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

Abstract

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.

Simulating the Poisson effect in lattice models of elastic continua

“…Similar formulations can be found elsewhere [20], [25] and are widely used to determine the spring constants of discrete methods. For a particular value of the Poisson ratio, it is possible to rewrite Eq.…”

“…Such relations are used herein for comparison purposes. Liu et al [25] derive similar relationships to model failure behaviors such as breaking displacement, shear resistance, and coefficient of friction. Alternatively, three-node discrete element models have been developed to accommodate a volumetric constitutive relation.…”

An Extended Grain‐Based Model for Characterizing Crystalline Materials: An Example of Marble

“…However, determining microparameters of a PFC model remains difficult, [14] and the difficulty multiplies with the increased number of model parameters. However, determining microparameters of a PFC model remains difficult, [14] and the difficulty multiplies with the increased number of model parameters.…”

Mechanism of formation of wiggly compaction bands in porous sandstone: 2. Numerical simulation using discrete element method

“…Similar to the normal force, the shear spring force is the product of the shear stiffness ( K s ) and the shear relative displacement ( X s ). An initial shear resistance ( Fs 0 , i.e., cohesion) exists between the elements, and the maximum shear force allowed by Coulomb friction is [ Liu et al ., ] where μ p is the interelement coefficient of friction. The intact bond will break when the shear force exceeds F Smax , and the maximum shear force of the broken bond ( F Smax ′) reduces to μ p · F n [ Hazzard et al , ].…”

“…Table gives the initial mechanical properties of the sandstone [ Bieniawski , ] and the calculated interelement mechanical properties. Mechanical properties of the close‐packed model determined by the conversion formulas are generally a bit lower than the theoretical values [ Liu et al ., ]. Therefore, the model was compressed or extended to test the effective mechanical properties.…”