Prediction of fatigue life improvement in natural rubber using configurational stress

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Prediction of fatigue life improvement in natural rubber

using configurational stress

Andri Andriyana, Erwan Verron

To cite this version:

Andri Andriyana, Erwan Verron. Prediction of fatigue life improvement in natural rubber using

configurational stress. International Journal of Solids and Structures, Elsevier, 2007, 44 (78),

pp.2079-2092. �10.1016/j.ijsolstr.2006.06.046�. �hal-01007193�

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Prediction of fatigue life improvement in natural rubber

using conﬁgurational stress

A. Andriyana, E. Verron

Institut de Recherche en Ge´nie Civil et Me´canique – GeM, UMR CNRS 6183, Ecole Centrale de Nantes, BP 92101, 44321 Nantes, France

To some extent, continua can no longer be considered as free of defects. Experimental observations on natural rubber revealed the existence of distributed microscopic defects which grow upon cyclic loading. However, these observations are not incorporated in the classical fatigue life predictors for rubber, i.e. the maximum principal stretch, the maximum prin-cipal stress and the strain energy. Recently, Verron et al. [Verron, E., Le Cam, J.B., Gornet, L., 2006. A multiaxial criterion for crack nucleation in rubber. Mech. Res. Commun. 33, 493–498] considered the configurational stress tensor to propose a fatigue life predictor for rubber which takes into account the presence of microscopic defects by considering that macro-scopic crack nucleation can be seen as the result of the propagation of microscopic defects. For elastic materials, it predicts privileged regions of rubber parts in which macroscopic fatigue crack might appear. Here, we will address our interest to a broader context. Rubber is assumed to exhibit inelastic behavior, characterized by hysteresis, under fatigue loading con-ditions. The configurational mechanics-based predictor is modified to incorporate inelastic constitutive equations. After-wards, it is used to predict fatigue life. The emphasis of the present work is laid on the prediction of the well-known fatigue life improvement in natural rubber under tension–tension cyclic loading.

Keywords: Conﬁgurational stress; Rubber; Fatigue; Crack initiation

1. Introduction

Despite large application of rubber in industrial parts, e.g. tires, vibration isolators, seals, medical devices, etc., it should be acknowledged that many aspects of fatigue failure in rubber remain incompletely understood. In order to prevent fatigue failure of rubber parts in service, a proper multiaxial fatigue life predictor is required. Two approaches are generally adopted to deﬁne end of life (Mars and Fatemi, 2002). The ﬁrst

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one is the crack growth approach in which fatigue life is defined as the number of cycles required by a preex-isting crack to grow to the point of failure. The idea of considering preexpreex-isting cracks or flaws was introduced byInglis (1913) and Griffith (1920), then firstly applied to rubber by Rivlin and Thomas (1953). The crack growth is determined by the calculation of the so-called tearing energy for given specimen and crack shapes and for prescribed loading conditions. The major difficulty of this approach is that initial crack shape and position should be known, which is not possible in most engineering problems. Moreover, for industrial parts having complex geometry under multiaxial loading, the computation of the tearing energy is revealed complicated.

The second approach is the crack nucleation approach, where fatigue life is defined as the number of cycles required to create a crack of a given size. This approach, which follows the work ofWo¨hler (1867)and was applied to rubber byCadwell et al. (1940), considers that fatigue life of rubbers can be determined from the history of strain and stress at each material point of the body. The three most widely used predictors for rub-ber are the maximum principal stretch, the maximum principal stress and the strain energy. However they fail to give satisfying predictions for multiaxial loading problems. Recently,Verron et al. (2006)used configura-tional mechanics theory to introduce a new fatigue life predictor which takes into account the presence of microscopic defects or flaws in rubber parts. This was motivated by the fact that a crack nucleation predictor should be formulated in terms of continuum mechanics quantities in order to be combined with standard finite element method in engineering applications. In this approach, it is considered that macroscopic crack nucle-ation can be seen as the consequence of microscopic defects growth. In fact it rnucle-ationalizes the work ofMars (2002)who proposed a so-called cracking energy density. First results on quasi-static loading problems are encouraging: the predictor is capable to predict privileged regions of loaded rubber body in which macro-scopic crack nucleation might appear.

