  The influence of the microstructural geometry on the behavior of porous media
is widely recognized, particularly in geomaterials, but also in biomaterials
and engineered materials. Recent advances in imaging techniques, such as X-ray
microcomputed tomography, and in modeling make it possible to capture the exact
morphometry of the microstructure with high precision. However, most existing
continuum theories only partially account for the morphometry. We propose here
a unifying approach to link the strength of porous materials with the necessary
and sufficient microstructural information, using Minkowski functionals, as per
Hadwiger's theorem. A morphometric strength law is inferred from synthetic
microstructures with a wide range of porosities and heterogeneities, through
qualitative 2D phase-field simulations. Namely, the damage is modeled at the
microstructural level by tracking the solid-pore interfaces under mechanical
loading. The strength is found to be best described by an exponential function
of the morphometers, thus generalizing early works on metals and ceramics. We
then show that the predictiveness of this relationship extends to real porous
media, including rocks and bones.
