# Design of Composite Plates of Extremal Rigidity

• L. V. Gibiansky
• A. V. Cherkaev
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 31)

## Abstract

We consider design problems for plates possessing an extremal rigidity. The plate is assumed to be assembled from two isotropic materials characterized by different values of their elastic moduli; the amount of each material is given. We look for a distribution of the materials which renders the plate’s rigidity for either its maximal or minimal value. The rigidity is defined here as work produced by an extremal load on deflection of the points of the plate. The optimal distribution of the materials is characterized by some infinitely often alternating sequences of domains occupied by each of the materials (see , ). This leads to the appearance of anisotropic composites; their structures are to be determined at each point of the plate.

The exact bounds on the elastic energy density are obtained; the micro-structures of the optimal composites are found. The effective properties of such composites (we call them matrix laminate composites) are explicitly calculated. These composites extend the set of available materials. The initial problems are formulated and solved for an extended set of design parameters.

We use the results to solve the problem of optimal design of the plate with variable thickness, given mass, and with additional restrictions on the range of the thickness values. A rule is found that allows us to distinguish the cases in which the optimal composite is assembled of elements having only maximal or minimal thickness.

The number of optimal design problems for the clamped square plate of maximal and minimal rigidity are solved numerically.

## Keywords

Optimal Design Matrix Composite Composite Plate Stiffness Tensor Optimal Design Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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