On the time dependence of viscoelastic variational solutions

Abstract

Thermodynamic operational-variational principles are employed in a study of the transient response of linear viscoelastic media with an arbitrary degree of anisotropy. Assuming displacements in the form of a series of products of space-dependent functions and time-dependent generalized coordinates, the (approximate) response is calculated by minimizing a functional which is analogous to the potential energy of an elastic body. Similarly, a principle analogous to the principle of minimum complementary energy of elasticity is used to deduce transient behavior of (approximate) stresses. The displacements are not required to satisfy equilibrium or stress boundary conditions, nor are stresses calculated from the complementary principle required to satisfy compatibility or displacement boundary conditions. It is found that when applied loads and displacements are step-functions of time, the transient component of stresses and displacements is given in most cases by a series of exponentials with negative, real arguments.