Prediction of fatigue life improvement in natural rubber ... · Mech. Res. Commun. 33, 493–498]...

International Journal of Solids and Structures 44 (2007) 2079–2092

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Prediction of fatigue life improvement in natural rubberusing configurational stress

A. Andriyana *, E. Verron

Institut de Recherche en Genie Civil et Mecanique – GeM, UMR CNRS 6183, Ecole Centrale de Nantes,

BP 92101, 44321 Nantes, France

Received 27 March 2006; received in revised form 28 June 2006Available online 5 July 2006

Abstract

To some extent, continua can no longer be considered as free of defects. Experimental observations on natural rubberrevealed the existence of distributed microscopic defects which grow upon cyclic loading. However, these observations arenot 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 criterionfor crack nucleation in rubber. Mech. Res. Commun. 33, 493–498] considered the configurational stress tensor to propose afatigue 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 predictsprivileged regions of rubber parts in which macroscopic fatigue crack might appear. Here, we will address our interest to abroader 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 fatiguelife improvement in natural rubber under tension–tension cyclic loading.� 2006 Elsevier Ltd. All rights reserved.

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 isrequired. Two approaches are generally adopted to define end of life (Mars and Fatemi, 2002). The first

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* Corresponding author. Fax: +33 2 40 37 25 66.E-mail address: [email protected] (A. Andriyana).

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2080 A. Andriyana, E. Verron / International Journal of Solids and Structures 44 (2007) 2079–2092

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 preexisting cracks or flaws was introducedby Inglis (1913) and Griffith (1920), then firstly applied to rubber by Rivlin and Thomas (1953). The crackgrowth is determined by the calculation of the so-called tearing energy for given specimen and crack shapesand for prescribed loading conditions. The major difficulty of this approach is that initial crack shape andposition should be known, which is not possible in most engineering problems. Moreover, for industrial partshaving complex geometry under multiaxial loading, the computation of the tearing energy is revealedcomplicated.

The second approach is the crack nucleation approach, where fatigue life is defined as the number of cyclesrequired to create a crack of a given size. This approach, which follows the work of Wohler (1867) and wasapplied to rubber by Cadwell et al. (1940), considers that fatigue life of rubbers can be determined from thehistory 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 failto 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 ofmicroscopic defects or flaws in rubber parts. This was motivated by the fact that a crack nucleation predictorshould be formulated in terms of continuum mechanics quantities in order to be combined with standard finiteelement 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 rationalizes the work of Mars(2002) who proposed a so-called cracking energy density. First results on quasi-static loading problems areencouraging: 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 of Verron et al. (2006) in predicting fatigue lifeimprovement of natural rubber is considered. When fatigue life of natural rubber, i.e. number of cyclesrequired to create a crack of a given size, is plotted as a function of stress amplitude and mean stress (a Haighdiagram), an interesting phenomenon is observed: under uniaxial tension–tension (TT) cyclic loading, for agiven stress amplitude, fatigue life increases with mean stress. Such a Haigh diagram is shown in Fig. 1. LinesN1, N2 and N3 represent iso-fatigue life lines, thus points A and F have the same fatigue life, N1, points B andE, N2, and points C and D, N3. This improvement is not observed for metallic materials. Even if the physicalorigin of this phenomenon is not well-known, it is often attributed to strain-induced crystallization, see forexample the works of Cadwell et al. (1940), Gent (1994) or Andre et al. (1999). Concerning crystallization,interesting results were experimentally obtained by Toki et al. (2000) and Trabelsi et al. (2003) who measuredthe real-time evolution of crystallization of natural rubber during stretching and retracting. Authors relatedthe hysteretic response, classically observed during mechanical experiments, with the kinetics of crystalliza-tion. Here, motivated by this observation, a simple phenomenological non-linear viscoelastic constitutiveequation which captures the hysteretic response of rubber is adopted to represent inelastic behavior. Otherinelastic phenomena such as stress-softening and permanent set are not taken into account. The purpose of

Fig. 1. Typical Haigh diagram of natural rubber.

