A numerical method for the study of shear band propagation in soft rocks

This paper investigates the possibility of interpreting progressive shear failure in hard soils and soft rocks as the result of shear propagation of a pre‐existing natural defect. This is done through the application of the principles of fracture mechanics, a slip‐weakening model (SWM) being used to simulate the non‐linear zone at the tips of the discontinuity. A numerical implementation of the SWM in a computation method based on the boundary element technique of the displacement discontinuity method (DDM) is presented. The crack and the non‐linear zone at the advancing tip are represented through a set of elements, where the displacement discontinuity (DD) in the tangential direction is determined on the basis of a friction law. A residual friction angle is assumed on the crack elements. Shear resistance decreases on elements in the non‐linear zone from a peak value at the tip, which is characteristic of intact material, to the residual value. The simulation of a uniaxial compressive test in plane strain conditions is carried out to exemplify the numerical methodology. The results emphasize the role played by the critical DD on the mechanical behaviour of the specimen. A validation of the model is shown through the back analysis of some experimental observations. The results of this back analysis show that a non‐linear fracture mechanics approach seems very promising to simulate experimental results, in particular with regards to the shear band evolution pattern. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical modelling of concrete flow: homogeneous approach

The aim of this paper is to model numerically concrete flow inside formworks like the Lbox. For this purpose, we use a finite element method with Lagrangian integration points (FEMLIP). We are able to follow in time and space material motion with any type of material behaviour, including non‐linear and time‐dependent ones. We also can deal with free surfaces or material interfaces. Bingham's rheology is used for fresh concrete behaviour. In order to compare with experiments, we have considered three concretes (OC, HPC and SCC) with contrasted rheologies. Their yield stress is identified by experimental slump tests and also compared with the value given by a formulation concrete software. Experimental data are found to be quite close to numerical predictions. We have also made some experimental flow tests in a LBOX. We measured the flow speed and the flow shape in the final stage. The numerical modelling of these experiments is very encouraging and shows the capability of the FEMLIP using the Bingham's law to model concrete flow and filling properties. Copyright © 2005 John Wiley & Sons, Ltd.     

Modelling crack propagation in concrete structures with a two scale approach

A simplified computational technique based on a refined global–local method is applied to the failure analysis of concrete structures. The technique distinguishes the scale of the structure, modelled with large size finite elements, from the scale at which material non‐linearity occurs due to progressive cracking and macro‐crack propagation. The finite element solution is split into two parts: a linear elastic analysis on a coarse mesh over the entire structure and a non‐linear analysis over a small part of the structure where a dense finite element grid is employed. In the non‐linear calculation, a non‐local damage model is implemented. These two computations are coupled with the help of an iterative scheme. The size and location of the region where a non‐linear analysis is performed, is adapted to follow the development of the damage zone. Numerical examples of mode I fracture of concrete specimens with straight and curved cracks are presented. Copyright © 2003 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.     

Crack propagation in sandstone: Combined experimental and numerical approach

Summary A combined experimental and numerical approach is adopted to investigate crack propagation in sandstone. Experiments on two types of sandstones show a simular behaviour as found in tests on concrete specimens. The heterogeneity of the material in combination with the stress situation, as a result of the applied load, governs the direction of crack propagation. Cracks that develop are not continuous, but overlaps exist mainly around the grain particles in the material. A simple lattice model, in which the material is schematized as a network of small beams, is adopted to simulate the experiments. Using the simulations carried out with the lattice model, the control parameter for stable displacement controlled four-point-shear tests was determined. The crack patterns obtained with the model are in good agreement with the experimental observations. However further study is needed to predict the load-displacement response correctly.     

