Effect of inherent microcrack populations on rock tensile fracture behaviour: numerical study based on embedded discontinuity finite elements

Inherent microcrack populations have a significant effect on the fracture behaviour of natural rocks. The present study addresses this topic in numerical simulations of uniaxial tension and three-point bending tests. For this end, a rock fracture model based on multiple intersecting embedded discontinuity finite elements is developed. The inherent (pre-existing) microcrack populations are represented by pre-embedded randomly oriented discontinuity populations. Crack shielding (through spurious locking) is prevented by allowing a new crack to be introduced, upon violation of the Rankine criterion, in an element with an initial crack unfavourably oriented to the loading direction. Rock heterogeneity is accounted for by random clusters of triangular finite elements representing different minerals of granitic numerical rock. Numerical simulations demonstrate the strength lowering effect of initial microcrack populations. This effect is substantially stronger under uniaxial tension, due to the uniform stress state, than in semicircular three-point bending having a non-uniform stress state with a clear local maximum of tensile stress.

     

Characteristics of the unloading process of rocks under high initial stress

The unloading process of rocks under high initial stress is complex, and verifying the mechanism of the unloading process in the field or in a laboratory is not straightforward. In this study, the unloading process of rocks under high initial stress was characterised by a mathematical physics model, which was then implemented in the finite element program LS-DYNA for analysis. In particular, the implicit and explicit methods were performed in sequence in the finite element simulation of rocks with initial stresses. In the numerical simulation, the characteristics of the dynamic unloading process of rocks were investigated for various peak initial stresses, initial stress release paths and initial stress release rates (ISRRs). The numerical results indicated that the rock failure could be induced by the release of the initial stress; furthermore, there is a relationship between the magnitude of the unloading failure and the peak initial stresses, the initial stress release paths and the ISRRs. When the initial stresses were at the same level, the equivalent initial stress release rate (EISRR) was introduced to quantitatively describe the characteristics of the unloading process. Using the numerical results, the unloading failure process was characterised, and a method for the static stress initialisation-dynamic unloading of rock was developed.     

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.     

岩石裂纹损伤及围压对剪切破坏影响研究

岩石内部细观裂纹的存在,对压缩作用下岩石剪切断裂的宏观现象有着重要的影响。然而,能够通过解析解阐释细观裂纹几何特性、围压等影响因素对压缩作用下剪切断裂面角度变化趋势的研究很少。基于Ashby模型中提出的裂纹尖端应力强度因子,提出了一种改进的考虑裂纹角度影响的应力强度因子表达式。利用该改进的应力强度因子表达式,推出了一个可以预测岩石峰值强度的裂纹扩展、应变与应力之间的本构关系。结合本构关系的峰值强度与摩尔-库仑失效准则,得到了岩石损伤与内摩擦角、黏聚力、剪切强度及失效断裂面角度之间的理论关系;讨论了围压、裂纹尺寸、角度及摩擦系数对岩石宏观剪切断裂面角度的影响,通过试验结果验证了模型合理性。结果表明：随着损伤增大,内摩擦角、黏聚力及剪切强度不断减小;随着围压增大、摩擦系数增大和初始裂纹尺寸减小,剪切断裂面角度不断增大;随着裂纹角度增大,剪切断裂纹面角度先减小后增大。     

卸围压下砂岩力学特性及细观机制颗粒流分析

鉴于试验砂岩内摩擦角较大及峰后脆性强的特点,选用颗粒流程序PFC2D中的cluster单元进行模拟,在常规三轴压缩试验校正的基础上得到一组能够真实反映砂岩宏观力学行为的细观参数,模拟了砂岩两种卸围压路径,结果表明其峰值强度随初始轴向应力的增大而不断增大,其规律与试验相同。选取了两种卸围压路径微裂纹数目随应力-应变的演化趋势进行分析,结果表明卸围压对试样造成的损伤较加轴压大,同时初始轴向应力增大一定程度上增加了试样承受破坏的能力。通过对不同应力路径裂纹两侧位移场的分析可知卸围压会造成颗粒在横向产生位移不连续,导致试样破坏。     

