Hysteresis effects studied by numerical simulations: Cyclic loading-unloading of a realistic sand model

When Hertz-Mindlin force laws are considered in the context of the effective-medium theory, the predictions yield a constant Poisson coefficient and bulk/shear elastic moduli that scale with pressure with a 1/3 power law exponent (P^1/3). This prediction contradicts early and recent experimental findings that conclude moduli grow faster with a 1/2 power law exponent (P^1/2). Such a conclusion is also reached by recent second-order corrections to linear elastic theory. In this work we use a discrete-particle method to study the elastic response of a model of sand that is unconsolidated because of cyclic loading. We use a detailed molecular dynamics simulation that accounts for Hertz-type grain interactions and history-dependent shear forces. The porous sand model is constructed from spherical particles whose size distribution mimics well-sorted unconsolidated sands. The geometry of the model is obtained by simulating critical processes in sedimentary rock formations. Hysteretic behavior and relations between the sample bulk modulus, strain, and stress are obtained. The simulated sample reproduces experimental transient and stationary loading-unloading behavior. We find good correspondence of pressure and strain dependence of elastic moduli in our model with semilinear elasticity theory predictions. Simple arguments explain low coordination numbers observed on force-transmitting samples and the tendency to reduce dissipation under cyclic loading. Our approach clearly shows that a Hertz-Mindlin grain interaction is not inconsistent with the experimental P^1/2 behavior of the bulk modulus.