Analytical solutions and numerical tests of elastic and failure behaviors of close‐packed lattice for brittle rocks and crystals

[1] Analytical solutions of elastic properties and failure modes of a two-dimensional close-packed discrete element model are proposed. Based on the assumption of small deformation, the conversion formulas between five inter-particle parameters of the lattice model and rock mechanical properties were derived. Using the formulas, the inter-particle parameters can be determined by Young's modulus (E), Poisson's ratio (v), tensile strength (Tu), compressive strength (Cu), and coefficient of intrinsic friction (μi). The lattice defined by the parameters simulates the elastic and failure behaviors of rocks and crystals and therefore can be used to investigate the initiation and development of geological structures quantitatively. Furthermore, the solutions also provide a theoretical basis for the calibration of parameters of random discrete assemblies. The model of quartz was used as an example to validate the formulas and test the errors. The simulated results show that E and v converge to theoretical values when particle number increases. These elastic properties are almost constant when the magnitude of strain is lower than 10−3. The simulated Tu and Cu of a single three-element unit are also consistent with the formulas. However, due to the boundary effects and stress concentrations, Tu and Cu of lattices with multiple units are lower than the values predicted by the formulas. Therefore, greater Tu and Cu can be used in the formulas to counteract this effect. The model is applicable to the simulation of complicated structures that involve deformation and failure at different scales.