Towards realizable free material optimization via energy bound hierarchy

Abstract

We study free orthotropic material optimization for two-dimensional plane-stress compliance minimization with two well-ordered isotropic phases. To connect free material optimization and composite realizability, we construct a hierarchy of admissible sets using zeroth-order, Voigt, and Hashin--Shtrikman energy bounds. The zeroth-order and Voigt models are convex, with admissible sets that coincide at global volume-budget endpoints and satisfy strict inclusion at intermediate volume budgets. The Voigt model reduces to isotropic variable-thickness-sheet optimization, while the single-loadcase continuum zeroth-order problem admits an orthotropic optimum. The classical Hashin--Shtrikman bounds yield a tighter admissible set that is nonconvex jointly in the elasticity tensor and volume fraction and whose convex hull equals the Voigt set. We derive reduced formulations by explicitly characterizing the minimum volume fraction compatible with these bounds. For orthotropic tensors, this volume-fraction estimator reduces to a one-dimensional search over the strain trace. In the single-loadcase continuum setting, the Hashin--Shtrikman-constrained optimal value equals that of the classical Allaire--Kohn relaxation, attained by a homogenized design with sequential-laminate effective tensors. For multiple loadcases, the bounds remain necessary but are generally insufficient for realizability. Numerical results illustrate the compliance hierarchy and close agreement with single-loadcase laminate references.

arXiv comment54 pages, 17 figures