Introduction to the Variational Formulation in Mechanics

4Hyperelastic Materials at Infinitesimal Strains

4.1 Introduction

Chapter 3 was devoted to the presentation of the foundational aspects of the mechanics regardless of the specifics of the constituent material that forms the bodies.

In this chapter, we will illustrate the application of these principles of the mechanics to the case of bodies undergoing infinitesimal deformations and featuring hyperelastic material behavior. To achieve this goal, we will first discuss the definition and properties of such hyperelastic constitutive equations, and then the attention will be on the variational formulations and extremal principles that can be constructed for such problems.

4.2 Uniaxial Hyperelastic Behavior

When performing uniaxial tests, a linear elastic material is characterized by a relation of the following kind,

(4.1)

where is the stress, is the deformation, and is the Young's modulus, or the elastic modulus.

In what follows, we will point out some features that, even if they appear to be obvious, will be useful in the forthcoming presentation.

As it is clear from (4.1), , that ...