Numerical prediction of blast‐induced stress wave from large‐scale underground explosion

This paper presents a numerical model for predicting the dynamic response of rock mass subjected to large‐scale underground explosion. The model is calibrated against data obtained from large‐scale field tests. The Hugoniot equation of state for rock mass is adopted to calculate the pressure as a function of mass density. A piecewise linear Drucker–Prager strength criterion including the strain rate effect is employed to model the rock mass behaviour subjected to blast loading. A double scalar damage model accounting for both the compression and tension damage is introduced to simulate the damage zone around the charge chamber caused by blast loading. The model is incorporated into Autodyn3D through its user subroutines. The numerical model is then used to predict the dynamic response of rock mass, in terms of the peak particle velocity (PPV) and peak particle acceleration (PPA) attenuation laws, the damage zone, the particle velocity time histories and their frequency contents for large‐scale underground explosion tests. The computed results are found in good agreement with the field measured data; hence, the proposed model is proven to be adequate for simulating the dynamic response of rock mass subjected to large‐scale underground explosion. Extended numerical analyses indicate that, apart from the charge loading density, the stress wave intensity is also affected, but to a lesser extent, by the charge weight and the charge chamber geometry for large‐scale underground explosions. Copyright © 2004 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.     

A micromechanical finite element model for linear and damage‐coupled viscoelastic behaviour of asphalt mixture

This study presents a finite element (FE) micromechanical modelling approach for the simulation of linear and damage‐coupled viscoelastic behaviour of asphalt mixture. Asphalt mixture is a composite material of graded aggregates bound with mastic (asphalt and fine aggregates). The microstructural model of asphalt mixture incorporates an equivalent lattice network structure whereby intergranular load transfer is simulated through an effective asphalt mastic zone. The finite element model integrates the ABAQUS user material subroutine with continuum elements for the effective asphalt mastic and rigid body elements for each aggregate. A unified approach is proposed using Schapery non‐linear viscoelastic model for the rate‐independent and rate‐dependent damage behaviour. A finite element incremental algorithm with a recursive relationship for three‐dimensional (3D) linear and damage‐coupled viscoelastic behaviour is developed. This algorithm is used in a 3D user‐defined material model for the asphalt mastic to predict global linear and damage‐coupled viscoelastic behaviour of asphalt mixture. For linear viscoelastic study, the creep stiffnesses of mastic and asphalt mixture at different temperatures are measured in laboratory. A regression‐fitting method is employed to calibrate generalized Maxwell models with Prony series and generate master stiffness curves for mastic and asphalt mixture. A computational model is developed with image analysis of sectioned surface of a test specimen. The viscoelastic prediction of mixture creep stiffness with the calibrated mastic material parameters is compared with mixture master stiffness curve over a reduced time period. In regard to damage‐coupled viscoelastic behaviour, cyclic loading responses of linear and rate‐independent damage‐coupled viscoelastic materials are compared. Effects of particular microstructure parameters on the rate‐independent damage‐coupled viscoelastic behaviour are also investigated with finite element simulations of asphalt numerical samples. Further study describes loading rate effects on the asphalt viscoelastic properties and rate‐dependent damage behaviour. Copyright © 2006 John Wiley & Sons, Ltd.     

A visco‐damage model for brittle materials under monotonic and sustained stresses

A theoretical model is proposed to describe the evolution of damage in brittle materials, such as concrete and masonry, subjected to increasing or sustained stresses of relatively high intensity. The model is based on the introduction of suitable damage variables in a rheological model. In this way, it is possible to describe the material behaviour under stresses either increasing or constant in time. The capabilities of the model in describing the mechanical response of material elements subjected to different stress histories are illustrated. Some correlations with experimental data from tests performed on masonry and concrete specimens are presented, to assess the reliability of the theoretical predictions. The results of some numerical applications to non‐proportional stress paths are also illustrated. Finally, the limitations of the proposal are discussed and possible further improvements are envisaged. Copyright © 2005 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.     

