A new damage model based on non‐local displacements

A new non‐local damage model is presented. Non‐locality (of integral or gradient type) is incorporated into the model by means of non‐local displacements. This contrasts with existing damage models, where a non‐local strain or strain‐related state variable is used. The new model is very attractive from a computational viewpoint, especially regarding the computation of the consistent tangent matrix needed to achieve quadratic convergence in Newton iterations. At the same time, its physical response is very similar to that of the standard models, including its regularization capabilities. All these aspects are discussed in detail and illustrated by means of numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Non‐local damage modelling of concrete: a procedure for the determination of model parameters

This paper presents a procedure for the determination of parameters of non‐local damage models. This is to assure a consistent response of a non‐local damage model, as choice of the internal length and other parameters of the model are varied. Correlations between the internal length and other parameters governing the local constitutive behaviour of the model are addressed and exploited. Focus is put on the relationship between the internal length of the non‐local model and the width of the fracture process zone. Numerical examples are used to demonstrate the rigour of the proposed method. Copyright © 2006 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.     

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.     

Review and enhancement of 3D concrete models for large‐scale numerical simulations of concrete structures

The present paper focuses on selected plasticity and damage‐plasticity models for describing the 3D material behavior of concrete. In particular, a plasticity model and a damage‐plasticity model are reviewed and evaluated. Based on the results of the evaluation, enhancements are proposed, aiming at improving the correspondence between predicted and observed material behavior and aiming at implementing a robust and efficient stress update algorithm in a finite element program for performing large‐scale 3D numerical simulations of concrete structures. The capabilities of the concrete models are demonstrated by 3D numerical simulations of benchmark tests with combined bending and torsional loading and combined compression and shear loading and by a large‐scale 3D finite element analysis of a model test of a concrete arch dam. Copyright © 2011 John Wiley & Sons, Ltd.     

A bounding surface elasto‐viscoplastic constitutive model for non‐isothermal cyclic analysis of asphaltic materials

An elasto‐viscoplastic constitutive model for asphaltic materials is presented within the context of bounding surface plasticity theory, taking into account the effects of the stress state, void binder degree of saturation, temperature and strain rate on the material behaviour. A stress state dependent non‐linear elasticity model is introduced to represent time‐independent recoverable portion of the deformation. The consistent visco‐plasticity framework is utilised to capture the rate‐dependent, non‐recoverable strain components. The material parameters introduced in the model are identified, and their determination from conventional laboratory tests is discussed. The capability of the model to reproduce experimentally observed response of asphaltic materials is demonstrated through numerical simulations of several laboratory test data from the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

A non‐coaxial critical state soil model and its application to simple shear simulations

The yield vertex non‐coaxial theory is implemented into a critical state soil model, CASM (Int. J. Numer. Anal. Meth. Geomech. 1998; 22 :621–653) to investigate the non‐coaxial influences on the stress–strain simulations of real soil behaviour in the presence of principal stress rotations. The CASM is a unified clay and sand model, developed based on the soil critical state concept and the state parameter concept. Without loss of simplicity, it is capable of simulating the behaviour of sands and clays within a wide range of densities. The non‐coaxial CASM is employed to simulate the simple shear responses of Erksak sand and Weald clay under different densities and initial stress states. Dependence of the soil behaviour on the Lode angle and different plastic flow rules in the deviatoric plane are also considered in the study of non‐coaxial influences. All the predictions indicate that the use of the non‐coaxial model makes the orientations of the principal stress and the principal strain rate different during the early stage of shearing, and they approach the same ultimate values with an increase in loading. These ultimate orientations are dependent on the density of soils, and independent of their initial stress states. The use of the non‐coaxial model also softens the shear stress evolutions, compared with the coaxial model. It is also found that the ultimate shear strengths by using the coaxial and non‐coaxial models are dependent on the plastic flow rules in the deviatoric plane. Copyright © 2006 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.     

