Estimation of crack opening from a two‐dimensional continuum‐based finite element computation

Damage models are capable of representing crack initiation and mimicking crack propagation within a continuum framework. Thus, in principle, they do not describe crack openings. In durability analyses of concrete structures however, transfer properties are a key issue controlled by crack propagation and crack opening. We extend here a one‐dimensional approach for estimating a crack opening from a continuum‐based finite element calculation to two‐dimensional cases. The technique operates in the case of mode I cracking described in a continuum setting by a nonlocal isotropic damage model. We used the global tracking method to compute the idealized crack location as a post‐treatment procedure. The original one‐dimensional problem devised in Dufour et al. [4] is recovered as profiles of deformation orthogonal to the idealized crack direction are computed. An estimate of the crack opening and an error indicator are computed by comparing finite element deformation profiles and theoretical profiles corresponding to a displacement discontinuity. Two estimates have been considered: In the strong approach, the maxima of the profiles are assumed to be equal; in the weak approach, the integrals of each profile are set equal. Two‐dimensional numerical calculations show that the weak estimates perform better than do the strong ones. Error indicators, defined as the distance between the numerical and theoretical profiles, are less than a few percentages. In the case of a three‐point bending, test results are in good agreement with experimental data, with an error lower than 10% for widely opened crack (> 40µm). Copyright © 2011 John Wiley & Sons, Ltd.     

An original semi‐discrete approach to assess gas conductivity of concrete structures

For civil engineering structures with a tightness role, structural permeability is a key issue. In this context, this paper presents a new proposition of a numerical modelling of leakage rate through a cracked concrete structure undergoing mode I cracking. The mechanical state of the material, considered in the framework of continuum mechanics based on finite element modelling, is described by means of the stress‐based nonlocal damage model which takes into account the stress state and provides realistic local mechanical fields. A semi‐discrete method based on the strong discontinuity approach to estimate crack opening is then considered in the post‐treatment phase. Using a Poiseuille's like relation, the coupling between the mechanical state of the material and its dry gas conductivity is performed. For validation purposes, an original experimental campaign is conducted on a dry concrete disc loaded in a splitting setup. During the loading, gas conductivity and digital image correlation analysis are performed. The comparison with the 3D experimental mechanical global response highlights the performance of the mechanical model. The comparison between crack openings measured by digital image correlation and estimated by the strong discontinuity method shows a good agreement. Finally, the results of the semi‐discrete approach coupled with the gas conductivity compared with experimental data show a good estimation of the structural conductivity. Consequently, if the mechanical problem is well modelled at the global scale, then the proposed approach provides good estimation of gas conductivity. Copyright © 2016 John Wiley & Sons, Ltd.     

Constitutive modelling of aggregate interlock in concrete

A model for formed cracks in concrete is presented. The model can be used in isolation for existing cracks or linked with other damage or plasticity models and applied once a crack has fully formed. It can be applied directly to interface finite elements or used to control the behaviour of crack planes in more general constitutive models that are applied to 2D and 3D continuum elements. The focus of the present development is on aggregate interlock and crack closing behaviour, which is examined from available experimental data. A contact function is derived and is used to differentiate between three contact states. These states are named open, where there is no contact, interlock, for which the stresses depend upon the nearest distance to the contact surface and closed, for which the stresses depend upon the relative displacements directly. The model is developed within an elasto‐plastic framework using effective stresses, which are related to the total stresses via a contact proportion function. The relationship between the effective normal and shear yield stresses is found to be parabolic and the yield shear stress to be dependent upon the opening and embedment displacements. The performance of the model is assessed against experimental data from shear‐normal tests and it is concluded that the model is able to represent the key characteristics of the behaviour of formed cracks in concrete. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Comparison of continuous and discontinuous constitutive models to simulate concrete behaviour under mixed‐mode failure conditions

