Modelling of anisotropic damage in brittle rocks under compression dominated stresses

A new model for describing induced anisotropic damage in brittle rocks is proposed. Although phenomenological, the model is based on physical grounds of micromechanical analysis. Induced damage is represented by a second rank tensor, which is related to the density and orientation of microcracks. Damage evolution is related to the propagation condition of microcracks. The onset of microcrack coalescence leading to softening behaviour is also considered. The effective elastic compliance of the damaged material is obtained from a specific form of Gibbs potential. Irreversible damage‐related strains due to residual opening of microcracks after unloading are also captured. All the model's parameters could be determined from conventional triaxial compression tests. The proposed model is applied to a typical brittle rock. Comparison between test data and numerical simulations shows an overall good agreement. The proposed model is able to describe the main features related to induced microcracks in brittle geomaterials. Copyright © 2002 John Wiley & Sons, Ltd.     

Micromechanical Modeling of Anisotropic Damage-Induced Permeability Variation in Crystalline Rocks

This paper presents a study on the initiation and progress of anisotropic damage and its impact on the permeability variation of crystalline rocks of low porosity. This work was based on an existing micromechanical model considering the frictional sliding and dilatancy behaviors of microcracks and the recovery of degraded stiffness when the microcracks are closed. By virtue of an analytical ellipsoidal inclusion solution, lower bound estimates were formulated through a rigorous homogenization procedure for the damage-induced effective permeability of the microcracks-matrix system, and their predictive limitations were discussed with superconducting penny-shaped microcracks, in which the greatest lower bounds were obtained for each homogenization scheme. On this basis, an empirical upper bound estimation model was suggested to account for the influences of anisotropic damage growth, connectivity, frictional sliding, dilatancy, and normal stiffness recovery of closed microcracks, as well as tensile stress-induced microcrack opening on the permeability variation, with a small number of material parameters. The developed model was calibrated and validated by a series of existing laboratory triaxial compression tests with permeability measurements on crystalline rocks, and applied for characterizing the excavation-induced damage zone and permeability variation in the surrounding granitic rock of the TSX tunnel at the Atomic Energy of Canada Limited’s (AECL) Underground Research Laboratory (URL) in Canada, with an acceptable agreement between the predicted and measured data.     

Coupling between anisotropic damage and permeability variation in brittle rocks

In this paper, a coupled constitutive model is proposed for anisotropic damage and permeability variation in brittle rocks under deviatoric compressive stresses. The formulation of the model is based on experimental evidences and main physical mechanisms involved in the scale of microcracks are taken into account. The proposed model is expressed in the macroscopic framework and can be easily implemented for engineering application. The macroscopic free enthalpy of cracked solid is first determined by approximating crack distribution by a second‐order damage tensor. The effective elastic properties of damaged material are then derived from the free enthalpy function. The damage evolution is related to the crack growth in multiple orientations. A pragmatic approach inspired from fracture mechanics is used for the formulation of the crack propagation criterion. Compressive stress induced crack opening is taken into account and leads to macroscopic volumetric dilatancy and permeability variation. The overall permeability tensor of cracked material is determined using a micro–macro averaging procedure. Darcy's law is used for fluid flow at the macroscopic scale whereas laminar flow is assumed at the microcrack scale. Hydraulic connectivity of cracks increases with crack growth. The proposed model is applied to the Lac du Bonnet granite. Generally, good agreement is observed between numerical simulations and experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

An anisotropic damage–plasticity model for saturated quasi‐brittle materials

This article is devoted to numerical modeling of anisotropic damage and plasticity in saturated quasi‐brittle materials such as rocks and concrete. The damaged materials are represented by an isotropic poroelastic matrix containing a number of families of microcracks. Based on previous works, a discrete thermodynamic approach is proposed. Each family of microcracks exhibits frictional sliding along crack surfaces as well as crack propagation. The frictional sliding is described by a Coulomb–Mohr‐type plastic criterion by taking into account the effect of fluid pressure through a generalized effective stress concept. The damage evolution is entirely controlled by and coupled with the frictional sliding. The effective elastic properties as well as Biot's coefficients of cracked porous materials are determined as functions of induced damage. The inelastic deformation due to frictional sliding is also taken into account. The procedure for the identification of the model's parameters is presented. The proposed model is finally applied to study both mechanical and poromechanical responses of a typical porous brittle rock in drained and undrained compression tests as well as in interstitial pressure controlled tests. The main features of material behaviors are well reproduced by the model. Copyright © 2012 John Wiley & Sons, Ltd.     

