A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies

Abstract

In the past a number of different linearized mathematical formulations of the infinitesimal incremental deformations of continuous bodies under initial stress have been proposed. The best-known formulations are reviewed, tabulated, and subjected to a comparative study. It is demonstrated that they can be derived as special cases of a unified general formulation, and are all correct and mutually equivalent. In each formulation, the incremental elasticity constants and the incremental material stress tensor have a different significance. Their mutual relationships are established. Thus the analysis of a problem which has already been solved according to one formulation need not be repeated for another formulation. Furthermore, the connections to the various definitions of the objective stress rate are shown. The arbitrariness of choice between the infinitely many possible forms of incremental equilibrium equations corresponds to the arbitrariness in the definitions of (a) the finite strain tensor, (b) the material stress tensor, (c) the objective stress rates, (d) the stability criterion, and (e) the elastic material in finite strain. For demonstration of the differences, the problems of surface buckling of an orthotropic half space and a column with shear are studied. It is shown that the predicted buckling stresses can differ almost by a ratio of 1:2 if the proper distinction between various formulations is not made.