A. Andriyana, E. Verron / International Journal of Solids and Structures 44 (2007) 2079–2092 2081

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 withemphasis on the finite strain inelastic response of solids is considered. A brief recall on the configurationalmechanics-based predictor as proposed by Verron et al. (2006) is presented in Section 3 with its extensionto 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 microscopic defects configuration(which can lead to crack nucleation), is governed by the effective part of the configurational stress as canbe shown by considering the second law of thermodynamics. Note that the effective part is designated asthe non-dissipative part in Andriyana and Verron (2005). In Section 4, the constitutive equation which modelshysteretic response is derived. Comparison between results obtained with the proposed and the classical pre-dictors are shown in Section 5. Concluding remarks are given in Section 6.

2. Theory of configurational mechanics

The theory of configurational mechanics was introduced by Eshelby (1951) when he proposed the conceptof the energy-momentum tensor and configurational forces in continuum mechanics of solids. While severalauthors consider this as basic objects of new concepts of mechanics (Gurtin, 2000), some others demonstratethat it is only an extension of the classical continuum mechanics (Maugin, 1995). This theory is designated asthe Eshelbian Mechanics by Maugin (1993), and the Mechanics in Material Space by Kienzler and Herrmann(2000) by contrast to the Newtonian Mechanics or the Mechanics in Physical Space, respectively. The rapiddevelopment of the configurational mechanics is partially due to difficulties of describing the evolution ofinternal material structure or configuration, that generally follows material movement and deformation, inthe physical space.

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

DivPþ B ¼ 0 and divrþ �b ¼ 0 ð1Þ

where P and r are the first Piola–Kirchoff and Cauchy stress tensors, B and �b are the physical body forcesper unit volume of the reference and the current configurations. In general, material deformation in thephysical 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 thephysical space is not an easy task (Steinmann, 2000). To overcome this difficulty, the balance of physicallinear 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 ofthe body (Truesdell and Noll, 1965). Such reference configuration may not be an actual configuration everoccupied by the body. As remarked by Maugin (1995), components of the equation of motion in theLagrangian description are still in the physical space, so that it is not an intrinsic formulation: the pull-backoperation 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 moreoperation 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 (Grosset al., 2003). In the present work, the latter is adopted. It is accomplished simply by multiplying the left-handside of the equation of motion in Lagrangian description by the transpose of the deformation gradient FT,thus we have:

FT DivPþ FTB ¼ 0 ð2Þ
After some algebraic manipulations, this equation becomes (Maugin, 1995)
DivRþG ¼ 0 ð3Þ


with

R ¼ W I� FTP ð4ÞG ¼ Ginh þGext þGine þGth ð5Þ

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

Ginh ¼ � oWoX

��explicit

; Gext ¼ �FTB;

Gine ¼ AGrada; Gth ¼ 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 theabsolute temperature and X is the placement, i.e. a representation of material inhomogeneity. Here internalvariables 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 densityper unit volume in the reference configuration and A is the thermodynamical driving force associated witha. Together with P, they are obtained by means of standard thermodynamical arguments (Coleman andGurtin, 1967):

P ¼ oWoF

; S ¼ � oWoT

; A ¼ � oWoa

ð7Þ

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

G ¼ Gine ¼ A Grada 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 ofthe configurational stress tensor. For the linear theory, the physical significance of the Cartesian componentsof this tensor were identified by Kienzler 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 theprincipal values and directions of the tensor in two dimensions. More recently, the same authors introducelocal fracture criteria for small strain problems based on the components of configurational stress (Kienzlerand Herrmann, 2002). In two dimensions, principal values and a von-Mises-like value of this tensor are relatedto the classical stress intensity factors in linear fracture mechanics.