In this study, the application of the previous approach ofVerron et al. (2006) in predicting fatigue life improvement of natural rubber is considered. When fatigue life of natural rubber, i.e. number of cycles required to create a crack of a given size, is plotted as a function of stress amplitude and mean stress (a Haigh diagram), an interesting phenomenon is observed: under uniaxial tension–tension (TT) cyclic loading, for a given stress amplitude, fatigue life increases with mean stress. Such a Haigh diagram is shown inFig. 1. Lines N1, N2and N3represent iso-fatigue life lines, thus points A and F have the same fatigue life, N1, points B and

E, N2, and points C and D, N3. This improvement is not observed for metallic materials. Even if the physical

origin of this phenomenon is not well-known, it is often attributed to strain-induced crystallization, see for example the works ofCadwell et al. (1940), Gent (1994) or Andre´ et al. (1999). Concerning crystallization, interesting results were experimentally obtained byToki et al. (2000) and Trabelsi et al. (2003)who measured the real-time evolution of crystallization of natural rubber during stretching and retracting. Authors related the hysteretic response, classically observed during mechanical experiments, with the kinetics of crystalliza-tion. Here, motivated by this observation, a simple phenomenological non-linear viscoelastic constitutive equation which captures the hysteretic response of rubber is adopted to represent inelastic behavior. Other inelastic phenomena such as stress-softening and permanent set are not taken into account. The purpose of

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the present paper is to demonstrate that the fatigue life improvement in rubber can be reproduced by a well-chosen fatigue life predictor and a simple phenomenological non-linear viscoelastic constitutive equation.

The paper is organized as follows. In Section 2, the general theory of configurational mechanics with emphasis on the finite strain inelastic response of solids is considered. A brief recall on the configurational mechanics-based predictor as proposed by Verron et al. (2006) is presented in Section 3 with its extension to inelastic behavior of rubber under fatigue loading. Here, we assume that the inelastic flow, which in micro-scopic viewpoint can also be regarded as the change or rearrangement of micromicro-scopic defects configuration (which can lead to crack nucleation), is governed by the effective part of the configurational stress as can be shown by considering the second law of thermodynamics. Note that the effective part is designated as the non-dissipative part inAndriyana and Verron (2005). In Section4, the constitutive equation which models hysteretic response is derived. Comparison between results obtained with the proposed and the classical pre-dictors are shown in Section5. Concluding remarks are given in Section6.

2. Theory of conﬁgurational mechanics

The theory of configurational mechanics was introduced byEshelby (1951)when he proposed the concept of the energy-momentum tensor and configurational forces in continuum mechanics of solids. While several authors consider this as basic objects of new concepts of mechanics (Gurtin, 2000), some others demonstrate that it is only an extension of the classical continuum mechanics (Maugin, 1995). This theory is designated as the Eshelbian Mechanics byMaugin (1993), and the Mechanics in Material Space byKienzler and Herrmann (2000)by contrast to the Newtonian Mechanics or the Mechanics in Physical Space, respectively. The rapid development of the configurational mechanics is partially due to difficulties of describing the evolution of internal material structure or configuration, that generally follows material movement and deformation, in the physical space.

In the classical Newtonian Mechanics, the motion of continuum is based on the postulates of balance of physical linear and angular momenta (Steinmann, 2000). The balance of physical linear momentum can be expressed in local Lagrangian and Eulerian descriptions. For static case, it is given respectively by