The role of non‐coaxiality in the simulation of strain localization based on classical and Cosserat continua

It is normally accepted that materials inside the shear band undergo severe rotation of the principal stress direction, which causes non‐coaxiality between the principal stress and principal plastic strain rate. However, classical plasticity flow theory implicitly assumes that the principal stress and the principal plastic strain rate are coaxial; thus, it may not correctly predict the onset of the shear band. In addition, classical continuum does not contain any internal length scales; as a result, it cannot provide a reasonable shear band thickness. In this study, the original vertex non‐coaxial plastic model based on the classical continuum is extended to the Cosserat continuum. The corresponding codes are implemented via the interface of the user defined element subroutine in ABAQUS. Through a simple shear test, the effectiveness of the user's codes is verified. Through a uniaxial compression test, the influence of non‐coaxiality on the onset, the orientation, and the thickness of the shear band is investigated. Results show that the onset of the shear localization is delayed, and the thickness of the shear band is widened when the non‐coaxial degree increases, while the orientation of the shear band is little affected by the non‐coaxial degree. In addition, it is found that the non‐coaxiality can weaken the micro‐polar effect to some extent; nonetheless, the Cosserat non‐coaxial model still has its advantage over the classical non‐coaxial model in capturing the pre‐bifurcation as well as the post‐bifurcation behaviors of strain localization. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermal recovery from a fractured medium in local thermal non‐equilibrium

Thermal recovery from a hot dry rock (HDR) reservoir viewed as a deformable fractured medium is investigated with a focus on the assumption of local thermal non‐equilibrium (LTNE). Hydraulic diffusion, thermal diffusion, forced convection and deformation are considered in a two‐phase framework, the solid phase being made by impermeable solid blocks separated by saturated fractures. The finite element approximation of the constitutive and field equations is formulated and applied to obtain the response of a generic HDR reservoir to circulation tests. A change of time profile of the outlet fluid temperature is observed as the fracture spacing increases, switching from a single‐step pattern to a double‐step pattern, a feature which is viewed as characteristic of established LTNE. A dimensionless number is proposed to delineate between local thermal equilibrium (LTE) and non‐equilibrium. This number embodies local physical properties of the mixture, elements of the geometry of the reservoir and the production flow rate. All the above properties being fixed, the resulting fracture spacing threshold between LTNE and LTE is found to decrease with increasing porosity or fluid velocity. The thermally induced effective stress is tensile near the injection well, illustrating the thermal contraction of the rock, while the pressure contribution of the fracture fluid is negligible during the late period. Copyright © 2012 John Wiley & Sons, Ltd.     

Experimental study and constitutive modelling of elasto‐plastic damage in heat‐treated mortar

This study investigates the effect of a heat‐treatment upon the thermo‐mechanical behaviour of a model cement‐based material, i.e. a normalized mortar, with a (w/c) ratio of 0.5. First, a whole set of varied experimental results is provided, in order to either identify or validate a thermo‐mechanical constitutive model, presented in the second paper part. Experimental responses of both hydraulic and mechanical behaviour are given after different heating/cooling cycling levels (105, 200, 300, 400^°C). The reference state, used for comparison purposes, is taken after mass stabilization at 60^°C. Typical uniaxial compression tests are provided, and original triaxial deviatoric compressive test responses are also given. Hydraulic behaviour is identified simultaneously to triaxial deviatoric compressive loading through gas permeability K_gas assessment. K_gas is well correlated with volumetric strain evolution: gas permeability increases hugely when ε_v testifies of a dilatant material behaviour, instead of contractile from the test start. Finally, the thermo‐mechanical model, based on a thermodynamics approach, is identified using the experimental results on uniaxial and triaxial deviatoric compression. It is also positively validated at residual state for triaxial deviatoric compression, but also by using a different stress path in lateral extension, which is at the origin of noticeable plasticity. Copyright © 2009 John Wiley & Sons, Ltd.     

