Dilatant Granular Materials Research Articles

Simulating intersection angles between conjugate faults in sea ice with different viscous–plastic rheologies

Abstract. Recent high-resolution pan-Arctic sea ice simulations show fracture patterns (linear kinematic features or LKFs) that are typical of granular materials but with wider fracture angles than those observed in high-resolution satellite images. Motivated by this, ice fracture is investigated in a simple uni-axial loading test using two different viscous–plastic (VP) rheologies: one with an elliptical yield curve and a normal flow rule and one with a Coulombic yield curve and a normal flow rule that applies only to the elliptical cap. With the standard VP rheology, it is not possible to simulate fracture angles smaller than 30∘. Further, the standard VP model is not consistent with the behavior of granular material such as sea ice because (1) the fracture angle increases with ice shear strength; (2) the divergence along the fracture lines (or LKFs) is uniquely defined by the shear strength of the material with divergence for high shear strength and convergent with low shear strength; (3) the angle of fracture depends on the confining pressure with more convergence as the confining pressure increases. This behavior of the VP model is connected to the convexity of the yield curve together with use of a normal flow rule. In the Coulombic model, the angle of fracture is smaller (θ=23∘) and grossly consistent with observations. The solution, however, is unstable when the compressive stress is too large because of non-differentiable corners between the straight limbs of the Coulombic yield curve and the elliptical cap. The results suggest that, although at first sight the large-scale patterns of LKFs simulated with a VP sea ice model appear to be realistic, the elliptical yield curve with a normal flow rule is not consistent with the notion of sea ice as a pressure-sensitive and dilatant granular material.

Micromechanical modelling of granular materials and FEM simulations

We present a micro-mechanical continuum model used for the description of dilatant granular materials with incompressible rotating grains for which the kinetic energy, in addition to the usual translational one, consists of other two terms owing to microstructural motions: in particular, it includes the dilatational expansions and contractions of the granules relative to one another, as well as the rotation movements of each grain compared to the others. Next, we propose a linear theory in which the representations of constitutive functionals are linear with respect to both the volume fraction and the micro-rotation gradients, and to the dissipative variables. At the end we test the linear model on a two-dimensional domain, in which the arising system of partial differential equations is solved using the finite element method; thus we obtain a numerical solution in the case of a simplified granular micromechanics. The obtained computations of the early granular dynamics are consistent with theoretical insights as deduced from the proposed model. Viscous and rotational contributions to the granular dynamics have been identified and compared each others.

Extended granular micromechanics

We analyze an extended mechanical model in the conservative case in order to describe a dilatant granular material with rotating grains for which the kinetic energy, in addition to the usual translational one, consists of three terms owing to microstructural motions: in particular, it includes the rotation of the grains, the dilatational expansion and contraction of the individual granules and of the granules relative to one another. Hence, we model the body as a continuum with a peculiar microstructure; after we follow classical procedures and define a variational principle of local type for a perfect fluid with microstructure, in accordance with the fluid-like behavior of granular materials: the motion equations are in good agreement with those obtained by other authors. At the end the particular case of a suspension of rotating rigid granules puts in evidence the possibility for granular materials to support shear stresses through the generation of micro-rotational gradients.

Propagation of sech2-type solitary waves in hierarchical KdV-type systems

A hierarchical Korteweg–de Vries-type evolution equation is applied for modelling wave propagation in dilatant granular materials. The model equation is integrated numerically under sech 2-type initial conditions using the Fourier transform-based pseudospectral method. Numerical simulations are carried out over a wide range of material parameters (two dispersion parameters and one microstructure parameter) and amplitudes of the initial wave. The analysis of the time–space behaviour of solutions results in five solution types. Besides typical KdV-like solitonic structures, wave packets were detected for some domains of material parameters.

Exact solutions of the hierarchical Korteweg–de Vries equation of microstructured granular materials

The problems under consideration are related to wave propagation in microstructured materials, characterized by higher-order nonlinear and higher-order dispersive effects; particularly, the wave propagation in dilatant granular materials. In the present paper the model equation is solved analytically by exact methods. The types of solutions are defined and discussed over a wide range of material parameters (two dispersion parameters and one microstructure parameter). The dispersion properties and the relation between group and phase velocities of the model equation are studied. The diagrams are drawn to illustrate the physical properties of the exact solutions.