Some Effects of Shearing Velocity on the Shear Stress-Deformation Behaviour of Hard–Soft Artificial Material Interfaces

Shear behaviour of the joints formed by the interface of two different material types, such as rock and cemented paste backfill, rock and concrete or two different rock types, have practical importance in many rock engineering activities. This paper presents the results of an experimental investigation into the shear behaviour of these special joints under pseudo-static shear velocity. Direct shear tests on concrete–plaster interfaces were carried out under boundary conditions of constant normal load and constant normal stiffness. Shearing velocities of the performed tests were in the range of 0.3–30 mm/min. The results of the shear tests conducted on the planar and rough artificial prepared joints showed that the shearing velocity has a significant influence on the shear strength, friction angle and shear stiffness of the hard–soft material interface. So that, these parameters were decreased when shear velocity was increased. Also, comparison of the tests results that performed on the concrete–plaster joints with those from tests on the plaster–plaster and concrete–concrete interfaces showed that the shear behaviour of concrete–plaster interface is governed mainly by the shear parameters of the plaster block (namely softer material).     

基于剪切梁层间失效模型的断层冲击地压分析

将剪切梁层间失效模型用于分析断层冲击地压的发生,在不考虑断层的弹性时对断层冲击地压进行了解析分析,得到了剪切梁的位移和应力分布规律和断层冲击地压发生时撕开裂纹长度和临界载荷。分析了撕开裂纹长度和临界载荷与各种力学参数之间的关系。研究结果表明：当采深越大,剪切梁的剪切弹性模量越小,剪切梁厚度越厚,断层带剪切软化模量越大;断层带厚度、滑动摩擦系数、剪应力峰值强度越小时,断层冲击地压容易发生。采深、剪切梁的剪切弹性模量、断层带剪切软化模量、剪应力峰值强度、断层带厚度这些参数对断层冲击地压的发生影响较大。其结果为预测、防治断层冲击地压提供了理论依据。     

Modelling the time‐dependent behaviour of granular material with hypoplasticity

This paper presents a constitutive model for time‐dependent behaviour of granular material. The model consists of 2 parts representing the inviscid and viscous behaviour of granular materials. The inviscid part is a rate‐independent hypoplastic constitutive model. The viscous part is represented by a rheological model, which contains a high‐order term denoting the strain acceleration. The proposed model is validated by simulating some element tests on granular soils. Our model is able to model not only the non‐isotach behaviour but also the 3 creep stages, namely, primary, secondary, and tertiary creep, in a unified way.     

A numerical study of the influence from pre‐existing cracks on granite rock fragmentation at percussive drilling

The aim of this study is to investigate the effect of pre‐existing, or structural, cracks on dynamic fragmentation of granite. Because of the complex behavior of rock materials, a continuum approach is employed relying upon a plasticity model with yield surface locus as a quadratic function of the mean pressure in the principal stress space coupled with an anisotropic damage model. In particular, Bohus granite rock is investigated, and the material parameters are chosen based on previous experiments. The equation of motion is discretized using a finite element approach, and the explicit time integration method is employed. The pre‐existing cracks are introduced in the model by considering sets of elements with negligible tensile strength that leads to their immediate failure when loaded in tension even though they still carry compressive loads as crack closure occurs because of compressive stresses. Previously performed edge‐on impact tests are reconsidered here to validate the numerical model. Percussive drilling is simulated, and the influence of the presence of pre‐existing cracks is studied. The results from the analysis with different crack lengths and orientations are compared in terms of penetration stiffness and fracture pattern. It is shown that pre‐existing cracks in all investigated cases facilitate the drilling process. Copyright © 2014 John Wiley & Sons, Ltd.     

水力裂缝穿越非连续面扩展时的断裂过程研究

非连续面发育是非常规油气储层的显著地质特征之一,水力裂缝能否穿越非连续面扩展会关系到压裂的改造效果。为研究水力裂缝穿越非连续面扩展时断裂过程区（fracture process zone,简称FPZ）发育特征,采用自主设计的可视化压裂试验装置对含预制摩擦界面的砂岩平板试件开展水力压裂试验。基于数字图像相关法实时监测了水力裂缝正交穿越界面扩展过程中的位移及应变场特征。试验结果表明,水力裂缝穿越界面扩展之前,断裂过程区已经开始跨越界面发育;裂缝能否穿越界面扩展在FPZ的初始发育阶段已经注定,不受FPZ内应力软化过程影响。基于Renshaw-Pollard准则建立了考虑FPZ边界范围的裂缝穿越非连续面扩展准则,并通过前人及文中试验数据进行了可靠性验证。相比而言,改进准则更准确地考虑了裂缝前端线弹性断裂力学的适用范围。研究发现FPZ长宽比对裂缝穿越界面扩展准则有显著影响,相同条件下,FPZ长宽比越大,裂缝正交穿越界面扩展所需要的摩擦系数下限值越小。     

