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.

本研究提出了一种新的方法，用于利用分析发展和数值模拟来估计沿晶界边界含有裂纹的固体的有效性质。后者是基于这种类型的虚拟微观结构的生成，通过叠加 Voronoï 镶嵌对颗粒网络进行建模，在 3-D 中随机分散重叠球体或在 2-D 中的圆盘，用于定位裂纹处晶间边界。本文研究了这种微观结构模型的不同特征，特别是由改变晶粒和球体/圆盘之间的尺寸比引起的形态效应。通过全场模拟，估计生成的微观结构的有效热导率，并与均匀弱化固体（呈现均匀裂纹分散）的有效热导率进行比较。对于后者的微观结构，Ponte-Castañeda 和 Willis (1995) 的上限结果接近于全场结果。此外，全场计算表明，晶间裂纹的空间分布会导致有效热导率的急剧下降。仅修改 Ponte Castañeda 和 Willis 界的数学表达式中的边界裂纹密度，提供了对被晶间裂纹削弱的固体的有效电导率的相关分析估计。该截止裂纹密度仅取决于微观结构参数。这种新的估计表明改进了 Sevostianov 和 Kachanov（2019）基于差分方案得出的估计值，至少对于此处考虑的微观结构而言。最后，还提供了对在晶间边界处弱化的各向同性裂纹固体的弹性模量的新估计。由此估计的 3-D 固体的有效体积模量显示保持在上限以下，这也可以通过将全场计算预测的有效电导率注入经典交叉性能关系中来生成。