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European Journal of Mechanics A/Solids 30 (2011) 1028e1039

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

Statistical approach for a hyper-visco-plastic model for filled rubber:Experimental characterization and numerical modeling

J.M. Martinez a, A. Boukamel a,*, S. Méo b, S. Lejeunes a

a Laboratoire de Mécanique et d’Acoustique, CNRS UPR-7051, 31 chemin Joseph-Aiguier, 13402 Cedex 20, Marseille, Franceb Laboratoire de Mécanique et Rhéologie, EPU Dept Productique, BP 407, F37204 Cedex 3, Tours, France

a r t i c l e i n f o

Article history:Received 31 May 2010Accepted 24 June 2011Available online 2 July 2011

Keywords:Hyper-visco-plastic behaviorRheological modelPayne effectGent-Fletcher effectFilled elastomer

* Corresponding author.E-mail address: [email protected]: http://www.lma.cnrs-mrs.fr

0997-7538/$ e see front matter � 2011 Elsevier Masdoi:10.1016/j.euromechsol.2011.06.013

a b s t r a c t

This paper presents a campaign of experimental tests performed on a silicone elastomer filled with silicaparticles. These tests were conducted under controlled temperatures (ranging from �55 �C to þ70 �C)and under uniaxial tension and in shearing modes. In these two classes of tests, the specimens weresubjected to cyclic loading at various deformation rates and amplitudes and relaxation tests at variouslevels of deformation. A statistical hyper-visco-elasto-plastic model is then presented, which coversa wide loading frequency spectrum and requires indentifying only a few characteristic parameters. Themethod used to identify these parameters consists in performing several successive partial identifica-tions with a view to reducing the coupling effects between the parameters. Lastly, comparisons betweenmodeling predictions and the experimental data recorded under harmonic loading, confirm the accuracyof the model in a relatively wide frequency range and a large range of deformations.

� 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

Elastomers belong to the high polymer family i.e. they consist ofmacromolecular chains of various lengths, with and without ramifi-cations. This structure confers on these materials a low level ofrigidity and a high level of deformability. In addition, the reinforce-ment of these materials with fillers accentuates their dissipativebehavior. Because of these properties, especially their dampingcapacity, thesematerials arewidely used in industry. The applicationon which this study focuses is that of the drag dampers for helicop-ters. These parts connect helicopter blades to the rotor and attenuatethe dragmovement. Designing these parts,which are often related tosafety, imposes a guarantee of high reliability under extreme oper-ating conditions (dynamic loading with multi-frequency and largeamplitudes, thermal constraints, etc). Meeting these specificationsrequires good knowledge of the mechanical behavior of the consti-tutivematerials. In addition, the behavior of an elastomer can dependheavily on the temperature, the degree of cross-linking and the typeof particles incorporated (carbon or silica), etc.

During recent decades, several approaches have been used byprevious authors tomodel various behavioral aspects of elastomers:

(A. Boukamel).

son SAS. All rights reserved.

� To describe the static behavior of the material, a hyperelasticapproach was used in: Treloar (1943, 1957) where statisticalmodels were proposed; and Mooney (1940); Rivlin (1958);Hart-Smith (1966); Ogden (1972), which involved the use ofphenomenological approaches.

� Some authors have used the damage mechanics approach todescribe the softening behavior occurring during the firstloading cycles, which is known as the Mullins effect, seeMullins (1947); Harwood et al. (1967). A theoretical frameworkwas proposed by Govindjee and Simo (1991,1992) in the case ofa hyperelastic behavior. A similar approach was described inSimo (1987); Miehe (1995) in the case of a visco-elasticmaterial.

� In Holzapfel and Reiter (1995); Holzapfel and Simo (1996a,b);Lion (1997), a thermomechanical coupling model was devel-oped, which takes into account the temperature dependence ofthe mechanical characteristics and describes the temperaturechanges resulting from the mechanical dissipation.

� Furthermore, to model the visco-elastic effects of these mate-rials, a framework of the Finite Non-Linear Visco-elasticity hasbeen proposed by several authors. These models can be clas-sified in the following groups, depending on the type offormulation used:

Those using an integral approach, which was mainly developedfor modeling non-linear materials with evanescent memory. Theseapproaches describe the behavior of the material using equations

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Fig. 1. Double-shearing specimen.

1 For relaxation tests, normalized stress is obtained by dividing the total stress bythe instantaneous stress.

J.M. Martinez et al. / European Journal of Mechanics A/Solids 30 (2011) 1028e1039 1029

giving the stress tensor in terms of the strain history. Rivlin (1958);Coleman (1964); Christensen (1971); Coleman and Noll (1961);Lianis (1963); Chang et al. (1978); Morman (1988).

Those using a differential approach, based on the concept ofintermediate states commonly used to describe finite elastice-plastic deformations (see Sidoroff (1973, 1974)). Defining interme-diate states provides the internal variables needed to describe thebehavior. This approach can be said to be an extension of rheo-logical models in the case of large strains: Sidoroff (1977); Le Tallec(1990); Le Tallec and Rahier (1994); Leonov (1992). The local statemethod, Lemaître and Chaboche (1996), provides the theoreticalframework of this formulation, and the internal variables areprovided by the intermediate states.

