I. Vardoulakis

Bifurcation Analysis in Geomechanics

Introduction. Basic concepts from continuum mechanics. Incremental continuum mechanics. Buckling of layered elastic media. Mechanics of water-saturated granular materials. Plasticity theory for granular materials. Bifurcation analysis of element tests. Shear band bifurcation in granular media. Cosserat continuum model for granular materials. Second grade plasticity theory for geomaterials. Stability of undrained deformations. Index.

A gradient flow theory of plasticity for granular materials

SummaryA flow theory of plasticity for pressure-sensitive, dilatant materials incorporating second order gradients into the flow-rule and yield condition is suggested. The appropriate extra boundary conditions are obtained with the aid of the principle of virtual work. The implications of the theory into shear-band analysis are examined. The determination of the shear-band thickness and the persistence of ellipticity in the governing equations are discussed.

Torsional surface waves in a gradient-elastic half-space

Abstract The present work deals with torsional wave propagation in a linear gradient-elastic half-space. More specifically, we prove that torsional surface waves (i.e. waves with amplitudes exponentially decaying with distance from the free surface) do exist in a homogeneous gradient-elastic half-space. This finding is in contrast with the well-known result of the classical theory of linear elasticity that torsional surface waves do not exist in a homogeneous half-space. The weakness of the classical theory, at this point, is only circumvented by modeling the half-space as having material properties variable with depth (E. Meissner, Elastische Oberflachenwellen mit Dispersion in einem inhomogenen Medium, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich 66 (1921) 181–195; I. Vardoulakis, Torsional surface waves in inhomogeneous elastic media, Internat. J. Numer. Anal. Methods Geomech. 8 (1984) 287–296; G.A. Maugin, Shear horizontal surface acoustic waves on solids, in: D.F. Parker, G.A. Maugin (Eds.), Recent Developments in Surface Acoustic Waves, Springer Series on Wave Phenomena, vol. 7, Springer, Berlin, 1988, pp. 158–172), as a layered structure (Maugin, 1988; E. Reissner, Freie und erzwungene Torsionsschwingungen des elastischen Halbraumes, Ingenieur-Archiv 8 (1937) 229–245) or by considering couplings with electric and magnetic fields for different types of materials (Maugin, 1988). The theory employed here is the simplest possible version of Mindlin’s (R.D. Mindlin, Micro-structure in linear elasticity, Arch. Rat. Mech. Anal. 16 (1964) 51–78) generalized linear elasticity. A simple wave-propagation analysis based on Hankel transforms and complex-variable theory was done in order to determine the conditions for the existence of the torsional surface motions and to derive dispersion curves and cut-off frequencies. Also, we notice that, up to date, no other generalized linear continuum theory (including the integral-type non-local theory) has successfully been proposed to predict torsional surface waves in a homogeneous half-space.

The asymmetry of stress in granular media

Here, we show that the average stress in granular media, which is defined from virtual work, may be asymmetric in the absence of contact moments. We specify the circumstances and amplitude of stress asymmetry, and calculate the corresponding couple stress and first stress moment. We also show that the average stress is always symmetric, when it is alternately defined by using statics and no contact moment. The stress asymmetry, which results from external moments, has an amplitude that decreases with the volume size. The present analysis applies to two- and three-dimensional particles of arbitrary shapes. The asymmetric stress, couple stress and first stress moment are analytically calculated in a particular example with cylindrical and spherical particles.

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain-softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C-1 continuity element. The required higher-order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill-posedness caused by strain-softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post-peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear-band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values.

Hydro-mechanical aspects of the sand production problem

This paper examines the hydro-mechanical aspect of the sand production problem and sets the basic frame of the corresponding mathematical modelling. Accordingly, piping and surface erosion effects are studied on the basis of mass balance and particle transport considerations as well as Darcys law. The results show that surface erosion is accompanied by high changes of porosity and permeability close to the free surface. Quantities which can be measured in experiment, like the amount of produced solids or fluid discharge, can be used in an inverse way to determine the constitutive parameters of the problem.

Vibration isolation using open or filled trenches Part 2: 3-D homogeneous soil

The isolation of structures from ground transmitted waves by open and infilled trenches in a three-dimensional context is numerically studied. The soil medium is assumed to be elastic or viscoelastic, homogeneous and isotropic. Waves generated by the harmonic motion of a surface rigid machine foundation are considered in this work. The formulation and solution of the problem is accomplished by the boundary element method in the frequency domain. The infinite space fundamental solution is used requiring discretization of the trench surface, the soil-foundation interface and some portion of the free soil surface. The proposed methodology is first tested for accuracy by solving three characteristic wave propagation problems with known solutions and then applied to several vibration isolation problems involving open and concrete infilled trenches. Three-dimensional graphic displays of the surface displacement pattern around the trenches are also presented.

Modelling of localisation and scale effect in thick-walled cylinders with gradient elastoplasticity

We model the progressive localisation of deformation which causes failure around thick-walled cylinders under external radial pressure. The study is based on a second-gradient elastoplastic model developed to regularise the ill-posedness caused by material strain-softening behaviour. The stress increment is related to both the strain increment and its Laplacian. The gradient terms introduce an internal length scale to the material allowing robust modelling of its post-peak behaviour. The numerical implementation is based on a C-1 finite element displacement formulation. Mesh insensitivity in terms of load-displacement and failure mechanism is demonstrated. The internal length in the constitutive equations enables modelling of the scale effect in thick-walled cylinders, according to which the load required to induce failure appears to be much larger for small holes than for large holes.

Gradient dependent dilatancy and its implications in shear banding and liquefaction

SummaryA gradient dependent dilatancy condition is assumed in order to capture the heterogeneous character of deformation in granular soils. This assumption is incorporated into the structure of classical deformation and flow theories of plasticity and its implications in two interesting examples of patterning instability, that is shear banding and liquefaction, are examined. Shear banding is considered within a modified gradient dependent deformation theory, while liquefaction is studied within a modified gradient dependent flow theory of plasticity. In both cases a deformation induced length scale is obtained near the instability, and this is identified with the thickness of the shear band or the spacing of the liquefying strips.ÜbersichtLokale Inhomogenitäten bei der Verformung granularer Materialien sind hier mit Hilfe einer gradientenabhängigen Dilatanzbeziehung beschrieben. Diese Annahme ist in die Struktur einer klassischen Deformations- und Fließtheorie der Plastizität eingebaut, und die hiermit einhergehenden Konsequenzen werden anhand zweier interessanten Fälle der Musterbildung in Böden, nämlich Scherfugenbildung und Verflüssigung, studiert. Scherfugenbildung ist innerhalb einer modifizierten gradientenabhängigen Deformationstheorie untersucht, wogegen Bodenverflüssigung innerhalb einer modifizierten Fließtheorie analysiert wird. In beiden Fällen bekommt man in der Umgebung des Instabilitätszustandes einen deformationsinduzierten Längenmaßstab, der sich als Scherfugendicke oder als Abstand der verflüssigten Bodenstreifen manifestiert.

Cracks in gradient elastic bodies with surface energy

In the present paper the effect of higher order gradients on the structure of line-crack tips is determined. In particular we introduce in the constitutive equations of the linear deformation of an elastic solid a volumetric energy term, which includes the contribution of the strain gradient, and a surface energy gradient dependent term and then determine the effect of these terms on the structure of the mode-III crack tip and the associated stress and strain fields. By making use of the solution in terms of Fourier transform of the equation of elastic equilibrium we solve the half-plane boundary value problems of: (a) specified tractions, and (b) prescribed displacements, along the crack surface, respectively.