Particle swarm optimization for numerical bifurcation analysis in computational inelasticity

An efficient and robust algorithm to numerically detect material instability or bifurcation is of great importance to the understanding and simulation of material failure in computational and applied mechanics. In this work, an intelligence optimizer, termed the particle swarm optimization, is introduced to the numerical solution of material bifurcation problem consisting of finding the bifurcation time as well as the corresponding bifurcation directions. The detection of material bifurcation is approached as a constrained minimization problem where the determinant of the acoustic tensor is minimized. The performance of the particle swarm optimization method is tested through numerical bifurcation analysis on both small and finite deformation material models in computational inelasticity with increasing complexity. Compared with conventional numerical approaches to detect material bifurcation, the proposed method demonstrates superior performance in terms of both computational efficiency and robustness. Copyright © 2016 John Wiley & Sons, Ltd.     

Bifurcations in finite element models with a non-associated flow law

A numerical approach to bifurcation problems in soil mechanics is described. After locating the bifurcation point by a combination of an incremental-iterative loading procedure and an eigenvalue analysis of the tangent stiffness matrix, the solution is continued on the localization path by a suitable combination of the fundamental solution and the eigenvector belonging to the lowest eigenvalue. The procedures are applied in a bifurcation analysis of a cohesionless soil in a biaxial testing device. The results suggest that a diffuse bifurcation mode with a short wavelength is encountered whereupon a shear band gradually develops. The inclination angle of the shear band compares well with analytical formulae and with empirical data.     

不同应力路径下水土耦合超固结黏土分叉分析

基于超固结黏土三维弹塑性本构模型,导出水-土耦合部分排水条件下饱和黏土应变局部化的萌生条件,求出在不同应力路径下,单相和水-土耦合条件下超固结土分叉的三维解析解和数值解。理论分析表明,应力罗德角 为-30°、15°和30°时,无论是单相条件,还是水土耦合两相介质条件,该模型均无分叉现象产生;应力罗德角为0°时,单相条件下该模型有分叉现象产生,水土耦合不排水条件下该模型无分叉现象出现,水-土耦合部分排水条件下分叉与否则与参数选取有关;应力罗德角为-15°时,无论是单相条件,还是水-土耦合条件,该模型均有分叉现象产生。利用嵌入了上述本构模型的有限元软件ABAQUS,对单相和水-土耦合条件下的多单元立方体应变局部化分叉现象进行数值分析。分析表明,在应力罗德角为 30°时,单相条件和水土耦合条件下都无分叉现象产生;应力罗德角为15°时,单相条件下有分叉现象产生,水-土耦合不排水条件下无分叉现象出现,水-土耦合部分排水条件下何时分叉则与渗透系数大小有关;应力罗德角为-15°和0°时,单相条件和水土耦合条件下该模型均有分叉现象产生。同一应力路径下数值解相对于理论解更易出现分叉。     

边坡稳定分析中应变局部化的简化计算

采用分岔理论作为局部化判断条件,提出通过弹粘塑性软化模型来捕捉水工结构中的应变局部化剪切带的有限元简化算法。方法的特点是在每个时步,可以得到作为分岔理论所必须的弹塑性矩阵,仅用其作局部化判断,而不影响程序本身的粘塑性迭代。与其他方法相比,该方法理论简洁,易于程序的实现。通过对边坡的稳定分析,结果表明：该方法是合理可行的。     

三维应力状态下砂的临界状态本构模型分叉分析

就服从非关联流动法则软化型砂的临界状态本构模型,在不同应力路径下的分叉进行了理论和数值分析。理论分析表明：分叉现象强烈依赖于应力路径,即当平均应力p一定时,应力路径在洛德角在-25o～15o之间时,均会在应力-应变关系硬化阶段出现分叉现象,而其他应力路径下不会产生分叉。并且分叉时对应的应力比、主应变及剪切带倾角都随洛德角的增加而变化,其值是先增加而后减小。采用回映应力更新算法,编写了该本构模型材料子程序,借助有限元软件ABAQUS及材料子程序,通过数值计算方法预测到了分叉点所对应的应力状态,表明了分叉现象在数值计算中的存在性,并通过数值预测值和理论解的对比,两者结果基本一致。     

