# 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, 2007, 44 (78), pp.2079-2092. 10.1016/j.ijsolstr.2006.06.046. hal-01007193

HAL Id: hal-01007193

https://hal.science/hal-01007193

Submitted on 16 Jun 2014

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Prediction of fatigue life improvement in natural rubber using configurational stress

A. Andriyana, E. Verron

Institut de Recherche en Génie Civil et Mécanique – 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 principal 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 macroscopic 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 conditions. The configurational mechanics-based predictor is modified to incorporate inelastic constitutive equations. Afterwards, 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
one is the crack growth approach in which fatigue life is defined as the number of cycles required by a preexisting crack to grow to the point of failure. The idea of considering preexisting cracks or flaws was introduced by Inglis (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 of Wöhler (1867) and was applied to rubber by Cadwell 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 rubber 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 configurational 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 nucleation 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 are encouraging: the predictor is capable to predict privileged regions of loaded rubber body in which macroscopic crack nucleation might appear.

In this study, the application of the previous approach of Verron 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 in Fig. 1. Lines $N_1$, $N_2$ and $N_3$ represent iso-fatigue life lines, thus points A and F have the same fatigue life, $N_1$, points B and E, $N_2$, and points C and D, $N_3$. 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 of Cadwell et al. (1940), Gent (1994) or André et al. (1999). Concerning crystallization, interesting results were experimentally obtained by Toki 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 crystallization. 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

<figure><img src="image_2.png" /><figcaption>Fig. 1. Typical Haigh diagram of natural rubber.</figcaption></figure>
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 microscopic 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 can be shown by considering the second law of thermodynamics. Note that the effective part is designated as the non-dissipative part in Andriyana and Verron (2005). In Section 4, the constitutive equation which models hysteretic response is derived. Comparison between results obtained with the proposed and the classical predictors 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 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 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 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

$$ \mathrm{Div} \Pi + \bar{\mathbf{B}} = 0 \quad \text{and} \quad \mathrm{div} \sigma + \bar{\mathbf{b}} = 0 \qquad (1) $$

where $\Pi$ and $\sigma$ are the first Piola-Kirchoff and Cauchy stress tensors, $\bar{\mathbf{B}}$ and $\bar{\mathbf{b}}$ 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 defects, 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 material 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 $\Pi$ from $\sigma$ and $\bar{\mathbf{B}}$ from $\bar{\mathbf{b}}$ is only partial, having for sole purpose to write the corresponding equation per unit volume and unit area of the reference configuration, but the resulting expression 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 manifold. 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 $\mathbf{F}^T$, thus we have:

$$ \mathbf{F}^T \mathrm{Div} \Pi + \mathbf{F}^T \bar{\mathbf{B}} = 0 \qquad (2) $$

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

$$ \mathrm{Div} \Sigma + \mathbf{G} = 0 \qquad (3) $$
with

$$ \Sigma = W\mathbf{I} - \mathbf{F}^T\mathbf{\Pi} \qquad (4) $$

$$ \mathbf{G} = \mathbf{G}^{\text{inh}} + \mathbf{G}^{\text{ext}} + \mathbf{G}^{\text{ine}} + \mathbf{G}^{\text{th}} \qquad (5) $$

where $\Sigma$ is the configurational stress tensor or the energy momentum tensor with respect to the reference configuration as introduced by Eshelby (1951, 1975) and Chadwick (1975). We refer $\mathbf{G}^{\text{inh}}$, $\mathbf{G}^{\text{ext}}$, $\mathbf{G}^{\text{ine}}$, $\mathbf{G}^{\text{th}}$ as 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

$$ \mathbf{G}^{\text{inh}} = -\left.\frac{\partial W}{\partial \mathbf{X}}\right|_{\text{explicit}}, \quad \mathbf{G}^{\text{ext}} = -\mathbf{F}^T\bar{\mathbf{B}}, \qquad (6) $$

$$ \mathbf{G}^{\text{ine}} = A \text{Grad} \alpha, \quad \mathbf{G}^{\text{th}} = S \text{Grad} T $$

where $W = W(\mathbf{F}, \mathbf{X}, T, \alpha)$ is the strain energy density per unit volume of the reference configuration, $T$ is the absolute temperature and $\mathbf{X}$ is the placement, i.e. a representation of material inhomogeneity. Here internal variables $\alpha$ 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 $\alpha$. Together with $\mathbf{\Pi}$, they are obtained by means of standard thermodynamical arguments (Coleman and Gurtin, 1967):

$$ \mathbf{\Pi} = \frac{\partial W}{\partial \mathbf{F}}, \quad S = -\frac{\partial W}{\partial T}, \quad A = -\frac{\partial W}{\partial \alpha} \qquad (7) $$

