水力劈裂问题的态型近场动力学建模

将态型近场动力学理论引入水力劈裂问题的模拟。构建了能反映岩土类材料准脆性断裂特征的态型近场动力学本构模型,并在物质点间相互作用力模型中加入等效水压力项,以实现在新生裂纹面上跟踪施加水压力。同时,考虑裂纹面间的接触,引入物质点间的短程排斥力作用,并设计了相应的接触算法。通过自编程序将模型和算法应用于含初始裂纹、不含初始裂纹以及含坝基软弱结构面的混凝土重力坝在高水头作用下的水力劈裂过程模拟,并与扩展有限元等模拟结果对比,验证了本文模型和算法的可行性和准确性。     

SBFEM与非局部宏微观损伤模型相结合的准脆性材料开裂模拟

混凝土是一种被广泛应用于土木和水利工程中的准脆性材料, 在各种内外部因素的作用下, 开裂是混凝土结构最为普遍的破坏形式, 准确模拟结构的开裂过程, 对于结构的安全评估至关重要. 将比例边界有限元与非局部宏微观损伤模型相结合提出一种准脆性材料开裂模拟新方法. 以比例边界有限元子域的比例中心作为物质点, 通过两比例中心(物质点对)之间的物质键的正伸长率来定义微细观损伤, 将某点影响域内物质键的微细观损伤加权平均得到该点的宏观拓扑损伤. 再引入能量退化函数, 将宏观拓扑损伤嵌入到比例边界有限元的基本框架中. 充分利用比例边界有限元网格允许存在悬挂节点的优势, 采用四叉树网格离散技术进行快速、高质量的多级网格划分与过渡. 通过一个I型开裂与一个混合型开裂的两个典型算例, 验证了该方法可捕获结构裂纹扩展路径与荷载变形曲线. 与现有的方法相比, 本文的损伤模型可得到更准确的局部开裂损伤带, 结果更为合理, 且具有更高的计算精度和计算效率. 当损伤过程区网格尺寸小于影响域半径的1/5时, 计算结果不存在网格敏感性问题.      

Stress-based nonlocal damage model

Progressive microcracking in brittle or quasi-brittle materials, as described by damage models, presents a softening behavior that in turn requires the use of regularization methods in order to maintain objective results. Such regularization methods, which describe interactions between points, provide some general properties (including objectivity and the non-alteration of a uniform field) as well as drawbacks (damage initiation, free boundary).A modification of the nonlocal integral regularization method that takes the stress state into account is proposed in this contribution. The orientation and intensity of nonlocal interactions are modified in accordance with the stress state. The fundamental framework of the original nonlocal method has been retained, making it possible to maintain the method’s advantages. The modification is introduced through the weight function, which in this modified version depends not only on the distance between two points (as for the original model) but also on the stress state at the remote point.The efficiency of this novel approach is illustrated using several examples. The proposed modification improves the numerical solution of problems observed in numerical simulations involving regularization techniques. Damage initiation and propagation in mode I as well as shear band formation are analyzed herein.     

A thermo-mechanical cohesive zone formulation for ductile fracture

The paper addresses the possibility to project both mechanical and thermal phenomena pertinent to the fracture process zone into a cohesive zone. A wider interpretation of the notion cohesive zone is thereby suggested to comprise not only stress degradation due to micro-cracking but also heat generation and energy transport. According to our experience, this widening of the cohesive zone concept allows for a more efficient finite element simulation of ductile fracture. The key feature of the formulation concerns the thermo-mechanical cohesive zone model, evolving within the thermo-hyperelastoplastic continuum, allowing for the concurrent modelling of both heat generation, due to the fracture process, and heat transfer across the fracture process zone. This is accomplished via thermodynamic arguments to obtain the coupled governing equation of motion, energy equation, and constitutive equations. The deformation map is thereby defined in terms of independent continuous and discontinuous portions of the displacement field. In addition, as an extension of the displacement kinematics, to represent the temperature field associated with the discontinuous heat flux across the fracture interface, a matching discontinuous temperature field involving the interface (or band) temperature is proposed. In the first numerical example, concerning dynamic quasi-brittle crack propagation in a thermo-hyperelastoplastic material, we capture the initial increase in temperature close to the crack surface due to the energy dissipating fracture process. In the second example, a novel application of ductile fracture simulation to the process of high velocity (adiabatic) cutting is considered, where some general trends are observed when varying the cutting velocity.     

