Bifurcation Analysis in Geomechanics

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1 Bifurcation Analysis in Geomechanics I. VARDOULAKIS Department of Engineering Science National Technical University of Athens Greece and J. SULEM Centre d'enseignement et de Recherche en Mecanique des Sols Ecole Nationale des Ponts et Chaussees/LCPC France BLACKIE ACADEMIC & PROFESSIONAL An Imprint of Chapman & Hal! London Glasgow Weinheim New York Tokyo Melbourne Madras

2 Contents 1 Introduction A historical note Observational background The frame of geomaterials constitutive modeling Considered topics 11 Literature 12 2 Basic concepts from continuum mechanics Kinematic and static considerations Lagrangian description of the deformation Eulerian description of the deformation Deformation of surface and volume elements Static considerations Time derivatives and rates Material time derivative and velocity Relative deformation gradient and its rate Rigid-body or Jaumann derivative Convective time derivative Material derivative of volume integrals Balance equations Mass balance Balance of linear momentum Balance of angular momentum Energy balance Entropy inequalities and balance Discontinuous fields and wave fronts Geometric compatibility conditions Kinematic compatibility conditions Dynamic compatibility conditions Weak discontinuities 46 Literature 47 3 Incremental continuum mechanics Updated Lagrangian description Kinematical considerations Plane-strain deformations Deformation of line, surface and volume elements Stresses and stress increments Infinitesimal strain superimposed upon finite strain Plane rectilinear deformations Superposition of rectilinear deformations Superposition of pure shear Hypoelastic constitutive equations 67

3 3.3 Equilibrium bifurcation The principle of virtual work The zero moment condition Configuration-dependent loading The linear bifurcation problem Uniqueness theorems under dead loading Acceleration waves and stationary discontinuities 88 Literature 91 4 Buckling of layered elastic media Folding of elastic and viscoelastic media as a bifurcation problem Surface and interfacial instabilities in elastic media Buckling of a single layer under initial stress Buckling of a system of layers the transfer matrix technique Surface instability of a homogeneous half-space The problem of wavelength selection Interfacial instability Periodic elastic multilayered media The asymptotic averaging method Example: Surface instabilities in a multilayered periodic half-space Limitations of the asymptotic averaging method Elastic anisotropic Cosserat continuum Basic concepts The Cosserat model of a multilayered medium Example: Buckling of an homogeneous Cosserat half-space The effect of surface parallel Griffith cracks Analytical solution for a single crack Buckling of a half-space with a periodic array of coplanar cracks A Cosserat continuum representation Influence of the initial stress field on crack propagation Concluding remarks and discussion 138 Literature Mechanics of water-saturated granular materials Definitions Mass balance equations Static considerations: partial and 'effective' stresses The influence of grain and fluid compressibility Balance of linear momentum Laws governing fluid flow in porous media Darcy's law Biot's modification of viscous and inertial drag Forchheimer's extension of Darcy's law Brinkman's and Aifantis' modification of Darcy's law The incremental initial, boundary value problem Governing equations The incremental problem Linear stability analysis Compaction instabilities Grain crushing Stability of non-uniform compaction 180 Literature 182

4 XI 6 Plasticity theory for granular materials Micromechanical considerations Kinematics Statics Flow theory of plasticity The Mroz-Mandel non-associative elastoplasticity Stress-dependent elasticity Finite strain formulations The equation of thermoelastoplasticity Drucker's postulate Uniqueness theorems for elastoplastic solids Simple constitutive models for frictional materials Stress invariants The Drucker-Prager and Mohr-Coulomb models Data reduction and model calibration Lade's yield surface model Extensions of isotropic hardening plasticity Non-potential flow rules Yield surface modifications Modeling of strain softening D-constitutive model for sand Model justification Formulation Example of model calibration 246 Literature Bifurcation analysis of element tests Observational background Bifurcation analysis of the triaxial compression and extension tests Problem statement A deformation theory of plasticity Governing equations Bifurcation condition Example of triaxial compression test on medium dense Karlsruhe sand Bifurcation analysis of the biaxial test Formulation of the diffuse bifurcation problem Classification of regimes and bifurcation condition Example: Biaxial compression test on a Dutch sand 274 References Shear-band bifurcation in granular media Equilibrium bifurcation and stability The Thomas-Hill-Mandel shear-band model Mandel's dynamic stability analysis Shear-band formation in element tests Shear-band analysis in plane strain rectilinear deformations Analysis of a biaxial compression test on sand Imperfection sensitivity of the biaxial test Spontaneous versus progressive localization Shear banding in sands: experiment versus theory Influence of porosity Influence of confining pressure Influence of anisotropy Influence of grain size and shape 306

5 Xll 8.4 Non-coaxial plasticity model Localization in inhomogeneous stress field The cavity inflation test Global bifurcation analysis of the cavity inflation test Progressive failure 325 Literature Cosserat continuum model for granular materials Micromechanical considerations Motivation Kinematical considerations Static considerations Basic concepts from Cosserat continuum mechanics Kinematics of 2D Cosserat continuum Dynamics and statics Principles of virtual work The boundary-layer effect The Miihlhaus-Vardoulakis Cosserat plasticity model Definitions Elastic strains Plastic strains Constitutive equations Prediction of the shear-band thickness Governing equations Shear-band solution Analytical and experimental results Discussion and numerical implications 373 Literature Second-grade plasticity theory for geomaterials Mindlin's formalism of microstructure Kinematics The principle of virtual work Example: Gradient elasticity theory with surface energy Second-grade plasticity theory for granular rocks Observational background Constitutive modeling Constitutive equations Formulation of the rate-boundary value problem Well-posedeness of the rate-boundary value problem Bifurcation analysis deep boreholes Problem statement Bifurcation analysis The scale effect A 2D-gradient model for granular media Constitutive equations Shear-band analysis 419 Literature Stability of undrained deformations Monotonic biaxial tests on water-saturated sand Experimental basis Simulation and discussion Theoretical implications 438

6 Xlll 11.3 Bifurcation and stability Undrained shear banding Linear stability analysis Regularization Globally undrained shear banding Grain size and shape effect 454 Literature 458 Index 461

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