裂隙岩体损伤局部化破坏分岔模型及其应用

采用概率统计方法分析节理裂隙岩体的几何特征，定义了反映裂隙岩体几何特征的组构张量．基于不可逆热力学理论，通过裂纹扩展的细观分析，得出了损伤的发展机理和演化方程，把损伤演化和裂隙的几何特征的变化联系起来，建立了弹塑性损伤本构关系．为分析含有节理裂隙岩体在发生局部化破坏时的特征，通过对发生局部化时的裂隙岩体的分析。构造了适用于节理裂隙岩体局部化分析的不连续分岔模型．利用非线性规划数值解法，可以得出局部化破坏的方向特征．在有限元方法中，根据该模型给出了节理裂隙岩体相关的算例，分析表明该模型用于分析裂隙岩体的局部化破坏是有效的．     

薄膜破坏过程数值模拟的MPM方法

将连续与不连续本构模型相结合用于薄膜破坏过程的模拟计算. 采用von Mises关联塑性本构模型描述材料的弹塑性变形过程，在塑性硬化和软化阶段通过检查分岔 是否发生, 从而判断是否有破坏出现，即材料出现应变局部化现象. 一旦出现破坏，则在发生 破坏的区域采用位移不连续(decohesion)本构模型进行模拟, 直到破坏面两侧材料完全分离. 分离阶段采用相应的分离算法同时运用黏性边界条件以提高计算效率. 结果表明结合连续与 不连续本构模型的方法, 能很好地模拟材料从局部化到完全破坏的过程，体现了材料破坏连续 建模的思想，可用于预测材料发生破坏的环境和形式, 从而通过改变材料工作环境提高其使用 寿命；同时数值算例也显示了物质点法(material-point-method, 简称MPM)的健壮性和有 效性.     

非贯通裂隙岩体损伤演化率相关性及变形特征

含非贯通裂隙岩体是自然界中岩体的主要赋存形式，其裂隙几何特征对岩体的强度及变形均产生显著影响。应变率对岩体的损伤演化及黏滞效应也具有显著的率相关性。首先，运用模型元件的方法，将非贯通裂隙岩体动态破坏过程视为具复合损伤、静态弹性特性、动态黏滞特性的非均质点组成，对黏弹性响应的Maxwell体进行改进，将细观损伤体与裂隙损伤演化的宏观损伤体根据等效应变假设并联组成宏细观复合损伤体，构建综合考虑岩体宏细观缺陷的动态损伤模型；其次，基于断裂力学及应变能理论，对岩体宏观裂隙动态扩展的能量机制进行分析，综合考虑初始裂隙应变能、裂隙动态损伤演化过程应变能、裂隙闭合应变能，得到裂隙岩体宏观动态损伤变量计算公式；最后，将模型计算结果与实验结果进行比较，模型计算结果与实验结果吻合较好，证明了模型的合理性，同时利用模型讨论了裂隙倾角、应变率、岩石性质对岩体变形特征的影响规律。     

从压杆的离散化模型看LS方法

通过压杆两自由度离散化模型形象地说明了 Lyapunov-Scbmidt方法的基本思想;用分岔理论的观点描述了压杆的失稳过程．     

三维加锚弹塑性损务模型在溪洛渡地下厂房工程中的应用

本文根据断续裂隙岩体的损伤机制,建立了三维弹塑性损伤本构模型反映裂隙岩体的损伤变形特性。考虑断续裂隙岩体的岩锚支护效应,建立了空间损伤锚柱单元模型模拟锚杆的支护效果。最后将建立的模型应用于溪洛渡水电站地下厂房,进行了洞室群开挖弹塑性损伤及岩锚支护三维非线性有限元计算,获得了一些有益的工程结论。     

静态分岔点的数值计算

本文讨论了古典分岔理论与分岔的奇异性理论的联系与区别，并针对分岔点的计算方法与所依赖的理论体系的关系进行了简单综述。     

碰撞振动系统强共振下的两参数动力学分析

建立了两自由度碰撞振动系统的动力学模型及其周期运动的Poincaré映射,当Jacobi矩阵存在一对共轭复特征值在单位圆上并满足强共振(λ40=1)条件时,通过中心流型-范式方法将四维映射转变为二维范式映射。理论分析了系统两参数开折的局部动力学行为,扩展了单参数分岔理论,给出了n-1周期运动产生Hopf分岔和次谐分岔的条件。数值仿真验证了所得出的理论,证明系统在共振点附近存在稳定的Hopf分岔不变环面和次谐分岔4-4周期运动。     

