Graph Approach to the Constitutive Modelling of Heterogeneous Solids

Recent decade has seen an increasing use of the concepts of graph theory in the analysis of solids with heterogeneous microstructures. Generally, two approaches can be identified depending on whether the mechanical microscale properties are assumed deterministic or random. In the first case, work was focused primarily on the possibility of finding the effective continuum representation in terms of the microscale quantities whereby the random geometry of microstructure is admitted. Satake formulated the equilibrium and compatibility relations at the level of a single grain [1], and later gave the relations between the microscopic (e.g. contact force) and macroscopic (e.g. stress) quantities [2]. It was established in that reference and in [3,4] that the key role in the formulation of the effective stress formula in terms of contact forces is played by the fabric tensor.