A constitutive model for the Mullins effect with changes in material symmetry

https://doi.org/10.1016/j.ijnonlinmec.2012.05.004Get rights and content

Abstract

When an unfilled or particle reinforced rubber is subjected to cyclic loading–unloading with a fixed amplitude from its natural reference configuration, the stress required on reloading is less than on the initial loading for a deformation up to the maximum value of the stretches achieved. The stress differences in successive loading cycles are largest during the first and second cycles and become negligible after about 4–6 cycles. This phenomenon is known as the Mullins effect. In this paper new experimental data are reported showing the change in material symmetry for an initially undamaged and isotropic material subjected to uniaxial and biaxial extension tests. The effect of preconditioning in one direction on the mechanical response when loaded in a perpendicular direction is discussed. A simple phenomenological model is derived to account for stress softening and changes in material symmetry. The formulation is based on the theory of pseudo-elasticity, the basis of which is the inclusion of scalar variables in the energy function. When active, these variables modify the form of the energy function during the deformation process and therefore change the material response. The general formulation is specialized to pure homogeneous deformation in order to fit the new data. The numerical results are in very good agreement with the experimental data.

Highlights

► Experimental characterization of particle filled elastomer. ► Uniaxial and planar biaxial tension tests to assess deformation induced anisotropy. ► The theory of pseudo-elasticity is used, the basis of which is the inclusion of damage variables. ► The constitutive model captures isotropic softening and generation of preferred directions.

Introduction

There has been an increase recently in the number of publications proposing constitutive laws to describe the Mullins effect in particle reinforced elastomers and the deformation induced transition from isotropic to anisotropic material behavior. From a scientific point of view, this renewed interest is justified by the fact that a number of fundamental issues in rubber elasticity theory remain unresolved.

In this paper we first describe new experimental results that illustrate the influence of stress-softening during uniaxial and biaxial extension tests on the change of the material symmetry group. The data are then used to formulate a new constitutive model for the isotropic, transversely isotropic and orthotropic material responses.

Consider an isotropic and elastic material in a stress-free undeformed reference configuration (natural configuration). Deformation induces anisotropy and, following Noll's rule, the material symmetry group relative to the deformed configuration changes. The material properties however are unchanged and after deformation the material symmetry relative to the natural configuration is reestablished, i.e. material response is isotropic after unloading.

For particle reinforced elastomers, on the other hand, deformation induces a permanent change in the material properties. Even if there is no residual strain and the original natural configuration is unchanged, the damage caused, by uniaxial tension for example, must induce a change in material symmetry relative to that configuration since a preferred direction has been generated. This is the direction of the extension, which is recorded by the material and influences the subsequent response, which in this case will be transversely isotropic. More generally, when a residual stress arises, the natural configuration changes and the material response relative to a new reference configuration has a symmetry group different from that relative to the original natural configuration. Again, for uniaxial deformation this will generate transverse isotropy. In general, however, these two aspects of induced anisotropy are different. For the constitutive model proposed in this paper we assume that the reference configuration is unchanged and therefore the theory is restricted in the sense that the residual strain is not accounted for. An overview of the theory describing the change of symmetry group in elastic materials, known as Noll's rule, is given by Ogden [33]. For a discussion on deformation induced damage, including a change in natural configuration, we refer to Horgan et al. [19] and Dorfmann and Ogden [9].

Mullins and co-workers, starting in the late 1940s, investigated stress softening of both unfilled and particle-reinforced rubbers and published their results in a series of papers [29], [30], [31], [14], [15], [16], [17]. Since then much experimental and theoretical research has been done to further characterize this phenomenon resulting in a large number of excellent publications. Most of these focus on a simplified representation of stress softening, where the material response is isotropic even after unloading and the accumulation of residual strain is not considered. These models are known to represent an idealized Mullins effect. We do not list the contributions made by individual authors, instead refer to the excellent overview given by Göktepe and Miehe [11] and to the recent review article by Diani et al. [7].

In evaluating experimental data, Mullins [29] observed that a tensile test produces stress softening in all directions, however, the degree of softening is different in different directions. Therefore, uniaxial loading–unloading of an initially undamaged and isotropic material restricts the symmetry group to a subgroup where the mechanical response exhibits a preferred direction. Horgan et al. [19] expand upon this observation and suggest a constitutive theory to account for the transition from isotropy to transverse isotropy. Residual strain is not considered and thus the original reference configuration is unchanged. Dorfmann and Ogden [9] were the first to propose a model that accounts for the Mullins effect and the permanent set accumulated during uniaxial loading–unloading. The natural reference configuration is then no longer stress free and the material response becomes transversely isotropic.

The tests performed by Mullins [29] to investigate anisotropy consist of two phases. First, a large rectangular sheet of rubber is stretched in one direction. Following unloading, two dog-bone shaped specimens are cut both along and perpendicular to the direction of stretch. The two specimens are then subjected to uniaxial extension and the stress–stretch data are compared. Mullins [29] found that softening in the direction perpendicular to the direction of stretch was less than half the softening produced in the direction of stretch. The test protocol used by Mullins [29] was used more recently by Machado et al. [22], [23], Hanson et al. [13], Merckel et al. [25], [26], and Diani et al. [5], [6] to investigate the degree of stress softening in different directions following a uniaxial tensile test. Muhr et al. [28] preconditioned two rectangular shaped specimens, one in pure shear the other in uniaxial tension, the latter in the direction perpendicular to the pure shear extension. Both specimens are then subjected to pure shear to evaluate the induced anisotropy. Dargazany and Itskov [3] used a cruciform shaped specimen to perform a series of uniaxial loading–unloading cycles in one direction with different but constant maximum stretch. Subsequently the same loading routine was repeated in the perpendicular direction and the corresponding results compared.