Div Pþ B ¼ 0 and div rþ b¼ 0 ð1Þ where P and r are the first Piola–Kirchoff and Cauchy stress tensors, B and b are the physical body forces per unit volume of the reference and the current configurations. In general, material deformation in the physical space is followed by a change or rearrangement of internal material structure, e.g. microscopic de-fects, dislocation or boundary phase. Describing the rearrangement of internal material configuration in the physical space is not an easy task (Steinmann, 2000). To overcome this difficulty, the balance of physical linear momentum has to be written onto the material space or material manifold (Maugin, 1995). The mate-rial space is an abstract notion of the space which consists of particles defining a reference configuration of the body (Truesdell and Noll, 1965). Such reference configuration may not be an actual configuration ever occupied by the body. As remarked by Maugin (1995), components of the equation of motion in the Lagrangian description are still in the physical space, so that it is not an intrinsic formulation: the pull-back operation that introduced P from r and B from b is only partial, having for sole purpose to write the cor-responding equation per unit volume and unit area of the reference configuration, but the resulting expres-sion are still containing components in the current configuration. Thus, it is necessary to make one more operation that allows to project canonically the equation of motion onto the material space or material man-ifold. This can be performed either by following the Noether’s theorem or by using a direct method (Gross et al., 2003). In the present work, the latter is adopted. It is accomplished simply by multiplying the left-hand side of the equation of motion in Lagrangian description by the transpose of the deformation gradient FT, thus we have:

FTDiv Pþ FT_B_{¼ 0} _ð2Þ

After some algebraic manipulations, this equation becomes (Maugin, 1995)

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with

R¼ W I FTP ð4Þ

G¼ Ginh_{þ G}ext_{þ G}ine_{þ G}th _ð5Þ

where R is the configurational stress tensor or the energy momentum tensor with respect to the reference con-figuration as introduced byEshelby (1951, 1975) and Chadwick (1975). We refer Ginh, Gext, Gine, Gthas the configurational forces per unit volume of the reference configuration associated with inhomogeneity, body force, inelastic response and thermal effects, respectively as underlined by Maugin (1995), Cleja-Tigolu and Maugin (2000). They are defined by

Ginh¼ oW oX _explicit; Gext¼ FTB; Gine¼ A Grad a; Gth¼ S Grad T

ð6Þ

where W = W(F, X, T, a) is the strain energy density per unit volume of the reference conﬁguration, T is the absolute temperature and X is the placement, i.e. a representation of material inhomogeneity. Here internal variables a are introduced as origin of irreversible processes, together with material inhomogeneity. They can be either scalar, vector or tensor, depending on considered phenomena. S is the entropy density per unit volume in the reference conﬁguration and A is the thermodynamical driving force associated with a. Together with P, they are obtained by means of standard thermodynamical arguments (Coleman and Gurtin, 1967): P¼oW oF; S ¼ oW oT ; A¼ oW oa ð7Þ

For isothermal processes or when the temperature is uniform, Gthvanishes. Furthermore, if the material is assumed to be homogeneous and not subjected to body force, the total conﬁgurational force is simply reduced to its inelastic component:

G¼ Gine¼ A Grad a ð8Þ

Various applications of the conﬁgurational forces can be found in the literature. The reader can refer to Maugin (1995)in which this concept is applied to thermoelastic fracture, locally dissipative solids, electromag-netic deformable bodies and elastodynamics.

Opposite to the case of configurational forces, only few studies are concerned with the peculiar properties of the configurational stress tensor. For the linear theory, the physical significance of the Cartesian components of this tensor were identified byKienzler and Herrmann (1997), who explain that [the ij-component] of the con-figurational stress tensor is the change in the total energy density at a point of an elastic continuum due to a mate-rial unit translation in xj-direction of a unit surface with normal in xi-direction. Authors also investigated the

principal values and directions of the tensor in two dimensions. More recently, the same authors introduce local fracture criteria for small strain problems based on the components of conﬁgurational stress (Kienzler and Herrmann, 2002). In two dimensions, principal values and a von-Mises-like value of this tensor are related to the classical stress intensity factors in linear fracture mechanics.

As underlined in Section1, an efficient fatigue life predictor should reflect the physical phenomena which take place during crack initiation and growth. Furthermore, it should be written in terms of continuum mechanics in order to be combined with the finite element method in engineering applications. Thus, based on the above theoretical summary, we consider that the configurational stress, which focuses on the behavior of defects in the material manifold appears to be the appropriate tool to develop a relevant fatigue life pre-dictor for rubber.