The paper presents detailed FE simulation results of concrete elements under mixed‐mode failure conditions according to the so‐called shear‐tension test by Nooru‐Mohamed, characterized by curved cracks. A continuous and discontinuous numerical two‐dimensional approach was used. In order to describe the concrete's behaviour within continuum mechanics, two different constitutive models were used. First, an elasto‐plastic model with isotropic hardening and softening was assumed. In a compression regime, a Drucker–Prager criterion with a non‐associated flow rule was used. In turn, in a tensile regime, a Rankine criterion with an associated flow rule was adopted. Second, an isotropic damage constitutive model was applied with a single scalar damage parameter and different definitions of the equivalent strain. Both constitutive laws were enriched by a characteristic length of micro‐structure to capture properly strain localization. As an alternative approach, the extended finite element method was used. Our results were compared with the experimental ones and with results of other FE simulations reported in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Three‐dimensional simulations of tensile cracks in geomaterials by coupling meshless and finite element method

Failure in geotechnical engineering is often related to tension‐induced cracking in geomaterials. In this paper, a coupled meshless method and FEM is developed to analyze the problem of three‐dimensional cracking. The radial point interpolation method (RPIM) is used to model cracks in the smeared crack framework with an isotropic damage model. The identification of the meshless region is based on the stress state computed by FEM, and the adaptive coupling of RPIM and FEM is achieved by a direct algorithm. Mesh‐bias dependency, which poses difficulties in FEM‐based cracking simulations, is circumvented by a crack tracking algorithm. The performance of our scheme is demonstrated by two numerical examples, that is, the four‐point bending test on concrete beam and the surface cracks caused by tunnel excavation. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive finite element analysis of mode I fracture in cement‐based materials

Microscopic studies using advanced experimental techniques have provided better insight into the fracture mechanisms in cement‐based materials. A clear understanding of fracture mechanisms is critical for the development of rigorous computational models for analysing fracture. Fracture analysis is usually carried out by finite element method. Accuracy of FE analysis depends upon the choice of mesh and for the predictions to be reliable, discretization errors are to be minimized. In cohesive crack approach, the non‐linearity is limited to the boundary conditions along the geometric discontinuity while the bulk of the material retains its elastic nature. The paper presents a mesh‐adaptive strategy based on ZZ error estimator to model discrete crack propagation in cement‐based materials. Examples of simulations have demonstrated the potential of the mesh‐adaptive technique in modelling the evolution of the localized strain profiles as well as failure of concrete test specimen. Copyright © 2001 John Wiley & Sons, Ltd.     

基于数值流形法的重力坝多裂纹扩展研究

传统数值流形法（NMM）在处理非连续变形问题时,仅限于几何构型不发生破坏的情况。针对这一不足,通过在裂纹尖端附近的物理片上增加用于模拟应力奇异性的增强位移函数,进一步发展了可用于几何构型破坏的扩展的高阶NMM。然后,将其应用到重力坝由连续到非连续的破坏过程分析中。首先,针对一含单裂纹的重力坝模型进行了敏感性分析,结果表明,在不同的扩展长度或网格密度下,其扩展路径基本相同且与文献结果保持一致。进而在此模型基础上又开展了多裂纹扩展分析,结果仅一条主导裂纹发生扩展,与文献结果基本一致。最后,针对印度的Koyna重力坝,通过设置不同的漫顶高度研究了其裂纹扩展路径的变化。结果表明,随着漫顶高度的增大,裂纹扩展路径逐渐趋向于水平方向扩展,而且坝体抵抗破坏的能力逐渐减弱。总体表明,NMM在求解实际工程问题时具有很好的数值稳定性和鲁棒性。     