As underlined in Section 1, an efficient fatigue life predictor should reflect the physical phenomena whichtake place during crack initiation and growth. Furthermore, it should be written in terms of continuummechanics in order to be combined with the finite element method in engineering applications. Thus, basedon the above theoretical summary, we consider that the configurational stress, which focuses on the behaviorof 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 vanishingstress and strain, stable configuration of minimum energy as shown in Fig. 2. Suppose that no body force acts

Fig. 2. Rubber body under mechanical loading.


on B. Under mechanical loading (thermal effects are not considered), the body deforms and occupies a timesequence of physical configurations. Let ðCtÞ 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. Whenloading is removed, after any time-dependent effect is disappeared, the body will occupy a new stress free con-figuration ðC00Þ, defined by the inverse motion gradient f (see Fig. 2).

Such stress free configuration is referred to as a natural configuration by Rajagopal and Srinivasa (2005).According to the authors, many bodies possess more than one natural configuration. The physical reasons fora 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 crackpropagation or cavitations for defects growth. As stated by the same authors, the configurational forcesare the thermodynamical driving forces associated with the evolution of natural configurations, i.e. internalmicrostructural rearrangements, from ðC0Þ to ðC00Þ in our case. These forces are the natural contributors tothe equation of motion in the material manifold equation (3). Thus, microstructural rearrangements arerelated 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), itcan be considered that this tensor contains phenomenological quantities which represent a part of the energyavailable to change the properties of the microstructure in a given material direction. So, for a purely elasticbody with no irreversible microstructural change under loading, the configurational force associated withinelasticity, Gine, is equal to zero and ðC00Þ and ðC0Þ are identic. In this case, both the configurational forceassociated with inhomogeneity, Ginh, and the configurational stress tensor have non-zero component in eachmaterial point. Moreover, if the body is homogenous with no singularity, e.g. no crack, Ginh is equal to zero.However, it does not imply that the configurational stress tensor is null. In this case, the energy available in thematerial, due to loading, to change microstructural properties is completely recovered during unloading.

As observed by Le 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 andposition. Furthermore, when applied loading is sufficiently large, microscopic defects growth can lead to ini-tiation (nucleation) of a macroscopic crack. For Andre et al. (1999), fatigue life Nf is defined by the number ofcycles 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 formacroscopic crack occurrence. The formulation in the purely elastic framework is discussed. Afterwards,extension of the proposed predictor for inelastic bodies is presented.


3.1. Elastic framework

Our attention focuses on the material point P of the rubber body B in the reference configuration ðC0Þ asshown in Fig. 3. Following experimental observation of Le Cam et al. (2004), this material point could be amicroscopic defect, e.g. microcavity or inclusion. As the defect is subjected to loading, it changes its configura-tion to ðCtÞ. If the microstructural change induced by loading is reversible, the defect will recover its initial con-figuration ðC0Þ by unloading. Otherwise, the microstructural change is irreversible and the defect configurationbecomes ðC00Þ upon unloading. It might be confusing to consider irreversible process, i.e. dissipation, in an elasticmaterial. However, as underlined by Gross et al. (2003), the presence of microscopic defects in the loaded bodyyields 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 in ðC0Þ and a material surface of outwardnormal N in material space as shown in Fig. 4. In this material point, energetic properties of defect behaviorare completely defined by R. Following the definition of Kienzler and Herrmann (1997) recalled above, for agiven unit direction h, the scalar h Æ RN is the change of the total energy at the corresponding material pointdue to a material unit translation in direction h of the unit surface with normal N. R is symmetric, thus it hasthree real eigenvalues, denoted (Ri)i=1,2,3 in the following, and three eigenvectors (Vi)i=1,2,3 so that

(

(

R ¼ RiVi � Vi ð9Þ
These three vectors correspond to the directions in which the energy release rates tend to open or close the defectby material normal traction without material shear traction. Moreover, as the force on a defect is defined as thenegative 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 planecrack which normal is defined by the corresponding eigenvector. If it is positive, the three eigenvalues are positiveand 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[(Ri)i=1,2,3,0]jwhere (Ri)i=1,2,3 are the principal configurational stresses of Ri) if R* > 0, the microscopic defect will tend to grow in the plane

normal to V*, the eigenvector associated with �R*,ii) if R* = 0 the microscopic defect tends to close.