Numerical modelling of heat shock-assisted rock fracture

This paper presents a numerical investigation on the effects of thermal shock as a pretreatment of rock prior to comminution. More specifically, the effect of heat shock-induced cracks on the uniaxial compressive strength of rock is numerically studied. The chosen constitutive model of rock employs a (strong) embedded discontinuity finite element formulation to describe cracks. The thermomechanical problem that governs the heat shock pretreatment of rocks is considered as an uncoupled problem because of a highly dominating role of the external heat influx. Two solution methods of the global problem are presented: an explicit-explicit dynamic scheme and an implicit-implicit quasi-static scheme. The model performance is tested in simulations on heterogeneous numerical rock samples subjected first to a heat shock pretreatment and then to a mechanical compression test. According to the results, the compressive strength of intact granite rock having the axial splitting failure mode can be substantially reduced by heat shock pretreatment.     

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.     

基于细观力学脆性岩石剪切特性演化模型研究

压缩作用下岩石内部细观裂纹扩展导致岩石产生损伤,其对岩石变形、强度等力学特性有着重要影响;然而,岩石内部裂纹扩展与剪切特性（黏聚力、内摩擦角及剪切应力）动态演化关系很少被研究。基于裂纹扩展机制推出的岩石应力-应变本构模型,并结合摩尔-库仑失效准则,推出了在岩石应力-应变关系峰值应力（对应岩石压缩强度）状态时,本构模型细观力学参数与岩石黏聚力、内摩擦角及剪切强度之间的状态关系。然后,引入岩石应力-应变本构关系塑性变形阶段服从摩尔-库仑屈服准则的力学流动规律,进而将已推出的应力-应变关系峰值状态点所满足的细观力学参数与黏聚力、内摩擦角关系,推广到岩石进入塑性变形后,岩石内部裂纹扩展（或应变）与黏聚力、内摩擦角及剪切应力动态演化的理论关系。随着裂纹扩展或应变增加,黏聚力、内摩擦角及剪切应力先增大,达到一个峰值点后减小,该结果与应力-应变本构曲线变化趋势相对应。通过试验结果验证了所提出理论结果的合理性。并讨论了初始裂纹之间摩擦系数对黏聚力、内摩擦角及剪切应力随裂纹扩展或应变演化规律的影响。     

Finite element modelling of frictional instability between deformable rocks

Earthquakes are recognized as resulting from a stick–slip frictional instability along faults. Based on the node‐to‐point contact element strategy (an arbitrarily shaped contact element strategy applied with the static‐explicit algorithm for modelling non‐linear frictional contact problems proposed by authors), a finite element code for modelling the 3‐D non‐linear friction contact between deformable bodies has been developed and extended here to analyse the non‐linear stick–slip frictional instability between deformable rocks with a rate‐ and state‐dependent friction law. A typical fault bend model is taken as an application example to be analysed here. The variations of the normal contact force, the frictional force, the transition of stick–slip instable state and the related relative slip velocity along the fault between the deformable rocks and the stress evolution in the total bodies during the different stages are investigated, respectively. The calculated results demonstrate the usefulness of this code for simulating the non‐linear frictional instability between deformable rocks. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

基于单元劈裂法的岩体黏结型结构面数值模拟

节理岩体中富含结构面等不连续体,其中很多结构面之间具有一定黏结强度,在冲击荷载作用下会发生脱结（debond）行为,从而消耗岩体应变能。在该结构面之间的黏结性质在很大程度上决定着岩体的破坏行为,发生脱结后,结构面之间不再具有黏结强度,岩体的力学性能发生弱化。为了对该类结构面进行模拟,在改进的Xu-Needleman势函数基础上推导出结构面单元,并将其嵌入到单元劈裂法中,模拟结构面的开裂过程,当结构面完全脱结后结构面单元就转化为一般节理单元。相应地,在数值实现过程中只是将结构面单元替换为一般节理单元即可。该结构面单元与单元劈裂法相结合,能够有效地模拟节理岩体的破坏过程。     

