We now turn our attention to the deformation and flow of a continuous body and in this chapter develop the concepts and mathematical expressions necessary for their quantification. We will be considering two types of body, elastic and fluid, depending on its behavior when subject to a non-hydrostatic or deviatoric stress (which we haven't defined yet). The displacement of an elastic body remains finite, while the displacement of a fluid body increases without bound. Whether a body behaves elastically and fluidly depends on several factors, including its composition, temperature and the state of stress. As a material is heated it tends to behave more like a fluid. And if the applied stresses are sufficiently great, a body will either begin to flow or break. Once a body has broken, it no longer is a continuous body and we would need a different mathematical formalism to describe its behavior.

Deformation is used primarily in reference to elastic bodies, while flow typically refers to fluid bodies, but they are closely related. Deformation is quantified by the relative change of position, i.e., strain, while flow is quantified by the time rate of change of position, i.e., the rate of strain. Equations governing strain and rate of strain are developed in the following sections.

Orientation to Kinematics

The form of the kinematic constraint on deformation and flow depends whether the body is elastic or fluid. An elastic body is one that can resist applied forces without continuing displacement (that is, x can be independent of t), while a fluid body experiences continuous displacement unless the internal (contact) forces are of a particular form (hydrostatic). In either case, these bodies must deform such that the amount of mass is conserved, and no voids form. The constraints on the deformation of elastic bodies and the flow of fluid bodies are developed in § 3.3 and § 3.4, respectively, leading to separate versions of the equation of conservation of mass for elastic and fluid bodies. Two versions of the equation of conservation of mass arise in part because elastic and fluid bodies are observed differently.