And those using micro-physically motivated models for filledelastomers, which are often based on hypotheses about the inter-actions between the agglomerates of fillers and the gum matrix:Drozdov (2001a,b; Drozdov and Dorfmann (2002, 2003); Drozdovet al. (2004)), or about the mechanisms underlying the deforma-tion and rearrangement of the macromolecular network: Tanakaand Edwards (1992); Drozdov (1998, 2000); Reese (2003).

In this paper, a meso-physically motivated approach is used tomodel the response of the material, in large strain and at variousfrequencies and temperatures. A statistical approach is thenproposed to develop a model based on the generalization of anassembly of rheological models. The advantage of this statisticalrheological model is that it can be used to simulate the behavior ofthe material in a wide frequency range while requiring only a fewparameters to be identified.

Firstwe present the results of a series of experimental tests,whichwere carried out on a silicone elastomer filled with silica. These testswere uniaxial tension and shear tests and were performed undercontrolled temperature (ranging from �55 �C to þ70 �C) and undervarious loading conditions (Relaxation tests, quasi-static and dynamicloading at various strain rates). The results show the dependence ofthe behavior of the material on the temperature, as well as on thestrain rate (Fletcher-Gent effect, see Fletcher andGent (1953)) and theamplitude of the strain (Payne effect, see Harwood et al. (1967)). Theconstitutivemodel is then developed on the basis of the fundamentalprinciple of thermodynamics of continuous media, adapted to finitestrain theory. Using the concept of intermediate configurations(multiplicative decomposition of the deformation gradient) and inlinewith the theory of thermodynamics of irreversible processes, andunder the hypothesis of the normal dissipation dependingonly on theinternal variables, the constitutive equation and the flow rules areobtained. A statistical approach is then applied, in order to extend thisrheologicalmodel to awide rangeof strain rates and to account for theplasticbehaviorof thematerial. In the following section, this statisticalhyper-visco-plasticmodel is analyzed in the caseof simple loads,witha view to propose a strategy for identifying its parameters. For thispurpose, analytical solutions are developed to simulate the relaxationresponse and the hardening test, respectively. In the case of cyclicloading, a semi-analytical responsewasobtainedusinga symbolic andnumeric computation software. These identificationswereperformedat various temperatures. Lastly, using the semi-analytical solutionunder sinusoidal shear loading conditions at various frequencies andamplitudes, the effect of parameters such as the temperature,frequency and loading amplitude on the harmonic response of theelastomer are analyzed.

2. Experimental analysis

2.1. Description of the experimental tests

An experimental campaign was conducted on a silicone(dimethyl-vinyl-siloxan vulcanized by peroxide) reinforced with

silica particles. The glass transition temperature of this elastomer isapproximately �105 �C. The following tests were carried out:

� Uniaxial tensile tests on specimens with a dumbbell shape (H2according to standard NF T46-002), to determine the quasi-static behavior and the relaxation response of the material.

� Shear tests on Double-Shearing specimens (DS, see Fig. 1).These specimens were successively subjected to: a quasi-staticloading-unloading cycle; relaxation tests at various shearinglevels; triangular cyclic loading, at various strain rates (from0.03 s�1, 0.03 s�1 to 10 s�1) and various amplitudes (12.5%, 25%and 50%).

All these tests were performed under controlled temperatures(ranging from �55 �C to þ70 �C) in a climatic chamber cooled byinjecting nitrogen and heated with an electrical resistance and theairflow. In the tensile tests, monitoring and deformationmeasurements were performed with a laser extensometer.

Remark 1 (Mullins effect) To eliminate the Mullins effect (seeMullins (1947); Harwood et al. (1967)) and therefore to characterizethe behavior of the stabilized material, a softening process was firstinduced by applying about ten cycles with an amplitude greaterthan the maximum strain imposed during the series of tests.

Remark 2 (Temperature stabilization) To avoid errors in thetemperature measurement, a waiting period of 10 min was fixedbetween each characterization test to allow the temperature toreach equilibrium inside the specimen. The characterization timewas sufficiently short to avoid a too strong self-heating phenomenain the specimen.

2.2. Experimental results

Relaxation tests: In the relaxation tests, the specimen was sub-jected to various strain levels: 25%, 50% and 100% under tensionloading; 20%, 30% and 54% under shear strain. The response of thematerial is described by the evolution of the normalized stress1

versus time. The curves presented in Fig. 2(a) and Fig. 2(c) showthat at temperatures above the ambient temperature, the evolutionof these stresses during relaxation was always independent of thestrain amplitude. At these temperatures, the relaxation mechanismseems independent of the strain level, under both tension and shear

a b

dc

Fig. 2. Response of the material in relaxation tests: first Piola-Kirchoff stress vs time. The vertical axis corresponds to the total stress divided by the instantaneous stress (normalizedstress).

Fig. 3. Quasi-static responses recorded in loading-unloading shear tests: first Piola-Kirchoff stress vs shear strain (shear rate: _g ¼ 0:03s�1, temperature: T ¼ 25 �C).

J.M. Martinez et al. / European Journal of Mechanics A/Solids 30 (2011) 1028e10391030

loading; whereas at lower temperatures, the responses doesn’tshow the same linearity of the stress depending on the deformation,especially in the case of uniaxial tension tests (see Fig. 2(b)). Thegraphs in Fig. 2(b) and Fig. 2(d) show the dependence of the relax-ation response on the temperature. The relaxation response wastherefore more sensitive to the temperature in the [�55 25 �Cto �25 �C] range than at higher temperatures (above 25 �C).