具有局部非线性动力系统周期解及稳定性方法

对于具有局部非线性的多自由度动力系统，提出一种分析周期解的稳定性及其分岔的方法该方法基于模态综合技术，将线性自由度转换到模态空间中，并对其进行缩减，而非线性自由度仍保留在物理空间中在分析缩减后系统的动力特性时，基于Ｎｅｗｍａｒｋ法的预估－校正－局部迭代的求解方法，与Ｐｏｉｎｃａｒé映射法相结合，推导出一种确定周期解，并使用Ｆｌｏｑｕｅｔ乘子判定其稳定性及分岔的方法     

二阶微分方程含有时变参数的非完全分岔的分析

用新方法研究二阶微分方程含有时变参数的非完全分岔问题。当分岔参数随时间线性慢变分别经过定常跨临界分岔值，叉型分岔值和鞍结分岔值时，分析了非完全分岔参数和时变参数的变化率对分岔转移迁的滞后和跃迁现象的影响，并给出分岔转迁发生的一般条件。通过数值计算给出分岔转迁区和分岔转迁值，还讨论了解对初值和参数的敏感性问题。     

弹塑性数值流形方法在边坡稳定分析中的应用

针对现有数值流形方法只能进行弹性计算的不足,建立了一个能够反映完整岩块弹塑性变形特征的本构模型,并借助VC++开发了内置该本构模型的弹塑性数值流形程序。利用该程序模拟了含单节理岩样的室内压缩试验,分析得到了其强度和变形特性,计算结果符合实际的物理现象,表明程序是正确有效的。考虑到数值流形方法本身能够有效模拟材料的不连续变形,新增的弹塑性分析功能又可以反映岩石的强度特性,将弹塑性数值流形程序应用于某含有不连续面的岩石边坡的稳定性分析。并结合锚杆单元的使用,对比分析了不同锚固方案的加固效果。程序提供的变形、应力等计算结果表明：预应力锚杆不仅可以防止不连续面发生剪切破坏,增强坡体的稳定性,限制塑性变形的发展;而且可以使不连续软弱层面对岩体变形的消极影响得以减弱,起到提高岩体整体性的作用。     

Simulation and interpretation of diffuse mode bifurcation of elastoplastic solids

The diffuse mode bifurcation of elastoplastic solids at finite strain is investigated. The multiplicative decomposition of deformation gradient and the hyperelasto-plastic constitutive relationship are adapted to the numerical bifurcation analysis of the elastoplastic solids. First, bifurcation analyses of rectangular plane strain specimens subjected to uniaxial compression are conducted. The onset of the diffuse mode bifurcations from a homogeneous state is detected; moreover, the post-bifurcation states for these modes are traced to arrive at localization to narrow band zones, which look like shear bands. The occurrence of diffuse mode bifurcation, followed by localization, is advanced as a possible mechanism to create complex deformation and localization patterns, such as shear bands. These computational diffuse modes and localization zones are shown to be in good agreement with the associated experimental ones observed for sand specimens to ensure the validity of this mechanism. Next, the degradation of horizontal sway stiffness of a rectangular specimen due to plane strain uniaxial compression is pointed out as a cause of the bifurcation of the first antisymmetric diffuse mode, which triggers the tilting of the specimen. Last, circular and punching failures of a footing on a foundation are simulated.     