3. A new rubber fatigue life predictor

Consider a rubber body B deﬁned by its reference conﬁguration ðC0Þ which corresponds to a vanishing

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on B. Under mechanical loading (thermal effects are not considered), the body deforms and occupies a time sequence of physical configurations. LetðCtÞ and F be the configuration and the deformation gradient,

respec-tively, at time t. An arbitrary material point P 2 B initially located at X in ðC0Þ is mapped to x in ðCtÞ. When

loading is removed, after any time-dependent eﬀect is disappeared, the body will occupy a new stress free con-ﬁgurationðC0

0Þ, deﬁned by the inverse motion gradient f (seeFig. 2).

Such stress free configuration is referred to as a natural configuration byRajagopal and Srinivasa (2005). According to the authors, many bodies possess more than one natural configuration. The physical reasons for a body to exist in different natural configurations can be very diverse. Indeed, it is the consequence of the inter-nal microstructural rearrangement, e.g. movement of dislocations for plasticity, cleavage fracture for crack propagation or cavitations for defects growth. As stated by the same authors, the configurational forces are the thermodynamical driving forces associated with the evolution of natural configurations, i.e. internal microstructural rearrangements, fromðC0Þ to ðC00Þ in our case. These forces are the natural contributors to

the equation of motion in the material manifold equation (3). Thus, microstructural rearrangements are related to the configurational stress tensor which is involved in this equation (Maugin, 1995). Furthermore, recalling physical significance of the configurational stress proposed by Kienzler and Herrmann (1997), it can be considered that this tensor contains phenomenological quantities which represent a part of the energy available to change the properties of the microstructure in a given material direction. So, for a purely elastic body with no irreversible microstructural change under loading, the configurational force associated with inelasticity, Gine, is equal to zero andðC0

0Þ and ðC0Þ are identic. In this case, both the conﬁgurational force

associated with inhomogeneity, Ginh, and the conﬁgurational stress tensor have non-zero component in each material point. Moreover, if the body is homogenous with no singularity, e.g. no crack, Ginhis equal to zero. However, it does not imply that the conﬁgurational stress tensor is null. In this case, the energy available in the material, due to loading, to change microstructural properties is completely recovered during unloading.

As observed byLe Cam et al. (2004), rubber bodies are not perfectly homogeneous. They contain distrib-uted microscopic defects. Upon fatigue loading, defects change their configurations, i.e. their size, form and position. Furthermore, when applied loading is sufficiently large, microscopic defects growth can lead to ini-tiation (nucleation) of a macroscopic crack. ForAndre´ et al. (1999), fatigue life Nfis defined by the number of

cycles required to create a 2 mm long crack.

To derive the expression of the proposed predictor, noted R*_{in the following, our present task is to}

deter-mine the energy release rate of all possible microscopic defects in order to exhibit the most propitious zone for macroscopic crack occurrence. The formulation in the purely elastic framework is discussed. Afterwards, extension of the proposed predictor for inelastic bodies is presented.

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3.1. Elastic framework

Our attention focuses on the material point P of the rubber body B in the reference conﬁguration ðC0Þ as

shown inFig. 3. Following experimental observation ofLe Cam et al. (2004), this material point could be a microscopic defect, e.g. microcavity or inclusion. As the defect is subjected to loading, it changes its conﬁgura-tion toðCtÞ. If the microstructural change induced by loading is reversible, the defect will recover its initial

con-ﬁgurationðC0Þ by unloading. Otherwise, the microstructural change is irreversible and the defect conﬁguration

becomesðC0

0Þ upon unloading. It might be confusing to consider irreversible process, i.e. dissipation, in an elastic

material. However, as underlined byGross et al. (2003), the presence of microscopic defects in the loaded body yields to the change of its total energy if these defects move relatively to the reference conﬁguration.

In order to determine the energy release rate, consider a defect inðC0Þ and a material surface of outward

normal N in material space as shown inFig. 4. In this material point, energetic properties of defect behavior are completely deﬁned by R. Following the deﬁnition ofKienzler and Herrmann (1997)recalled above, for a given unit direction h, the scalar h Æ RN is the change of the total energy at the corresponding material point due to a material unit translation in direction h of the unit surface with normal N. R is symmetric, thus it has three real eigenvalues, denoted (Ri)i=1,2,3in the following, and three eigenvectors (Vi)i=1,2,3so that

