Effective properties of an isotropic solid weakened by micro-cracks located at inter-granular boundaries

This study presents a new methodology for estimating the effective properties of solids containing cracks along the inter-granular boundaries, using analytical developments and numerical simulations. The latter are based on the generation of virtual microstructures of such type obtained by superimposing a Voronoi tessellation modeling the granular network with a random dispersion of overlapping spheres in 3-D, or disks in 2-D, which serve to locate the cracks at the inter-granular boundaries. The different features of this microstructure model are studied herein, especially the morphological effects induced by varying the size ratio between grains and spheres/disks. By means of full-field simulations, the effective thermal conductivities of the generated microstructures are estimated and compared with those of uniformly weakened solids (presenting uniform crack dispersion). For the latter microstructures, the Ponte-Castaneda & Willis (1995) upper bound turns out to be close to the full-field results. In addition, the full-field computations show that the spatial distribution of inter-granular cracks induces a dramatic degradation of the effective thermal conductivity. Modifying only the cutoff crack density in the mathematical expression of the Ponte Castaneda and Willis bound provides a relevant analytical estimate of the effective conductivity of solids weakened by inter-granular cracks. This cutoff crack density only depends on the microstructural parameters. This new estimate is shown to improve the one derived by Sevostianov & Kachanov (2019) and based on the differential scheme at least for the microstructures considered herein. Finally, new estimates of the moduli of elasticity for isotropic cracked solids weakened at inter-granular boundaries are also provided. The effective bulk modulus thus estimated for 3-D solids is shown to remain below the upper bound which can also be generated by injecting the effective conductivity predicted by full-field computations into the classical cross-property relations.