Energy dissipation and post‐bifurcation behaviour of granular soils

AbstractExtensive experimental data indicate that plastic flow in granular materials is non‐coaxial (i.e. the principal directions of the plastic strain increment tensor and the stress tensor do not coincide) for loadings involving principal stress rotation. The degree of non‐coaxiality depends on the magnitude of principal stress rotation, hence plastic flow is also incrementally non‐linear. The paper presents an analysis and discussion of the effects of non‐coaxial plastic flow on the post‐bifurcation response of dilatant granular materials. Significant differences are observed between the energy dissipation and stress–dilatancy response of soils before and after shear‐band formation. These differences are attributed to the non‐coaxiality of plastic flow during post‐bifurcation shearing. In turn, the non‐coaxiality is attributed to principal stress rotation within the shear band, which is consistent with the assumption that the materials inside the shear undergo simple shear deformation. Experimental data are presented to support the analytical results. Copyright © 2006 John Wiley & Sons, Ltd.

Incorporating the effects of fabric in the dilatant double shearing model for planar deformation of granular materials

The objective of this paper is to incorporate the effects of fabric and its evolution into the Dilatant Double Shearing Model [Mehrabadi, M.M., Cowin, S.C., 1978. Initial planar deformation of dilatant granular materials. J. Mech. Phys. Solids 26, 269–284] for granular materials in order to capture the anisotropic behavior and the complex response of granular materials in cyclic shear loading. An important consequence of considering the fabric is that one can have unequal shearing rates along the two slip directions. This property leads to the non-coaxiality of the principal axes of stress and strain rate, which is more appropriate for a material that exhibits initial and induced anisotropy. In addition, we employ a fabric-dependent elasticity tensor with orthotropic symmetry. The model developed in this paper also predicts one of the experimentally observed characteristics of granular materials: the gradual concentration of the contact normals towards the maximum principal stress direction. We implement the constitutive equations into ABAQUS/Explicit by writing a user material subroutine in order to predict the strength anisotropy of granular materials in a plane strain biaxial compression test and investigate the mechanical behavior of granular materials under the cyclic shear loading conditions. The predictions from this model show good quantitative agreement with the experiments of [Park, C.S., 1990. Anisotropy in deformation and strength properties of sands in plane strain compression, Masters Thesis, University of Tokyo; Park, C.S., Tatsuoka, F., 1994. Anisotropic strength and deformation of sands in plane strain compression. In: XIII ICSMFE, New Delhi, India; Okada, N., 1992. Energy dissipation in inelastic flow of cohesionless granular media. Ph.D. Thesis, University of California, San Diego].

On solitons in microstructured solids and granular materials

The problems under consideration are related to wave propagation in nonlinear dispersive media, characterised by higher-order nonlinear and higher-order dispersive effects. Particularly two problems — wave propagation in dilatant granular materials and wave propagation in shape-memory alloys — are studied. Model equations are KdV-like evolution equations in both cases. The types of solutions are determined and analysed. It has been found that waves in granular materials are composed by two concurrent ensembles of solitary waves and there exist a threshold parameter value governing the behaviour of stable waves in alloy type materials.

Two approximate analytical solutions for the kinematically determined velocity equations for granular solids

For axially symmetric flows of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming dilatant double shearing a closed set of three first-order partial differential equations are obtained. These supposedly simple equations are deceptive, because although they are simple in appearance, the determination of exact solutions is non-trivial. For one of the known families of solutions which has not been studied previously, the authors present the non-linear ordinary differential equation for the stress angle ψ and determine two small ψ approximations. Furthermore, the stream function and streamlines are obtained for ψ determined numerically and from the two small ψ approximations. For purposes of comparison, the streamlines for three further known exact solutions are also presented. In addition, we briefly examine the circumstances for which solutions of the velocity equations satisfy the principle of non-negative plastic work. For example, we are able to establish that in the case when the velocity equations are derived from a plastic potential, the solutions always satisfy the principle when the material has no cohesion.

Textural evidence for dilatant (shear thickening) rheology of magma at high crystal concentrations

Investigation of the rheology of magmas at high crystal concentrations by experimental means has proved problematic. An alternative approach is to study textures of igneous rocks that not only preserve evidence of the kinematics of magma flow, such as flow direction, but can also preserve evidence of rheology. Flow textures in multiply intruded trachyte dykes on Fraser Island, eastern Australia record evidence of dilatant flow during solidification. This conclusion is reached by interpretation of microscopic ductile shear zones that disrupt the groundmass of aligned feldspar laths. Detailed three-dimensional investigation demonstrates that the dihedral angle between conjugate micro-shear zones is approximately 65°. This conjugate angle is equivalent to that observed in dilatant granular materials such as sand. Dilatant behaviour is synonymous with shear thickening rheology indicating that the magma flow is time-dependent and resists high flow rates. Some of the dykes contain autobrecciation fragments that may represent localities where the ductile flow rate threshold was exceeded. Newtonian or pseudoplastic (shear thinning) rheology of crystal-poor magmas must progressively give way to shear thickening rheology during cooling and increasing crystal concentration.

