FINITE DISPLACEMENTS IN RHEOLOGICAL BODIES

Abstract

The study of the behavior of continuous matter is basic in many disciplines, such as in various branches of engineering and in the study of the Earth's interior. Herein, it is evidently necessary to have a sufficiently general mathematical formalism to encompass the behavior of any type of material under any mechanical conditions. Customary "rheological" theories suffer from various drawbacks; they are either (i) restricted to too specialized "ideal" materials, or (ii) restricted to too special displacements, or (iii) restricted to too specialized mathematical representations. The present paper attempts to fill the need for a summary of the representations of the dynamics of arbitrary materials. The displacement within the continuous medium is described by three "co-ordinate" functions as functions of three "parameters" and of time. Extensive use is made of the fact that, insofar as the expression of any physical statement is concerned, "co-ordinates" and "parameters" are entirely equivalent. Formulas are deduced which enable one to express the boundary conditions, the equations of motion, and any chosen rheological condition in either parameter space or co-ordinate space. The notion of finite strain is scrutinized.