Upscaling the Strength Domain of Heterogeneous Cohesive-Frictional Materials via a General FFT-Based Limit Analysis Approach

Limit analysis and yield design provide a well-defined mathematical framework for upscaling the strength properties of heterogeneous materials. These techniques can be incorporated into an FFT-based computational micromechanics framework to evaluate the strength of heterogeneous materials, based on images of their microstructure. However, the currently existing approaches are limited to rather specific forms of strength criteria, and their convergence for cohesive frictional materials was questioned. In the present contribution, we abstract and reformulate an existing kinematic upper-bound approach that relies on an Augmented Lagrangian and on Uzawa's algorithm, and link it to the traditional and damped ADMM. We show that the strength constraint may be enforced by means of an orthogonal projection operator, and provide the closed-form formulation of the projector for two cases: the Drucker-Prager and the Mohr-Coulomb strength criteria. We also discuss the link between the choice of the discretization and the bounding status of the solution. Finally, building upon the ADMM reformulation of the problem, we ascertain the sublinear convergence of the FFT-based strength homogenization method for a large class of convex strength criteria, which includes cohesive-frictional materials, provided that a projector is available, either analytically or algorithmically. A first set of numerical studies, performed on a toy example, highlights the role of the solver parameters and the discretization on the effectiveness and accuracy of the approach. The obtained FFT-based strength homogenization results are compared to state-of-the-art Finite-Element Limit-Analysis for validation and to discuss the bounding status of the solution. Then, computational experiments performed on industrial scale, potentially porous microstructures, show that FFT-based strength homogenization enables the analysis of the macroscopic strength anisotropy induced by the local geometric anisotropy of the microstructure using reasonable computational resources.