Poroelastic behaviour of saturated brittle rock with anisotropic damage

A new anisotropic poroelastic damage model is proposed for saturated brittle porous materials. The model is formulated in the framework of the continuum damage mechanics. A second‐rank symmetric tensor is used to characterize material damage due to oriented microcracks. The classic Biot poroelastic theory is then extended to include poroelastic damage coupling. Both the deterioration of elastic properties and poroelastic coefficients is taken into account. A suitable procedure for determination of model parameters from standard laboratory tests is presented. The validity of the model is tested through comparison between numerical predictions and experimental data in various loading conditions. The overall performance of the model is evaluated. The choice of relevant effective stress for the microcrack propagation criterion in saturated cohesive geomaterials is discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

A self‐consistent approach for micro–macro modeling of elastic–plastic deformation in polycrystalline geomaterials

This paper is devoted to multi‐scale modeling of elastic–plastic deformation of a class of geomaterials with a polycrystalline microstructure. We have extended and improved the simplified polycrystalline model presented in [Zeng T. et al., 2014. Mech. Mater. 69 (1):132–145]. A rigorous and fully consistent self‐consistent (SC) scheme is proposed to describe the interaction among plastic mineral grains. We have also deeply discussed the numerical issues related to the numerical implementation of the proposed micromechanical model. The efficiency of the proposed model and the related numerical procedure is evaluated in several representative cases. We have compared the numerical results respectively obtained from the fully SC model and two simplified ones. It is found that the SC model produces a softer stress–strain response than that of the simplified models. The comparisons between the estimation of overall behavior of a granite in different loading conditions and experimental data are also conducted. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical simulation of bolt‐supported tunnels by means of a multiphase model conceived as an improved homogenization procedure

This paper examines the possibility of applying a homogenization procedure to analyze the convergence of a tunnel reinforced by bolts, regarded as periodically distributed linear inclusions. Owing to the fact that a classical homogenization method fails to account for the interactions prevailing between the bolts and the surrounding ground and thus tends to significantly overestimate the reinforcement effect in terms of convergence reduction, a so‐called multiphase model is presented and developed, aimed at improving the classical homogenization method. Indeed, according to this model, the bolt‐reinforced ground is represented at the macroscopic scale as the superposition of two mutually interacting continuous phases, describing the ground and the reinforcement network, respectively. It is shown that such a multiphase approach can be interpreted as an extension of the homogenization procedure, thus making it possible to capture the ground–reinforcement interaction in a proper way, provided the constitutive parameters of the model and notably those relating to the interaction law can be identified from the reinforced ground characteristics. The numerical implementation of this model in a finite element method‐based computer code is then carried out, and a first illustrative application is finally presented. Copyright © 2008 John Wiley & Sons, Ltd.     

An approach to the modeling of viscoelastic damage. Application to the long‐term creep of gypsum rock materials

A three‐dimensional phenomenological model is developed to describe the long‐term creep of gypsum rock materials. The approach is based on the framework of continuum damage mechanics where coupling with viscoelasticity is adopted. Specifically, a local damage model based on the concept of yield surface is proposed and deeply investigated. Among the many possibilities, we choose in this work its coupling with a generalized Kelvin–Voigt rheological model to formulate the whole behavior. Long‐term as well as short‐term relaxation processes can be integrated in the model by means of as many as necessary viscoelastic processes. The numerical discretization is described for an easy integration within a finite element procedure. Finally, a set of numerical simulations is given to show the possibilities of the presented model. It shows good agreement with some experimental results found in the literature. Copyright © 2012 John Wiley & Sons, Ltd.     

