Brief Review of Continuum Mechanics Theories

The classical theory of continuum mechanics has its roots in the nineteenth century, in the foundational work of Augustin-Louis Cauchy, although its rigorous, modern development has been built upon Noll’s axiomatic framework which allows for a unified study of deformable materials. In the mathematical description of a material’s response to mechanical loading there are two important basic assumptions which form the foundation of continuum mechanics: (1) the mechanical stress at a given material point at time t is determined by the past history of the deformation of a neighborhood of the considered point (the principle of determinism and local action), and (2) the response of a material is the same for all observers (the principle of material objectivity). These principles are however too general to properly characterize the nature of specific materials and further simplifications of the relationship between mechanical stress and deformation are necessary. Such simplifications arise, for instance, from assumptions of infinitesimal deformations or for finite deformations that a material is simple, homogeneous, non-aging, has preferred directions of deformation, and experiences internal constraints, (like incompressibility, inextensibility, rigidity). In this chapter we provide a brief review of these concepts, as well as specific constitutive laws that have been used in brain research. In addition, we will present some modern theories that generalize classical continuum mechanics and may prove very useful in future studies of brain biomechanics.