Strong discontinuities and continuum plasticity models: the strong discontinuity approach

The paper presents the Strong Discontinuity Approach for the analysis and simulation of strong discontinuities in solids using continuum plasticity models. Kinematics of weak and strong discontinuities are discussed, and a regularized kinematic state of discontinuity is proposed as a mean to model the formation of a strong discontinuity as the collapsed state of a weak discontinuity (with a characteristic bandwidth) induced by a bifurcation of the stress–strain field, which propagates in the solid domain. The analysis of the conditions to induce the bifurcation provides a critical value for the bandwidth at the onset of the weak discontinuity and the direction of propagation. Then a variable bandwidth model is proposed to characterize the transition between the weak and strong discontinuity regimes. Several aspects related to the continuum and, their associated, discrete constitutive equations, the expended power in the formation of the discontinuity and relevant computational details related to the finite element simulations are also discussed. Finally, some representative numerical simulations are shown to illustrate the proposed approach.     

Shear banding analysis of geomaterials by strain gradient enhanced damage model

Shear band localization is investigated by a strain-gradient-enhanced damage model for quasi-brittle geomaterials. This model introduces the strain gradients and their higher-order conjugate stresses into the framework of continuum damage mechanics. The influence of the strain gradients on the constitutive behaviour is taken into account through a generalized damage evolutionary law. A weak-form variational principle is employed to address the additional boundary conditions introduced by the incorporation of the strain gradients and the conjugate higher-order stresses. Damage localization under simple shear condition is analytically investigated by using the theory of discontinuous bifurcation and the concept of the second-order characteristic surface. Analytical solutions for the distributions of strain rates and strain gradient rates, as well as the band width of localised damage are found. Numerical analysis demonstrates the shear band width is proportionally related to the internal length scale through a coefficient function of Poisson’s ratio and a parameter representing the shape of uniaxial stress–strain curve. It is also shown that the obtained distributions of strains and strain gradients are well in accordance with the underlying assumptions for the second-order discontinuous shear band boundary and the weak discontinuous bifurcation theory.     

Discontinuous bifurcation analysis of thermodynamically consistent gradient poroplastic materials

In this work, analytical and numerical solutions of the condition for discontinuous bifurcation of thermodynamically consistent gradient-based poroplastic materials are obtained and evaluated. The main aim is the analysis of the potentials for localized failure modes in the form of discontinuous bifurcation in partially saturated gradient-based poroplastic materials as well as the dependence of these potentials on the current hydraulic and stress conditions. Also the main differences with the localization conditions of the related local theory for poroplastic materials are evaluated to perfectly understand the regularization capabilities of the non-local gradient-based one. Firstly, the condition for discontinuous bifurcation is formulated from wave propagation analyses in poroplastic media. The material formulation employed in this work for the spectral properties evaluation of the discontinuous bifurcation condition is the thermodynamically consistent, gradient-based modified Cam Clay model for partially saturated porous media previously proposed by the authors. The main and novel feature of this constitutive theory is the inclusion of a gradient internal length of the porous phase which, together with the characteristic length of the solid skeleton, comprehensively defined the non-local characteristics of the represented porous material. After presenting the fundamental equations of the thermodynamically consistent gradient based poroplastic constitutive model, the analytical expressions of the critical hardening/softening modulus for discontinuous bifurcation under both drained and undrained conditions are obtained. As a particular case, the related local constitutive model is also evaluated from the discontinuous bifurcation condition stand point. Then, the localization analysis of the thermodynamically consistent non-local and local poroplastic Cam Clay theories is performed. The results demonstrate, on the one hand and related to the local poroplastic material, the decisive role of the pore pressure and of the volumetric non-associativity degree on the location of the transition point between ductile and brittle failure regimes in the stress space. On the other hand, the results demonstrate as well the regularization capabilities of the non-local gradient-based poroplastic theory, with exception of a particular stress condition which is also evaluated in this work. Finally, it is also shown that, due to dependence of the characteristic lengths for the pore and skeleton phases on the hydraulic and stress conditions, the non-local theory is able to reproduce the strong reduction of failure diffusion that takes place under both, low confinement and low pore pressure of partially saturated porous materials, without loosing, however, the ellipticity of the related differential equations.     

