A new finite element formulation for one-dimensional analysis of elastic-plastic materials

It is well accepted that severe numerical difficulties arise when using the conventional finite element displacement method to analyse incompressible, or nearly incompressible, solids. These effects are caused by the kinematic constraints imposed on the nodal velocities by the constant volume condition. In elastic-plastic analysis, these effects are due to a conflict between the plastic flow rule and the finite element discretization. Although several methods have been proposed to cope with this problem, none has been based on the appropriate choice of displacement interpolation to minimise the constraints. In this paper, a new displacement interpolation, which is able to reduce the imposed constraints, is adopted. Comparisons of the results with those from a conventional linear displacement interpolation are made for predictions of cylindrical and spherical cavity expansion limit pressures in elastic-plastic solids. This study suggests that the proposed displacement interpolation is preferable to the conventional one in the elastic-plastic finite element analysis of one dimensional-axisymmetric problems which involve nearly incompressible material behaviour.     

Dynamic analysis of an offshore structure on an elastic-plastic foundation-soil system

The effect of foundation stiffness upon the dynamic response of an offshore structure is investigated. The non-linear deformation behaviour of a stratified seabed is included in the finite element simulation. The seabed is represented by a simplified spring model. In the linear elastic analyses the stiffnesses are derived using elastic half-space theory. For the non-linear problems, elastic-plastic finite element analyses are used to generate equivalent properties.An example of the method is given as applied to a three towered concrete gravity platform. Variation of natural frequency is used to indicate the significance of changes in parameters such as soil shear modulus, structural stiffness and deck mass. The forced response behaviour to random wave excitation is also investigated using a linear elastic analysis. The effect of the interaction of the foundation — soil with the structure on the dynamic response is demonstrated by the results.     

Numerical modelling of large deformations associated with driving of open‐ended piles

This paper presents a ‘Eulerian‐like’ finite element technique to simulate the large accumulated displacements of piles subjected to multiple hammer blows. For each hammer blow, results are obtained using a standard small strain finite element model and, at the end of each hammer blow, material flow is taken into account with reference to a fixed finite element mesh. Residual stresses calculated at the Gauss integration points of the deformed finite element mesh are mapped on to the fixed finite element mesh, and these stresses are used as initial stresses for the next hammer blow. At the end of each hammer blow, stiffness and mass matrices are recalculated for the volume of material remaining inside the fixed finite element mesh. Results obtained with and without allowing material to flow through the fixed mesh are compared for several hammer blows. Build up of residual stresses, soil flow and yielded points around the pile are presented for plugged, partially‐plugged and unplugged piles. Using the new finite element technique, the driving of a pile from the soil surface is studied. The ability to analyse this and other large deformation problems is the main advantage of the new finite element technique. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical analysis of rafts on visco-elastic media using eigenvector expansions

Problems of a raft on a visco-elastic continuum may be converted to equivalent elastic problems by means of a Laplace transformation. In this paper a numerical solution of the equivalent problem is obtained by finite element or other techniques. This solution is converted into the form of an eigenvector expansion and then transformed into a numerical solution to the original problem by inverting the Laplace transform. This form of solution has the advantage of applying to any visco-elastic model, and of requiring little additional computation as a result of changing the visco-elastic model, the relative raft-soil stiffness or the load pattern. The application of the method is illustrated by results for circular rafts, strip footings of finite length and rectangular rafts, and particular attention is paid to a realistic soil creep function which is asymptotic to a linear function of log time.     

