Capturing strain localization in dense sands with random density

Abstract

SUMMARYThis paper presents a three-invariant constitutive framework suitable for the numerical analyses of localization instabilities in granular materials exhibiting unstructured random density. A recently proposed elastoplastic model for sands based on critical state plasticity is enhanced with the third stress invariant to capture the difference in the compressive and extensional yield strengths commonly observed in geomaterials undergoing plastic deformation. The new three-invariant constitutive model, similar to its two-invariant predecessor, is capable of accounting for meso-scale inhomogeneities as well as material and geometric nonlinearities. Details regarding the numerical implementation of the model into a fully nonlinear finite element framework are presented and a closed-form expression for the consistent tangent operator, whose spectral form is used in the strain localization analyses, is derived. An algorithm based on the spectral form of the so-called acoustic tensor is proposed to search for the necessary conditions for deformation bands to develop. The aforementioned framework is utilized in a series of boundary-value problems on dense sand specimens whose density fields are modelled as exponentially distributed unstructured random fields to account for the effect of inhomogeneities at the meso-scale and the intrinsic uncertainty associated with them.

Diffuse bifurcations engraving diverse shear bands in granular materials

“…We use the 3-invariant representation of plasticity 29,30 to take account of the Lode angle dependency, while more common 2-invariant representation can be referred to Borja and Tamagnini 31 and Borja and Andrade. 32 The constitutive formulation herein is based on the framework of multiplicative hyperelasto-plasticity for finite strains.…”

“…This model has widely been used to model elastic property of cohesionless clays and sands. 30,32 The hyperelastic model for geomaterials used in the present study is an extended version proposed by Yamakawa et al 10 to include tensile strengths. The specific form of the hyperelastic energy function for this extended model is defined as…”

“…The values f and g can be different, but should be taken as f ⩽ g so as to ensure nonnegative plastic dissipation. If f ≠ g , it followsM( ) ≠Ñ( ) and thus f ≠ g. In this case, the plastic flow rule is nonassociative in a deviatoric sense, 30 leading to a nonsymmetric elastoplastic tangent moduli tensor. Further discussion based on the thermodynamical requirement can be referred to Andrade and Borja.…”

“…The explicit form of the derivatives p∕ , q∕ , and ∕ are given in, eg, equation 19 in Andrade and Borja. 30 Also useful is that tr( g ) appearing in the denominator in Equation 54 is given by g∕ p, which is associated to the direction of plastic volumetric strain rate. ‡ ‡…”

“…Diffuse mode bifurcation, 3,64 pointwise local material bifurcation, 39,53,65 and their competing mechanism in failure in geomaterials 66 are still on-going research subjects. In previous works, 30,32,[67][68][69] occurrence of material bifurcations during transitional courses of deformations was investigated. In these studies, spatially varying material property § § In these figures and hereafter, we use the logarithmic axial compressive strain a ∶= −log(H∕H 0 ) (compression positive) and the logarithmic volumetric strain v ∶= log(V∕V 0 ) (expansion positive), where H and V denote, respectively, the current height and volume of the specimen after deformation, and H 0 and V 0 the initial ones before deformation.…”