Granular materials: constitutive equations and strain localization

Strain localization into shear bands is commonly observed in natural soil masses, as well as in human-built embankments, footings, retaining walls and other geotechnical structures. Numerical predictions for the process of shear band formation are critically dependent on the constitutive equations employed. In this paper, the plane strain “double-shearing” constitutive model (e.g., Spencer, A.J.M., 1964. A theory of the kinematics of ideal soils under plane strain conditions. Journal of the Mechanics and Physics of Solids 12, 337–351; Spencer, A.J.M., 1982, Deformation of ideal granular materials. In: Hopkins, H.G., Sewell, M.J. (Eds.), Mechanics of Solids. Pergamon Press, Oxford and New York, pp. 607–652; Mehrabadi, M.M., Cowin, S.C., 1978. Initial planar deformation of dilatant granular materials. Journal of the Mechanics and Physics of Solids 26, 269–284; Nemat-Nasser, S., Mehrabadi, M.M., Iwakuma, T. 1981. On certain macroscopic and microscopic aspects of plastic flow of ductile materials. In: Nemat-Nasser, S. (Ed.), Three-dimensional Constitutive Relations and Ductile Fracture. North-Holland, Amsterdam, pp. 157–172; Anand, L., 1983. Plane deformations of ideal granular materials. Journal of the Mechanics and Physics of Solids 31, 105–122) is generalized to three dimensions including the effects of elastic deformation and pre-peak behavior. The constitutive model is implemented in a finite element program and is used to predict the formation of shear bands in plane strain compression, and plane strain cylindrical cavity expansion. The predictions from the model are shown to be in good quantitative agreement with the recent experiments of Han, C., Drescher, A., (1993. Shear bands in biaxial tests on dry coarse sand. Soils and Foundations 33, 118–132) and Alsiny, H., Vardoulakis, I., Drescher, A., (1992. Deformation localization in cavity inflation experiments on dry sand. Geotechnique 42, 395–410) on a dry sand. The constitutive model is also used to predict the stress state in a static sand pile — a topic which has occupied the attention of many investigators in recent years. In our simulations we model an initially loose sand mass as a cohesionless material with a mobilized internal friction coefficient which evolves from an initial value of zero to a saturation value. The formation of a sand pile is numerically modeled as a two-step process: (i) In the first step a conical sand mass is placed between a flat rigid surface and an axi-symmetric conical mold. Interaction between the sand mass and the rigid base plate is modeled using an interface friction coefficient which has the same value as the saturation value of the internal friction coefficient. The sand mass (which is confined between the base plate and the conical mold) is subjected to gravity loading, and the system is allowed to equilibrate. (ii) In the second step the conical mold is quickly lifted and the sand mass allowed to reach a new equilibrate, but slightly slumped configuration. In the process of slight slumping, there is non-homogeneous plastic deformation of the sand pile. This non-homogeneous plastic deformation, coupled with the evolving internal friction coefficient, naturally gives rise to a static stress state which exhibits the interesting feature that the vertical stress distribution at the base of the sand pile does not have a maximum under the apex of the cone, but shows a local dip there.

On the elasticity and plasticity of dilatant granular materials

We present a model which simulates the response of a granular material (made up of non-plastic grains) to small cyclic shear strain. The model is based upon a dissipation function representing energy dissipated as plastic deformation and a quadratic dilatancy rule describing the volume response which produces kinematic hardening. Elastic deformation is included in the model through a function describing the rate at which it is stored as deformation occurs. Two sets of experimental data (one for a shear test, the other for a circular loading test) are used to evaluate the accuracy of the model. For the shear test, both the shear strain–shear stress curve and the shear strain–volume strain curve are reproduced well. In the circular loading test, the principal strain response and volume response are modelled realistically.