Quasi-static shear response: The quasi-static test was a loading-unloading test, performed at low strain rate ( _g ¼ 0.03 s�1) and forthree shear amplitudes (gmax ¼ 12.5%, 25% and 50%). The stress-strain curves given in Fig. 3 show that even at low rates of defor-mation, the material shows dissipative behavior. It will therefore benecessary to take the plasticity into account when developing theconstitutive model.

Cyclic tests: Cyclic tests were carried out under triangularloading conditions with gradually increasing amplitude. At eachamplitude, a dozen cycles were applied. The values of the param-eters in this set of tests were as follows:

� Temperature (T): 70 �C, 40 �C, 25 �C,�25 �C,�40 �C and�55 �C.� Shear rate ( _g): 0.03 s�1, 0.1 s�1, 0.3 s�1, 1 s�1, 3 s�1 and 10 s�1.� Shear amplitude (gmax): 12.5%, 25% and 50%.

a b

Fig. 4. Cyclic responses in triangular shear tests: first Piola-Kirchoff stress vs shear strain (temperature: T 25 �C). The first cycles has been removed to keep only stabilized responses.


Fig. 4(a) and 5 show the effects of the loading amplitude on thestabilized response. Qualitatively, these responses show the strongnon-linearity at high amplitudes. In addition, it is worth noting thedecrease in the global stiffness observed when the strain amplitudeincreases. These results therefore clearly confirm that thisphenomenon, which is known as the Payne effect, is morepronounced at low temperatures (see Fig. 5(a)). In previous studies,this softening has often been attributed to the plastic behavior ofelastomers reinforced with fillers.

As with other visco-elastic polymer materials, the influence ofthe strain rate was more classical (see Fig. 4(b) and 6) at all thetemperatures tested: an increase in the global stiffness and thecyclic dissipation with the strain rate were clearly observed.

Lastly, the hysteresis loops presented in Fig. 7 show the stronginfluence of the temperature on the behavior of the material. It canbe seen in particular that: a softening and a decrease in the cyclicdissipation occurs as the temperature increases; at low tempera-tures the hysteresis loop shows a non-linear behavior, which ischaracterized by: the angular point, the stiffening observed at theend of the cycle and the contraction of the loop at zero strain. Thesenon-linear features, which are more pronounced at low tempera-tures, are consistent with plastic behavior. These results thereforeconfirm that plastic behavior begins to predominate when thematerial approaches the glass transition point.

In conclusion, the following aspects of the behavior have to betaken into account in the model:

a b

Fig. 5. Effects of the strain amplitude on the response of the material to tr

� The geometric non-linearities due to large strains.� The dissipative behavior induced by viscous effects whichshould be coupled to the hyperelasticity.

� The model must be able to describe the behavior in a widerange of strain rates, and in particular, to reflect the Feltcher-Gent effect.

� The effects of plasticity on the behavior, including the Payneeffect in particular.

Other phenomena, such as the Mullins effect and the self-heating of the material, were also observed during in thisexperiments. However, these aspects will not be integrateddirectly into the model developed in study, because their analysishas been widely discussed in the literature, see Mullins (1947,1956, 1959); Mullins and Tobin (1965); Govindjee and Simo(1991, 1992).

3. Constitutive equations

3.1. Some generalities about rheological modeling

Using the concept of the local intermediate configuration,introduced by Sidoroff (1974); Sidoroff (1975), the transformationgradient tensor F is split into a viscous and anelastic parts (Fig. 8):

F ¼ Fe$Fv (1)

iangular cyclic shearing tests for various temperatures ( _g ¼ �0:3s�1).

a b

Fig. 6. Effects of the strain rate on the response of the material under triangular cyclic shearing tests at various temperatures (gmax ¼ 50%).

Fig. 8. Intermediate configuration.


Then, assuming that the Clausius-Duhem inequality can bewritten in Eulerian terms as follows (neglecting the thermaleffects):

f ¼ s : D� J�1r0_j (2)

where r0 is the density in the initial configuration, f is the intrinsicdissipation, s is the Cauchy stress tensor and D represents theEulerian strain rate tensor:

D ¼ 12

�L þ LT

�with L ¼ _F$F�1 (3)

J denotes the determinant of the gradient tensor F, and j is thefree specific energy which is expressed as the sum:

jðB;BeÞ ¼ jvðBeÞ þ j0ðBÞ (4)

where B ¼ F$FT and Be ¼ Fe$FTe are sets of independent thermo-dynamic variables. So, one can express _j as follows:

_j ¼ vj0vB

: _Bþ vjv

vBe: _Be (5)

The time derivatives of the left Cauchy-Green tensor and thelocal changes in volume are given by:

_B ¼ L$Bþ B$LT (6)

Fig. 7. Effects of the temperature on the response of the the material under triangularcyclic shearing (gmax ¼ 50%, _g ¼ �10s�1).