Development of the distinct lattice spring model for large deformation analyses

This study develops the distinct lattice spring model (DLSM) for geometrically nonlinear large deformation problems. The formulation of a spring bond deformation under a large deformation is derived under the Lagrange framework using polar decomposition. The results reveal that the DLSM's stiffness matrix under small deformations is the tangent stiffness matrix of the DLSM under large deformations. The formulation of the spring bond internal force under a given configuration is also presented and can be used to calculate the unbalanced force. Using these formulations, three nonlinear solving methods (the Euler method, modified Euler method, and Newton method) are developed for the DLSM with which to tackle large deformation problems. To investigate the performance of the developed model, three numerical examples involving large deformations are presented, the results of which are also in good agreement with the analytical and finite element method solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

A material model for inelastic rock deformation with spatial variation of pore pressure

This paper sets forth the theoretical background and basic numerical expressions for the incorporation of elastic-plastic constitutive equations for ductile rock into a finite element computer code. The derivation of an expression for the total strain rate is performed both for a total stress formulation and for a formulation that employs the concept of effective stress for inelastic behaviour. Specific expressions for the incremental strain rate are presented for the case of a porous material having a quadratic initial yield surface and observing the associated flow rule with a special hardening law for subsequent plastic deformation. A final section of the paper summarizes the expressions required to insert the quadratic yield surface model into a finite element code.     

Parametric variational principle for elastic–plastic consolidation analysis of saturated porous media

Finite element procedures for numerical solution of various engineering problems are often based on variational formulations. In this paper, a parametric variational principle applicable to elastic-plastic coupled field problems in consolidation analysis of saturated porous media is presented. This principle can be used to solve problems where materials are inconsistent with Drucker's postulate of stability, such as in non-associated plasticity flow or softening problems. The finite element formulation was given, and it can be solved by either the conventional method or a parametric quadratic programming method.     

Performance of displacement finite elements for modelling incompressible materials

It is well established that severe numerical difficulties may arise when the displacement finite element method is used to analyse the behaviour of incompressible solids and this is particularly true for axisymmetric problems. These numerical difficulties are caused by excessive kinematic constraints and are reflected by strong oscillations in the calculated stress distribution and overestimations of collapse loads. The purpose of this paper is to present new displacement finite element formulations which are particularly suitable for axisymmetric analysis of incompressible materials. A direct comparison is made of the performance of various displacement finite elements in the analysis of elastic or plastic incompressible materials under axisymmetric loading conditions. Particular attention is focused on the performance of various axisymmetric displacement elements in predicting the stress field of incompressible or nearly incompressible materials. Copyright © 2000 John Wiley & Sons, Ltd.     

Strain parameters of discontinuities in rock for finite element calculation

The bulk strain induced by excavation of rooms for storage of highly radioactive waste is usually unimportant, but the change in aperture and shear displacement of permeable fractures may be of practical importance. Hence, the normal and shear of such discontinuities have to be considered and are commonly predicted by using special deformation moduli termed “normal” and “shear” stiffnesses in numerical calculations. For the use of some finite element methods, these have to be converted to compression and shear moduli ,which, in turn, requires that the thickness of the discontinuity is known. The normal and shear strain can be expressed in terms of soil mechanical parameters, yielding the compression modulus M and the shear modulus G, with the required form for finite element calculations. By definition they are functions of the normal and shear stiffness K_n and K_s. It is concluded that calculation of the normal and shear strain of discontinuities with clastic fillings by the use of finite element methods and deformation moduli derived from stiffness numbers is very uncertain except when the geometry of the weaknesses can be accurately defined.     

地基沉降大变形有限元分析的几何刚度效应

研究了基于完全拉格朗日（Total Lagrangian）描述的大变形有限元法分析地基沉降问题的几何刚度效应。在有限元列式的推导过程中严格考虑了土力学表述习惯的影响。通过算例分析,主要研究了几何刚度效应对荷载-沉降曲线的影响,并对比分析了不同率型大变形分析中的几何刚度效应问题。结果表明,几何刚度效应的存在减小了地基大变形有限元系统的刚度;忽略几何刚度效应将导致沉降计算结果偏小,在地基变形较大的情况下误差更明显,Truesdell率型大变形分析的最终沉降结果与小变形法的结果一致。几何刚度效应在地基大变形有限元分析中具有一定程度的影响,处理不当可能出现结构刚度增大的现象。大变形分析结果的性质偏于刚硬。     

