Generalized Clapeyron's theorem

TL;DR

This paper extends Clapeyron's theorem to nonlinear elasticity, deriving new integral relations that reveal symmetries and invariances in material responses, and demonstrating their applications through diverse case studies.

Contribution

It introduces the Generalized Clapeyron's Theorem, a nonlinear analog that captures symmetries in elastic energy functionals within Calculus of Variations.

Findings

New integral relations for nonlinear elasticity

Reinterpretation of classical energy-force relations

Applications across various mechanics problems

Abstract

Clapeyron's Theorem in classical linear elasticity provides a way to explicitly express the energy stored in an equilibrium configuration in terms of the work of the forces applied on the boundary. We derive several new integral relations which can be viewed as nonlinear analogs of this classical result, reinterpreting them as rather general statements within Calculus of Variations. In the framework of nonlinear elasticity these relations reflect various partial symmetries of the material response, for instance, scale-invariance or scaling homogeneity. In particular, when the energy functional is scale-free, the obtained result can be interpreted as the Generalized Clapeyron's Theorem (GCT). Remarkably, it combines rather naturally the work of physical and configurational forces. We present a series of illuminating case studies showing the variety of applications of various obtained relations in different seemingly unrelated problems of mechanics.