Shear banding analysis of geomaterials by strain gradient enhanced damage model

Shear band localization is investigated by a strain-gradient-enhanced damage model for quasi-brittle geomaterials. This model introduces the strain gradients and their higher-order conjugate stresses into the framework of continuum damage mechanics. The influence of the strain gradients on the constitutive behaviour is taken into account through a generalized damage evolutionary law. A weak-form variational principle is employed to address the additional boundary conditions introduced by the incorporation of the strain gradients and the conjugate higher-order stresses. Damage localization under simple shear condition is analytically investigated by using the theory of discontinuous bifurcation and the concept of the second-order characteristic surface. Analytical solutions for the distributions of strain rates and strain gradient rates, as well as the band width of localised damage are found. Numerical analysis demonstrates the shear band width is proportionally related to the internal length scale through a coefficient function of Poissons ratio and a parameter representing the shape of uniaxial stress–strain curve. It is also shown that the obtained distributions of strains and strain gradients are well in accordance with the underlying assumptions for the second-order discontinuous shear band boundary and the weak discontinuous bifurcation theory. 2005 Elsevier Ltd. All rights reserved.

[1]  Norman A. Fleck,et al.  Strain gradient crystal plasticity: size-dependentdeformation of bicrystals , 1999 .

[2]  M. Shi,et al.  Plastic flow localization in mechanism-based strain gradient plasticity , 2000 .

[3]  René Chambon,et al.  Plastic continuum with microstructure, Local Second Gradient Theories for Geomaterials : Localization Studies , 2001 .

[4]  Richard Bellman,et al.  Plastic Flow and Fracture in Solids , 1962 .

[5]  E. Aifantis.  On the Microstructural Origin of Certain Inelastic Models , 1984 .

[6]  Charles Jaeger,et al.  Rock mechanics and engineering: Author Index , 1979 .

[7]  P. Germain,et al.  The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure , 1973 .

[8]  D. Caillerie,et al.  One-dimensional localisation studied with a second grade model , 1998 .

[9]  H. Zbib,et al.  On the gradient-dependent theory of plasticity and shear banding , 1992 .

[10]  H. Zbib,et al.  Measurement of shear band characteristics in low carbon steel using photoelasticity , 1991 .

[11]  Ekkehard Ramm,et al.  Simulation of strain localization with gradient enhanced damage models , 1999 .

[12]  Elias C. Aifantis,et al.  Dislocations and Disclinations in Gradient Elasticity , 1999 .

[13]  Hans Muhlhaus,et al.  A variational principle for gradient plasticity , 1991 .

[14]  E. Aifantis.  SECTION 4.12 – Gradient Plasticity , 2001 .

[15]  Norman A. Fleck,et al.  A reformulation of strain gradient plasticity , 2001 .

[16]  Huajian Gao,et al.  Mechanism-based strain gradient plasticity— I. Theory , 1999 .

[17]  E. Aifantis,et al.  Recent Developments in Gradient Plasticity—Part II: Plastic Heterogeneity and Wavelets , 2002 .

[18]  R. Hill.  A general theory of uniqueness and stability in elastic-plastic solids , 1958 .

[19]  René de Borst,et al.  Material Instabilities in Solids , 1998 .

[20]  J. Rice,et al.  CONDITIONS FOR THE LOCALIZATION OF DEFORMATION IN PRESSURE-SENSITIVE DILATANT MATERIALS , 1975 .

[21]  J. Hadamard,et al.  Leçons sur la propagation des ondes et les equations de l'hydrodynamique , 2015 .

[22]  Hussein M. Zbib,et al.  On the structure and width of shear bands , 1988 .

[23]  Hussein M. Zbib,et al.  A Gradient-Dependent Flow Theory of Plasticity: Application to Metal and Soil Instabilities , 1989 .

[24]  Rhj Ron Peerlings,et al.  Gradient enhanced damage for quasi-brittle materials , 1996 .

[25]  J. Lemaître,et al.  Handbook of materials behavior models , 2001 .

[26]  Ioannis Vardoulakis,et al.  A gradient flow theory of plasticity for granular materials , 1991 .

[27]  John W. Rudnicki,et al.  The inception of faulting in a rock mass with a weakened zone , 1977 .

[28]  Balasingam Muhunthan,et al.  Gradient plasticity modelling of strain localization in granular materials , 2004 .

[29]  Nicolas Triantafyllidis,et al.  A gradient approach to localization of deformation. I. Hyperelastic materials , 1986 .

[30]  H. Askes,et al.  A classification of higher-order strain-gradient models in damage mechanics , 2003 .

[31]  R. D. Mindlin.  Second gradient of strain and surface-tension in linear elasticity , 1965 .

[32]  E. Aifantis,et al.  Recent Developments in Gradient Plasticity—Part I: Formulation and Size Effects , 2002 .

[33]  I. Vardoulakis,et al.  The thickness of shear bands in granular materials , 1987 .

[34]  E. Aifantis.  Update on a class of gradient theories , 2003 .

[35]  W. T. Koiter.  Couple-Stress in the Theory of Elasticity , 1964 .

[36]  Qiang Yang,et al.  Simulation of localization failure with strain‐gradient‐enhanced damage mechanics , 2002 .

[37]  K. Roscoe.  THE INFLUENCE OF STRAINS IN SOIL MECHANICS , 1970 .

[38]  Zenon Mróz,et al.  Finite element analysis of deformation of strain‐softening materials , 1981 .

[39]  E. Aifantis,et al.  Strain gradient and wavelet interpretation of size effects in yield and strength , 2003 .

[40]  N.,et al.  A PHENOMENOLOGICAL THEORY FOR STRAIN GRADIENT EFFECTS IN PLASTICITY , 2002 .

[41]  R. Toupin.  Elastic materials with couple-stresses , 1962 .

[42]  E. Aifantis,et al.  Element-Free Galerkin Implementation of Gradient Plasticity. Part II: Applications to 2D Strain Localization and Size Effects , 2003 .

[43]  E. Aifantis,et al.  Element-Free Galerkin Implementation of Gradient Plasticity. Part I: Formulation and Application to 1D Strain Localization , 2003 .

[44]  N. S. Ottosen,et al.  Properties of discontinuous bifurcation solutions in elasto-plasticity , 1991 .

[45]  René de Borst,et al.  Gradient-dependent plasticity: formulation and algorithmic aspects , 1992 .

[46]  N. Fleck,et al.  Strain gradient plasticity , 1997 .

[47]  Elias C. Aifantis,et al.  The physics of plastic deformation , 1987 .

[48]  R. D. Mindlin.  Micro-structure in linear elasticity , 1964 .