R¼ RiVi Vi ð9Þ

These three vectors correspond to the directions in which the energy release rates tend to open or close the defect by material normal traction without material shear traction. Moreover, as the force on a defect is deﬁned as the negative gradient of the total energy with respect to the change in position of the defect (Kienzler and Herrmann, 2000), the morphology of the defect evolves with respect to the direction opposite to the material force (Stein-mann et al., 2001). So, as the body tends to reduce its total energy, the minimum eigenvalue of R should be con-sidered in order to predict the evolution of the defect. If it is negative, the defect will tend to grow to form a plane crack which normal is deﬁned by the corresponding eigenvector. If it is positive, the three eigenvalues are positive and the defect will tend to shrink. Finally, the new crack nucleation predictor R*_{can be summarized as follows:}

Fig. 3. Material point in elastic rubber body under loading.

R*₌_jmin[(R

i)i=1,2,3,0]j

where (Ri)i=1,2,3are the principal conﬁgurational stresses of R

(i) if R*_{> 0, the microscopic defect will tend to grow in the plane}

normal to V*_{, the eigenvector associated with}_R*_,

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First results obtained with this predictor for uniaxial fatigue loading conditions are encouraging (Verron et al., 2006). It is capable to predict privileged regions of loaded rubber body in which macroscopic crack nucleation might appear. In the next section, the extension to the case of inelasticity is considered.

3.2. Extension to inelastic framework

Consider the same material point P in rubber body under loading as shown inFig. 5, but now the body is supposed to exhibit inelastic behavior. As the defect is subjected to loading, it changes its conﬁguration to ðCtÞ. Following the separation of the deformation gradient as proposed byLee (1969)for plasticity and

Sidor-oﬀ (1974)for viscoelasticity, we can also introduce an intermediate conﬁgurationðCiÞ which deformation

gra-dients satisfy F = FeFi. Here Fiis the inelastic part of the deformation gradient which transforms the body

fromðC0Þ to ðCiÞ. It is to note that this decomposition is not unique. It is a conceptual one, and can generally Fig. 4. Conﬁgurational stress at material point in the reference conﬁguration.

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not be determined experimentally (Huber and Tsakmakis, 2000). In our case,ðCiÞ is chosen as a stress-free

conﬁguration which can be obtained by an inﬁnitely fast unloading.

Loading removal atðCtÞ brings the material point P conﬁguration to ðC0iÞ after any time-dependent eﬀect is

disappeared. The conﬁguration corresponds to ðC00Þ in Fig. 3. Here we postulate that the inelastic ﬂow,

Li¼ _FiF1i , is the macroscopic representation of microscopic rearrangements, e.g. defect growth. In fact, Li

is driven by only a part of the configurational stress tensor. This part will be referred to as the effective part of the configurational stress, noted REF, and its expression will be defined in the following.

Considering hyperelasticity, the inelastic body is characterized by the existence of the strain energy density W = W(F, Fi). Note that any thermal eﬀect in W is neglected and Fireplaces a in Eqs.(7) and (8). Applying the

second law of thermodynamics, expression of dissipation per unit volume of the reference conﬁguration is given by (Gross et al., 2003)

D ¼ P : _F _W ¼ R : ðF1i F_iÞ ¼ ^R : _Fi¼ ~R : Li ð10Þ where R¼ W I FT_P ^ R¼oW oFi ¼ FTi R ~ R¼ ^RFT_i ¼ FT i RF T i ð11Þ

In this definition, ^Ris the two-field-point configurational stress relative to the intermediate and reference con-figurations, while ~Ris the configurational stress relative to the intermediate configuration. Clearly the latter is the thermodynamical driving force associated with the inelastic flow Li(see Eq.(10)). Following our

assump-tion that the inelastic ﬂow is only governed by REFand since rubber is classically considered as incompressible, for a quasi-static case we have:

REF¼ ~R¼ W I FTePF T

i ð12Þ

The choice to express R in the intermediate conﬁguration was also adopted in the case of ﬁnite plasticity (see for example works ofCermelli et al., 2001; Maugin, 1995; Cleja-Tigolu and Maugin, 2000 and Maugin, 2002).