The velocity field for the punch problem for dilatant granular materials

For the problem of indentation of a granular half-space by a smooth rigid punch, an alternative velocity field is obtained for shear-index yield functions, which is fully consistent with the dilatant double-shearing theory proposed for granular materials. The velocity field presented in (1 ), which although a bonafide solution of the governing equations, assumes that the flow is tangential to the bounding slip-line separating the deforming and rigid regions. In other words, in (1) it is assumed that there is no normal component of the flow across this bounding characteristic. This assumption, although physically well motivated, is not fully consistent with the dilatant double-shearing rule that an expansion accompanies any shearing deformation. In this latter situation the normal velocity is discontinuous and this implies the formation of voids in the material. Numerical results are presented comparing the two velocity fields and these conflicting issues are discussed.

Indentation of a double shearing dilatant granular material by a smooth rigid wedge

Presented herein is a complete solution to the normal plane strain indentation of a double shearing dilatant granular material by a smooth rigid wedge. The stress and velocity fields, resistance to penetration, and contact stress are compared with those of materials obeying (i) the incompressible double shearing theory, and (ii) the Mohr-Coulomb associated flow rule.

On a Family of Axially Symmetric Kinematically Determined Plastic Flows

For axially symmetric flows of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming either a plastic potential or dilatant double shearing, a set of three first-order partial differential equations is obtained. These equations turn out to be deceptive, because although they are simple in appearance, the determination of simple exact solutions is nontrivial. For one of the known families of solutions, the authors present a simple and straightforward asymptotic expansion for the stress angle and show that in a restricted range, this expansion provides an accurate solution of the nonlinear ordinary differential equation. In addition, for the same family of solutions, the authors examine a special case that permits one integration and enables the velocity components to be given explicitly in terms of the stress angle. This partial analytical solution may then be coupled either with the asymptotic expansion or with a full numerical solution of the governing first-order differential equation for the stress angle. In addition, the authors show that a special case of the nonlinear ordinary differential equation admits a simple first integral that has not been given previously. Finally, a detailed numerical comparison is made for various parameter values of the asymptotic expansion, the full numerical solution, and with known exact analytical solutions.

The velocity equations for dilatant granular flow and a new exact solution

For axially symmetric flow of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming dilatant double shearing a set of three first order partial differential equations are obtained. These equations turn out to be deceptive because although simple in appearance, the determination of simple exact solutions is non-trivial. Here we show that all the known functional forms of existing solutions also arise systematically by consideration of the "optimal systems" of the classical Lie symmetries which indicates that any further solution types most likely arise from non-classical symmetries. For one of the known families we present a special case which admits a particularly simple closed form expression, which has not been previously given in the literature. For this particular special case the integral curves (streamlines) can be readily obtained as well as a simple analytical "approximation" for the particle paths. The streamlines and the validity of the analytical approximation are shown graphically.

Dynamics and wave propagation in dilatant granular materials

The equations of motion for dilatant granular material are obtained from a Hamiltonian variational principle of local type in the conservative case. The propagation of nonlinear waves in a region with uniform state is studied by means of an asymptotic approach that has already appeared useful in an investigation on wave propagation in bubbly liquids and in fluid mixtures. When the grains are assumed to be incompressible, it is shown that the material behaves as a continuum with latent microstructure.

Localized bifurcation of pressure sensitive dilatant granular materials

Localized bifurcation of pressure sensitive dilatant granular materials

A derivation of the Mehrabadi-Cowin equations

An alternative derivation to that given by Mehrabadi and Cowin (1978) is presented here for a pair of kinematic equations governing a certain class of flows in the plastic deformation of dilatant granular materials. This class has been described by Spencer (1981) as double shearing flows. In their derivation Mehrabadi and Cowin (1978), prior to presenting the equations relative to rectangular Cartesian coordinates, obtained an intermediate pair of equations relative to a non-orthogonal network of characteristic coordinates. The essential difference between the original and present derivation is that here, the flow rule, expressed relative to rotating, rectangular Cartesian coordinates, is transformed directly to obtain the kinematic equations relative to fixed rectangular Cartesian coordinate axes, without the need to obtain the characteristic equations.

Plane deformations of ideal granular materials‡

The flow rule based on a double-shearing model developed by A. J. M. Spencer, and M. M. Mehrabadi and S. C. Cowin is intended for use in analyzing fully developed plane flows of rate-independent, isotropic, rigidperfectly plastic frictional and dilatant granular materials. With the view of extending the range of applicability of this physically based flow rule to the prepeak (prelocalization) regime, we systematically introduce isotropic hardening (softening) via the usual consideration of scalar hardening variables in the yield function. However, for this new model to be in accord with some recent experimental observations of I. Vardoulakis on shear-band localizations, it is found that we have to abandon the classical assumption of the coincidence of stress and velocity characteristics.