Sull'equilibrio isotermo dei corpi elastici con un vincolo di appoggio unilaterale liscio nel caso di deformazioni finite

Abstract

We study the analytical problem connected with the isothermal equilibrium of an elastic body with an unilateral supporting constraint on an indeformable frictionless surface in the case of« finite» deformations. As a first approach to investigate this boundary-value problem, we think of the external active forces as linearly dipending upon a parameter: this leads to consider a finite or infinite sequence of linear differential systems(« auxiliary» systems). In§ 2 we show how one can determine the (unilateral) boundary-value problem to be associated to each auxiliary system.§ 3 is dedicated to study the uniqueness of the solution of the auxiliary systems by means of the analysis of the conditions for the integrability of the successive auxiliary systems. The investigation of uniqueness leads to a classification into several types of the supporting surfaces and requires, for each of these types, a systematic study of the integrability conditions. In this manner we can verify that the boundary-value problem for the auxiliary systems has been« correctly set». ln§ 4 we deal with the existence problem for the generical auxiliary system; we show the possibility of a weak, integral-type formulation—expressed only in terms of the stress-components—for the boundary-value problem associated to the generical auxiliary system and we come to an existence theorem expressible only in terms of stress, though the boundary conditions concern also the displacement.