Two phenomenological models are proposed by Horgan et al. [19]. One is based on the network alteration concept proposed by Marckmann et al. [24] and assumes that the material response is isotropic even after unloading. The other considers the transition from isotropy to transverse isotropy for an initially isotropic material subjected to uniaxial tension. The formulation of the latter is based on the theory of pseudo-elasticity developed by Ogden and Roxburgh [34]. The theory of pseudo-elasticity is also used by Dorfmann and Ogden [9] to define a strain energy function that accounts separately for the stress softening and residual strain effects. A micro-mechanically based approach is used by Göktepe and Miehe [11] to formulate a two-step approach to describe the deformation induced anisotropic damage. Directional damage parameters are defined and the macroscopic material response is obtained by a numerical homogenization with respect to spatial orientations. The model proposed by Diani et al. [5], [6] uses a total energy obtained by summing the contributions over a finite number of material directions, which represents an anisotropic extension of the network alteration theory presented by Marckmann et al. [24]. The concept of directional models is based on the earlier works by Pawelski [36] and Diani et al. [4]. An energy function that depends explicitly on the principal stretches and directions is used by Shariff [38] to formulate an anisotropic model, which is validated using data by Mullins [30] and Muhr et al. [28]. Dargazany and Itskov [3] use a network model developed by Govindjee and Simo [12] to describe the evolution of a polymer chain network connecting adjacent carbon black aggregates. Very recently, Merckel et al. [25] use Taylor expansions of the inverse Langevin function and of a directional damage formulation to extend the works by Diani et al. [5], [6].

In this study, to assess deformation induced anisotropy in particle filled elastomers, a series of uniaxial and planar biaxial tension tests are performed. Each of the experiments performed consists of two phases. During phase 1 an isotropic undamaged sample is subjected to 5 cycles of uniaxial or biaxial tension tests. During phase 2, the effect of preconditioning is evaluated by loading the specimen in a direction perpendicular to earlier loadings. Based on these data a constitutive model is proposed that simulates stress softening and the transition from isotropy to anisotropy. The theory of pseudo-elasticity is used for this model, the basis of which is the inclusion of damage variables in the energy function in order to separately capture directional independent stress softening and generation of preferred directions. Considering these effects separately is consistent with the observations and findings given by Mullins [29]. The accuracy of the model is validated using the new experimental results.

The paper is organized as follows. In Section 2 we describe specimen shape, the experimental protocol and present new results showing stress–stretch data of uniaxial and biaxial tension tests. The effect of preconditioning in one direction on the mechanical response when loaded in a perpendicular direction is discussed. Section 3 summarizes the relevant equations of rubber elasticity theory. An overview of the theory of pseudo-elasticity is given in Section 4. The theory is used in Section 5 to formulate a new constitutive model for damage induced anisotropy. The model accounts for isotropic softening and deformation induced anisotropy. The general formulation is specialized in Section 6 and the value of the material parameters determined. Numerical results are presented and their accuracy validated using the data presented in Section 2. Some concluding remarks are given in Section 7.

Section snippets

Experimental methods

To assess deformation induced anisotropy in a particle-reinforced rubber, a series of uniaxial and planar biaxial tests are carried out using specimens cut from a 1.52 mm thick sheet. Pressure rolling of thin rubber sheets inevitably induces a preferred direction to the material rendering it slightly stiffer in the direction of calendering. For most applications this difference in material behavior is all but negligible. In this study, to reduce the influence of the manufacturing process on the

Basic equations

In this section we give a brief overview of the basic equations that are needed to describe the stress–deformation response of a rubberlike solid. For full details of the theory of elasticity summarized in this section the reader is referred to, for example, Holzapfel [18] and Ogden [33].

To describe the deformation, we denote the stress-free reference configuration of the body by B0 and identify a generic material point by its position vector X relative to an arbitrary chosen origin.

Pseudo-elasticity

The experimental data show that deformation induces a permanent change in the symmetry group of the material. Uniaxial loading–unloading, for example, generates transverse isotropy that needs to be reflected in the material behavior.

Dorfmann and Ogden [9] used the theory of pseudo-elasticity, developed by Ogden and Roxburgh [34], to account for stress softening and damage induced anisotropy in particle-reinforced rubber. In the theory of pseudo-elasticity the strain energy function W(F),

A constitutive model for damage induced anisotropy

The theory discussed in Section 4 is a very general framework used by Ogden and Roxburgh [34] to model the idealized Mullins effect. Dorfmann and Ogden [9] included two variables in the energy function in order to separately capture stress softening and residual strain effects. In this section we generalize the model in [9] to account for stress softening and induced anisotropy.

The material is again taken to be incompressible and initially isotropic. To reduce the number of material parameters,

Model selection

We now apply the theory derived in 4 Pseudo-elasticity, 5 A constitutive model for damage induced anisotropy to solve boundary value problems and validate the results using the experimental data presented in Section 2. For a thin sheet of material subjected to planar biaxial extension in the (1,2)-plane and with no forces applied in the third direction we have σ30. For simplicity, we also assume that deformation induced damage does not occur in the 3-direction. Then, specializing (66), (67)

Concluding remarks

This paper includes new experimental data to assess changes in material symmetry of a particle filled rubber when subjected to uniaxial or planar biaxial tension. The geometric shape and size of the specimen is important to evaluate stress softening and to show the transition from isotropy to anisotropy. To perform cyclic loading in two perpendicular directions we decided on a cruciform layout with each of the four arms connected to an independently controlled actuator. Real time strain control

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