Nonlinear Viscoelastic Solids

Abstract

1. Introduction. The classical theories of the mechanics of deformable materials are, for the most part, linear theories and ones in which the stress depends only on the kinematic situation existing at the time of measurement of the stress. For many materials, such as viscoelastic solids and liquids, these assumptions are not valid. Since the paper by Oldroyd [13], considerable advances have been made in formulating the continuum mechanics of such materials. In the present paper an account of this work will be given, particularly in relation to viscoelastic solids. Provided that any characteristic lengths which may exist iu the solid can be ignored, we may take as our starting point in the development of appropriate constitutive equations the assumption that the stress tensor: depends not only on the displacement gradients existing in the material considered at the instant at which the stress is measured, but also on those which existed in it at all previous times, i.e., the stress is a functional of the deformation gradient history up to the instant of measurement. If some reference state is adequately specified, the assumption that stress depends on the history of the deformation gradients and the assumption that stress depends on the history of the velocity or acceleration. gradients are quite equivalent, since either of the latter determine the deformation gradient history. The form of the functional dependence of the stress on the deformation gradient history cannot be arbitrarily chosen. It is subject to two major restrictions. The first of these, which is discussed in 2, arises from the fact that a simultaneous time-dependent rotation of the deformed body and reference system leaves the stress components unaltered. The second, which is discussed in 3 and 5, arises from any symmetry which the material may possess in its reference state. The important particular case when the material is isotropic in its reference state is discussed in 5. In certain circumstances, the history of the deformation gradients may be adequately described by a knowledge of the values of the deformation gradients and of the gradients of velocity, acceleration, second acceleration., etc., at the instant of measurement of the stress. The development of canonical forms for the constitutive equations in this case is discussed generally in 4 and for the particular circumstance when the material is isotropic in its reference state in 5. The constitutive equations which arise from these general studies are inevitably complicated. In 6 attention is drawn to the great simplifications whichmay arise