Following similar arguments as in the case of elasticity, the fatigue life predictor for inelastic constitutive equations is obtained by simply modifying the previous predictor as follows:

4. Constitutive equations

As mentioned in Section1, the present work deals with the prediction of fatigue life improvement in rubber under tension–tension cyclic loading using the proposed fatigue life predictor. Fatigue life improvement in rubber is often attributed to strain-induced crystallization (Cadwell et al., 1940; Gent, 1994 or Andre´ et al., 1999). A few phenomenological constitutive models for strain-induced crystallization can be found in the lit-erature, see for example inNegahban (2000), Ahzi et al. (2003) and Rao (2003). Nevertheless, motivated by experimental observations ofToki et al. (2000) and Trabelsi et al. (2003)which demonstrate the close relation-ship between the hysteretic response of rubber and the kinetics of crystallization, a simple phenomenological visco-hyperelastic model which captures the hysteretic response is adopted. This model is used to compute the proposed fatigue life predictor, R*_{, and classical fatigue life predictors widely used in the literature, i.e. the}

R*₌_jmin[(R

i)i=1,2,3,0]j

where (Ri)i=1,2,3are the principal conﬁgurational stresses of REF

(i) if R*_{> 0, the microscopic defect will tend to grow in the plane normal}

to V*_{, the eigenvector associated with}_R*_,

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maximum principal stretch, kmax, the maximum principal stress, rmax, and the strain energy, W. Results and

comparison between classical predictors and the proposed one are presented. 4.1. Governing equations

Consider a rectangular rubber bar with uniform cross section subjected to simple tension in the e1direction

(Fig. 6). Assuming the incompressibility of material, the deformation of the bar is defined by x1¼ kX1 x2¼ X2 ffiffiffi k p x3¼ X3 ffiffiffi k p ð13Þ

where k is uniform and equal to the ratio between deformed and undeformed length of the specimen. To mod-el ﬁnite strain inmod-elastic response of rubber which is characterized by hysteresis, a visco-hypermod-elastic rheological model is adopted as schematically shown inFig. 7.

The total deformation gradient tensor F acts both on networks A and B, i.e.:

F¼ FA¼ FB ð14Þ

The deformation gradient tensor on network B can further be decomposed into elastic and inelastic parts through a tensor product:

FB¼ FeFi ð15Þ

where transformation Fiintroduces an intermediate conﬁguration as emphasized previously. The left Cauchy–

Green strain tensor and its elastic part are simply: B¼ k2e1 e1þ 1 kðe2 e2þ e3 e3Þ Be¼ k2ee1 e1þ 1 ke ðe2 e2þ e3 e3Þ ð16Þ

The total Cauchy stress is the sum of equilibrium rAand non-equilibrium rBparts (Green and Tobolsky,

1946; Lion, 1996 and Bergstro¨om and Boyce, 1998), thus after considering that r22= r33= 0, we have:

r¼ rAþ rB¼ 2WI k2 1 k þ 2We I k 2 e 1 ke e1 e1 ð17Þ

Fig. 6. Rubber bar under simple tension.

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with WI=oWA/oI1and We_I ¼ oWB=oIe1. Here, I1and Ie1 are the ﬁrst invariants of the B and Be, respectively

while WAand WBare the strain energy density associated with networks A and B. Assuming a linear behavior

for the dashpot, its deformation rate is given by (Huber and Tsakmakis, 2000): Di¼ 1 gðF T er ex BF T e Þdeviatoric ð18Þ

where g represents the viscosity and rex

B is the extra-part of rBdeﬁned by

rex B ¼ 2W e Ik 2 ee1 e1 ð19Þ

Furthermore, we postulate that the strain energy density of the model has the form:

W ¼ WAðI1Þ þ WBðIe1Þ ¼ CA1ðI1 3Þ þ CA2ðI1 3Þ2þ CA3ðI1 3Þ3þ CBðIe1 3Þ ð20Þ

Next, the following material parameters are adopted:

CA1¼ 1 MPa; CA2¼ 102MPa; CA3¼ 1:5 104MPa

CB¼ 6 MPa; g¼ 2 MPa s

ð21Þ

and the strain rate is set to _k¼ 1 s1_{. The theoretical model response to the strain history deﬁned in}_{Fig. 8(a) is}

presented inFig. 8(b). For this calculation, the stretch amplitude, kamp, is ﬁxed while the mean stretch, km, is

varied. It is demonstrated that for given kamp, the size of the hysteretic loop decreases when kmincreases. A

similar trend is experimentally observed byLion (1996) and Abraham et al. (2005). It is important to note that the response obtained is only theoretical: it depends on the choice of the material parameters.