Bifurcation analysis of deep boreholes: II. Scale effect

This paper is a continuation of a previous one on bifurcation of deep boreholes under uniform stress at infinity,¹ and it focuses on the effect of borehole radius on borehole stability. Rock is described by the equations of a deformation theory of plasticity for cohesive-frictional, incompressible materials with Cosserat structure. The inhomogeneous bifurcation problem is formulated and the corresponding eigenvalue problem is solved numerically by the finite element method. It is demonstrated that the combined Cosserat and stress-gradient effects lead to an increase of the bifurcation stress with decreasing borehole radius (scale effect).     

Diffuse bifurcations engraving diverse shear bands in granular materials

Shear bands with characteristic spatial patterns observed in an experiment for a cubic or parallelepiped specimen of dry dense sand were simulated by numerical bifurcation analysis using the Cam‐clay plasticity model. By incorporating the subloading surface concept into the plasticity model, the model became capable of reproducing hardening/softening and contractive/dilative behavior observed in the experiment. The model was reformulated to be compatible with the multiplicative hyperelasto‐plasticity for finite strains. This enhanced constitutive model was implemented into a finite‐element code reinforced by a stress updating algorithm based on the return‐mapping scheme, and by an efficient numerical procedure to compute critical eigenvectors of elastoplastic tangent stiffness matrix at bifurcation points. The emergence of diamond‐ and column‐like diffuse bifurcation modes breaking uniformity of the materials, followed by the evolution of shear bands through strain localization, was observed in the analysis. In the bifurcation analysis of plane strain compression test, unexpected bifurcation modes, which broke out‐of‐plane uniformity and led to 3‐dimensional diamond‐like patterns, were detected. Diffuse bifurcations, which were difficult to observe by experiments, have thus been found as a catalyst creating diverse shear band patterns.     

地质灾害中的应变局部化现象

首先介绍应变局部化现象及其研究进展，然后介绍地质灾害中常见的应变局部化现象，即边坡失效、圆形洞室失效、地震中的局部化现象和流固耦合应变局部化现象。采用材料的分叉分析方法和变形局部化的数值模拟研究，可望对地质灾害的机理分析、预测和控制提供新的思路。     

基于分叉理论的岩体局部化现象研究

为了研究岩体局部化现象,基于轴向压杆稳定实例,运用数理和力学知识,根据Drucker-Prager（简称D-P）准则,推导了基于分叉理论的通用表达式。在此基础上,结合有限差分理论得出FLAC3D计算中岩体内发生分叉的判定条件,之后在自带的FISH编译语言的辅助下,在FLAC3D中建立数值分析模型,采用D-P本构关系,研究了岩体局部化现象及其发展过程。研究结果表明：在FLAC3D中以20 000步为间隔所得的不平衡力比率模型系统逐渐趋于稳定,不平衡力比率逐渐降低,局部化现象的规模逐渐扩大至稳定,清晰度逐渐增加,证实了分叉理论能够有效地分析研究岩体局部化现象。给出的岩体局部分叉判定条件和数值验证,较为科学合理地阐述了常用分叉判定条件的由来,为解释岩体局部化现象的机制和过程提供了理论支撑,从而对岩体局部化现象研究起到了一定的推进作用。     

Transient bifurcation condition of partially saturated porous media at finite strain

Bifurcation of unsaturated soils into a localized shear band is a ubiquitous failure mode of partially saturated soils. The density and degree of saturation have major impacts on the inception of localized deformations in unsaturated soils. Unsaturated fluid flow may dramatically change the density and degree of fluid saturation of unsaturated soils. Therefore, the unsaturated fluid flow is a potential trigger for shear banding in such materials. In this paper, we derive a simplified bifurcation condition of localized deformation in unsaturated soils under the local transient condition at finite strain. This transient bifurcation condition is implemented into a nonlinear finite element code to study the inception of localized deformation in unsaturated soil specimens. Numerical simulations are conducted to study the impact of soil fabrics of density, a ‘bonding’ variable, and intrinsic permeability on the inception of localized failures via the transient bifurcation criterion. Mesh sensitivity analysis is performed to demonstrate the viscosity effect of unsaturated fluid flow on the localized deformation. Numerical simulations demonstrate that the transient bifurcation condition can detect the localized deformation triggered by the internal unsaturated fluid flow process in unsaturated soils. Copyright © 2016 John Wiley & Sons, Ltd.     

