Constitutive Equations

Abstract

Introduction Constitutive equationsIn linearly viscoelastic materials, the constitutive equation relating time dependent stress r(t) and time dependent strain e(t), is a Boltzmann integral:where J(t) is the creep compliance as it depends on time t, and E(t) is the relaxation modulus. Many constitutive models have been developed to describe the nonlinear viscoelastic behavior of materials; see Schapery (1969) and Findley et al. (1976). Models described by Johnson et al. (1996) are for large deformations and Pioletti et al. (1998) take into account the strain rate. In the present study we consider a singleintegral form called nonlinear superposition. This allows the relaxation function to depend on strain level and creep to depend on stress:A particular form of this, due to Fung (1972), assumes the strain-dependent modulus is separable into the product of a function of time and a function of strain: Abstract Creep and stress relaxation are known to be interrelated in linearly viscoelastic materials by an exact analytical expression. In this article, analytical interrelations are derived for nonlinearly viscoelastic materials which obey a single integral nonlinear superposition constitutive equation. The kernel is not assumed to be separable as a product of strain and time dependent parts. Superposition is fully taken into account within the single integral formulation used. Specific formulations based on power law time dependence and truncated expansions are developed. These are appropriate for weak stress and strain dependence. The interrelated constitutive formulation is applied to ligaments, in which stiffness increases with strain, stress relaxation proceeds faster than creep, and rate of creep is a function of stress and rate of relaxation is a function of strain. An interrelation was also constructed for a commercial diecast aluminum alloy currently used in small engine applications.