The Distance to Cubic Symmetry Class as a Polynomial Optimization Problem

Generically, a fully measured elasticity tensor has no material symmetry. For single crystals with a cubic lattice, or for the aeronautics turbine blades superalloys such as Nickel-based CMSX-4, cubic symmetry is nevertheless expected. It is in practice necessary to compute the nearest cubic elasticity tensor to a given raw one. Mathematically formulated, the problem consists in finding the distance between a given tensor and the cubic symmetry stratum. It has recently been proved that closed symmetry strata are affine algebraic sets (for any tensorial representation of the rotation group): they are defined by polynomial equations without requirement to polynomial inequalities. Such equations have furthermore been derived explicitly for the closed cubic elasticity stratum. We propose to make use of this mathematical property to formulate the distance to cubic symmetry problem as a polynomial (in fact quadratic) optimization problem, and to derive its quasi-analytical solution using the technique of Gröbner bases. The proposed methodology also applies to cubic Hill elasto-plasticity (where two fourth-order constitutive tensors are involved).