Evaluation of nonlocal approaches for modelling fracture near nonconvex boundaries

Integral-type nonlocal damage models describe the fracture process zones by regular strain profiles insensitive to the size of finite elements, which is achieved by incorporating weighted spatial averages of certain state variables into the stress–strain equations. However, there is no consensus yet how the influence of boundaries should be taken into account by the averaging procedures. In the present study, nonlocal damage models with different averaging procedures are applied to the modelling of fracture in specimens with various boundary types. Firstly, the nonlocal models are calibrated by fitting load–displacement curves and dissipated energy profiles for direct tension to the results of mesoscale analyses performed using a discrete model. These analyses are set up so that the results are independent of boundaries. Then, the models are applied to two-dimensional simulations of three-point bending tests with a sharp notch, a V-type notch, and a smooth boundary without a notch. The performance of the nonlocal approaches in modelling of fracture near nonconvex boundaries is evaluated by comparison of load–displacement curves and dissipated energy profiles along the beam ligament with the results of meso-scale simulations. As an alternative approach, elastoplasticity combined with nonlocal and over-nonlocal damage is also included in the comparative study.     

准脆性材料断裂过程区中的圆饼微裂纹损伤计算

准脆性工程材料及结构在外力作用下，不仅引起内部缺陷变化和微裂纹的出现及发展，且使得其结构承载能力降低或性能劣化．在其材料失效过程中常存在裂缝与断裂损伤过程区．为研究材料细观缺陷或微裂纹与宏观破坏的规律，通过细观力学方法，对于代表性体积单元RVE中的圆饼型微裂纹的尺寸与密度变化，探讨其宏观断裂过程区力学参量与损伤之间的量化关系．借助宏观断裂过程区的黏聚裂纹模型，将损伤单元RVE嵌入到宏观裂缝端部的断裂过程区中，对其进行联接细观损伤到宏观破坏的力学多尺度研究．文中也通过实验数据，对其理论计算结果进行了算例的讨论与分析．     

Hysteretic deteriorating model for quasi-brittle materials based on micromechanical damage approach

A damage model, which is based on the stochastic modeling of the microstructures, is developed for the quasi-brittle materials subjected to repeated loading. According to this model, the overall response of the material is represented with a series of micro-elements joined in parallel. A combined model is proposed for the micro-element considering the fracture as well as the hysteretic energy dissipation. To account for the progressive failure, the random fracture strains are assigned to the micro-elements. Therefore the overall parallel bundle is considered as a stationary random field. Then by averaging the microscopic random field, the overall loading, unloading and reloading curves are derived analytically. Two hysteretic rules are derived from the proposed model, and the overall hysteretic deteriorating behaviors could be well reproduced. To demonstrate the validity of the present model, the numerical results are shown against the stochastic simulated curves as well as the experimental data. The present model provides an alternative approach for the efficient modeling of the hysteretic deteriorating behaviors for the quasi-brittle materials.     

Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour

An improved gradient-enhanced approach for softening elasto-plasticity is proposed, which in essence is fully nonlocal, i.e. an equivalent integral nonlocal format exists. The method utilises a nonlocal field variable in its constitutive framework, but in contrast to the integral models computes this nonlocal field with a gradient formulation. This formulation is considered ‘implicit’ in the sense that it strictly incorporates the higher-order gradients of the local field variable indirectly, unlike the common (explicit) gradient approaches. Furthermore, this implicit gradient formulation constitutes an additional partial differential equation (PDE) of the Helmholtz type, which is solved in a coupled fashion with the standard equilibrium condition. Such an approach is particularly advantageous since it combines the long-range interactions of an integral (nonlocal) model with the computational efficiency of a gradient formulation. Although these implicit gradient approaches have been successfully applied within damage mechanics, e.g. for quasi-brittle materials, the first attempts were deficient for plasticity. On the basis of a thorough comparison of the gradient-enhancements for plasticity and damage this paper rephrases the problem, which leads to a formulation that overcomes most reported problems. The two-dimensional finite element implementation for geometrically linear plain strain problems is presented. One- and two-dimensional numerical examples demonstrate the ability of this method to numerically model irreversible deformations, accompanied by the intense localisation of deformation and softening up to complete failure.     

Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials: A 3D study

In a recent publication (Yang et al., 2009. Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials. Int. J. Solids Struct. 46 (17) 3222–3234), we developed a finite element method capable of modelling complex two-dimensional (2D) crack propagation in quasi-brittle materials considering random heterogeneous fracture properties. The present study extends the method to model three-dimensional (3D) problems. First, 3D cohesive elements are inserted into the initial mesh of solid elements to model potential crack surfaces by a specially designed, flexible and efficient algorithm and corresponding computer program. The softening constitutive laws of the cohesive elements are modelled by spatially-varying 3D Weibull random fields. Monte Carlo simulations are then carried out to obtain statistical information of structural load-carrying capacity. A concrete cube under uniaxial tension was analysed as an example. It was found that as the 2D heterogeneous model, the 3D model predicted realistic, complicated fracture processes and load-carrying capacity of little mesh-dependence. Increasing heterogeneity in terms of the variance in the tensile strength random fields resulted in lower mean and higher standard deviation of peak loads. Due to constraint effects and larger areas of unsmooth, non-planar fracture surfaces, 3D modelling resulted in higher mean and lower standard deviation of peak loads than 2D modelling.     

A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids

In order to simulate quasi-brittle failure in porous elastic solids, a continuum damage model has been developed within the framework of strain gradient elasticity. An essential ingredient of the continuum damage model is the local strain energy density for pure elastic response as a function of the void volume fraction, the local strains and the strain gradients, respectively. The model adopts Griffith’s approach, widely used in linear elastic fracture mechanics, for predicting the onset and the evolution of damage due to evolving micro-cracks. The effect of those micro-cracks on the local material stiffness is taken into account by defining an effective void volume fraction. Thermodynamic considerations are used to specify the evolution of the latter. The principal features of the model are demonstrated by means of a one-dimensional example. Key aspects are discussed using analytical results and numerical simulations. Contrary to other continuum damage models with similar objectives, the model proposed here includes the effect of the internal length parameter on the onset of damage evolution. Furthermore, it is able to account for boundary layer effects.     

Interaction-based non-local damage model for failure in quasi-brittle materials

The purpose of this paper is to present a new macroscopic approach to describe the evolving non-local interactions which are produced at the mesoscale during damage and failure in quasi-brittle materials. A new-integral type non-local model is provided where the weight function is directly built from these interactions, and therefore takes into account their evolution during the material failure intrinsically.     

Analysis on the cohesive stress at half infinite crack tip

The nonlinear fracture behavior of quasi-brittle materials is closely related with the cohesive force distribution of fracture process zone at crack tip. Based on fracture character of quasi-brittle materials, a mechanical analysis model of half infinite crack with cohesive stress is presented. A pair of integral equations is established according to the superposition principle of crack opening displacement in solids, and the fictitious adhesive stress is unknown function . The properties of integral equations are analyzed, and the series function expression of cohesive stress is certified. By means of the data of actual crack opening displacement, two approaches to gain the cohesive stress distribution are proposed through resolving algebra equation. They are the integral transformation method for continuous displacement of actual crack opening, and the least square method for the discrete data of crack opening displacement. The calculation examples of two approaches and associated discussions are give     

基于近场动力学理论的准脆性材料的本构力函数的构建

近场动力学理论(PD)是基于非局部思想的连续介质力学新理论,用于研究材料破坏问题。根据准脆性材料破坏的线性和非线性的力学行为,在初始微观弹脆性材料(PMB)的本构力函数中引入了键的损伤模型,将键的断裂过程分成了线性的弹性变形阶段和非线性的损伤变形阶段,以此构建了准脆性材料的本构力函数的基本形式。以典型的准脆性材料为例构建了其本构力函数,通过在压缩载荷下对含预制不同角度单裂纹缺陷的类岩材料的裂纹扩展进行PD数值模拟仿真,裂纹起裂位置和扩展方向与试样试验结果在一定程度上保持了一致,证明了该基于近场动力学理论的典型准脆性材料的本构力函数可用于该类材料的破坏分析。     