地震作用下丰满重力坝的塑性损伤

基于混凝土塑性损伤本构模型,将可以表征材料和结构渐进破坏的损伤变量作为内变量,考虑了混凝土拉压异性的损伤变量,利用有限元软件ABAQUS,对丰满混凝土重力坝在经受地震作用下的结构损伤区域的发展变化进行安全性评价。在数值计算过程中,对于受地震荷载作用所引起的地面水平运动而导致大坝水位水平运动对坝体的影响,通过扩充后的广义Westergaard公式予以计算模拟。通过分析得到了丰满大坝在地震作用下的塑性损伤破坏情况,并评价了丰满大坝的安全性,亦说明利用损伤力学对大坝进行安全性评价是一个可行的方法。     

Coupled damage and plasticity modelling in transient dynamic analysis of concrete

In a concrete structure subjected to an explosion, for example a concrete slab, the material is subjected to various states of stress which lead to many modes of rupture. Closer to the explosive, a state of strong hydrostatic compression is observed. This state of stress produces an irreversible compaction of the material. Away from the zone of explosion, confinement decreases and the material undergoes compression with a state of stress, which is slightly triaxial. Finally, the compression wave can be reflected on a free surface and becomes a tensile wave, which by interaction with the compression wave, produces scabbing. We present, in this paper, a model aimed at describing these three failure modes. It is based on visco‐plasticity and rate dependent damage in which a homogenization method is used in order to include the variation of the material porosity due to compaction. The model predictions are compared with several experiments performed on the same concrete. Computations of split Hopkinson tests on confined concrete, a tensile test with scabbing, and an explosion on a concrete slab are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical simulations of simple shear with non‐coaxial soil models

This paper investigates the effects of a non‐coaxial model on simulated stress–strain behaviour of granular materials subject to simple shearing under various initial conditions. In most cases, a significant difference of predictions between coaxial and non‐coaxial modelling is found during the early stage in shearing. With the increase in shearing, non‐coaxial simulations approach and tend to coincide with coaxial simulations. It is also found that the roles of non‐coaxial modelling in simulating simple shear behaviour are considerably influenced by hardening rules, flow rules, initial static lateral pressure coefficients. In some cases, the non‐coaxial modelling gives a similar simulation as the coaxial modelling. In other cases, the non‐coaxial modelling decreases the hardening response or softening response of materials, compared with the coaxial modelling. Under certain conditions, the predicted peak strength of materials with non‐coaxial modelling is larger than that for coaxial modelling. Some of these observations can be attributed to the amount of principal stress rotation in various cases analysed. Others can be attributed to the difference between the directions of the non‐coaxial plastic flow and those for coaxial plastic flow. Copyright © 2005 John Wiley & Sons, Ltd.     

Preconditioned IDR(s) iterative solver for non‐symmetric linear system associated with FEM analysis of shallow foundation

Non‐associated flow rule is essential when the popular Mohr–Coulomb model is used to model nonlinear behavior of soil. The global tangent stiffness matrix in nonlinear finite element analysis becomes non‐symmetric when this non‐associated flow rule is applied. Efficient solution of this large‐scale non‐symmetric linear system is of practical importance. The standard Krylov solver for a non‐symmetric solver is Bi‐CGSTAB. The Induced Dimension Reduction [IDR(s)] solver was proposed in the scientific computing literature relatively recently. Numerical studies of a drained strip footing problem on homogenous soil layer show that IDR(s = 6) is more efficient than Bi‐CGSTAB when the preconditioner is the incomplete factorization with zero fill‐in of global stiffness matrix K_ep (ILU(0)‐K_ep). Iteration time is reduced by 40% by using IDR(s = 6) with ILU(0)‐K_ep. To further reduce computational cost, the global stiffness matrix K_ep is divided into two parts. The first part is the linear elastic stiffness matrix K_e, which is formed only once at the beginning of solution step. The second part is a low‐rank matrix Δ, which is re‐formed at each Newton–Raphson iteration. Numerical studies show that IDR(s = 6) with this ILU(0)‐K_e preconditioner is more time effective than IDR(s = 6) with ILU(0)‐K_ep when the percentage of yielded Gauss points in the mesh is less than 15%. The total computation time is reduced by 60% when all the recommended optimizing methods are used. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of undrained cavity expansion in elasto‐plastic soils with non‐linear elasticity

