Variational and Lagrangian Methods in Viscoelasticity

Abstract

The time history of a thermodynamic system perturbed from equilibrium under the assumption of linearity obeys certain differential equations. Starting from Onsager’s reciprocity relations we have shown [1] how they may be derived from generalized concepts of free energy and dissipation function. This provides a most fruitful link between physical chemistry, thermodynamics, and mechanics, and leads to a very general formulation of relations between stress and strain in linear viscoelasticity in operational form. The outline of this development is given in section 2. The matrix relating stress and strain is formally identical with the matrix of twenty-one distinct coefficients in the theory of Elasticity. In linear viscoelasticity the elements of this matrix are functions of the differential time operator. The form of this operator is also derived from the theory. Variational formulation of deformation and stress field problems are outlined in section 3 and lead to generalizations of Lagrange’s equations with operational coefficients. We also introduce a general principle expressing the formal correspondence between problems in viscoelasticity and Elasticity. By the latter it is possible to carry over almost all solutions of the theory of Elasticity into that of a corresponding problem of viscoelasticity. Thus we uncover in one stroke a vast area of solved problems for viscoelastic media. This approach provides a compact and synthetic formulation of linear viscoelasticity. As an example we derive in section 4 some general properties of a medium with a uniform relaxation spectrum. An outline is also given of a new approach to the dynamics of plates or shells for isotropic or anisotropic media. It includes the classical theories for elastic materials as first order approximation. The method is also, of course, applicable to improving the theory of plates and shells in the purely elastic case when the effect of increasing thickness is taken into account.