Fourier-based strength homogenization of porous media

An efficient numerical method is proposed to upscale the strength properties of heterogeneous media with periodic boundary conditions. The method relies on a formal analogy between strength homogenization and non-linear elasticity homogenization. The non-linear problems are solved on a regular discretization grid using the Augmented Lagrangian version of Fast Fourier Transform based schemes initially introduced for elasticity upscaling. The method is implemented for microstructures with local strength properties governed either by a Green criterion or a Von Mises criterion, including pores or rigid inclusions. A thorough comparison with available analytical results or finite element elasto-plastic simulations is proposed to validate the method on simple microstructures. As an application, the strength of complex microstructures such as the random Boolean model of spheres is then studied, including a comparison to closed-form Gurson and Eshelby based strength estimates. The effects of the microstructure morphology and the third invariant of the macroscopic stress tensor on the homogenized strength are quantitatively discussed.