The influence of residual stress on finite deformation elastic response

Abstract

This paper is concerned with the effect of residual stress on the elastic behaviour of materials undergoing finite elastic deformations. The theory is based on a general constitutive framework for hyperelastic materials with residual stress. Several simple problems, whose solutions are known in the situation where there is no residual stress, are analyzed in order to elucidate the influence of residual stress, and the results are illustrated for two prototype constitutive laws.

Introduction

Residual stresses may be present in unloaded solid materials and can have an important influence on material response under applied loads compared with situations where there is no residual stress. Here, following Hoger [8], we consider a residual stress to be a stress maintained in equilibrium in a body in the absence of loads. Residual stresses are non-homogeneous and the response of a material with residual stress is therefore necessarily non-homogeneous. Residual stresses differ from stresses that arise from processes such as ‘shrink-fit’ which involve fixed boundaries on which loads are generated.

For the development of constitutive laws there is no need to distinguish between residual stresses and initial stresses, i.e. stresses that also satisfy equilibrium but may be associated with loads on the body, may be homogeneous and may be accompanied by finite strains (and then often referred to as pre-stresses). In the context of linear elasticity a general theory of materials with initial stress was developed by Biot [1] and applied to problems of wave propagation by Biot [2]; see also the monograph Biot [3]. In a more general setting, elastic constitutive theory of materials with residual stress has been developed in a series of papers by Hoger (see, for example, [8], [9], [10], [11], [12], [17]), in part motivated by applications to soft tissue mechanics, and used by Man and Lu [19] and Man [18] among others. Further results have been obtained by Shams et al. [28] and Singh and Ogden [25] with particular reference to constitutive theory and small amplitude wave propagation.

In engineering materials residual stresses can arise during manufacture and can be designed to improve material performance or can develop unintentionally, possibly compromising the integrity of the material and leading to premature failure. Typically, engineering materials support small strains in service and most of the literature dealing with theoretical aspects of residual stress effects has been based on the linear theory of elasticity, and in this context the linear theory is adequate. However, with the continuing development of engineering materials capable of large elastic deformation, such as elastomers (including magneto- and electro-sensitive elastomers), there is a need to examine the effects of residual stress in problems of finite elastic deformation. Moreover, many soft biological tissues exhibit residual stresses when excised from, for example, arteries [4], [5], [30] or the heart [24], [6], and such tissues also have elastic behaviour that requires finite elasticity for its description. For reviews of different aspects of the application of finite elasticity to arterial wall mechanics, see Holzapfel et al. [13] and Holzapfel and Ogden [14]. In biological tissues residual stresses typically arise from processes of growth and development (see, for example, the review by Humphrey [15]).

Despite these developments there are relatively few papers where the effects of residual stresses on the finite deformation of elastic materials have been analyzed theoretically. The purpose of this paper is therefore to provide some insight into the influence of residual stress in some basic boundary-value problems, in particular for soft materials capable of large elastic deformations.

In Section 2 we summarize the basic equations required for the description of residual stress and the constitutive equation for the non-linear (hyper-) elastic response relative to a residually stressed, but unstrained, configuration. This is then specialized in Section 3 to a general plane strain formulation, which considerably simplifies the equations, and then applied in Section 3.1 to a brief discussion of homogeneous simple shear deformation with homogeneous initial stress. In particular, it is noted that the usual Poynting effect can be reversed by the presence of residual stress. We then go on to consider, in Section 3.2, the azimuthal shear (locally a simple shear) of a circular cylindrical tube of residually stressed material (with radial and azimuthal components of residual stress). Radial inflation of the same tube under an internal pressure is then examined in Section 3.3. For each of the above problems a prototype strain-energy function and a simple residual stress distribution are used to obtain numerical results, and it is found that a negative radial residual stress makes it easier to deform the material as compared with the same problems in the absence of residual stress. In Section 4 the three-dimensional problem of extension and torsion of a solid circular cylinder is examined, again with a prototype strain-energy function and residual stress distribution. In this case a negative radial residual stress makes it harder to apply a deformation (in this case a torsion) compared with the situation without residual stress.

For different treatments of the problems considered in this paper for the situation in which there is no residual stress, see, for example, Truesdell and Noll [29] and Ogden [21].