A local constitutive model with anisotropy for ratcheting under 2D axial-symmetric isobaric deformation

Abstract

A local constitutive model for anisotropic granular materials is introduced and applied to isobaric (homogeneous) axial-symmetric deformation. The simplified model (in the coordinate system of the bi-axial box) involves only scalar values for hydrostatic and shear stresses, for the volumetric and shear strains as well as for the new ingredient, the anisotropy modulus.The non-linear constitutive evolution equations that relate stress and anisotropy to strain are inspired by observations from Discrete Element Method (DEM) simulations. For the sake of simplicity, parameters like the bulk and shear modulus are set to constants, while the shear stress ratio and the anisotropy evolve with different rates to their critical state limit values when shear deformations become large.When applied to isobaric deformation in the bi-axial geometry, the model shows ratcheting under cyclic loading. Fast and slow evolution of anisotropy with strain (relative to the evolution of anisotropy in stress) lead to dilatancy and contractancy, respectively. Furthermore, anisotropy acts such that it works "against" the strain/stress, e.g., a compressive strain builds up anisotropy that creates additional stress acting against further compression.

The critical-state yield stress (termination locus) of adhesive powders from a single numerical experiment

“…The evolution of the shear stress towards the steady state was the subject of study in 2D [30,31] and 3D [12], but will not be addressed here. A model that connects, in particular, the evolution of stress and structure is in development and will be published elsewhere [32,33].…”

Deformation Modes for Assemblies of Frictionless Polydisperse Spheres

“…We use the factor 2 in the denominator of Eqs. (6) and (7) following the definition of the second stress invariant, even though different factors have been proposed in literature [6], which result in a change in the maximum anisotropy values obtained. The deviatoric stress ߪ ா quanitfies the (stress) anisotropy between the compression/de-compression and inactive directions.…”

Micro-macro and jamming transition of polydisperse granular materials

“…[29,43,64,142,143,189,193,208] and references therein, consider explicitly the influence of the micro-mechanical structure on the elastic stiffness, plastic flow-rule or non-coaxiality of stress and strain. In a similar fashion, the anisotropy model proposed in [128,131] postulates the split of isotropic and deviatoric stress, strain and fabric and includes the microstructural anisotropy as a kinematic variable, assumed to be independent of stress, whose behavior is described by an evolution equation.…”

“…The former are purely straincontrolled, while the latter (3) is mixed strain-and-stress-controlled either with constant side stress [128] or constant pressure [131]. The isotropic and deviatoric modes 0 and 2 are pure modes, which both take especially simple forms.…”

“…Furthermore, the extensive set of DEM simulations is used to calibrate the anisotropic constitutive model, as proposed in Refs. [128,131]. After calibration through isotropic [69] and volume conserving pure shear simulations, the derived relations between the parameters and volume fraction are used to predict uniaxial deformations.…”