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 Voronoï 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-Castañeda and 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 cut-off crack density in the mathematical expression of the Ponte Castañeda and Willis bound provides a relevant analytical estimate of the effective conductivity of solids weakened by inter-granular cracks. This cut-off crack density only depends on the microstructural parameters. This new estimate is shown to improve the one derived by Sevostianov and 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.