_J ¼ Jð1 : LÞ (7)

where 1 is the identity tensor, and:

_Be ¼ L$Be þ Be$LT � 2Ve$Dov$Ve (8)

where Ve is the purely elastic strain tensor (i.e. coming from thepolar decomposition Fe ¼ Ve$Re) and

Dov ¼ Re$Dv$RT

e (9)

is the objective measure of the anelastic strain rate. By injectingequations (6), (7) and (8) in (5), the variation of the free energy canbe written as follows:

_j ¼�2B$

vj0vB

�: Dþ

�2Be$

vjv

vBe

�: D�

�2Ve$

vjv

vBe$Ve

�: Do

v

(10)

with the incompressibility conditions:

D : 1 ¼ 0; Dov : 1 ¼ 0 (11)

Using the assumption of the normal dissipation (we choosea quadratic pseudo-potential of dissipation, 4v, depending only onDov ), the constitutive equation and the evolution law are obtained as

follows2:

2 The symbol D stands for the deviatoric operator.

a b

Fig. 9. Statistical hyper-visco-plastic model.

3 The symbol + denotes a Legendre-Fenchel transformation.


s ¼ s0 þ sy � p1 with

8>>>><>>>>:

s0 ¼�2r0J�1B$

vjo

vB

�D

sy ¼�2r0J�1Be$

vjy

vBe

�D (12)

v4v

vDov

¼�2r0J

�1Ve$vjv

vBe$Ve

�D(13)

where p is the hydrostatic pressure due to the local incompressi-bility condition. Equations (12) and (13) can be said to be a gener-alization of the classical Zener rheological model to the case offinite strain.

3.2. A statistical approach for a hyper-visco-plastic model

The constitutive model must first reflect the behavior of thematerial in awide range of strain rates, but it also has to account forthe effects of the plasticity, such as the Payne effect, and thebehavior of the material at low temperatures. Previous studies haveshown that plasticity gives good agreement between the experi-mental data and the model at low temperatures (Boukamel et al.(2005)), and that some rheological models are suitable formodeling the behavior under a given range of loads (Olsson andAustrell (2001); Miehe and Keck (2002); Nedjar (2002)). Undermore complex loading conditions, rheological models with severalbranches (see Fig. 9(a)) seem to account satisfactorily for thebehavior of the material. However, the disadvantage of thesemodels is that they require identifying a large number ofparameters.

In order to overcome this difficulty, a statistical approach wasdeveloped, whereby the assembly of discrete rheological branchesis extended to a continuous model with an infinite number ofbranches. The advantage of this method is that it covers a widerange of retardation times (or a large frequency spectrum). Thisapproach gives the advantages of a multi-branch assembly withoutincreasing the number of parameters in the model.

The model presented here, can be motivated in micro-physicalterms by the heterogeneity of reinforced elastomers, especially in

the case of a silicone elastomer slightly filled with silica particles. Inorder to account for this heterogeneity, it is therefore assumed thatthe elastomeric matrix is dense and that the inclusions, whichconsist of particles of silica agglomerated together with a thinrubber bond, are supposed to be slightly reticulated (see forinstance Drozdov and Dorfmann (2003)). Based on these assump-tions, the behavior of the elastomer can be defined as follows:

� The behavior of elastomeric matrix is hyperelastic;� The behavior of the inclusions is hyper-visco-elastic (anextension of the Maxwell model to large strains);

� The inclusion/matrix interfaces is assumed to have a hyper-elasto-plastic behavior (an extension of the Saint-Venant modelto large strains).Two statistic quantities are introduced, namely:� ui, which denotes the activation energy of the mechanisminclusion/matrix, i.e. the energy required to break the links atthe interface (Drozdov (2000)),

� and Pi which represents the probability of that a population ofinclusion corresponds to the activation energy ui.

The discrete form of the statistical model can be written:

8>>><>>>:

j ¼ j0ðBÞ þPNi¼1

~jyðui;BeðuiÞÞPi þ jp�Bep

4 ¼ PN

i¼1

~4y

�ui;Do

y ðuiÞPi þ 4+

p ðsPÞ(14)

where Dop denotes the anelastic objective strain rate of the elasto-

plastic branch (same definition as in equation (9)), j0 denotes thespecific free energy associated with the matrix, whereas ~jv and jp

are the free energies associated with the inclusions and the inclu-sions/matrix interfaces, respectively. ~4v and 4+

p are the pseudo-potential of dissipation3 corresponding to the inclusions and tothe interface, respectively.

a b

Fig. 10. Evolution of statistical functions: predominance of the instantaneous elasticity.


Using the continuous statistical model in Fig. 9(b) to generalizethis formulation, we obtain:

8>>>>>><>>>>>>:

j ¼ j0ðBÞ þZN0

~jvðu;BeðuÞÞPðuÞduþ jp�Bep

4 ¼ZN0

~4vðu;Dov ðuÞÞPðuÞduþ 4+

p�sp

(15)

where u is a random variable associated with the activation energyof a relaxation micromechanism, and PðuÞ is the probability thata population of fillers has the given value u.

Substituting the potentials (15) in equations (12) give theconstitutive equation of the statistical model4:

s ¼ s0 þZN0

~syðuÞPðuÞduþ sP

� r1 with

8>>>>>>>><>>>>>>>>:

s0 ¼ 2r0J�1�B$

vj0vB

�D

~syðuÞ ¼ 2r0J�1

"BeðuÞ$v

~jyðuÞvBeðuÞ

#D

sP ¼ 2r0J�1�Bep$

vjp

vBep

�D(16)

and the following evolution laws:

v~4vðuÞvDo

v ðuÞ¼ 2r0J

�1

"BeðuÞ$v

~jvðuÞvBeðuÞ

#D(17)

Dop ¼

"v4+

p

vsp

#D(18)

The various forms of the potentials can be chosen so that: theneo-Hookean incompressible hyperelastic model for the matrixbehavior; the neo-Hookean form, for the hyperelasticity of theinclusions and the inclusions/matrix interface; a quadratic form forthe pseudo-potential of viscous dissipation of the inclusions; anda perfectly plastic pseudo-potential at the interface. These choicescan be written as follows:

4 In the hyper-elasto-plastic branch, F is split into an elastic part Fep and a plasticpart FPP (F ¼ Fep$Fpp). The objective measure of the plastic strain rate Do

p istherefore defined in the same way as Do

v (see 9).

