -165.79 151.04 -201.04 168.70

data std dev [MPa] 2.482 6.685 5.863 9.782 8.459time [s] 5 500 6 500 7 500 8 500 9 500mean value [MPa] -199.22 206.82 -213.21 218.62 -223.23

model maximum [MPa] -193.92 216.20 -204.93 229.47 -211.95minimum [MPa] -207.90 200.21 -223.23 208.78 -235.95

responses std dev [MPa] 2.183 2.717 3.401 4.150 4.914opt vector [MPa] -199.71 206.91 -212.85 217.77 -221.88mean value [MPa] -210.41 207.03 -224.34 218.92 -234.61

mean value maximum [MPa] -183.32 216.82 -196.72 230.18 -208.31minimum [MPa] -225.32 185.17 -242.33 195.50 -252.65

data std dev [MPa] 11.669 8.514 11.715 8.667 11.231

The mean value of the distances between the model responses and the measured data ofall 132 experiments with respect to the target function (13) is 9.0%, which is better thanthe corresponding value obtained from the Chaboche model, which is 9.51%. This resultcoincides with the one obtained in Section 4, where the extended Chaboche model alsoyields the best model fits.

From the calculation of the correlation matrices of the model parameters it turns out thathigh correlations appear only in a few cases. The parameters of the two Chaboche modelsare the yield stress Y , the parameters of the flow rule K and n, the parameters of thekinematic hardening equations a1, c1, a2, c2 with two additional parameters p0 and v forthe extended Chaboche model, the parameters b and q of the isotropic hardening variable,and the relaxation parameters 1,m1, 2,m2, r,mr. Among the correlation coefficientsof the parameters of each model we find only three relevant ones concerning the Chabocheand four relevant ones concerning the extended Chaboche model. The highly correlatedparameters and their correlation coefficients are stated in Table 14. All other correlationcoefficients are relatively small and thus not of importance.

The parameters a2 and c2 of the Armstrong and Frederick equation of the Chabochemodel and of the Haupt, Kamlah, and Tsakmakis equation of the extended Chaboche



Table 14 Relevant correlation coefficients of the parameters of the Chaboche model (left)and of the extended Chaboche model (right).

Chaboche extended ChabocheParameters Correlation Parameters Correlation

a2, c2 0.893 a2, c2 0.688q, b -0.817 2, m2 -0.701

r, mr -0.891 q, b -0.861r, mr -0.655

model, respectively, show positive correlations. When the Armstrong and Frederick equa-tion without recovery term is integrated for e. g. a tension experiment, the solution

X2(p) =

c2

a2 (1 exp(a2 p)) (24)

is obtained. This explains the positive correlation of the parameters, since a2 controls thespeed of saturation but the limit for increasing plastic strain is c2

a2. Thus, for large a2 the

parameter c2 must be large as well (and vice versa) in order to reach the same limit. For theextended Chaboche model this correlation is less than it is for the Chaboche model, sinceinstead of the parameter a2 we have a term containing another inner variable as describedin equation (9). A similar explanation as above for the kinematic hardening can be givenfor the negative correlation of the parameters b and q of the isotropic hardening equation.The solution of the differential equation (7) without recovery term is

R(p) = q (1 exp(b p)). (25)Thus, the limit for increasing accumulated plastic strain p is q and the speed of saturationis given by b. For large values of b the values of q must get smaller, since otherwise theresulting values of the isotropic hardening equation would be too high especially for smallplastic strains. A deficiency of our experimental data is the lack of saturating cycles forthe tension-compression tests. This means particularly, that the limit q of the isotropichardening cannot be determined accurately. Thus, the parameter q can also be used bythe optimization procedure for an adjustment of the model response at low plastic strainsand thus especially for an improvement of the description accuracy. For the negative cor-relations of the two recovery parameters r and mr (and also 2 and m2 of the extendedChaboche model) an analogous explanation as above can be given. In the recovery termr R(t)mr an increase of the exponent mr must be compensated by a decrease of thefactor r and vice versa. This explains also the high variance of the parameter r, since achange by one standard deviation as stated in Table 6 and in Table 10 can be compensatedby a change of one standard deviation of mr.

6 ConclusionIn this work two viscoplastic constitutive models are investigated and their material param-eters are determined by stochastic optimization methods. In Section 4 the optimal param-eters obtained from fits to all available data are stated. Furthermore, the description accu-racy of the two models are discussed and fits to single experiments as well as to all tension,creep, and cyclic experiments are presented. It is demonstrated that the applied methodswork appropriately, although up to 17 parameters have to be identified simultaneously. Thesame integration and optimization methods have been employed before in [22, 37, 39] forthe parameter identification of the Chan, Bodner, and Lindholm model. The constitutivemodels describe the test data of single experiments accurately. The extended Chaboche


GAMM-Mitt. 30, No. 2 (2007) 427

model yields the best fits to the experimental data and it turns out that particularly cyclicexperiments are described much better by this constitutive model. This is due to the kine-matic hardening variable of Haupt, Kamlah, and Tsakmakis [25], since similar results cannot be achieved by the Chaboche model without this hardening equation.

Since there is not enough experimental data available for statistical investigations, wepresent a method of stochastic simulation. This method enables us to generate a largeamount of artificial data with the same stochastic properties as the original experimentaldata. In [20, 21] it is shown by applying nonparametric statistical tests that the resultsfor the parameter fits from experimental data and from artificial data coincide well. Thus,for the analysis of the parameter estimates in Section 5 we apply only fits to artificialdata generated by the stochastic simulations. It turns out, that relaxation periods are notdescribed appropriately by both models, when other kinds of experiments are fitted simul-taneously. This result was also found for other models in e. g. [15] and in [24], wherea similar Chaboche model is investigated. The description accuracy of the cyclic exper-iments is the best for the extended Chaboche model, which describes these experimentsvery precisely. The presented mean values of the 250 parameter fits coincide well with theones of the optimal parameter vectors, although simultaneous fits to only one experimentfrom each kind have been performed. The standard deviations of the identified parametersgive a good idea of the obtainable precision, i. e. they show the influence of natural devia-tions in measured data on the determined parameters. The yield stress Y can be estimatedquite precisely, whereas e. g. some parameters of the fast saturating kinematic hardeningvariable X1 can not be determined with high precision. The analysis in Section 5 is closedwith a brief discussion of the correlation coefficients of the parameters obtained from the250 parameter fits. The parameters a2 and c2 of the Armstrong and Frederick variable inthe Chaboche model and of the Haupt, Kamlah, and Tsakmakis equation in the extendedChaboche model are strongly correlated. The same holds for the parameters b and q of theisotropic hardening equation and the recovery parameters r and mr. The yield stress Yfor instance is not highly correlated to the other parameters. In [20, 21] a principal compo-nent analysis is presented which yields similar correlation results. Another application forthe stochastic simulations is presented in [20] where a design of experiments is performedin order to find a combination of experiments which yields accurate parameter estimates bylow test expenses.

Acknowledgements The authors are very indebted to Hans-Dieter Alber for discussions on theexistence and uniqueness of solutions of the constitutive equations given in Section 2. Particularly, weare very grateful to him for deriving the existence and uniqueness results in both cases, the Chabocheand the extended Chaboche model, from a result in [1]. Furthermore, we would like to thank DFGfor financial support and Franz Gustav Kollmann for a very stimulating co-operation in SFB 298.Additionally, we are grateful to a reviewer for his valuable comments and suggestions for furtherresearch.

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409_ftp chaboche with stochastic methods

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