Automatic building of a numerical simplified constitutive law for Ceramic Matrix Composites using Singular Value Decomposition

Abstract

In this work, a technique for reducing the dimensionality of a complex constitutive law for Ceramic Matrix Composites (CMC) using Singular Value Decomposition (SVD) is presented. An exploration of methods for low-rank matrix approximations begins by proposing a decomposition of the internal state variables (fourth-order tensors) of the reference thermodynamic potential in a certain range of loading conditions. The difference with a previous methodology that introduces a modified potential with a reduced set of state variables is that here, SVD is performed directly on the state variables of the reference model. The performance of the specific low-rank approximations are illustrated by thorough investigations concerning the “energy” captured by the singular vectors, as well as, by the computation of the relative error in the energy potential, in the elastic strain tensor components and in the energy dissipation. Levels of the relative error are compared over a wide range of stress loading conditions for different rank-k approximations. The reduced model is shown to be efficient in representing the reference model, and is well suited for the application and implementation in standard finite element codes.