A micromechanics-based strain gradient damage model for fracture prediction of brittle materials – Part II: Damage modeling and numerical simulations

In this paper, we established a strain-gradient damage model based on microcrack analysis for brittle materials. In order to construct a damage-evolution law including the strain-gradient effect, we proposed a resistance curve for microcrack growth before damage localization. By introducing this resistance curve into the strain-gradient constitutive law established in the first part of this work (Li, 2011), we obtained an energy potential that is capable to describe the evolution of damage during the loading. This damage model was furthermore implemented into a finite element code. By using this numerical tool, we carried out detailed numerical simulations on different specimens in order to assess the fracture process in brittle materials. The numerical results were compared with previous experimental results. From these studies, we can conclude that the strain gradient plays an important role in predicting fractures due to singular or non-singular stress concentrations and in assessing the size effect observed in experimental studies. Moreover, the self-regularization characteristic of the present damage model makes the numerical simulations insensitive to finite-element meshing. We believe that it can be utilized in fracture predictions for brittle or quasi-brittle materials in engineering applications.     

Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-brittle materials

The aim of this paper is to develop a thermodynamically consistent micromechanical concept for the damage analysis of viscoelastic and quasi-brittle materials. As kinematical damage variables a set of scalar-, vector-, and tensor-valued functions is chosen to describe isotropic and anisotropic damage. Since the process of material degradation is governed by physical mechanisms on levels with different length scale, the macro- and mesolevel, where on the mesolevel microdefects evolve due to microforces, we formulate in this paper the dynamical balance laws for macro- and microforces and the first and second law of thermodynamics for macro- and mesolevel.Assuming a general form of the constitutive equations for thermo-viscoelastic and quasi-brittle materials, it is shown that according to the restrictions imposed by the Clausius–Duhem inequality macro- and microforces consist of two parts, a non-dissipative and a dissipative part, where on the mesolevel the latter can be regarded as driving forces on moving microdefects. It is shown that the non-dissipative forces can be derived from a free energy potential and the dissipative forces from a dissipation pseudo-potential, if its existence can be assured.The micromechanical damage theory presented in this paper can be considered as a framework which enables the formulation of various weakly nonlocal and gradient, respectively, damage models. This is outlined in detail for isotropic and anisotropic damage.     

固体结构损伤破坏统一相场理论、算法和应用

固体开裂引起的损伤和断裂是工程材料和结构最为普遍的破坏形式. 为了防止这种破坏, 结构设计首先必须了解裂缝在固体内如何萌生、扩展、分叉、汇聚甚至破碎; 更重要的是, 还需要准确量化这些裂缝演化过程对于结构完整性和安全性降低的不利影响. 针对上述固体结构损伤破坏问题, 本工作系统地介绍了笔者提出的统一相场理论、算法及其应用. 作为一种裂缝正则化变分方法, 统一相场理论将基于强度的裂缝起裂准则、基于能量的裂缝扩展准则以及满足变分原理的裂缝路径判据纳入同一框架内. 不仅常用的脆性断裂相场模型是该理论的特例, 还自然地给出了一类同时适用于脆性断裂和准脆性破坏的相场正则化内聚裂缝模型即 PF-CZM. 该模型非常便于通过有限元等方法加以数值实现; 为了求解有限元空间离散后得到的非线性方程组, 还介绍了几种常用的数值算法, 其中整体BFGS拟牛顿迭代算法的计算效率最高. 静力、动力和多场耦合条件下若干二维和三维代表性算例表明: 相场正则化内聚裂缝模型 PF-CZM能够高精度地再现复杂裂缝演化导致的脆性和准脆性固体损伤破坏; 特别是, 所有情况下, 模型的数值结果不依赖于裂缝尺度参数和有限元网格. 因此, 该模型具有相当好的预测能力, 有望在工程结构的损伤破坏分析方面发挥重要作用. 最后建议了若干值得进一步开展的研究课题.      