For isothermal processes or when the temperature is uniform, $\mathbf{G}^{\text{th}}$ vanishes. 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:

$$ \mathbf{G} = \mathbf{G}^{\text{ine}} = A \text{Grad} \alpha \qquad (8) $$

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, electromagnetic 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 by Kienzler and Herrmann (1997), who explain that [the ij-component] of the configurational stress tensor is the change in the total energy density at a point of an elastic continuum due to a material unit translation in $x_i$-direction of a unit surface with normal in $x_i$-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 Section 1, 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 predictor for rubber.

### 3. A new rubber fatigue life predictor

Consider a rubber body $\mathcal{B}$ defined by its reference configuration ($\mathcal{C}_0$) which corresponds to a vanishing stress and strain, stable configuration of minimum energy as shown in Fig. 2. Suppose that no body force acts
<figure><img src="image_3.png" /><figcaption>Fig. 2. Rubber body under mechanical loading.</figcaption></figure>

on $\mathcal{B}$. Under mechanical loading (thermal effects are not considered), the body deforms and occupies a time sequence of physical configurations. Let $(\mathcal{C}_t)$ and $\mathbf{F}$ be the configuration and the deformation gradient, respectively, at time $t$. An arbitrary material point $P \in \mathcal{B}$ initially located at $\mathbf{X}$ in $(\mathcal{C}_0)$ is mapped to $\mathbf{x}$ in $(\mathcal{C}_t)$. When loading is removed, after any time-dependent effect is disappeared, the body will occupy a new stress free configuration $(\mathcal{C}'_0)$, defined by the inverse motion gradient $\mathbf{f}$ (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 for a body to exist in different natural configurations can be very diverse. Indeed, it is the consequence of the internal 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, from $(\mathcal{C}_0)$ to $(\mathcal{C}'_0)$ 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, $\mathbf{G}^{\text{ine}}$, is equal to zero and $(\mathcal{C}'_0)$ and $(\mathcal{C}_0)$ are identical. In this case, both the configurational force associated with inhomogeneity, $\mathbf{G}^{\text{inh}}$, and the configurational stress tensor have non-zero component in each material point. Moreover, if the body is homogeneous with no singularity, e.g. no crack, $\mathbf{G}^{\text{inh}}$ is 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 by Le Cam et al. (2004), rubber bodies are not perfectly homogeneous. They contain distributed 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 initiation (nucleation) of a macroscopic crack. For André et al. (1999), fatigue life $N_f$ is defined by the number of cycles required to create a 2 mm long crack.

To derive the expression of the proposed predictor, noted $\Sigma^*$ in the following, our present task is to determine 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.
### 3.1. Elastic framework

Our attention focuses on the material point P of the rubber body $\mathcal{B}$ in the reference configuration ($\mathcal{C}_0$) as shown in Fig. 3. Following experimental observation of Le 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 configuration to ($\mathcal{C}_i$). If the microstructural change induced by loading is reversible, the defect will recover its initial configuration ($\mathcal{C}_0$) by unloading. Otherwise, the microstructural change is irreversible and the defect configuration becomes ($\mathcal{C}'_0$) upon unloading. It might be confusing to consider irreversible process, i.e. dissipation, in an elastic material. However, as underlined by Gross 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 in ($\mathcal{C}_0$) and a material surface of outward normal N in material space as shown in Fig. 4. In this material point, energetic properties of defect behavior are completely defined by $\Sigma$. Following the definition of Kienzler and Herrmann (1997) recalled above, for a given unit direction $\theta$, the scalar $\theta \cdot \Sigma N$ is the change of the total energy at the corresponding material point due to a material unit translation in direction $\theta$ of the unit surface with normal N. $\Sigma$ is symmetric, thus it has three real eigenvalues, denoted ($\Sigma_i$)$_{i=1,2,3}$ in the following, and three eigenvectors ($V_i$)$_{i=1,2,3}$ so that

$$ \Sigma = \Sigma_i V_i \otimes V_i \qquad (9) $$

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 (Steinmann et al., 2001). So, as the body tends to reduce its total energy, the minimum eigenvalue of $\Sigma$ should be considered 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 $\Sigma^*$ can be summarized as follows:

$$ \Sigma^* = |\min[(\Sigma_i)_{i=1,2,3,0}]| $$

where $(\Sigma_i)_{i=1,2,3}$ are the principal configurational stresses of $\Sigma$