Fig. 4. Configurational stress at material point in the reference configuration.


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 nucleationmight 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 in Fig. 5, but now the body issupposed to exhibit inelastic behavior. As the defect is subjected to loading, it changes its configuration toðCtÞ. Following the separation of the deformation gradient as proposed by Lee (1969) for plasticity and Sidor-off (1974) for viscoelasticity, we can also introduce an intermediate configuration ðCiÞ which deformation gra-dients satisfy F = FeFi. Here Fi is the inelastic part of the deformation gradient which transforms the bodyfrom ðC0Þ to ðCiÞ. It is to note that this decomposition is not unique. It is a conceptual one, and can generally

Fig. 5. Material point in inelastic rubber body under loading.


not be determined experimentally (Huber and Tsakmakis, 2000). In our case, ðCiÞ is chosen as a stress-freeconfiguration which can be obtained by an infinitely fast unloading.

Loading removal at ðCtÞ brings the material point P configuration to ðC0iÞ after any time-dependent effect isdisappeared. The configuration corresponds to ðC00Þ in Fig. 3. Here we postulate that the inelastic flow,Li ¼ _FiF

�1i , 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 Fi replaces a in Eqs. (7) and (8). Applying thesecond law of thermodynamics, expression of dissipation per unit volume of the reference configuration isgiven by (Gross et al., 2003)

(

(

D ¼ P : _F� _W ¼ �R : ðF�1i

_FiÞ ¼ �R : _Fi ¼ �~R : Li ð10Þ
where
R ¼ W I� FTP

R ¼ oWoFi

¼ F�Ti R

~R ¼ RFTi ¼ F�T

i RFTi

ð11Þ

In this definition, R is the two-field-point configurational stress relative to the intermediate and reference con-figurations, while ~R is the configurational stress relative to the intermediate configuration. Clearly the latter isthe thermodynamical driving force associated with the inelastic flow Li (see Eq. (10)). Following our assump-tion that the inelastic flow is only governed by REF and since rubber is classically considered as incompressible,for a quasi-static case we have:

REF ¼ ~R ¼ W I� FTe PFT

i ð12Þ
The choice to express R in the intermediate configuration was also adopted in the case of finite plasticity
(see for example works of Cermelli 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 constitutiveequations is obtained by simply modifying the previous predictor as follows:

R* = jmin[(Ri)i=1,2,3,0]jwhere (Ri)i=1,2,3 are the principal configurational stresses of REF

i) if R* > 0, the microscopic defect will tend to grow in the plane normalto V*, the eigenvector associated with �R*,

ii) if R* = 0 the microscopic defect tends to close.

4. Constitutive equations

As mentioned in Section 1, the present work deals with the prediction of fatigue life improvement in rubberunder tension–tension cyclic loading using the proposed fatigue life predictor. Fatigue life improvement inrubber 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 in Negahban (2000), Ahzi et al. (2003) and Rao (2003). Nevertheless, motivated byexperimental observations of Toki 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 phenomenologicalvisco-hyperelastic model which captures the hysteretic response is adopted. This model is used to compute theproposed fatigue life predictor, R*, and classical fatigue life predictors widely used in the literature, i.e. the


maximum principal stretch, kmax, the maximum principal stress, rmax, and the strain energy, W. Results andcomparison 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 e1 direction(Fig. 6). Assuming the incompressibility of material, the deformation of the bar is defined by

x1 ¼ kX 1 x2 ¼X 2ffiffiffi

kp x3 ¼

X 3ffiffiffikp ð13Þ

where k is uniform and equal to the ratio between deformed and undeformed length of the specimen. To mod-el finite strain inelastic response of rubber which is characterized by hysteresis, a visco-hyperelastic rheologicalmodel is adopted as schematically shown in Fig. 7.