On the linear elastic responses of the 2D bonded discrete element model

The bonded discrete element model (DEM) is a numerical tool that is becoming widely used when studying fracturing, fragmentation, and failure of solids in various disciplines. However, its abilities to solve elastic problems are usually overlooked. In this work, the main features of the 2D bonded DEM which influence Poisson's ratio and Young's modulus, and accuracy when solving elastic boundary value problems, are investigated. Outputs of numerical simulations using the 2D bonded DEM, the finite element method, a hyper elasticity analysis, and the distinct lattice spring model (DLSM) are compared in the investigation. It is shown that a shear interaction (local) factor and a geometric (global) factor are two essential elements for the 2D bonded DEM to reproduce a full range of Poisson's ratios. It is also found that the 2D bonded DEM might be unable to reproduce the correct displacements for elastic boundary value problems when the represented Poisson's ratio is close to 0.5 or the long-range interaction is considered. In addition, an analytical relationship between the shear stiffness ratio and the Poisson's ratio, derived from a hyper elasticity analysis and applicable to discontinuum-based models, provides good agreement with outputs from the 2D bonded DEM and DLSM. Finally, it is shown that the selection of elastic parameters used the 2D bonded DEM has a significant effect on fracturing and fragment patterns of solids.     

Nodal-based three-dimensional discontinuous deformation analysis (3-D DDA)

This paper presents a new numerical model that can add a finite element mesh into each block of the three-dimensional discontinuous deformation analysis (3-D DDA), originally developed by Gen-hua Shi. The main objectives of this research are to enhance DDA block’s deformability. Formulations of stiffness and force matrices in 3-D DDA with conventional Trilinear (8-node) and Serendipity (20-node) hexahedral isoparametric finite elements meshed block system due to elastic stress, initial stress, point load, body force, displacement constraints, inertia force, normal and shear contact forces are derived in detail for program coding. The program code for the Trilinear and Serendipity hexahedron elements have been developed, and it has been applied to some examples to show the advantages achieved when finite element is associated with 3-D DDA to handle problems under large displacements and deformations. Results calculated for the same models by use of the original 3-D DDA are far from the theoretical solutions while the results of new numerical model are quite good in agreement with theoretical solutions; however, for the Trilinear elements, more number of elements are needed.     

一种基于有限元的岩石圈长期变形数值计算方法

有限单元法以其灵活性和精确性,成为固体地球科学中广为使用的数值方法。从短周期的地震活动到长周期的岩石圈变形、地幔对流,甚至行星演化,有限单元法几乎在固体地球科学的各个领域都占据着十分重要的位置。随着研究的深入,某些特定的地学问题给有限元计算带来挑战,尤其是岩石圈尺度大变形的数值计算,比如俯冲带的演化、剪切带中塑性流变导致的应力集中。基于显式有限元,尝试考虑粘弹塑性岩石圈大变形过程的数值计算。应用Marker-In-Cell（MIC）方法处理物质迁移。在描述基本原理和流程的基础上,对粘弹性变形、弹塑性变形、大变形过程及热传递过程等核心模块分别做了基准测试,而这四个模块是模拟岩石圈长期变形的关键。由测试结果和其他学者的（解析或数值）研究结果比对情况来看,受测试的核心模块基本达到了测试要求。可以预见,现有的基本算法可以满足研究岩石圈大变形的需要,进一步的具体研究工作将探讨这类问题。从科学问题层面讲,逐渐复杂的科学问题有利于数值模型的成熟。已达到基准测试的数值方法对下一步开展一些具体的地球动力学数值模拟研究有实际意义。      

