Some Remarks on the Morphology of Non-Unique Solutions in Nonlinear Elastostatics

Abstract

Elastostatics is concerned with the study of equilibrium states of elastic structures and may be regarded as a subtheory of elastodynamics, when velocities and accelerations are zero. A reliable prediction of the non-unique response of a suitably supported structure subject to outer loads, tractions and other boundary conditions relies decisively on the consideration of all possible solution points corresponding to a whole set (domain) of outer control quantities. The required prediction of the ambiguous behaviour of the structure and the nonlocal character of some singular phenomena, such as e. g. snap-through from one to another stable equilibrium state illustrate the urgent necessity to develop mathematical methods which are suitable for a rigorous nonlocal qualitative and quantitative analysis of multivalued solutions. In this paper such a method called morphology analysis is briefly outlined and applied to a shallow arch and to a shallow shell problem.