Embedded beam element with interaction surface for lateral loading of piles

This paper presents a numerical formulation of a three dimensional embedded beam element for the modeling of piles, which incorporates an explicit interaction surface between soil and pile. The formulation is herein implemented for lateral loading of piles but is able to represent soil–pile interaction phenomena in a general manner for different types of loading conditions or ground movements. The model assumes perfect adherence between beam and soil along the interaction surface. The paper presents a comparison of the results obtained by means of the present formulation and by means of a previously formulated embedded pile element without interaction surface, as well as reference semi‐analytical solutions and a fully 3D finite element (FE) model. It is seen that the proposed embedded element provides a better convergence behavior than a previously formulated embedded element and is able to reproduce key features of a full 3D FE model. Copyright © 2015 John Wiley & Sons, Ltd.     

Explicit integration of a porosity‐dependent hydro‐mechanical model for unsaturated soils

An adaptive substepping explicit integration scheme is developed for a porosity‐dependent hydro‐mechanical model for unsaturated soils. The model is referred to as the modified σ –Θ model in this paper, which features the employment of the subloading surface plasticity and the stress–saturation approach. On numerical aspects, convex/nonconvex subloading surfaces in the σ –Θ space may result in incorrect loading–unloading decisions during the integration. A new loading–unloading decision method is developed here to solve the problem and then embedded into the explicit integration scheme for the modified σ –Θ model. In addition, to enhance the accuracy of the explicit integration, local errors from both hydraulic and mechanical components are included in the error control for each substep. A drift correction method is also developed to ensure the state point lies on the subloading surface in the σ –Θ space within a set error level. The performance of the loading–unloading decision method for the modified σ –Θ model is discussed through comparing it with the conventional loading–unloading decision method. The importance of involving the hydraulic component in the error control is also demonstrated. The accuracy and efficiency of the proposed adaptive substepping explicit integration scheme for the modified p–Θ model are also studied via several numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermal shock weakening of granite rock under dynamic loading: 3D numerical modeling based on embedded discontinuity finite elements

This paper presents a numerical study of thermal shock weakening of granite rock under dynamic loading. A fully 3D numerical scheme based on a combined continuum viscodamage-embedded discontinuity model and an explicit scheme to solve the underlying thermomechanical problem was developed and validated through numerical examples. First, the dynamic Brazilian disc test is simulated on intact numerical rock. Then, thermal shock-induced cracking due to a moving external heat flux boundary condition, mimicking experiments based on plasma jet treatment, is numerically predicted. Finally, numerical Brazilian disc test is conducted on the thermal shocked numerical samples. The predicted and experimental weakening effects are in good agreement demonstrating that the present modeling approach has good predictive capabilities. The practical significance of the results is that heat shock pretreatment can substantially enhance rock gravel and rubble crushing.     

Paleoproterozoic Granitoids on Liaodong Peninsula,North China Craton

Paleoproterozoic granitoids are an important constituent of the Jiao–Liao–Ji Belt(JLJB). The spatial-temporal distribution and types of Paleoproterozoic granitoids are closely related to the evolution of the JLJB. In this paper, we review the field occurrence, petrography, geochronology, and geochemistry of Paleoproterozoic granitoids on Liaodong Peninsula, northeast China. The Paleoproterozoic granitoids can be divided into pre-tectonic(～2.15 Ga; peak age=2.18 Ga) and post-tectonic(～1.85 Ga) granitoids. The pre-tectonic granitoids are magnetite and hornblende–biotite monzogranites and granodiorites. Pre-tectonic monzogranites are widespread in the JLJB and have A_2-type affinities. In contrast, pretectonic granodiorites are only present in the Simenzi area and have adakitic affinities. The post-tectonic granitoids consist of porphyritic monzogranite, syenite, diorite, granodiorite, quartz monzonite, monzogranite, and granitic pegmatite, which are adakitic rocks and I-, S-, and A_2-type granitoids. The assemblage of pre-tectonic A_2-type granitoids and adakitic rocks indicates the initial tectonic setting of the JLJB was a continental back-arc basin. The assemblage of post-tectonic adakitic rocks and I-, S-, and A_2-type granitoids indicates a post-collisional setting. The 2.20–2.15 Ga A_2-type granitoids and adakitic rocks were associated with the initial stage of back-arc extension, and the peak of back-arc extension is inferred from the subsequent(2.15–2.10 Ga) mafic intrusive activity. The ～1.90 Ga adakitic rocks mark the beginning of the postcollisional stage, which was followed by the intrusion of low-temperature S-and I-type granitoids. High-to low-pressure granitoids(S-type) were generated during the peak of post-collisional lithospheric delamination and asthenospheric upwelling. The emplacement of later granitic pegmatites occurred during the waning of the orogeny.     

