On the existence of solutions to some problems of optimal design for bars and plates

Abstract

The problem of the optimal control of the material characteristics of continuous media necessitates an extension of the initial class of materials to the set of composites assembled from elements belonging to the initial class. Such an extension guarantees the existence of an optimal control and is equivalent to the construction of theG-closureGU of the initial setU. In this paper, we consider some problems of constructingG-closures for the operators ▽·² D·▽ and ▽·▽·⁴ D··▽▽, where² D and⁴ D denote self-adjoint tensors of 2nd and 4th rank, respectively, their components belonging to bounded sets ofL _∞. These operators arise in the theory of the torsion of bars and in the theory of bending of thin plates. A procedure is suggested that provides estimates of some sets Σ containingGU. These estimates are expressed through weak limits of certain functions of the elements of theU-set. The estimates are based on the weak convergence of the elastic energy and, for operators of 4th order, also on the weak convergence of the second invariant of deformation,

$$I_2 (e) = w_{xx} w_{yy} - w_{xy}^2 - I_2 (e^0 ) = w_{xx}^0 w_{yy}^0 - (w_{xy}^0 )^2 .$$

For operators of 2nd order, we consider the problem of the control of the orientation of the principal axes of² D. The set Σ is constructed, and it is shown that its elements correspond, in a sense, to media with some well-determined microstructure (layered media). A number of problems for operators of 4th order are also considered. For these problems, the Σ-set is constructed for the control problem of the modulus of dilatationk, if the shear modulus μ is fixed and also for the control problem of the shear modulus μ for fixed value ofk. The problem of the orientation of the principal axes of elasticity is analyzed for a medium with cubic symmetry (semi-isotropic medium). The description of theGU-set, with the aid of the weak limits of theU-elements, allows one to prove the existence of optimal solutions to a fairly wide class of problems with weakly continuous cost functionals. Some examples are considered in detail.