Hypo-Elasticity

Abstract

The most common way of treating the mechanics of continuous bodies and, in particular, the mechanics of elastic bodies, is based on the assumption of the existence of a free equilibrium state, which generally is an unstressed state. That is, one postulates the existence of an equilibrium state without external forces and, generally, without stress. The deformations are measured from this state C*. In effect, assuming the existence of a natural state of equilibrium [without stress] determines a first property of the thermodynamic potential: Y _rs derived from it must be zero at C*. At any rate, dropping the condition that C* be unstressed, it is recognized that the stress is necessarily a uniform pressure if the body is homogeneous and isotropic in C*, while it is a uniform stress if the body is anisotropic. It may sometimes be convenient to measure the deformations from a state for which the stress is known but depends on the coordinates in a complicated manner, as happens, for example, in the case of Volterra’s dislocations. It is often convenient to assume such a stressed reference state and to measure deformations from it. The common way of studying elastic problems is acceptable and convenient when it is easy and useful to assume an unstressed state as reference.