(i) if $\Sigma^* > 0$, the microscopic defect will tend to grow in the plane normal to $V^*$, the eigenvector associated with $-\Sigma^*$,

(ii) if $\Sigma^* = 0$ the microscopic defect tends to close.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Material point in elastic rubber body under loading.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Configurational stress at material point in the reference configuration.</figcaption></figure>

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 in Fig. 5, but now the body is supposed to exhibit inelastic behavior. As the defect is subjected to loading, it changes its configuration to ($\mathcal{C}_i$). Following the separation of the deformation gradient as proposed by Lee (1969) for plasticity and Sidoroff (1974) for viscoelasticity, we can also introduce an intermediate configuration ($\mathcal{C}_i$) which deformation gradients satisfy $\mathbf{F} = \mathbf{F}_e \mathbf{F}_i$. Here $\mathbf{F}_i$ is the inelastic part of the deformation gradient which transforms the body from ($\mathcal{C}_0$) to ($\mathcal{C}_i$). It is to note that this decomposition is not unique. It is a conceptual one, and can generally

<figure><img src="image_6.png" /><figcaption>Fig. 5. Material point in inelastic rubber body under loading.</figcaption></figure>
not be determined experimentally (Huber and Tsakmakis, 2000). In our case, ($\mathcal{C}_i$) is chosen as a stress-free configuration which can be obtained by an infinitely fast unloading.

Loading removal at ($\mathcal{C}_i$) brings the material point P configuration to ($\mathcal{C}'_i$) after any time-dependent effect is disappeared. The configuration corresponds to ($\mathcal{C}'_0$) in Fig. 3. Here we postulate that the inelastic flow, $L_i = \dot{F}_i F_i^{-1}$, is the macroscopic representation of microscopic rearrangements, e.g. defect growth. In fact, $L_i$ 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 $\Sigma^{\text{EF}}$, and its expression will be defined in the following.

Considering hyperelasticity, the inelastic body is characterized by the existence of the strain energy density $W = W(\mathbf{F}, \mathbf{F}_i)$. Note that any thermal effect in $W$ is neglected and $\mathbf{F}_i$ replaces $\alpha$ 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)

$$ \mathcal{D} = \Pi : \dot{\mathbf{F}} - \dot{W} = -\Sigma : (\mathbf{F}_i^{-1} \dot{\mathbf{F}}_i) = -\hat{\Sigma} : \dot{\mathbf{F}}_i = -\tilde{\Sigma} : \mathbf{L}_i \quad (10) $$

where

$$ \begin{aligned} \Sigma &= W\mathbf{I} - \mathbf{F}^T\Pi \\ \hat{\Sigma} &= \frac{\partial W}{\partial \mathbf{F}_i} = \mathbf{F}_i^{-T}\Sigma \\ \tilde{\Sigma} &= \hat{\Sigma}\mathbf{F}_i^T = \mathbf{F}_i^{-T}\Sigma\mathbf{F}_i^T \end{aligned} \quad (11) $$

In this definition, $\hat{\Sigma}$ is the two-field-point configurational stress relative to the intermediate and reference configurations, while $\tilde{\Sigma}$ is the configurational stress relative to the intermediate configuration. Clearly the latter is the thermodynamical driving force associated with the inelastic flow $L_i$ (see Eq. (10)). Following our assumption that the inelastic flow is only governed by $\Sigma^{\text{EF}}$ and since rubber is classically considered as incompressible, for a quasi-static case we have:

$$ \Sigma^{\text{EF}} = \tilde{\Sigma} = W\mathbf{I} - \mathbf{F}_c^T\Pi\mathbf{F}_i^T \quad (12) $$