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.     

Embedded discontinuity approach for coupled hydromechanical analysis of fractured porous media

In this paper, a new continuum approach for the coupled hydromechanical analysis of fractured porous media is proposed. The methodology for describing the hydraulic characteristics invokes an enriched form of Darcy's law formulated in the presence of an embedded discontinuity. The constitutive relations governing the hydromechanical response are derived by averaging the fluid pressure gradient and the discontinuous displacement fields over a selected referential volume of the material, subject to some physical constraints. The framework incorporates an internal length scale which is explicitly embedded in the definition of gradient operators. The respective field equations are derived following the general form of balance equations in interacting continua. The conventional finite element method is then employed for the spatial discretization, and the generalized Newmark scheme is used for the temporal discretization. The proposed methodology is verified by some numerical examples dealing with a steady-state flow through fractured media as well as a time-dependent consolidation in the presence of a discontinuity.     

Subduction and exhumation mechanisms of ultra‐high and high‐pressure oceanic and continental crust at Makbal (Tianshan,Kazakhstan and Kyrgyzstan)

The Makbal Complex in the northern Tianshan of Kazakhstan and Kyrgyzstan consists of metasedimentary rocks, which host high‐P (HP) mafic blocks and ultra‐HP Grt‐Cld‐Tlc schists (UHP as indicated by coesite relicts in garnet). Whole rock major and trace element signatures of the Grt‐Cld‐Tlc schist suggest a metasomatized protolith from either hydrothermally altered oceanic crust in a back‐arc basin or arc‐related volcaniclastics. Peak metamorphic conditions of the Grt‐Cld‐Tlc schist reached ~580 °C and 2.85 GPa corresponding to a maximum burial depth of ~95 km. A Sm‐Nd garnet age of 475 ± 4 Ma is interpreted as an average growth age of garnet during prograde‐to‐peak metamorphism; the low initial εΝd value of ?11 indicates a protolith with an ancient crustal component. The petrological evidence for deep subduction of oceanic crust poses questions with respect to an effective exhumation mechanism. Field relationships and the metamorphic evolution of other HP mafic oceanic rocks embedded in continentally derived metasedimentary rocks at the central Makbal Complex suggest that fragments of oceanic crust and clastic sedimentary rocks were exhumed from different depths in a subduction channel during ongoing subduction and are now exposed as a tectonic mélange. Furthermore, channel flow cannot only explain a tectonic mélange consisting of various rock types with different subduction histories as present at the central Makbal Complex, but also the presence of a structural ‘dome’ with UHP rocks in the core (central Makbal) surrounded by lower pressure nappes (including mafic dykes in continental crust) and voluminous metasedimentary rocks, mainly derived from the accretionary wedge.     

Failure pattern of brittle rock governed by initial microcrack characteristics

Fracturing processes of rock are simulated using a micro- and fracture mechanical-based numerical modeling approach. The numerical model considers material heterogeneity and initial microcrack distributions. The gradual formation of macroscopic fractures by coalesced microcracks is successfully reproduced. Distinct failure modes are observed in the model under different loadings. Agreement is shown between the numerical results and laboratory observations. The influence of microcrack orientations on the fracture patterns is quantified by the numerical models. Possible mechanisms describing the splitting failure of hard brittle rocks under uniaxial compression are proposed and discussed.