8>>>>>>>>>>>><>>>>>>>>>>>>:

r0j0 ¼ C1ðI1ðBÞ � 3Þr0

~jvðuÞ ¼ GðuÞðI1ðBeðuÞÞ � 3Þ~4vðuÞ ¼ hðuÞ

2Dov : Do

v

r0jp ¼ Ap�I1�Bep

� 3

4+p ¼<

sp� c>

(19)

where C1 is the coefficient of the neo-Hookean density, GðuÞ andhðuÞ are two functions of the random variable u, c and Ap are theparameters involved in the hyper-elasto-plastic branch, < : >

denotes the Mac-Cauley brackets and sp ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffisp : sp

p .To define the distribution function PðuÞ, a classical Gaussian

distribution centered at the origin and characterized by the stan-dard deviation U was adopted:

PðuÞ ¼ 1ZN0

PðuÞdu

��uU

�2�(20)

To choose the functions GðuÞ and hðuÞ, which describe thevariations in the hyperelastic and viscous characteristics dependingon u, several forms were tested. The functions giving the bestmatch with the experimental data were of the form:

8<:

GðuÞ ¼ G0½u�;hðuÞ ¼ hN

�ln� ffiffiffiffi

up þ 1

u

þ 1�

(21)

In fact, these expressions for the hyperelastic and viscouscharacteristics lead to a decreasing evolution of the retardationtime depending on u. Combining this variation with the distribu-tion function (20) makes it possible to focus on the instantaneouselastic response rather than on the delayed response (Fig. 10).

Lastly, by injecting (19) in (17), we obtain the constitutiveequation:

s ¼ s0 þZN0

~syðuÞPðuÞduþ sP

� p1 with

8>>>><>>>>:

s0 ¼ 2C1BD

~syðuÞ ¼ 2GðuÞBeðuÞD�sP ¼ 2ApBep

D (22)

Table 1Identification strategy: C1, Ap and c are identified on quasi-static and delayed responses, G0, U and hN are obtained by fitting the values to the instantaneous or cyclic responsesat various strain rates.

Loading Model analysis Parameters identified

Table 2Parameters identified at various temperatures.

T(�C) c1(MPa) ap(MPa) c(Mpa) a0Mpa) hN (MPa.s) U Error (%)

3 s�1 10 s�1

�55 0.538 2.424 0.107 0.151 0.14 1.56 10.84 12.43�40 0.486 2.109 0.088 0.136 0.105 1.56 12.45 12.17�25 0.451 1.767 0.075 0.121 0.096 1.56 15.28 11.8225 0.386 1.23 0.04 0.0614 0.063 1.56 7.98 8.7540 0.359 1.024 0.031 0.059 0.056 1.56 7.78 6.7970 0.342 0.855 0.02 0.051 0.053 1.56 10.15 6.86


and by using (19), (17) and (18) in (8), we obtain the followingflow rules:

_BeðuÞ ¼ L$BeðuÞ þ BeðuÞ$LT � 4GðuÞhðuÞBeðuÞ$BeðuÞD (23)

_Bep ¼ L$Bep þ Bep$LT � 2 <sp

� c>sp

kspk$Bep (24)

The statistical hyper-visco-plastic model given by expressions(22) therefore includes 6 parameters which have to be identified, 5of which are determinist parameters (C1, G0, hN, Ap, c) and one ofwhich is a statistical parameter (U).

a b

Fig. 11. Comparison between stabilized hysteretic cycles at various temperatures and strain rates. The solid line show the model response, the points show the experimental results.


4. Identification of the model parameters

To identify the model parameters by fitting the response of themodel to the experimental data, an algorithm implemented in theMATHEMATICA software was used. This algorithm is based ona minimization of the sum of the squared differences between theexperimental data and the analytical or semi-analytical responses.The latter are obtained by simulating the uniaxial tension tests andthe double shear tests, in which the responses are assumed to behomogeneous and incompressible.

4.1. Analytical forms of tension responses

Under uniaxial tension, the elastic and anelastic gradients of thetransformation are written as follows, respectively:

F ¼

266664l 0 00

1ffiffiffil

p 0

0 01ffiffiffil

p

377775; Fa ¼

2666664

la 0 0

01ffiffiffiffiffila

p 0

0 01ffiffiffiffiffila

p

3777775 (25)

Substituting these expressions into the constitutive equationsand the complementary law 22, the response of the material undervarious loading modes can be obtained using an analytical form orafter a numerical solving.