The choice to express $\Sigma$ 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 constitutive equations is obtained by simply modifying the previous predictor as follows:

$$ \Sigma^* = \min[(\Sigma_i)_{i=1,2,3,0}] $$

where $(\Sigma_i)_{i=1,2,3}$ are the principal configurational stresses of $\Sigma^{\text{EF}}$

(i) if $\Sigma^* > 0$, the microscopic defect will tend to grow in the plane normal to $\mathbf{V}^*$, the eigenvector associated with $-\Sigma^*$,

(ii) if $\Sigma^* = 0$ 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 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 André et al., 1999). A few phenomenological constitutive models for strain-induced crystallization can be found in the literature, see for example in Negahban (2000), Ahzi et al. (2003) and Rao (2003). Nevertheless, motivated by experimental observations of Toki et al. (2000) and Trabelsi et al. (2003) which demonstrate the close relationship 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, $\Sigma^*$, and classical fatigue life predictors widely used in the literature, i.e. the
maximum principal stretch, $\lambda_{max}$, the maximum principal stress, $\sigma_{max}$, 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 $e_1$ direction (Fig. 6). Assuming the incompressibility of material, the deformation of the bar is defined by

$$x_1 = \lambda X_1 \quad x_2 = \frac{X_2}{\sqrt{\lambda}} \quad x_3 = \frac{X_3}{\sqrt{\lambda}} \qquad (13)$$

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

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

$$F = F_A = F_B \qquad (14)$$

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

$$F_B = F_c F_i \qquad (15)$$

where transformation $F_i$ introduces an intermediate configuration as emphasized previously. The left Cauchy-Green strain tensor and its elastic part are simply:

$$\mathbf{B} = \lambda^2 \mathbf{e}_1 \otimes \mathbf{e}_1 + \frac{1}{\lambda} (\mathbf{e}_2 \otimes \mathbf{e}_2 + \mathbf{e}_3 \otimes \mathbf{e}_3) \\ \mathbf{B}_c = \lambda_c^2 \mathbf{e}_1 \otimes \mathbf{e}_1 + \frac{1}{\lambda_c} (\mathbf{e}_2 \otimes \mathbf{e}_2 + \mathbf{e}_3 \otimes \mathbf{e}_3) \qquad (16)$$

The total Cauchy stress is the sum of equilibrium $\sigma_A$ and non-equilibrium $\sigma_B$ parts (Green and Tobolsky, 1946; Lion, 1996 and Bergströom and Boyce, 1998), thus after considering that $\sigma_{22} = \sigma_{33} = 0$, we have:

$$\sigma = \sigma_A + \sigma_B = \left( 2W_I \left( \lambda^2 - \frac{1}{\lambda} \right) + 2W_I^c \left( \lambda_c^2 - \frac{1}{\lambda_c} \right) \right) \mathbf{e}_1 \otimes \mathbf{e}_1 \qquad (17)$$

<figure><img src="image_7.png" /><figcaption>Fig. 6. Rubber bar under simple tension.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Rheological model.</figcaption></figure>
with $W_I = \partial W_A / \partial I_1$ and $W_I^e = \partial W_B / \partial I_1^e$. Here, $I_1$ and $I_1^e$ are the first invariants of the **B** and **B_e**, respectively while $W_A$ and $W_B$ are 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):

$$ \mathbf{D}_i = \frac{1}{\eta} (\mathbf{F}_e^T \boldsymbol{\sigma}_B^{\text{ex}} \mathbf{F}_e^{-T})_{\text{deviatoric}} \qquad (18) $$

where $\eta$ represents the viscosity and $\boldsymbol{\sigma}_B^{\text{ex}}$ is the extra-part of $\boldsymbol{\sigma}_B$ defined by

$$ \boldsymbol{\sigma}_B^{\text{ex}} = 2W_I^e \lambda_c^2 \mathbf{e}_1 \otimes \mathbf{e}_1 \qquad (19) $$

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

$$ W = W_A(I_1) + W_B(I_1^e) = C_{A1}(I_1 - 3) + C_{A2}(I_1 - 3)^2 + C_{A3}(I_1 - 3)^3 + C_B(I_1^e - 3) \qquad (20) $$

