Localization of the deformation in tests on sand sample

Diffuse bifurcations engraving diverse shear bands in granular materials

“…Suppose that there could be 2 distinct solutions of the incremental displacement Δu (0) and Δu (1) for an incremental external force ΔF ext imposed on an equilibrium state. Then, let the difference between the 2 solutions be denoted by Δu (*) ∶= Δu (1) − Δu (0) .…”

“…*** We denote vector/matrix expressions of discretized variables by bold-Roman characters to distinguish from tensorial expressions. a depends on the directions of the incremental deformations induced by Δu (0) and Δu (1) . Therefore, we need to consider all possible loading/unloading branches and choose the most critical branch for a conservative estimation of bifurcations.…”

“…Herein, 2 distinct solutions of the velocity field v (0) and v (1) are supposed to exist. The velocity gradient tensors corresponding to the 2 velocity fields are then given by l (0) ∶= gradv (0) and l (1) ∶= gradv (1) . The nominal strain tensors corresponding to l (0) and l (1) are given by the rate form constitutive Equation 53 2 , ie,…”

“…where the tangent moduli tensors (1∕J)a (0) and (1∕J)a (1) depend, respectively, on the loading direction l (0) and l (1) , and, as shown in Equation 54, have 2 branches of the elastic loading/unloading case and the plastic loading case. Let Δ * (•) denote the difference between the 2 values corresponding to v (0) and v (1) , namely, Δ * (•) ∶= (•) (1) − (•) (0) .…”

“…Figure 1, for example, shows shear bands on cubic sand specimens displaying apparent geometrical patterns. 1 Spatially periodic hexagonal and stripe patterns on uniform granular materials have been observed and understood as a consequence of pattern formation. 2,3 On the other hand, the direct bifurcating patterns of a cubic domain were obtained as *…”