4.2. Fatigue predictors

Until now we have derived the instantaneous fatigue life predictor R*_{. In order to take into account the}

loading history, the increment of the conﬁgurational stress should be integrated over the entire history of the material. For fatigue loading, it involves several thousands cycles that it is impossible to perform this accu-mulation. However, it is known that rubbers exhibit a steady state cyclic response under fatigue loading con-ditions (Andre´ et al., 1999; Abraham et al., 2005). This response is characterized by a stabilized hysteretic response. Thus, it is considered that the evaluation of the proposed fatigue predictor over only one stabilized cycle would be suﬃcient to predict fatigue life.

Results obtained by the proposed fatigue life predictor are compared with those obtained with classical pre-dictors, i.e. the maximum principal stretch, kmax, the maximum principal stress, rmaxand the strain energy, W.

These predictors are also evaluated over a cycle. The following method is adopted to compute each predictor over a cycle: 0 2 4 6 8 10 12 0.8 1 1.2 1.4 1.6 1.8 2 λ t 0.8 1 1.2 1.4 1.6 1.8 2 −6 −4 −2 0 2 4 6 λ Π a b

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• Maximum principal stretch: kmax¼ Z cycle jdkj ¼ Z cycle j _kj dt ð22Þ

• Maximum principal stress:

The stress component which tends to open microscopic defects is in tension direction, e1. However, as

shown in Fig. 8(b), during unloading this component has negative value for certain deformation states. Thus, we have to exclude contributions of stress because it closes the defect. In other words, we set dr = 0 when r < 0. The predictor is given by

rmax¼ Z cycle jdrj ¼ Z cycle drA dk þ 1 ki 1k _ki _kki ! drB dke " # _k dt ð23Þ • Strain energy: W ¼ Z cycle jdW j ¼ Z cycle dWA dI1 dI1 dkþ 1 ki 1k _ki _kki ! dWb dIe 1 dIe₁ dke " # _k dt ð24Þ • Proposed predictor:

The instantaneous eﬀective conﬁgurational stress is given by REF¼ R ¼ WAþ WB 2WI k2 1 k 2We I k 2 e 1 ke e1 e1þ ðWAþ WBÞðe2 e2þ e3 e3Þ ð25Þ

To take into account only the component which tends to open microscopic defects (which is in the tension direction, e1), we set dR = dR

EF

= 0 when R¼ REF

11 >0. Then the predictor is simply computed by

R¼ Z cycle jdRj ¼ Z cycle dRA dk þ 1 ki 1k _ki _kki ! dRB dke " # _k dt ð26Þ

5. Results and discussion

The formulation of each fatigue predictor has been derived above. To predict fatigue life improvement, dif-ferent strain-controlled loading conditions are considered, i.e. the inﬂuence of both mean strain and strain amplitude is investigated in order to build theoretically the Haigh diagram. This diagram is classically plotted in terms of stress amplitude, ramp, and mean stress, rm. However, it will be presented here in the terms of kamp

and km. Moreover, iso-fatigue life lines are replaced by iso-value lines of each fatigue life predictor for diﬀerent

loading. For given kampand km, the values of fatigue predictors over a cycle are computed. Two results are

given for each predictor. The ﬁrst one is the evolution of fatigue predictors as functions of kmin. This curve

gives the minimum stretch needed to enhance fatigue life, if any, for given kamp. The second one is the

theo-retical Haigh diagram. These results are presented inFigs. 9–12.

ObservingFig. 9, it is obvious that the maximum principal stretch can not be used to predict fatigue life improvement: fatigue life always decreases (fatigue life predictor increases) when either kmor kampincreases.

