A deformation or a motion is said to be a universal solution if it satisfies the balance equations with zero body force for all materials in a given class, and is supported in equilibrium by suitable surface tractions alone. On the other hand for a given deformation or motion, a local universal relation is an equation relating the stress components and the position vector which holds at any point of the body and which is the same for any material in a given class. Universal results of various kinds are fundamental aspects of the theory of finite elasticity and they are very useful in directing and warning experimentalists in their exploration of the constitutive properties of real materials. The aim of this chapter is to review universal solutions and local universal relations for isotropic nonlinearly elastic materials.

Introduction

One of the main problems encountered in the applications of the mechanics of continua is the complete and accurate determination of the constitutive relations necessary for the mathematical description of the behavior of real materials.

After the Second World War the enormous work on the foundations of the mechanics of continua, which began with the 1947 paper of Rivlin on the torsion of a rubber cylinder published in the Journal of Applied Physics, has allowed important insights on the above mentioned problem (Truesdell and Noll, 1965).