Next, the following material parameters are adopted:

$$ \begin{aligned} C_{A1} &= 1 \text{ MPa}, & C_{A2} &= -10^{-2} \text{ MPa}, & C_{A3} &= 1.5 \times 10^{-4} \text{ MPa} \\ C_B &= 6 \text{ MPa}, & \eta &= 2 \text{ MPa s} \end{aligned} \qquad (21) $$

and the strain rate is set to $\dot{\lambda} = 1 \text{ s}^{-1}$. The theoretical model response to the strain history defined in Fig. 8(a) is presented in Fig. 8(b). For this calculation, the stretch amplitude, $\lambda_{\text{amp}}$, is fixed while the mean stretch, $\lambda_m$, is varied. It is demonstrated that for given $\lambda_{\text{amp}}$, the size of the hysteretic loop decreases when $\lambda_m$ increases. A similar trend is experimentally observed by Lion (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 $\Sigma^*$. 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 accumulation. However, it is known that rubbers exhibit a steady state cyclic response under fatigue loading conditions (André 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 predictors, i.e. the maximum principal stretch, $\lambda_{\text{max}}$, the maximum principal stress, $\sigma_{\text{max}}$ and the strain energy, $W$. These predictors are also evaluated over a cycle. The following method is adopted to compute each predictor over a cycle:

<figure><img src="image_9.png" /><figcaption>Fig. 8. (a) Arbitrary strain history and (b) model response.</figcaption></figure>
* Maximum principal stretch:

$$ \lambda_{\max} = \int_{cycle} |d\lambda| = \int_{cycle} |\dot{\lambda}| dt \qquad (22) $$

* Maximum principal stress:

The stress component which tends to open microscopic defects is in tension direction, $e_1$. 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 $d\sigma = 0$ when $\sigma < 0$. The predictor is given by

$$ \sigma_{\max} = \int_{cycle} |d\sigma| = \int_{cycle} \left| \left[ \frac{d\sigma_A}{d\lambda} + \frac{1}{\lambda_i} \left( 1 - \frac{\lambda_i \dot{\lambda}_i}{\lambda \lambda_i} \right) \frac{d\sigma_B}{d\lambda_e} \right] \dot{\lambda} \right| dt \qquad (23) $$

* Strain energy:

$$ W = \int_{cycle} |dW| = \int_{cycle} \left| \left[ \frac{dW_A}{dI_1} \frac{dI_1}{d\lambda} + \frac{1}{\lambda_i} \left( 1 - \frac{\lambda_i \dot{\lambda}_i}{\lambda \lambda_i} \right) \frac{dW_B}{dI_1^c} \frac{dI_1^c}{d\lambda_e} \right] \dot{\lambda} \right| dt \qquad (24) $$

* Proposed predictor:

The instantaneous effective configurational stress is given by

$$ \Sigma^{\text{EF}} = \Sigma = \left( W_A + W_B - 2W_I \left( \lambda^2 - \frac{1}{\lambda} \right) - 2W_I^c \left( \lambda_e^2 - \frac{1}{\lambda_e} \right) \right) e_1 \otimes e_1 + (W_A + W_B)(e_2 \otimes e_2 + e_3 \otimes e_3) \qquad (25) $$

To take into account only the component which tends to open microscopic defects (which is in the tension direction, $e_1$), we set $d\Sigma = d\Sigma^{\text{EF}} = 0$ when $\Sigma = \Sigma_{11}^{\text{EF}} > 0$. Then the predictor is simply computed by

$$ \Sigma^* = \int_{cycle} |d\Sigma| = \int_{cycle} \left| \left[ \frac{d\Sigma_A}{d\lambda} + \frac{1}{\lambda_i} \left( 1 - \frac{\lambda_i \dot{\lambda}_i}{\lambda \lambda_i} \right) \frac{d\Sigma_B}{d\lambda_e} \right] \dot{\lambda} \right| dt \qquad (26) $$