Similar results are obtained for the strain energy as shown inFig. 10. In fact, during the transition from ten-sion–compression (TC) to tension–tension (TT), a small increase in fatigue life is detected. However the origin of this enhancement is not easy to explain.

The maximum principal stress gives fairly good results in predicting fatigue life improvement as illustrated inFig. 11. This is quite surprising as experimental results give evidence that energy controls fatigue life rather than strain or stress (Abraham et al., 2005). In our opinion, this is due to the fact that the stress is cumulated over a cycle and only the stress component which corresponds to crack opening is taken into account, which is not the case when stress predictors are considered in published works (Abraham et al., 2005; Andre´ et al.,

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1999). Better results are obtained with the proposed predictor as shown inFig. 12. The fatigue life improve-ment is clearly highlighted. However, it has to be remembered that this improveimprove-ment, as well as in the case of the maximum principal stress, depends on the choice of the constitutive equation. Both the maximum

princi-a b 0 1 2 3 4 1 2 3 4 5 6 7 8 λ_min λ_max λ_amp=0.6 λ_amp=0.8 λ_amp=0.4 λ_amp=0.2 1.5 2 2.5 3 0.2 0.3 0.4 0.5 0.6 0.7 0.8 λ_m λ_amp λmax

.

=2.9 λmax

.

=3.6 λ_max

.

=3.4 λ

.

_max=4.2 λ_max

.

=4

Fig. 9. (a) Evolution of kmaxand (b) Haigh diagram in kmax.

0 1 2 3 4 0 10 20 30 40 50 λ_min σ_max λ_amp=0.8 λ_amp=0.6 λ_amp=0.4 λ_amp=0.2 1.5 2 2.5 3 0.2 0.3 0.4 0.5 0.6 0.7 0.8 λ_m λ_amp σ_max

.

=20

.

σ_max=17

.

σ_max=27

.

σ_max=25

.

σ_max=19 σ_max

.

=19

.

σ_max=26 a b

Fig. 11. (a) Evolution of rmaxand (b) Haigh diagram in rmax.

0 1 2 3 4 0 5 10 15 20 25 λ_min W λ_amp=0.6 λ_amp=0.8 λ_amp=0.4 λ_amp=0.2 1.5 2 2.5 3 0.2 0.3 0.4 0.5 0.6 0.7 0.8 λ_m λ_amp W=4

.

W=7

.

W=6

.

W=10

.

W=8

.

a b

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pal stress and the proposed approach predict that the fatigue improvement occurs at kmin> 1 which is a

nat-ural consequence of the use of a visco-hyperelastic model, i.e. time-dependent hyperelasticity. In general, the Haigh diagram is experimentally plotted as a function of rampand rm, and initial improvement is observed to

start at rmin= 0 during the transition between TC and TT. In our knowledge, no experimental data in terms of

kampand kmare available in literature, so the value of kminwhich corresponds to initial improvement can not

be directly evaluated. Finally, it is important to note that no attempt is made to correlate R*_{to number of}

cycles at end of life and until now only problems which admit analytical solutions have been considered. 6. Conclusion

In this paper, we have introduced an extension of the configurational mechanics-based fatigue predictor, proposed initially by Verron et al. (2006), to the case of inelasticity. The extended predictor is then used to predict the improvement of fatigue life observed experimentally in natural rubber. Rubber is assumed to exhi-bit inelastic response characterized only by hysteresis, thus a simple phenomenological visco-hyperelastic con-stitutive equation is chosen. In this study, evaluation of a single stabilized cycle is considered to be sufficient. Moreover it is assumed that the inelastic flow, which in microscopic viewpoint can also be regarded as the evolution of microscopic defects (which can lead to crack nucleation) is only governed by the effective part of the configurational stress. This part is obtained by projecting the configurational stress tensor onto the intermediate configuration. Results show that this simple method can be used to predict qualitatively well the improvement of fatigue life in natural rubber under tension–tension cyclic loading despite no attempt has been made to correlate a direct relationship between R*_{and number of cycles at end of life.}

To close this paper, it is to note that we have used the intermediate configuration to express the effective part of the configurational stress tensor. Nevertheless, as the decomposition of F = FeFiis not unique, further

investigations are necessary to determine the most relevant formulation. References

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