弹塑性材料的平面应力非连续分岔

基于平面应力非连续分岔特性的一般描述,运用统一强度理论,得出了非相关流动情形的弹塑性材料平面应力非连续分岔的起始方位角以及相应的最大硬化模量的统一解析解,并且分析了材料拉压异性以及不同程度的中间应力对结果的影响,进而发现所得的结果一强度准则的选取有关,揭示了在分岔研究中正确选取符合材料特性的强度准则的重要性。最后,同特线理论比较发现平面应力剪切带型非连续分岔同平面应力特征线重合。     

Discontinuous Galerkin finite element discretization of a strongly anisotropic diffusion operator

The discretization of a diffusion equation with a strong anisotropy by a discontinuous Galerkin finite element method is investigated. This diffusion term is implemented in the tracer equation of an ocean model, thanks to a symmetric tensor that is composed of diapycnal and isopycnal diffusions. The strong anisotropy comes from the difference of magnitude order between both diffusions. As the ocean model uses interior penalty terms to ensure numerical stability, a new penalty factor is required in order to correctly deal with the anisotropy of this diffusion. Two penalty factors from the literature are improved and established from the coercivity property. One of them takes into account the diffusion in the direction normal to the interface between the elements. After comparison, the latter is better because the spurious numerical diffusion is weaker than with the penalty factor proposed in the literature. It is computed with a transformed coordinate system in which the diffusivity tensor is diagonal, using its eigenvalue decomposition. Furthermore, this numerical scheme is validated with the method of manufactured solutions. It is finally applied to simulate the evolution of temperature and salinity due to turbulent processes in an idealized Arctic Ocean. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamics of inelastic deformation of porous rocks and formation of localized compaction zones studied by numerical modeling

The paper presents a numerical analysis of the inelastic deformation process in porous rocks during different stages of its development and under non-equiaxial loading. Although numerous experimental studies have already investigated many aspects of plasticity in porous rocks, numerical modeling gives valuable insight into the dynamics of the process, since experimental methods cannot extract detailed information about the specimen structure during the test and have strong limitations on the number of tests. The numerical simulations have reproduced all different modes of deformation observed in experimental studies: dilatant and compactive shear, compaction without shear, uniform deformation, and deformation with localization. However, the main emphasis is on analysis of the compaction mode of plastic deformation and compaction localization, which is characteristic for many porous rocks and can be observed in other porous materials as well. The study is largely inspired by applications in petroleum industry, i.e. surface subsidence and reservoir compaction caused by extraction of hydrocarbons and decrease of reservoir pressure. Special attention is given to the conditions, evolution, and characteristic patterns of compaction localization, which is often manifested in the form of compaction bands. Results of the study include stress-strain curves, spatial configurations and characteristics of localized zones, analysis of bifurcation of stress paths inside and outside localized zones and analysis of the influence of porous rocks properties on compaction behavior. Among other results are examples of the interplay between compaction and shear modes of deformation.To model the evolution of plastic deformation in porous rocks, a new constitutive model is formulated and implemented, with the emphasis on selection of adequate functions defining evolution of yield surface with deformation. The set of control parameters of the model is kept as short as possible; the parameters are carefully selected to have simple and intuitive physical interpretation whenever possible. Results demonstrate that evolution of the yield surface with deformation has major influence on the resulting characteristics of deformation patterns, which is not sufficiently acknowledged in the literature.     

Meshfree analysis of softening elastoplastic solids using variational multiscale method

A meshfree multiscale method is presented for efficient analysis of elastoplastic solids. In the analysis of softening elastoplastic solids, standard finite element methods or meshfree methods typically yield mesh-dependent results. The reason for this well-known effect is the loss of ellipticity of the boundary value problem. In this work, the scale decomposition is carried out based on a variational form of the problem. A coarse scale is designed to represent global behavior and a fine scale to represent local behavior. A fine scale region is detected from the local failure analysis of an acoustic tensor to indicate a region where deformation changes abruptly. Each scale variable is approximated using a meshfree method. Meshfree approximation is well-suited for adaptivity. As a method of increasing the resolution, a partition of unity based extrinsic enrichment is used. In particular, fine scale approximations are designed to appropriately represent local behavior by using a localization angle. Moreover, the regularization effect through the convexification of non-convex potential is embedded to represent fine scale behavior. Each scale problem is solved iteratively. The proposed method is applied to shear band problems. In the results of analysis about pure shear and compression problems, straight shear bands can be captured and mesh-insensitive results are obtained. Curved shear bands can also be captured without mesh dependency in the analysis of indentation problem.