A discrete thermodynamic approach for anisotropic plastic–damage modeling of cohesive‐frictional geomaterials

A discrete plastic–damage model is developed for cohesive‐frictional geomaterials subjected to compression‐dominated stresses. Macroscopic plastic strains of material are physically generated by frictional sliding along weakness planes. The evolution of damage is related to the evolution of weakness planes physically in connection with the propagation of microcracks. A discrete approach is used to account for anisotropic plastic flow and damage evolution, by introducing two stress invariants and one plastic hardening variable for each family of sliding weakness planes. Plastic flow in each family is coupled with damage evolution. The proposed model is applied to typical geomaterials and comparisons between numerical predictions and experimental data are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

岩石弹脆性分维损伤本构模型

本文定义岩石构元中破裂面的分维值为各向同性损伤变量,而各个方向上裂纹面的累加量定义为各向异性损伤变量,并根据裂纹发育特征提出了损伤变量演化方程,从而建立起岩石脆性变形破坏过程的分维损伤本构模型。最后,利用该模型对大理岩单轴压缩应力应变曲线进行了模拟,结果说明本文提出的模型是较为合理的。     

Numerical modeling of fracturing in permeable rocks via a micromechanical continuum model

The paper presents a micromechanical approach to describe the failure of low-permeability brittle rocks as a multiscale fracturing process based on a poroelastic microcrack-damage model. Failure is formulated deep down at the fine pore scale as a material degradation phenomenon driven by microcrack growth that also impacts upon hydromechanical properties. A set of damage tensors describes the effect of dual-scale porosities (nanopores and microcracks) on both the hydraulic and poroelastic rock properties. Essentially, the multiscale model reconstructs the coupling effect of hydromechanical forces at the continuum level from the ground up through the upscaling of multiphase interactions at the fundamental structural level of the material. As a result, many macroscopic characteristics emerge naturally such as friction angle, fracture properties, and most importantly, Biot's coefficient taking on a tensorial form that is generally anisotropic. The model is validated within the framework of finite elements to illustrate various baseline constitutive features such as the effect of microcrack growth on the nonlinear stress-strain response and the induced anisotropy in the context of lab experimental tests and boundary value problems. Heterogeneities of the rock samples were incorporated by choosing material properties to be stochastic following Weibull and lognormal distributions. Numerical results appropriately replicated typical experimental observations where fracture localization and propagation are shown to be a multiscale phenomenon emerging from microcrack growth and coalescence at the microscale, with concomitant enhancement in fluid conductivity.     

Constitutive model for tailing soils subjected to freeze–thaw cycles based on meso-mechanics and homogenization theory

A constitutive model is proposed for tailing soils subjected to freeze–thaw cycles based on the meso-mechanics and homogenization theory. The evolution of meso-structure upon loading was analyzed within the framework of breakage mechanism. When the new model is formed, tailing soils are idealized as composite materials composed of bonded elements described by an elastic brittle model and frictional elements described by a double hardening model. Based on meso-mechanics and homogenization theory, the nonuniform distributions of stress and strain within the representative volume element are given by introducing a structure parameter of breakage ratio with the derivation of the strain coefficient tensor, which connects the strains of the bonded elements and the representative volume element. The methods for determining model parameters are suggested based on the available tested results. The model proposed here can predict the deformation properties of tailing soils experiencing freeze–thaw cycles with acceptable accuracy. The strain-hardening and post-peak strain-softening behaviors of tailing soils under various confining pressures as well as different numbers of freeze–thaw cycles are well captured, and the dilatancy and contraction features are also adequately represented.

     

A rate‐dependent cohesive crack model based on anisotropic damage coupled to plasticity

In quasi‐brittle material the complex process of decohesion between particles in microcracks and localization of the displacement field into macrocracks is limited to a narrow fracture zone, and it is often modelled with cohesive crack models. Since the anisotropic nature of the decohesion process in separation and sliding is essential, it is particularly focused in this paper. Moreover, for cyclic and dynamic loading the unloading, load reversal (including crack closure) and rate dependency are essential features that are included in a new model. The modelling of degradation is based on a ‘localized’ version of anisotropic continuum damage coupled to inelasticity. The concept of strain energy equivalence between the states in the effective and nominal settings is adopted in order to define the free energy of the interface. The proposed fracture criterion is of the Mohr type, with a smooth transition of the failure and kinematics (slip and dilatation) characteristics between tension and shear. The chosen potential, of the Lemaitre‐type, for evolution of the dissipative processes is additively decomposed into plastic and damage parts, and non‐associative constitutive equations are obtained. The constitutive equations are integrated by applying the backward Euler rule and by using Newton iteration. The proposed model is assessed analytically and numerically and a typical calibration procedure for concrete is proposed. Copyright © 2006 John Wiley & Sons, Ltd.