4.1.1. Relaxation testTo obtain the relaxation response, the instantaneous and

delayed stresses are written in terms of the axial component of thefirst Piola-Kirchoff stress tensor P (s ¼ P11):

ly ¼

8>>>>>>>>><>>>>>>>>>:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14��b3

�3s

3

vuut þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14��b3

�3s

3

vuut if b � 3ffiffiffiffiffiffi433

p

2

ffiffiffib3

rCos

26413

0B@p� Acos

0B@� 1

2

�3b

�32

1CA1CA375 if b � 3ffiffiffiffiffiffi

433p

wit

8>>>>>>>>>>><>>>>>>>>>>>:

s0 ¼ 2�l3 � 1

�240@C1 þ

ZN0

GðuÞPðuÞdu1A 1

l2

35 þ

ffiffiffi32

rc

l

sN ¼ 2C1l2

�l3 � 1

�þ

ffiffiffi32

rc

l

_sjt¼0 ¼ �83

�2l3 � l � 1

l3

� ZN0

GðuÞ2hðuÞ PðuÞdu

(26)

where s0 is the instantaneous stress response and sN is the infinite(long-time) stress response.

4.1.2. Hardening testThe axial stress is written here in the quasi-static case, in the

form:

sðlÞ ¼ 2l

"C1l

�l3 � 1

�þ Ap

l3e � 1le

#(27)

The plastic and elastic elongations lp and le are given by:

_lp ¼D_lEH�

l� lyand le ¼ l

lp(28)

where H is a hardening function which is obtained from eq. (24) lyis the elongation corresponding to the plastic yield:

h b ¼ c

2Ap

ffiffiffi32

r(29)

Fig. 12. Evolution of parameters vs temperature.


Lastly, the residual strain at zero stress l0 corresponds to theequation sðl0Þ ¼ 0 when _l < 0, which can be written as a functionof ðC1;Ap;cÞ and the maximum strain.

4.2. Shear responses

In the case of shear tests, the gradient tensors are taken to be asfollows:

F ¼241 g 00 1 00 0 1

35; Fa ¼

2664la1 ga 00 la2 0

0 01

la1la2

3775 (30)

4.2.1. Relaxation testWith the approximation, lai ¼ 1, the instantaneous and

delayed stress relaxation terms (s ¼ P12) are given by:

8>>>>>>>>>>><>>>>>>>>>>>:

s0 ¼ 2g

0@C1 þ

ZN0

GðuÞPðuÞdu1A þ

ffiffiffi2

p

2c

sN ¼ 2C1gþffiffiffi2

p

2c

_st¼0 ¼ �8gZN0

GðuÞ2hðuÞ PðuÞdu

(31)

a b

Fig. 13. Comparison of the global response of the model (solid lin

4.2.2. Hardening testThe response of the material under quasi-static loading/

unloading can be approximated as follows. The shearing stress iswritten:

sðgÞ ¼ 2C1gþ 2Apge (32)

The plastic and elastic shear strain gp and ge are given by:

_gp ¼ �_g�H�

g� gy

�and ge ¼ g� gp (33)

where gy is the shear strain corresponding to the plastic yield:

gy ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9þ 3c2

2A2p

s� 3

2

vuuuutx

ffiffiffi2

pc

4Ap(34)

4.2.3. Cyclic testMore generally, based on expressions 30, the response to a cyclic

shear test can be obtained bywriting the constitutive equation (22).This leads to a system of differential equations, the solutions ofwhich are fga; la1 ; la2g. These systems can be solved using a Runge-Kutta scheme.

4.3. Identification algorithm

The identification of the parameters of the modelX ¼ fC1;G0; hN;Ap;c;Ug can be reduced to the minimization ofthe difference between the experimental curves fðli; siÞ; i ¼ 1;NTgand fðgi; siÞ; i ¼ 1;NSg and the theoretical responses ðl; sðl;XÞÞand ðg; sðg;XÞÞ. This difference is characterized by the least squaredistance:

EðXÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNT

i¼1

xiðsi � sðli;XÞÞ2þXNS

i¼1

hiðsi � sðgi;XÞÞ2vuut (35)

where xi and hi are the weights associated with the tensile andshear tests, respectively.

This minimization problem is solved using Powell’s iterativealgorithm (see Fletcher (1987)), which is a conjugate directionmethod without gradient calculation. This algorithm is combinedwith a one-dimensional minimization procedure in each directionwhich is based on a quadratic interpolation of the function to beminimized.

In short, the identification procedure consists of:

1. determining of the response of the material with a set of modelparameters, under a given loading mode;

e) and experimental data (points) in bi-harmonic shear test.

a b

c d

Fig. 14. Comparisons between modeling predictions (solid line) and experimental data (points).


2. calculating the least square difference between the modelingpredictions and the experimental data;

3. applying the iterative procedure to minimize the least squaredifference.

4.4. Identification strategy

Given the complexity of the model, the number of parameterswhich have to be identified and themultiplicity of the experimentaldata required to identify these parameters, it was necessary todevelop a strategy for decoupling the various stages in the identi-fication procedure. This identification strategy was based on thedistinction between the instantaneous and delayed responses, aswell as between the effects of viscosity and plasticity. Based on theanalytical or semi-analytical results outlined in the previous para-graphs, the following strategy was therefore adopted:

1. Quasi-static and delayed responses are used to identify theparameters C1, Ap and c.

2. The other parameters, G0, U and hN, can be obtained by fittingthe values to the instantaneous or cyclic responses at variousstrain rates.

Table 1 summarizes the successive steps in the identificationstrategy.

