On Optimal Shapes in Materials and Structures

In the micromechanics design of materials, as well as in the design of structural connections, the boundary shape plays an important role. The objective may be the stiffest design, the strongest design or just a design of uniform energy density along the shape. In an energy formulation it is proven that these three objectives have the same solution, at least within the limits of geometrical constraints, including the parametrization. Without involving stress/strain fields, the proof holds for 3D-problems, for power-law nonlinear elasticity and for anisotropic elasticity.

To clarify the importance of parametrization, the problem of material/hole design for maximum bulk modulus is analysed. A simple optimality criterion is derived and with a simple superelliptic parametrization, agreement with Hashin-Shtrikman bounds are found. More general examples including nonequal principal strains, nonlinear elasticity and orthotropic elasticity show the versatility of the optimality criterion approach. In spite of this, the mathematical programming approach will be used in the future study of the multiparameter and/or multipurpose problems.