A stress-based formulation of the free material design problem with the trace constraint and single loading condition

The problem to find an optimal distribution of elastic moduli within a given plane domain to make the compliance minimal under the condition of a prescribed value of the integral of the trace of the elastic moduli tensor is called the free material design with the trace constraint. The present paper shows that this problem can be reduced to a new problem of minimization of the integral of the stress tensor norm over stresses being statically admissible. The eigenstates and Kelvin’s moduli of the optimal Hooke tensor are determined by the stress state being the minimizer of this problem. This new problem can be directly treated numerically by using the Singular Value Decomposition (SVD) method to represent the statically admissible stress fields, along with any unconstrained optimization tool, e.g.: Conjugate Gradient (CG) or Variable Metric (VM) method in multidimensions.