2. Constitutive Equations in Hereditary Integral Form

Materials respond to external load by deforming and straining, and by developing stresses. The internal stresses corresponding to a given set of strains depend on the constitution of the material itself. For this reason, the rules that permit calculation of internal stresses from known strains, or vice versa, are called constitutive laws or constitutive equations. There are two equivalent ways to describe the mathematical relationships between stresses and strains for viscoelastic materials. One form uses integrals to define the constitutive relations, while the other relates stresses and strains by means of differential equations. Starting from Boltzmann’s superposition principle, this chapter develops the integral form of the one-dimensional constitutive equations for linearly viscoelastic materials. This is followed by a discussion of the principle of fading memory, which helps to define the acceptable analytical forms of the material property functions. It is then shown that the closed-cycle condition (i.e., that the steady-state response of a non-aging viscoelastic material to a periodic excitation be periodic) requires that the material property functions depend only on the difference of their arguments. The chapter also examines the relationships between the relaxation modulus and creep compliance functions in the physical time domain as well as in Laplace-transformed space. Various alternative forms of the integral constitutive equations often encountered in practice are discussed as well.