5. Results and discussion

5.1. Identification results

The six model parameters (C1, G0, hN, Ap, c, U) were identifiedsuccessively, at various temperatures (�55 �C, �40 �C, �25 �C,25 �C, 40 �C and 70 �C) and at various strain rates (3 s�1,10 s�1). Thevalues obtained are given in Table 2.

The Table 2 also gives the relative identification error obtainedfor each temperature and each strain rate. A maximum of 15% oferror is obtained and the Fig. 11 shows two examples of identifi-cation results. A good agreement between the predictions of themodel and the experimental data is observed in the shear tests, atvarious temperatures and strain rates.

Lastly, Fig. 12 shows the evolution of the model parametersidentified (normalized) versus the temperature. These curvessuggest an exponential decay of all the parameters with theincrease of the temperature, except for U, which has been fixed atall temperatures, since it characterized only the range of retarda-tion times to be covered by the model. This result shows that thepresent model seems consistent as the evolution of the parametersbetween the temperatures is monotonous in the temperature rangeconsidered.

5.2. Relevance of the model

Comparisons between the results of the model predictions andthe experimental data (which has not been used for identification),obtained under sinusoidal shear loading conditions, show that themodel accurately predicts the effects observed experimentally,namely, the Payne effect (see Harwood et al. (1967)), as shown inFig. 14(c), and The Gent-Fletcher effect (see Fletcher and Gent(1953)) as shown in Fig. 14(a). In addition, Fig. 14(b) and (d) showthe existence of good agreement between the simulated and theexperimental data on the cyclic dissipation depending on thefrequency and the amplitude.

Other comparisons made in multi-harmonic loading in sheartests also shows the ability of this model to accurately simulate thebehavior of materials subjected to a combination of several loads atdifferent frequencies (see Fig. 13). These results show that thepresent model can successfully predict the complex behavior ofa highly dissipative silicone rubber for a large range of strain


amplitudes and strain rates with a few number of materialparameters.

6. Conclusion

In this study, a statistical approach was used to develop a hyper-visco-plastic model covering awide frequency range. This approachhas the same advantages of classical multi-branch models, such asthe ability to simulate the behavior of material for several decadesof retardation time, but it do not show the same inconvenient asonly a few number of material parameters are required (6 in thepresent model).

A series of experiments were conducted under various loadingand temperature conditions, in order to identify the parametersinvolved in the model, using an algorithm developed with theMATHEMATICA software library. To optimize this identificationprocedure, a relevant strategy was adopted, which consisted indistinguishing between the various stages in the procedure andthus reducing the effects of coupling between the parameters.

The results obtained at various temperatures show the ability ofthis model to simulate the behavior of the material in a wide rangeof temperatures. In addition, the comparisons between themodeling predictions and the experimental data recorded atvarious frequencies and strain amplitudes have shown a goodagreement. The present model is therefore capable to reproducethe complex behavior of filled rubber in particular the Gent-Fletcher and Payne effects.

References

Boukamel, A., Martinez, J., Méo, S., 2005. A statistical approach for a hyper-visco-elasto-plastic model for a filled elastomer. In: Constitutive Models for RubberIV. In: Proceedings of the 4th European Conference for Constitutive Models forRubber, pp. 191e196.

Chang, W., Bloch, R., Tschoegl, N., 1978. The behaviour of rubber-like materials inmoderatly large deformations. J. Rheology 22, 1e32.

Christensen, R., 1971. Theory of viscoelasticity, an introduction. Academic Press Inc.,New York.

Coleman, B., 1964. Thermodynamics of materials with memory. Arch. RationalMech. Anal. 17, 1e46.

Coleman, B., Noll, W., 1961. Foundations of linear viscoelasticity. Rev. Mod. Phys. 33,239e249.

Drozdov, A., 1998. A model for the nonlinear viscoelastic response in polymers atfinite strains. Int. J. Solids Struct. 35 (18), 2315e2347.

Drozdov, A., 2000. A model of cooperative relaxation in finite viscoelasticity ofamorphous polymers. Int. J. Non-Linear Mech. 35, 897e909.

Drozdov, A., 2001a. A model for the viscoelastic and viscoplastic responses of glassypolymers. Int. J. Solids Struct. 38, 8285e8304.

Drozdov, A., 2001b. Modelling nonlinear viscoelasticity and damage in amorphousglassy polymers. Math. Comput. Model. 33, 883e893.

Drozdov, A., Agarwal, S., Gupta, R., 2004. Linear thermo-viscoelasticity of isotacticpolypropylene. Comput. Mater. Sci. 29, 195e213.

Drozdov, A., Dorfmann, L., 2002. The nonlinear viscoelastic response of carbonblack-filled naturals rubbers. Int. J. Solids Struct. 39, 5699e5717.

Drozdov, A.D., Dorfmann, L., 2003. A micro-mechanical model for the response offilled elastomers at finite strains. Int. J. Plasticity 19, 1037e1067.

Fletcher, R., 1987. Pratical methods of optimization. John Wiley & Sons.Fletcher, W., Gent, A., 1953. Non-linearity in the dynamic properties of vulcanised

rubber compounds. I.R.I Transctions, 266e280.Govindjee, S., Simo, J., 1991. A micro-mechanically based continumm damage model

for carbon black-filled rubbers incorporating mullins’effect. J. Mech. Phys. Solids39, 87e112.