Fuzzy parameters analysis of time‐dependent fracture of concrete dam models

In order to apply the mechanical properties (measured on material specimens or laboratory‐sized models) to large structures (such as concrete dams), a non‐linear theory able to predict the size‐scale effect has to be used. One of these theories was first proposed by Hillerborg and co‐workers (fictitious crack model) and is based on the earlier works by Barenblatt and Dugdale for metals (cohesive crack model). It is based on the existence of a fracture process zone (FPZ), where the material undergoes strain softening. The behaviour of the material outside the FPZ is linear elastic. A large number of short‐time laboratory tests were executed, by varying the load, under crack mouth opening displacement control. Since concrete exhibits a time‐dependent behaviour, an interaction between creep and micro‐crack growth occurs in the FPZ. Therefore, different testing conditions can be applied: rupture can be achieved by keeping the load constant before peak value (pre‐peak tests), or after peak value and after an unloading and reloading procedure (post‐peak tests). The crack propagation rate is shown to be small enough to neglect inertial forces and large enough to keep the time‐dependent behaviour of the process zone as dominant compared to the behaviour of the undamaged and viscoelastic zone. Due to the variability in material microstructure from one specimen to another, experimental data show large ranges of scatter. Well established methods in probability theory require sufficient experimental data in order to assume a probability density distribution. The objective of this study is to investigate the ranges of variation of the time response under constant load in simple structural elements associated with pre‐selected variation (fuzziness) in the main material parameters. For situations where the values of the material parameters are of a non‐stochastic nature, the fuzzy set approach to modelling variability has been proposed as a better and more natural approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Coupling between progressive damage and permeability of concrete: analysis with a discrete model

A significant increase of the permeability of concrete upon micro‐cracking and a good correlation between the evolution of damage (material stiffness) and permeability are observed experimentally. The present contribution investigates this correlation theoretically, with the help of lattice analyses. Scaling analysis of lattices which contain elastic brittle bonds has shown that the material degradation should be described by the evolution of the material stiffness, or compliance, in a continuum setting (damage models). This result is reviewed and further documented in the first part of the paper. In the second part, hydro‐mechanical problems are considered with the construction of a hydraulic lattice, dual to the mechanical one. We observe that the average permeability upon micro‐cracking is the lattice scale‐independent controlling variable in the hydraulic problem. Additionally, results show that in a continuum poro‐mechanical approach, the evolution of the material permeability ought to be related to the elastic unloading stiffness, described e.g. with the help of continuum damage variables. Copyright © 2005 John Wiley & Sons, Ltd.     

Interaction between a circular opening and fractures originating from its boundary in a piecewise homogeneous plane

The complex variables boundary element method (CVBEM) is used to study interaction between a circular opening and fractures originating from its boundary in a piecewise homogeneous plane. A new complex hypersingular equation for piecewise homogeneous media with a circular opening is obtained. The equation is solved using the CVBEM technique with circular and straight boundary elements and polynomial approximations (with square root asymptotics for crack tip elements) for the unknown functions. The algorithm is verified through comparison with known semi‐analytical and numerical solutions that involve interaction between a circular opening and specific systems of cracks or other openings. New numerical results concerning the interaction of the circular opening with the cracks and circular inclusions are presented. The method is applied to an important problem in the petroleum industry: modelling propagation of hydraulic fractures in the vicinity of a borehole. Copyright © 2000 John Wiley & Sons, Ltd.     

Computational mechanics of the steel–concrete interface

Concrete cracking in reinforced concrete structures is governed by two mechanisms: the activation of bond forces at the steel–concrete interface and the bridge effects of the reinforcement crossing a macro‐crack. The computational modelling of these two mechanisms, acting at different scales, is the main objective of this paper. The starting point is the analysis of the micro‐mechanisms, leading to an appropriate choice of (measurable) state variables describing the energy state in the surface systems: on the one side the relative displacement between the steel and the concrete, modelling the bond activation; on the other hand, the crack opening governing the bridge effects. These displacement jumps are implemented in the constitutive model using thermodynamics of surfaces of discontinuity. On the computational side, the constitutive model is implemented in a discrete crack approach. A truss element with slip degrees of freedom is developed. This degree of freedom represents the relative displacement due to bond activation. In turn, the bridge effect is numerically taken into account by modifying the post‐cracking behaviour of the contact elements representing discrete concrete cracks crossed by a rebar. First simulation results obtained with this model show a good agreement in crack pattern and steel stress distribution with micro‐mechanical results and experimental results. Copyright © 2001 John Wiley & Sons, Ltd.     