## 5. Results and discussion

The formulation of each fatigue predictor has been derived above. To predict fatigue life improvement, different 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, $\sigma_{\text{amp}}$, and mean stress, $\sigma_m$. However, it will be presented here in the terms of $\lambda_{\text{amp}}$ and $\lambda_m$. Moreover, iso-fatigue life lines are replaced by iso-value lines of each fatigue life predictor for different loading. For given $\lambda_{\text{amp}}$ and $\lambda_m$, 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 $\lambda_{\text{min}}$. This curve gives the minimum stretch needed to enhance fatigue life, if any, for given $\lambda_{\text{amp}}$. The second one is the theoretical 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 life improvement: fatigue life always decreases (fatigue life predictor increases) when either $\lambda_m$ or $\lambda_{\text{amp}}$ increases. Similar results are obtained for the strain energy as shown in Fig. 10. In fact, during the transition from tension-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 in Fig. 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; André et al.,
<figure><img src="image_10.png" /><figcaption>Fig. 9. (a) Evolution of $\lambda_{max}$ and (b) Haigh diagram in $\lambda_{max}$.</figcaption></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. (a) Evolution of $W$ and (b) Haigh diagram in $W$.</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. (a) Evolution of $\sigma_{max}$ and (b) Haigh diagram in $\sigma_{max}$.</figcaption></figure>

<figure><img src="image_15.png" /></figure>

1999). Better results are obtained with the proposed predictor as shown in Fig. 12. The fatigue life improvement is clearly highlighted. However, it has to be remembered that this improvement, as well as in the case of the maximum principal stress, depends on the choice of the constitutive equation. Both the maximum princi-
<figure><img src="image_16.png" /><figcaption>Fig. 12. (a) Evolution of $\Sigma^*$ and (b) Haigh diagram in $\Sigma^*$.</figcaption></figure>

pal stress and the proposed approach predict that the fatigue improvement occurs at $\lambda_{min} > 1$ which is a natural 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 $\sigma_{amp}$ and $\sigma_m$, and initial improvement is observed to start at $\sigma_{min} = 0$ during the transition between TC and TT. In our knowledge, no experimental data in terms of $\lambda_{amp}$ and $\lambda_m$ are available in literature, so the value of $\lambda_{min}$ which corresponds to initial improvement can not be directly evaluated. Finally, it is important to note that no attempt is made to correlate $\Sigma^*$ 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 exhibit inelastic response characterized only by hysteresis, thus a simple phenomenological visco-hyperelastic constitutive 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 $\Sigma^*$ 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 $F = F_cF_i$ is not unique, further investigations are necessary to determine the most relevant formulation.

## References

* Abraham, F., Alshuth, T., Jerrams, S., 2005. The effect of minimum stress and stress amplitude on the fatigue life of non-strain crystallising elastomers. Mater. Design 26, 239-245.

* Ahzi, S., Makradi, A., Gregory, R.V., Edie, D.D., 2003. Modeling of deformation behavior and strain-induced crystallization in poly (ethylene terephthalate) above the glass transition temperature. Mech. Mater. 35, 1139-1148.

* André, N., Cailletaud, G., Piques, R., 1999. Haigh diagram for fatigue crack initiation prediction of natural rubber components. Kautsch. Gummi Kunstst. 52, 120-123.

* Andriyana, A., Verron, E., 2005. Effect of the hysteretic response of elastomers on the fatigue life. In: Austrell, P.E., Kari, L. (Eds.), Constitutive Models for Rubber IV. A.A. Balkema Publishers, pp. 31-36.

* Bergströom, J.S., Boyce, M.C., 1998. Constitutive modeling of the large strain time-dependent behavior of elastomers. J. Mech. Phys. Solids 46 (5), 931-954.
Cadwell, M., Merril, R., Sloman, C., Yost, F., 1940. Dynamic fatigue life of rubber. Ind. Eng. Chem. 12 (1), 19-23.

Cermelli, P., Fried, E., Sellers, S., 2001. Configurational stress, yield and flow in rate-independent plasticity. Proc. R. Soc. Lond. 457, 1447-1467.

Chadwick, P., 1975. Applications of an energy-momentum tensor in non-linear elastostatics. J. Elast. 5 (3-4), 249-258.

Cleja-Tigolu, S., Maugin, G.A., 2000. Eshelby's stress tensors in finite elastoplasticity. Acta Mech. 139, 231-249.

Coleman, B., Gurtin, M., 1967. Thermodynamics with internal state variables. Chem. Phys. 47 (2), 597-613.

Eshelby, J.D., 1951. The force on an elastic singularity. Phil. Trans. R. Soc. Lond. A 244, 87-112.

Eshelby, J.D., 1975. The elastic energy-momentum tensor. J. Elast. 5 (3-4), 321-335.

Gent, A.N., 1994. Science and Technology of Rubber, second ed. Academic Press, New York, chapter 10, p. 471.