A unified method to predict diffuse and localized instabilities in sands

A simplified method to analyse diffuse and localized bifurcations of sand under drained and undrained conditions is presented in this paper. This method utilizes results from bifurcation analysis and critical state plasticity theory to detect the onset of pure and dilatant shear band formation, static liquefaction and drained shear failures systematically. To capture the soil collapse observed in experiments, the instability state line concept originated by Chu, Lo and Lee in 1993 is adopted. Emphasis is given to examine how the presence of pore-fluid may facilitate or delay instability after yielding occurs. The predictions of instabilities are compared with experimental data from triaxial compression tests on Toyoura and Changi sands.     

含裂隙组岩体的稳定问题

借鉴单晶体塑性理论,构建了岩体的多屈服面模型,用以模拟岩体中初始存在的多组裂隙。通过重构屈服函数,及简单的两种情况表明,可以采用常用的屈服准则作为屈服函数。随后,使用该多屈服面模型,分析了包含初始裂隙组岩体的稳定问题。在算例中研究了硬化模量随局部化条带方向的变化规律,并分析了不同破坏模式对临界硬化模量和临界局部化条带方向的影响。分析表明,该模型用于分析裂隙岩体的稳定问题是可行的。     

真三维状态下土体强度及变形分叉影响

通过引入应力Lode角建立合理的角隅函数描述偏平面形状,对三维Mohr-Coulmb强度准则进行修正,使之能合理反映中主应力比对土体峰值内摩擦角的影响。修正后的准则在偏平面上的强度线与Lade-Duncan准则相似,但更具灵活性,预测结果更符合松砂真三轴试验结果。由于土体在加载过程中常发生应变局部化现象,导致其强度得不到充分发挥。基于非共轴塑性模型的分叉分析研究了应变局部化发生对土体强度发挥的影响,通过与密砂真三轴试验结果对比表明,分叉强度能合理反映在不同中主应力比条件下应变局部化发生对土体强度的降低作用。     

隧道开挖中围岩损伤演化分析及力学参数预测

数值方法所得出结论的可靠性很大程度上取决于岩体力学参数的选取,因此基于损伤演化分析,对损伤变量与围岩力学参数之间关系进行了研究。在各向同性损伤的假定下,通过分析开挖过程中围岩损伤演化,建立了采用损伤变量对弹性模量、泊松比、黏聚力、摩擦角等围岩材料参数进行估算的方法。采用弹性纵波波速定义初始损伤,根据壁城隧道现场监测资料建立ANSYS渐进性数值模型,反演典型断面开挖过程中围岩的宏观力学参数。分析结果表明,该方法可以有效地确定围岩力学参数的变化情况,与实际情况吻合较好。     

Buckling of layered elastic media: A cosserat-continuum approach and its validation

In this paper the stability of layered, anisotropic, hypoelastic media under initial stress is investigated. The bifurcation problem is formulated and solved numerically by use of a semi-analytical transfer matrix method. An alternative analytical approach in dealing with inhomogeneous media is developed by representing a periodically layered medium as a homogeneous Cosserat continuum. The Cosserat-continuum approach is validated against the numerical transfer matrix solution.     

Uniqueness and existence in plasticity models for unsaturated soils

The paper investigates the mathematical structure of plasticity models for unsaturated soils and provides a strategy to capture the loss of uniqueness of the incremental solution upon loading and/or wetting paths. To derive bifurcation conditions in simple analytical form, the analysis is restricted to isotropic stress states. This choice has allowed the inspection of the most common classes of constitutive models through a unified notation, as well as the study of different forms of coupling between plasticity and state of saturation. It is shown that, similar to saturated soil plasticity, the loss of admissibility of the plastic solution is governed by critical values of the hardening modulus. At variance with the classical case, however, these moduli can be positive even if the plastic flow rule is associated (bifurcation in the hardening regime). The paper shows that such non-trivial features derive from hydro-mechanical coupling, i.e. they depend on the approach used to reproduce suction effects and evolving retention properties. In other words, although the problem of loss of uniqueness affects all classes of plasticity models for unsaturated soils, different constitutive assumptions may not have the same outcome in terms of bifurcation potential. As a result, new concepts are introduced to compare the mathematical robustness of the different constitutive approaches, as well as to interpret their predictions in the light of precise bifurcation criteria.     