An isotropic unilateral damage model coupled with frictional sliding for quasi-brittle materials

In this paper, we present an original extension of an isotropic damage model for quasi-brittle materials and assess its predictive capabilities. The proposed model accounts not only for unilateral behavior related to the opening and closure of microcracks but also for inelastic strains reflecting the frictional sliding along closed microcracks. More importantly, owing to its careful mathematical formulation with a particular attention paid to the continuous differentiability of the underlined thermodynamic potential, the model ensures the continuity of the inelastic stress–strain response. First applications show that it is able to predict the asymmetric behavior and hysteretic response of microcracked materials such as concrete and some rocks     

基于结构化变形驱动的非局部宏-微观损伤模型的真Ⅱ型裂纹模拟

Ⅱ型载荷作用下裂纹变形模式也为Ⅱ型的破坏问题称为真Ⅱ型破坏.准确定量地把握真Ⅱ型破坏的全过程是具有挑战性的问题.本文采用结构化变形驱动的非局部宏-微观损伤模型对真Ⅱ型破坏问题进行了模拟.根据结构化变形理论将点偶的非局部应变分解为弹性应变与结构化应变两部分,进而利用Cauchy-Born准则与结构化应变计算点偶的结构化正伸长量.在本文中,结构化应变取为非局部应变的偏量部分.当点偶的结构化正伸长量超过临界伸长量时,微细观损伤开始在点偶层次发展.将微细观损伤在作用域中进行加权求和得到拓扑损伤,并通过能量退化函数将其嵌入到连续介质-损伤力学框架中进行数值求解.进一步地,本文采用Gauss-Lobatto积分格式计算点偶的非局部应变,将积分点数目降低到4个,显著降低了前处理和非线性分析的计算成本.通过对Ⅱ型加载下裂尖应变场的分析揭示了采用偏应变作为结构化应变的原因.对两个典型真Ⅱ型破坏问题的模拟结果表明,本文方法不仅可以把握Ⅱ型加载下的真Ⅱ型裂纹扩展模式,同时可以定量刻画加载过程中的载荷-变形曲线,且不具有网格敏感性.最后指出了需要进一步研究的问题.     

Tensile failure of two-dimensional quasi-brittle foams

Stress redistribution caused by damage onset and the subsequent local softening plays an important role in determining the ultimate tensile strength of a cellular structure. The formation of damage process zones with struts dissipating a finite amount of fracture energy will require the macroscopic stress to be increased in order to continue structural damage. The goal of this paper is to investigate the influence of the fracture energy of the solid on the tensile fracture strength and the strain to fracture in quasi-brittle two-dimensional foams using a microstructural model. We analyze the mesoscopic damage and failure mechanisms in uniaxial tension. Relative density, strut cross-sectional profile, solid’s fracture strain, and fracture energy are varied systematically. The effect of the specific fracture energy on the peak behavior has been shown to be captured by the ratio of the fracture energy to the stored elastic energy. We have also explored the net section strength variation in the presence of a central crack at two different fracture energies. Comparison is made between two structurally identical quasi-brittle and ductile strain hardening foams to identify the differences in the damage mechanisms.     

Influence of cracking on tensile behavior of brittle materials

This paper deals with mechanical behavior of quasi-brittle materials in tension, such as concrete. Two elementary LEFM models are presented. Attention is focused on the crack phenomenon, highlighting the influence of initial damage on structural response. The first model consists of a plate of finite size with a central crack. The second model corresponds to an infinite plate with an infinite set of collinear cracks of equal length and spaced at a constant distance apart. For both models, an analytical study is conducted which leads to a determination of the link between critical stress and displacement, initially considering only the incremental displacement due to the damage present in the material, and subsequently also taking into account the elastic compliance of the structure. An analysis is also performed in which the concomitance of brittle fracture and plastic collapse is considered, which reveals interesting scale effects.