Green, M.S., Tobolsky, A.V., 1946. A new approach for the theory of relaxing polymeric media. J. Chem. Phys. 14, 87-112.

Griffith, A., 1920. The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. A 221, 163-198.

Gross, D., Kolling, S., Mueller, R., Schmidt, I., 2003. Configurational forces and their application in solid mechanics. Eur. J. Mech. Solids A/Solids 22, 669-692.

Gurtin, M.E., 2000. Configurational Forces as Basic Concept of Continuum Physics. Springer, Berlin.

Huber, N., Tsakmakis, C., 2000. Finite deformation viscoelasticity laws. Mech. Mater. 32, 1-18.

Inglis, C.E., 1913. Stresses in plate due to the presence of cracks and sharp corners. Trans. Inst. Naval Architects 55 (41), 219.

Kienzler, R., Herrmann, G., 1997. On the properties of the Eshelby tensor. Acta Mech. 125, 73-91.

Kienzler, R., Herrmann, G., 2000. Mechanics in Material Space. Springer, Berlin.

Kienzler, R., Herrmann, G., 2002. Fracture criteria based on local properties of the Eshelby tensor. Mech. Res. Commun. 29, 521-527.

Le Cam, J.-B., Huneau, B., Verron, E., Gornet, L., 2004. Mechanism of fatigue crack growth in carbon black filled natural rubber. Macromolecules 37, 5011-5017.

Lee, E.H., 1969. Elastic-plastic deformation at finite strain. Trans. ASME J. Appl. Mech. 36, 1-6.

Lion, A., 1996. A constitutive model for carbon black filled rubber: experimental investigations and mathematical representation. Continuum Mech. Thermodyn. 8, 153-169.

Mars, W.V., 2002. Cracking energy density as a predictor of fatigue life under multiaxial conditions. Rubber Chem. Technol. 75, 1-17.

Mars, W.V., Fatemi, A., 2002. A literature survey on fatigue analysis approaches for rubber. Int. J. Fatigue 24, 949-961.

Maugin, G.A., 1993. Material Inhomogeneities in Elasticity. Chapman and Hall, London.

Maugin, G.A., 1995. Material forces: Concepts and applications. Appl. Mech. Rev. 48 (5).

Maugin, G.A., 2002. Remarks on the Eshelby thermomechanics of materials. Mech. Res. Commun. 29, 537-542.

Negahban, M., 2000. Modeling the thermomechanical effects of crystallization in natural rubber: I. The theoretical structure. Int. J. Solids Struct. 37, 2777-2789.

Rajagopal, K.R., Srinivasa, A.R., 2005. On the role of the Eshelby energy-momentum tensor in materials with natural configurations. Math. Mech. Solids 10, 3-24.

Rao, I.J., 2003. Effect of the rate of deformation on the crystallization behavior of polymers. Int. J. Non-linear Mech. 38, 663-676.

Rivlin, R.S., Thomas, A.G., 1953. Rupture of rubber. I. Characteristic energy for tearing. J. Polym. Sci. 10, 291-318.

Sidoroff, F., 1974. Un modèle viscoélastique non-linéaire avec configuration intermédiaire. J. Mécanique 13, 679-713.

Steinmann, P., 2000. Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Struct. 37, 7371-7391.

Steinmann, P., Ackermann, D., Barth, F.J., 2001. Application of material forces to hyperelastostatic fracture mechanics. II. Computational setting. Int. J. Solids Struct. 38, 5509-5526.

Toki, S., Fujimaki, T., Okuyama, M., 2000. Strain-induced crystallization of natural rubber as detected real-time by wide-angle X-ray diffraction technique. Polymer 41, 5423-5429.

Trabelsi, S., Albouy, P., Rault, J., 2003. Stress-induced crystallization properties of natural and synthetic cis-polyisoprene. Rubber. Chem. Technol. 77, 303-316.

Truesdell, C., Noll, W., 1965. Handbuch der Physik Bd. III/3: The Non-linear Field Theories of Mechanics. Springer-Verlag, Berlin.

Verron, E., Le Cam, J.B., Gornet, L., 2006. A multiaxial criterion for crack nucleation in rubber. Mech. Res. Commun. 33, 493-498.

Wöhler, A., 1867. Wöhler experiments on the strength of metals. Engineering 2, 160.