4. Strain and Stress

The mathematical theories of elasticity, viscosity, and plasticity all follow the same course in continuum mechanics. Firstly, the notions of stress and strain are developed; secondly, a stress–strain relationship between these quantities or their derivatives is assumed which idealizes the behavior of actual materials; and finally, using this relationship, equations of motion or equilibrium are set up which enable the state of stress or strain to be estimated when a body is subjected to prescribed forces. In this chapter the analysis of stress and strain will be developed in detail. We will start with three dimensional theories and leave the 2D theories to classical textbooks such as Jaeger (1969a, b). The three dimensional theory in Sect. 1.2 leads to the Mohr’s representation, which provides a simple geometrical construction for the stress across any plane. The analysis of strain is essentially inherited in the deformation of granular material: an assemblage of particles of multiple dimensions. We will start from elastic (infinitesimal strain) theories and evolve to finite strain theory, applicable to brittle upper tectonic materials on slopes which experience large strain.