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

F ¼ FA ¼ FB ð14Þ
The deformation gradient tensor on network B can further be decomposed into elastic and inelastic parts
through a tensor product:

FB ¼ FeFi ð15Þ
where transformation Fi introduces an intermediate configuration as emphasized previously. The left Cauchy–Green strain tensor and its elastic part are simply:
B ¼ k2e1 � e1 þ1

kðe2 � e2 þ e3 � e3Þ

Be ¼ k2ee1 � e1 þ

1

ke

ðe2 � e2 þ e3 � e3Þð16Þ

The total Cauchy stress is the sum of equilibrium rA and non-equilibrium rB parts (Green and Tobolsky,1946; Lion, 1996 and Bergstroom and Boyce, 1998), thus after considering that r22 = r33 = 0, we have:

r ¼ rA þ rB ¼ 2W I k2 � 1

k

� �þ 2W e

I k2e �

1

ke

� �� e1 � e1 ð17Þ

Fig. 6. Rubber bar under simple tension.

Fig. 7. Rheological model.


with WI = oWA/oI1 and W eI ¼ oW B=oIe

1. Here, I1 and Ie1 are the first invariants of the B and Be, respectively

while WA and WB are the strain energy density associated with networks A and B. Assuming a linear behaviorfor the dashpot, its deformation rate is given by (Huber and Tsakmakis, 2000):

Di ¼1

gðFT

e rexB F�T

e Þdeviatoric ð18Þ

where g represents the viscosity and rexB is the extra-part of rB defined by

rexB ¼ 2W e

Ik2ee1 � e1 ð19Þ

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

W ¼ W AðI1Þ þ W BðIe1Þ ¼ CA1ðI1 � 3Þ þ CA2ðI1 � 3Þ2 þ CA3ðI1 � 3Þ3 þ CBðIe

1 � 3Þ ð20Þ
Next, the following material parameters are adopted:
CA1 ¼ 1 MPa; CA2 ¼ �10�2 MPa; CA3 ¼ 1:5� 10�4 MPa

CB ¼ 6 MPa; g ¼ 2 MPa sð21Þ

and the strain rate is set to _k ¼ 1 s�1. The theoretical model response to the strain history defined in Fig. 8(a) ispresented in Fig. 8(b). For this calculation, the stretch amplitude, kamp, is fixed while the mean stretch, km, isvaried. It is demonstrated that for given kamp, the size of the hysteretic loop decreases when km increases. Asimilar trend is experimentally observed by Lion (1996) and Abraham et al. (2005). It is important to note thatthe 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 theloading history, the increment of the configurational stress should be integrated over the entire history ofthe 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 hystereticresponse. Thus, it is considered that the evaluation of the proposed fatigue predictor over only one stabilizedcycle 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, rmax and the strain energy, W.These predictors are also evaluated over a cycle. The following method is adopted to compute each predictorover a cycle:

0 2 4 6 8 10 120.8

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

Fig. 8. (a) Arbitrary strain history and (b) model response.


• Maximum principal stretch:

W

RE

R�

kmax ¼Z

cycle

jdkj ¼Z

cycle

j _kjdt ð22Þ

• Maximum principal stress:

The stress component which tends to open microscopic defects is in tension direction, e1. However, asshown 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 setdr = 0 when r < 0. The predictor is given by

ax ¼Z

cycle

jdrj ¼Z

cycle

drA

dkþ 1

ki1� k _ki

_kki

!drB

dke

" #_k

��dt ð23Þ
rm
• Strain energy:

¼Z

cycle

jdW j ¼Z

cycle

dW A

dI1

dI1

dkþ 1

ki1� k _ki

_kki

!dW b

dI e1

dI e1

dke

" #_k

��dt ð24Þ

• Proposed predictor:

The instantaneous effective configurational stress is given by

F ¼ R ¼ W A þ W B � 2W I k2 � 1

k

� �� 2W e

I k2e �

1

ke

� �� e1 � e1 þ ðW A þ W BÞðe2 � e2 þ e3 � e3Þ ð25Þ

To take into account only the component which tends to open microscopic defects (which is in the tensiondirection, e1), we set dR = dREF = 0 when R ¼ REF

11 > 0. Then the predictor is simply computed by

¼Z

cycle

jdRj ¼Z

cycle

dRA

dkþ 1

ki1� k _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 strainamplitude is investigated in order to build theoretically the Haigh diagram. This diagram is classically plottedin 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 differentloading. For given kamp and km, the values of fatigue predictors over a cycle are computed. Two results aregiven for each predictor. The first one is the evolution of fatigue predictors as functions of kmin. This curvegives 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 in Figs. 9–12.

