On some properties of optimal thermoelastic designs in the case of fixed stress and deformation fields

Abstract

Sets of distributions of elastic moduli sare studied, giving rise to identical deformation or stress fields in a thermoelastic body. The affine character of the sets is proved and their interdependence is studied. It is shown that the problem of the distribution of the elastic parameters realizing a minimal stress level for the given deformation field represents a problem of convex programming. The case of the optimal design of a thermoelastic beam is discussed. In view of the wide range of possibilities of controlling new technological methods of creating new materials and structures, the problem arises of optimal design, i.e. of constructing such fields of elastic parameters, which would ensure the stress-strain field most suitable for use. The problem of the optimal distribution of the Lame'parameters in an elastic body was studied in /1/ using the condition of least work done by the external forces. The distribution of the Lame'parameters maximizing the torsional rigidity of a prismatic rod was studied in /2/. The problem of the optimal distribution of Young's modulus in a rod was solved in /3/ using the condition of maximum critical load causing loss of stability. The distribution of Young's modulus in a prestressed beam was obtained in /4/ from the requirement that the highest first eigenfrequency be realized. In many cases it is important not only to improve the rigidity characteristics of the constructions, but also to reduce the stress level. Below the authors investigate the general properties of the sets of elastic moduli distributions ensuring, for fixed loads and the temperature field, the realization of one and the same deformation or stress field. It is established that the sets have an affine structure, intersect at a unique point, and, that the tangent spaces of these sets are mutually complementary.