STATIC SHAPE CONTROL OF FORCE-INDUCED INFINITESIMAL DEFORMATIONS SUPERIMPOSED ON LARGE DEFORMATIONS OF THERMOELASTIC BODIES

Abstract

The present paper is concerned with the geometrically nonlinear static theory of anisotropic thermoelastic solids and structures. We consider infinitesimal incremental deformations superimposed on a given state of possibly large strain, the latter being called the intermediate state. Our goal is to derive a distribution of incremental thermal actuation stresses, which, when applied to the intermediate state together with a given set of incremental body forces and surface tractions, give zero incremental displacements everywhere in the body under consideration. This problem belongs to the field of static shape control, a notion originally introduced by Haftka and Adelman, who developed a procedure for determining temperatures in control elements to minimize the infinitesimal distortion of a large space antenna from its original shape. The present paper is concerned with the extension of shape control to infinitesimal force-induced static distortions from a large pre-deformation. Referring to the intermediate configuration as the reference configuration, we show that, in order to compensate the incremental force-induced deformations everywhere within the body, the incremental thermal actuation stress tensor must be equal to any statically admissible incremental first-order Piola–Kirchhoff stress tensor, a relation that is derived under the assumption that the intermediate state is infinitesimally superstable. We also discuss under which conditions it is possible to work with an isotropic thermal actuation stress. Finally, we present a formulation for shape control of infinitesimal deflections superimposed on a state of possibly large deflections of a slender beam.