Introduction to the Variational Formulation in Mechanics

12High‐Order Continua

12.1 Introduction

The theory underlying the mechanics of the so‐called micromorphic materials was systematically developed in the mid the 20th century by A.C. Eringen and was compiled in [80, 81]. Within the framework of this theory the continuum body is regarded as composed of deformable particles, which are characterized by an inner structure with finite size. The motivation to provide a richer description than that given by classical continuum mechanics is rooted in the possibility of taking into account the mechanical phenomenology occurring at a smaller scale. This can be accomplished, within the context of continuum mechanics, by exploiting high‐order theories of continua. By high‐order it must be understood that the kinematical description of the particles that form the body possess additional entities which account for high‐order motion modes than a simple velocity vector field.

The main drawback of this kind of theory lies in the construction of adequate constitutive laws. From the experimental point of view, the development of experimental settings to quantify the relation between generalized strain measures and generalized internal stresses results in a more than arduous task.

In this chapter we will once again make use of the variational structure provided by the Method of Virtual Power through the formulation of the Principle of Virtual Power (PVP) presented in Chapter 3. Hence, by exploiting duality arguments between generalized strain rates ...