Quasi-static incremental behavior of granular materials: Elastic–plastic coupling and micro-scale dissipation

Abstract

The paper addresses a common assumption of elastoplastic modeling: that the recoverable, elastic strain increment is unaffected by alterations of the elastic moduli that accompany loading. This assumption is found to be false for a granular material, and discrete element (DEM) simulations demonstrate that granular materials are coupled materials at both microand macro-scales. Elasto-plastic coupling at the macro-scale is placed in the context of thermomechanics framework of Tomasz Hueckel and Hans Ziegler, in which the elastic moduli are altered by irreversible processes during loading. This complex behavior is explored for multi-directional loading probes that follow an initial monotonic loading. An advanced DEM model is used in the study, with non-convex non-spherical particles and two different contact models: a conventional linear-frictional model and an exact implementation of the Hertz-like Cattaneo-Mindlin model. Orthotropic true-triaxial probes were used in the study (i.e., no direct shear strain), with tiny strain increments of 2 × 10 −6 . At the micro-scale, contact movements were monitored during small increments of loading and load-reversal, and results show that these movements are not reversed by a reversal of strain direction, and some contacts that were sliding during a loading increment continue to slide during reversal. The probes show that the coupled part of a strain increment, the difference between the recoverable (elastic) increment and its reversible part, must be considered when partitioning strain increments into elastic and plastic parts. Small increments of irreversible (and plastic) strain and contact slipping and frictional dissipation occur for all directions of loading, and an elastic domain, if it exists at all, is smaller than the strain increment used in the simulations.

Multi-directional behavior of granular materials and its relation to incremental elasto-plasticity

“…We note, again, that the reversible strain is not the "recoverable strain," as irreversible processes in the form of contact slip, contact micro-slip, and particle rearrangement occur at all stages and in all directions of a loading-unloading cycle in stress-space. The technique for isolating the reversible part of a strain increment was used in [24] to demonstrate that granular materials exhibit elastic-plastic coupling and that during strain hardening, contact slip -a distinctly irreversible mechanism -accompanies loading in any direction. Another irreversible mechanism, the non-dissipative gain and loss of contacts, can also occur during deformation [43], and this mechanism was not precluded in the AB probes.…”

“…With each principle, we more fully ascertain granular behavior and the nature of the nonconformity or agreement. The first principle was the focus of a second article by the authors [24], which demonstrated that elastic strain increments, although recoverable, are not reversible, a finding that is more fully described below in the context of elastic-plastic coupling [26,27]. As such, the elastic-plastic partition of strain increments -the basis of the first principle -is replaced with a reversible-irreversible partition, and the terms "reversible" and "irreversible", defined below, will replace elastic and plastic in this work (see also [28,29]).…”

“…Yet principle 3 was affirmed in several other studies. In regard to principle 4, some studies have found that the elastic and plastic regions in incremental stress-space are distinct and are separated by a flat hyperplane (i.e., a yield surface 2 Lewin & Burland [9] shale powder triaxial --10 kPa Tatsuoka & Ishihara [10] sand triaxial -E-P 0.001-0.006 Bardet [11] disks linear DEM E-P ≈ 1×10 −3 Anandarajah et al [12] sand -triaxial E-P ≈ 1×10 −4 Royis & Doanh [13] sand -triaxial E-P ≈ 1×10 −4 Kishino [14] spheres linear GEM E-P 1 kPa Alonso-Marroquín [15,16] polygons linear DEM E-P 10 kPa ≈ 1×10 −5 Calvetti et al [17,18] spheres linear 3 DEM R-I ≈ 2×10 −4 Sibille et al [19] spheres linear DEM R-I -Plassiard et al [20] spheres linear 4 DEM R-I ≈ 2×10 −5 Froiio and Roux [21] disks linear DEM 5 -≈ 4×10 −6 Harthong & Wan [22] spheres linear DEM E-P 0.1kPa Wan & Pinheiro [23] spheres -DEM R-I & E-P 0.1 kPa Kuhn & Daouadji [24] sphere-clusters linear & Hertz DEM R-I & E-P 2×10 −6 Current work sphere-clusters Hertz DEM R-I 2×10 −6 1 GEM = Granular Element Method; triaxial = laboratory experiments 2 R-I = reversible-irreversible; E-P = elastic-plastic (see Section 2.2) 3 with rotational restraint of particles 4 with rolling friction of contacts 5 hybrid method: assembly loaded with DEM; probes applied with method similar to GEM [14,25] Table 2: Results of previous 2D simulations and their conformance with the six principles of conventional elasto-plasticity: Y = conforms with the principle, N = contradicts the principle.…”

“…In regard to principle 4, some studies have found that the elastic and plastic regions in incremental stress-space are distinct and are separated by a flat hyperplane (i.e., a yield surface 2 Lewin & Burland [9] shale powder triaxial --10 kPa Tatsuoka & Ishihara [10] sand triaxial -E-P 0.001-0.006 Bardet [11] disks linear DEM E-P ≈ 1×10 −3 Anandarajah et al [12] sand -triaxial E-P ≈ 1×10 −4 Royis & Doanh [13] sand -triaxial E-P ≈ 1×10 −4 Kishino [14] spheres linear GEM E-P 1 kPa Alonso-Marroquín [15,16] polygons linear DEM E-P 10 kPa ≈ 1×10 −5 Calvetti et al [17,18] spheres linear 3 DEM R-I ≈ 2×10 −4 Sibille et al [19] spheres linear DEM R-I -Plassiard et al [20] spheres linear 4 DEM R-I ≈ 2×10 −5 Froiio and Roux [21] disks linear DEM 5 -≈ 4×10 −6 Harthong & Wan [22] spheres linear DEM E-P 0.1kPa Wan & Pinheiro [23] spheres -DEM R-I & E-P 0.1 kPa Kuhn & Daouadji [24] sphere-clusters linear & Hertz DEM R-I & E-P 2×10 −6 Current work sphere-clusters Hertz DEM R-I 2×10 −6…”

“…Recent DEM simulations have shown, however, that micro-scale movements are irregular and non-affine, with particles seemingly darting and dodging in a highly irregular manner, and that contact movements are highly diverse, even when one considers a subset of contacts having the same orientation [57,58,59]. Moreover, a fraction of contacts are persistently non-elastic and continue to undergo frictional slip even when the direction of bulk deformation is reversed [24].…”