Propagation of a hydraulic fracture parallel to a free surface

This paper analyses the plane strain problem of a fracture, driven by injection of an incompressible viscous Newtonian fluid, which propagates parallel to the free surface of an elastic half‐plane. The problem is governed by a hyper‐singular integral equation, which relates crack opening to net pressure according to elasticity, and by the lubrication equations which describe the laminar fluid flow inside the fracture. The challenge in solving this problem results from the changing nature of the elasticity operator with growth of the fracture, and from the existence of a lag zone of a priori unknown length between the crack tip and the fluid front. Scaling of the governing equations indicates that the evolution problem depends in general on two numbers, one which can be interpreted as a dimensionless toughness and the other as a dimensionless confining stress. The numerical method adopted to solve this non‐linear evolution problem combines the displacement discontinuity method and a finite difference scheme on a fixed grid, together with a technique to track both crack and fluid fronts. It is shown that the solution evolves in time between two asymptotic similarity solutions. The small time asymptotic solution corresponding to the solution of a hydraulic fracture in an infinite medium under zero confining stress, and the large time to a solution where the aperture of the fracture is similar to the transverse deflection of a beam clamped at both ends and subjected to a uniformly distributed load. It is shown that the size of the lag decreases (to eventually vanish) with increasing toughness and compressive confining stress. Copyright © 2005 John Wiley & Sons, Ltd.     

Influence of size on the constitutive equations of concrete or rock dowels

The numerical fracture analysis of non‐homogeneous rock or concrete dowels subjected to shear and compression is described in detail. The method of analysis allows the consideration of scale and rate effects due to material non‐homogeneity and fracture. The proposed approach is verified by comparing numerical predictions with experimental results reported in the literature for a series of small rock samples, since experimental evidence for large bodies is not yet available (2007). Results generated by Monte Carlo simulation using the so‐called discrete element method to model the dowels suggest that a simple three parameters law can be used to predict the relationship between tangential stress at the base and lateral distortion. It is observed that the larger the size of the cubes, the smaller both the peak tangential stress and the rupture distortion. Size effects are also evaluated in samples with vertical restraint. The influence of loading rate is likewise numerically assessed for two sample sizes. The effect is compatible with experimental evidence available for concrete using small samples. Copyright © 2008 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.     

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.     

Numerical determination of the tensile response and the dissipated fracture energy of concrete: role of the mesostructure and influence of the loading rate

At the mesoscopic scale, concrete can be considered as a mix of coarse aggregates with a mortar paste matrix. In this paper, we investigate numerically the influence of aggregates arrangements and loading rate on the tensile response of concrete. Each coarse aggregate is assumed to be circular with six different radiuses following the aggregates size distribution of real gravel. Rate‐independent cohesive elements are used to model failure within the mesostructure. Our results show that the spatial distribution of heterogeneities does not influence the peak strength, while it changes the post‐peak macroscopic response. This implies that our specimen size is large enough for strength computation but that larger mesostructures should be considered to obtain fully reliable toughness predictions. Although the cohesive approach is able to capture the transition from one macro‐crack in quasi‐static to multiple micro‐cracks in fast dynamics, which increases the dissipated fracture energy, our results suggest that the full extent of the high‐rate strengthening of concrete observed experimentally for loading rates greater than 1/s cannot be captured with rate independent constitutive laws.Copyright © 2013 John Wiley & Sons, Ltd.