层状边坡岩体的屈曲和溃屈性态研究

在对层状边坡岩体结构进行稳定性分析的过程中,屈曲和溃屈是两个主要问题。针对边坡岩体结构的赋存特点和分岔特性,以及研究方法仅采用线性理论和已有研究成果只停留在结构屈曲层面上的状况,文章应用初始后屈曲理论和突变理论探讨了岩体结构的屈曲性态和溃屈性态,给出了边坡岩体上部滑动段滑动的条件,得出了边坡岩体的分岔方程和溃屈方程,指出了分岔方程的适用条件和溃屈方程的适用范围,确定了边坡岩体后屈曲状态下溃屈破坏的下限,建立了岩体结构溃屈破坏的判据,并以雅砻江下游的霸王山边坡为例进行研究。结果表明:边坡岩体的赋存状况决定了其势函数的分岔集,从边坡的分岔集可衍生出结构在分岔状态下的分岔载荷和分岔方程;分岔方程表达了边坡岩体结构的屈曲状态,也表达了结构出现屈曲的准则;边坡岩体的赋存状况还决定了岩体结构溃屈破坏时岩体材料屈服的取值范围,而岩体结构溃屈破坏的上限或下限则取决于岩体材料的屈服极限。     

Bifurcation and generalized mixed loading conditions in geomaterials

The concept of bifurcation has been widely discussed, based, for example, on homogeneous tests in soil mechanics. Essentially two‐dimensional (axisymmetric) conditions have been considered. This paper aims at deriving a bifurcation criterion in geomechanics, independently of the problem's dimension. This criterion is related to the vanishing of the determinant of the symmetric part of the material's (or the discrete system's) constitutive (or stiffness) matrix. The derivation is essentially based on the notion of loading parameters (controlling the loading). Basically, a bifurcation occurs when the existence of a unique solution to the quasistatic problem, involving a given set of loading parameters, is lost. Interestingly, the criterion is shown to be independent of the choice of loading parameters. As a possible extension, the context of structure mechanics is considered, and the close connection between both soil and structure analysis is discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Bifurcation modeling in geomaterials: From the second‐order work criterion to spectral analyses

The present paper investigates bifurcation analysis based on the second‐order work criterion, in the framework of rate‐independent constitutive models and rate‐independent boundary‐value problems. The approach applies mainly to nonassociated materials such as soils, rocks, and concretes. The bifurcation analysis usually performed at the material point level is extended to quasi‐static boundary‐value problems, by considering the stiffness matrix arising from finite element discretization. Lyapunov's definition of stability (Annales de la faculté des sciences de Toulouse 1907; 9 :203–274), as well as definitions of bifurcation criteria (Rice's localization criterion (Theoretical and Applied Mechanics. Fourteenth IUTAM Congress, Amsterdam, 1976; 207–220) and the plasticity limit criterion are revived in order to clarify the application field of the second‐order work criterion and to contrast these criteria. The first part of this paper analyses the second‐order work criterion at the material point level. The bifurcation domain is presented in the 3D stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. The relevance of this criterion, when the nonlinear constitutive model is expressed in the classical form (dσ = Mdε) or in the dual form (dε = Ndσ), is discussed. In the second part, the analysis is extended to the boundary‐value problems in quasi‐static conditions. Nonlinear finite element computations are performed and the global tangent stiffness matrix is analyzed. For several examples, the eigenvector associated with the first vanishing eigenvalue of the symmetrical part of the stiffness matrix gives an accurate estimation of the failure mode in the homogeneous and nonhomogeneous boundary‐value problem. Copyright © 2008 John Wiley & Sons, Ltd.