Computational model coupling mode II discrete fracture propagation with continuum damage zone evolution

We propose a numerical method that couples a cohesive zone model (CZM) and a finite element‐based continuum damage mechanics (CDM) model. The CZM represents a mode II macro‐fracture, and CDM finite elements (FE) represent the damage zone of the CZM. The coupled CZM/CDM model can capture the flow of energy that takes place between the bulk material that forms the matrix and the macroscopic fracture surfaces. The CDM model, which does not account for micro‐crack interaction, is calibrated against triaxial compression tests performed on Bakken shale, so as to reproduce the stress/strain curve before the failure peak. Based on a comparison with Kachanov's micro‐mechanical model, we confirm that the critical micro‐crack density value equal to 0.3 reflects the point at which crack interaction cannot be neglected. The CZM is assigned a pure mode II cohesive law that accounts for the dependence of the shear strength and energy release rate on confining pressure. The cohesive shear strength of the CZM is calibrated by calculating the shear stress necessary to reach a CDM damage of 0.3 during a direct shear test. We find that the shear cohesive strength of the CZM depends linearly on the confining pressure. Triaxial compression tests are simulated, in which the shale sample is modeled as an FE CDM continuum that contains a predefined thin cohesive zone representing the idealized shear fracture plane. The shear energy release rate of the CZM is fitted in order to match to the post‐peak stress/strain curves obtained during experimental tests performed on Bakken shale. We find that the energy release rate depends linearly on the shear cohesive strength. We then use the calibrated shale rheology to simulate the propagation of a meter‐scale mode II fracture. Under low confining pressure, the macroscopic crack (CZM) and its damaged zone (CDM) propagate simultaneously (i.e., during the same loading increments). Under high confining pressure, the fracture propagates in slip‐friction, that is, the debonding of the cohesive zone alternates with the propagation of continuum damage. The computational method is applicable to a range of geological injection problems including hydraulic fracturing and fluid storage and should be further enhanced by the addition of mode I and mixed mode (I+II+III) propagation. Copyright © 2016 John Wiley & Sons, Ltd.     

Smeared crack approach: back to the original track

This paper briefly reviews the formulations used over the last 40 years for the solution of problems involving tensile cracking, with both the discrete and the smeared crack approaches. The paper focuses on the smeared approach, identifying as its main drawbacks the observed mesh‐size and mesh‐bias spurious dependence when the method is applied ‘straightly’. A simple isotropic local damage constitutive model is considered, and the (exponential) softening modulus is regularized according to the material fracture energy and the element size. The continuum and discrete mechanical problems corresponding to both the weak discontinuity (smeared cracks) and the strong discontinuity (discrete cracks) approaches are analysed and the question of propagation of the strain localization band (crack) is identified as the main difficulty to be overcome in the numerical procedure. A tracking technique is used to ensure stability of the solution, attaining the necessary convergence properties of the corresponding discrete finite element formulation. Numerical examples show that the formulation derived is stable and remarkably robust. As a consequence, the results obtained do not suffer from spurious mesh‐size or mesh‐bias dependence, comparing very favourably with those obtained with other fracture and continuum mechanics approaches. Copyright © 2006 John Wiley & Sons, Ltd.     