A large strain analysis of undrained expansion of a spherical/cylindrical cavity in a soil modelled as non‐linear elastic modified Cam clay material is presented. The stress–strain response of the soil is assumed to obey non‐linear elasticity until yielding. A power‐law characteristic or a hyperbolic stress–strain curve is used to describe the gradual reduction of soil stiffness with shear strain. It is assumed that, after yielding, the elasto‐plastic behaviour of the soil can be described by the modified Cam clay model. Based on a closed‐form stress–strain response in undrained condition, a numerical solution is obtained with the aid of simple numerical integration technique. The results show that the stresses and the pore pressure in the soil around an expanded cavity are significantly affected by the non‐linear elasticity, especially if the soil is overconsolidated. The difference between large strain and small strain solutions in the elastic zone is not significant. The stresses and the pore pressure at the cavity wall can be expressed as an approximate closed‐form solution. Copyright © 2001 John Wiley & Sons, Ltd.     

Integration of a continuum damage model for shale with the cutting plane algorithm

The stability of integration is essential to numerical simulations especially when solving nonlinear problems. In this work, a continuum damage mechanics model proposed by the first author is implemented with an integration method named cutting plane algorithm (CPA) to improve the robustness of the simulation. This integration method is one type of return mapping algorithm that bypasses the need for computing the gradients. We compare the current integration method with the previous direct method, and the result shows that the cutting plane algorithm exhibits excellent performance under large loading rate conditions. To enhance accuracy of the new method, a control procedure is utilized in the implementation of the algorithm based on error analysis. Thereafter, the theory of poromechanics is utilized with the damage model to account for the effects of fluid diffusion. Laboratory tests simulated with finite element method illustrate distinct behaviors of shale with different loading rates and indicate the development of microcrack propagation under triaxial compression. Copyright © 2016 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.     

Elasto‐plastic model for cement‐treated sand

This paper presents an elasto‐plastic model for non‐linear analyses of cement‐treated sand. Various laboratory tests were systematically carried out to investigate the pre‐peak and post‐peak behaviours of a cement‐treated sand. On the basis of these experimental results, the new model was built within the framework of a relatively simple elasto‐plastic theory. Two failure criteria are employed to express tensile and shear failure characteristics observed in the experimental results of the cement‐treated sand. The proposed model can describe strain‐hardening and strain‐softening responses under both failure modes. In the strain‐softening rules, the smeared crack concept is used, and a characteristic length is considered to avoid the issue of mesh‐size dependency. Since the failure criterion and strain‐hardening/softening rules are based on the experimental evidences, the model is relatively easy to understand and the parameters used in the model have clear physical meaning. The proposed model was applied to simulate the behaviour of cement‐treated sand in various laboratory tests, allowing for a reasonable comprehensive evaluation. It was demonstrated that the proposed model is suitable for describing both the tensile and shear failure behaviours of cement‐treated sand. Copyright © 2006 John Wiley & Sons, Ltd.     

A non‐coaxial constitutive model for sand deformation under rotation of principal stress axes

A constitutive model for the simulation of non‐coaxiality, an aspect of anisotropic behavior of sand subjected to the rotation of the principal stress axes, is presented in this paper. Experimental studies have shown that non‐coaxiality or non‐coincidence of principal plastic strain increments with principal stress axes under loadings involving the rotation of principal stress axes may be considerable. Besides, the rotation of the principal stress axes results in dramatic effects on stiffness and dilatant behavior of sand. Therefore, the consequences of principal stress axes rotation on deformational behavior, dilatancy and soil stiffness must be taken into account in theoretical and practical problems. To this aim, the following steps are taken: (1) A general relationship for flow direction with respect to possibility of non‐coaxial flow is developed. Moreover, special circumstances linking non‐coaxiality to instantaneous interaction between loading and soil fabric are proposed. (2) Proposing novel expressions for plastic modulus and dilatancy function, the model is enforced to provide realistic simulations when sand is subjected to the rotation of the principal stress axes. Finally, with numerous examples and comparisons, the model capabilities are shown under various stress paths and drainage conditions. Copyright © 2007 John Wiley & Sons, Ltd.