Observing Fig. 9, it is obvious that the maximum principal stretch can not be used to predict fatigue lifeimprovement: fatigue life always decreases (fatigue life predictor increases) when either km or kamp increases.Similar results are obtained for the strain energy as shown in Fig. 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 originof this enhancement is not easy to explain.

The maximum principal stress gives fairly good results in predicting fatigue life improvement as illustratedin Fig. 11. This is quite surprising as experimental results give evidence that energy controls fatigue life ratherthan strain or stress (Abraham et al., 2005). In our opinion, this is due to the fact that the stress is cumulatedover a cycle and only the stress component which corresponds to crack opening is taken into account, which isnot the case when stress predictors are considered in published works (Abraham et al., 2005; Andre et al.,

a b

0 1 2 3 41

2

3

4

5

6

7

8

λmin

λmax

λamp

=0.6

λamp

=0.8

λamp

=0.4

λamp

=0.2

1.5 2 2.5 30.2

0.3

0.4

0.5

0.6

0.7

0.8

λm

λamp

λmax

=2.9.

λmax

=3.6.

λmax

=3.4.

λmax

=4.2.

λmax

=4.

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

0 1 2 3 40

10

20

30

40

50

λmin

σmax

λamp

=0.8

λamp

=0.6

λamp

=0.4

λamp

=0.2

1.5 2 2.5 30.2

0.3

0.4

0.5

0.6

0.7

0.8

λm

λamp

σmax

=20.

.σ

max=17

.σmax

=27

.σmax

=25

.σmax

=19.

σmax

=19

.σ

max=26

a b

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

0 1 2 3 40

5

10

15

20

25

λmin

Wλ

amp=0.6

λamp

=0.8

λamp

=0.4

λamp

=0.2

1.5 2 2.5 30.2

0.3

0.4

0.5

0.6

0.7

0.8

λm

λamp

W=4.

W=7.

W=6.

W=10.

W=8.

a b

Fig. 10. (a) Evolution of W and (b) Haigh diagram in W.


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

0 1 2 3 45

10

15

20

25

λmin

Σ*

λamp

=0.8

λamp

=0.6

λamp

=0.4

λamp

=0.2

1.5 2 2.5 30.2

0.3

0.4

0.5

0.6

0.7

0.8

λm

λamp

Σ*=14.

.Σ*=13

Σ*=13.

Σ*=16.Σ*=18.

.Σ*=15

.Σ*=12

a b

Fig. 12. (a) Evolution of R* and (b) Haigh diagram in R*.


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, theHaigh diagram is experimentally plotted as a function of ramp and rm, and initial improvement is observed tostart at rmin = 0 during the transition between TC and TT. In our knowledge, no experimental data in terms ofkamp and km are available in literature, so the value of kmin which corresponds to initial improvement can notbe directly evaluated. Finally, it is important to note that no attempt is made to correlate R* to number ofcycles 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 topredict 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 theevolution of microscopic defects (which can lead to crack nucleation) is only governed by the effective partof the configurational stress. This part is obtained by projecting the configurational stress tensor onto theintermediate configuration. Results show that this simple method can be used to predict qualitatively wellthe improvement of fatigue life in natural rubber under tension–tension cyclic loading despite no attempthas 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 effectivepart of the configurational stress tensor. Nevertheless, as the decomposition of F = FeFi is not unique, furtherinvestigations are necessary to determine the most relevant formulation.

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