Simulation of damage–permeability coupling for mortar under dynamic loads

The results reported in this paper deal with the simulation of damage in cohesive geomaterials such as rocks or concrete subjected to dynamic loads. The practical objective is to stimulate the production of tight gas reservoirs with a technique that is an alternative to hydraulic fracturing. The principle is that when subjected to dynamic loads, cohesive materials such as concrete, rocks or ceramics exhibit distributed micro‐cracking as opposed to localised cracking observed under static loads. Hence, a low permeability rock containing gas trapped into occluded pores can be fragmented with the help of dynamic loads, and gas can be extracted in a much more efficient way compared with hydraulic fracturing, where only large macro cracks are formed with very few connections between occluded pores. At the stage of laboratory development of this technique, compressive underwater shock waves have been used to increase the intrinsic permeability of concrete specimens. In a previous study, pressure waves generated by pulsed arc electrohydraulic discharges in water were used in order to induce micro‐cracking and an increase of average permeability of concrete hollow cylinders subjected to confinement stresses (equivalent to geostatic stresses). We discuss here a 3‐D anisotropic constitutive model aimed at describing the dynamic response of these specimens. It is based on rate‐dependent continuum damage constitutive relations. Crack closure effects and damage‐induced anisotropy are included in the model. The directional growth of damage is related to the directional growth of material intrinsic permeability. Numerical simulations of damage induced by shock waves show good agreement with the experiments for various confinement levels of the specimens. Copyright © 2013 John Wiley & Sons, Ltd.     

Particle manifold method (PMM): A new continuum‐discontinuum numerical model for geomechanics

Particle manifold method (PMM) is a new extension of the numerical manifold method (NMM). PMM uses a mathematical cover system to describe the motion and deformation of a particle‐based physical domain. By introducing the concept of particle into NMM, PMM takes the advantages of easy topological and contact operations with particles. In this article, the methodology, formulations and implementation of the method are presented, together with modelling examples for validation. It is found that good solutions for both continuous and discontinuous problems are obtained by the new developed PMM. Due to the underlying coupled continuum‐discontinuum property of PMM, it has great potential for modelling of geomechanical problems. Copyright © 2012 John Wiley & Sons, Ltd.     

EARTHQUAKE ANALYSIS OF GRAVITY DAMS BASED ON DAMAGE MECHANICS CONCEPT

In this paper, the seismic response analysis of concrete gravity dams is presented using the concept of Continuum Damage Mechanics. The analysis is performed using the finite element technique and a proper material degradation/damage model. The damage criterion used here is a second order tensor model based on elastic-brittle characterization and on a power function of the principal tensile stress. The methodology employed is shown to be computationally efficient and consistent in its treatment of both damage growth and propagation. Other important features considered in the analysis are: (1) dam–foundation interaction (2) appropriate modelling of joined rock mass using continuum damage mechanics, and (3) proper modelling of unbounded domain of foundation rock. The infinite media representation of the foundation material has been achieved by using doubly asymptotic approximation. The results of the analysis indicate that the seismic response of a damaged concrete dam could be significantly different from that of an undamaged one. In particular, the analysis shows that during a seismic event, the microstructure of a damaged zone can significantly change due to growth and propagation of microcracks.     

Numerical modelling of bit–rock fracture mechanisms in percussive drilling with a continuum approach

This paper presents a numerical method for continuum modelling of the dynamic bit–rock interaction process in percussive drilling. The method includes a constitutive model based on a combination of the recent viscoplastic consistency model, the isotropic damage concept and a parabolic compression cap. The interaction between the drill bit and rock is modelled using contact mechanics by treating the bit as a rigid body. As the bit–rock interaction in percussive drilling is a transient event, the method is implemented in explicit dynamics FEM. The rock strength heterogeneity is characterized at the mesoscopic level statistically using the Weibull distribution. The bit–rock interaction is simulated under axisymmetric conditions using cylindrical and hemispherical buttons. The choice of the quite complex constitutive model accounting, e.g. for plastic compaction, viscoplastic shear and tensile failure along with induced damage and rate dependency is justified by numerical simulations. Moreover, the quasi‐static and dynamic cases are compared in plane strain simulations. Finally, some results clarifying the discrepancy of opinions found in the literature concerning the side (lateral) crack formation are obtained. Copyright © 2010 John Wiley & Sons, Ltd.