Govindjee, S., Simo, J., 1992. Trasnsition from micro-mechanics to computationallyefficient phenomenology: carbon black filled rubbers incorporating mullins’effect. J. Mech. Phys. Solids 40, 213e233.

Hart-Smith, L., 1966. Elasticity parameters for finite deformations of rubber-likematerials. J. Appl. Math. Phys. 17, 608e626.

Harwood, J.,Mullins, L., Payne,A.,1967. Stress softening in rubbers: a review. J. IRI,17e27.Holzapfel, G., Reiter, G., 1995. Fully coupled thermomechanical behaviour of

viscoelastic solids treated with finite elements. Int. J. Eng. Sci. 33, 1037e1058.Holzapfel, G., Simo, J., 1996a. Entropy elasticity of isotropic rubber-like solids at

finite strains. Comput. Methods Apll. Mech. Eng. 132, 17e44.Holzapfel, G., Simo, J., 1996b. A new viscoelastic constitute model for continuous

media at finite thermomechanical changes. Int. J. Struct. 33, 3019e3034.Le Tallec, P., 1990. Numerical analysis of viscoelastic problems. Masson.Le Tallec, P., Rahier, C., 1994. Numerical models of steady rolling for non-linear

viscoelastic structures in finite deformations. Int. J. Numer. Methods Eng. 37,1159e1186.

Lemaître, J., Chaboche, J., 1996. Mécanique des matériaux solides. Dunod, Paris.Leonov, A., 1992. On thermodynamics and stability of general maxwell-like visco-

elastic constitutive equations, theoretical and applied rheology. In: XIth Int.Cong. On Rheology. pp. 97e99.

Lianis, G., 1963. Constitutive equations of viscoelastic solids under finite deforma-tion. Purdue University report A&ES 63-11.

Lion, A., 1997. On the large deformation behaviour of reinforced rubber at differenttemperatures. J. Mech. Phys. Solids 45, 1805e1834.

Miehe, C., 1995. Discontinuous and continuous damage evolution in ogden-typelarge-strain elastic materials. Eur. J. Mech. A/Solids 14, 697e720.

Miehe, C., Keck, J., 2002. Surimposed finite elastic-viscoelastic-plastoelastic stressresponse with damage in filled rubbery polymers. experiments, modelling andalgorithmic implementation. J. Mech. Phys. Solids 48, 323e365.

Mooney, M., 1940. A theory of large elastic deformation. J. Appl. Phys. 11, 582e592.Morman, K., 1988. Original contributions. an adaptation of finite linear viscoelas-

ticity theory for rubber-like by use of the generalised strain measure. Rheo-logica Acta 27, 3e14.

Mullins, L., 1947. Effect of strectching on the properties of rubber. J. Rubber Res. 16,275e289.

Mullins, L., 1956. Determination of degree of crosslinking in natural rubber vulca-nizates. part i. J. Poly. Sci. 19, 225e236.

Mullins, L., 1959. Determination of degree of crosslinking in natural rubber vulca-nizates. part iii. J. Appl. Poly. Sci. 2, 1e7.

Mullins, L., Tobin, N., 1965. Stress softening in rubber vulcanizates. part i, use ofa strain amplification factor to describe the elastic of filler-reinforced rubber.J. Appl. Poly. Sci. 9, 2933e3009.

Nedjar, B., 2002. Frameworks for finite strain viscoelastic-plasticity based onmultiplicative decomposition. part i: continuum formulations. Comput.Methods Appl. Mech. Eng. 191, 1541e1562.

Ogden, R., 1972. Large deformation isotropic elasticity, on the correlation of theoryand experiment for incompressible rubber-like solids. Proc. Roy. Soc. LondA326, 565e584.

Olsson, A., Austrell, P., 2001. A fitting procedure for a viscoelastic-elastoplasticmaterial. In: Constitutive Models for Rubber II. Proceedings of the 2nd Euro-pean Conference for Constitutive Models for Rubber.

Reese, S., 2003. A micromechanically motivated material model for the thermo-viscoelastic material behaviour of rubber-like polymers. Int. J. Plasticity 19,909e940.

Rivlin, R., 1958. Some topics in finite elasticity. In: 1st Symp. on Naval Struct. Mech.Sidoroff, F., 1973. The geometrical concept of intermediate configuration and elastic

finite strain. Arch. Mech. 25 (2), 299e309.Sidoroff, F., 1974. Un modèle viscoélastique non linéaire avec configuration inter-

médiaire. J. Méc 13 (4), 679e713.Sidoroff, F., 1975. Variables internes en viscoélasticité, 2. milieux avec configuration

intermédiaire. J. Méc 14 (4), 571e595.Sidoroff, F., 1977. Rhéologie non-linéaire et variables internes tensorielles. In:

Symposium franco-polonais, Cracovie.Simo, J., 1987. On a fully three-dimensional finite-strain viscoelastic damage model:

fomulation and computational aspects. Comput. Methods Apll. Mech. Eng. 60,153e163.

Tanaka, F., Edwards, S., 1992. Viscoelastic properties of physically cross-linkednetworks. transient network theory. Macromolecules 25, 1516e1523.

Treloar, L., 1943. The elasticity of a network of long chain molecules i. Trans. FaradaySoc. 39, 36e64.

Treloar, L., 1957. The present status of the theory of large elastic deformations. In:The rheology of elastomers.

statistical approach for a hyper-visco-plastic model for filled rubber: experimental...

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