基于显式时间积分的球颗粒DDA计算方法

非连续变形分析方法（DDA）是一种平行于有限元法的新型数值计算方法,该方法基于最小势能原理,把每个离散块体的变形、运动和块体之间的接触统一到平衡方程中进行隐式求解。然而,传统DDA方法在计算过程中需组装整体刚度矩阵并联立求解方程组,在用于大型岩土工程问题的三维数值模拟时占用内存较大、耗时较长、计算效率极低。因此,提出一种基于显式时间积分的三维球颗粒DDA方法。该方法在求解过程中不需要组装整体刚度矩阵,在求解加速度时,由于质量矩阵为对角矩阵,可存储为一维向量占用内存较少,且可分块逐自由度求解,效率较高,在接触判断上采用最大位移准则简化了接触算法,采用较小的时步,保证了计算的精确性;通过几个典型算例验证了该方法的准确性及计算效率。     

Large strain analysis of some geomechanics problems by the finite element method

This paper presents a rational approach to the finite strain analysis of elastic-plastic materials. An updated incremental finite element technique was applied to problems of shallow foundations of homogeneous as well as multilayer soils. This was based on a variational principle which is suitable for such problems.     

爆炸荷载作用下浅埋结构动态响应有限元计算

采用有限元数值计算方法分析了地下浅埋结构的非线性动力响应,得到了结构在不同时刻的动态响应模拟计算数据。该地下浅埋结构由金属壳体和上覆土体共同组成,对于土体和金属壳体分别采用Drucker-Prager和Mohr-Coulomb屈服准则以及弹塑性本构关系,分析其非线性动态位移场和应力场。根据爆炸试验数据,研究了爆炸冲击波随距离和时间的变化过程,并且将其作为作用在结构内部的爆炸荷载。有限元数值模拟结果表明,采用非线性有限元模拟爆炸冲击荷载作用下地下浅埋结构的弹塑性动力响应是有效的,并且证明所设计的地下防护结构是安全的。     

软土地区深基坑连续墙最大变形简易计算方法

采用有限单元法研究了影响软土地区地下连续墙最大侧向变形的主要参数。针对基坑开挖深度H、基坑开挖宽度B、单位宽度地下连续墙系统刚度S、支撑结构的轴向刚度 及黏土归一化的不排水抗剪强度 为不排水抗剪强度, 为有效垂直应力）5个参数进行分析研究,通过回归分析研究结果,给出地下连续墙最大侧向变形的简易计算方法。利用简易计算方法,计算实际工程中基坑案例的地下连续墙最大侧向变形,并与现场监测结果进行对比,验证了计算方法的准确性,可为以后预估地下连续墙最大侧向变形及检查设计提供参考。     

被动桩水平位移荷载施加位置的探讨

被动桩是指一种由于土体水平位移而发生挠曲变形的桩。目前不论是室内试验还是数值计算,常在模型边界对土体施加水平位移荷载（位移边界条件）形成土体的位移场,用来研究被动桩的变形特征,但不同研究者采用的水平位移施加位置（边界位置）与被动桩的距离存在较大差异。结合一个工程案例,采用有限单元法对水平位移加载位置进行了单变量参数分析。研究表明,随着水平位移加载位置与被动桩距离的增加,桩身变形显著减小。这表明了施加位移边界后地基中存在明显的应力扩散。当该距离小于5.5倍桩径时,在模型边界施加同等大小的水平位移所需应力显著增加。同时讨论了水平荷载加载位置与被动桩距离的合理范围,认为应在5～8倍桩径范围之内。     

基于节理网络有限元的柱状节理岩体等效变形模量结构效应研究

因其特殊的地质成因,柱状节理成为一种具有强烈非连续性及各向异性的特殊岩体结构。以白鹤滩水电工程设计、建设中所遇到的柱状节理玄武岩体为背景,针对柱状节理岩体的等效变形模量问题,在探讨了其结构效应表征参数的基础上,利用节理网络有限元为工具,研究各结构效应表征参数对柱状节理岩体等效变形模量的影响。研究结果表明：在柱轴线横向平面上,对于柱体不规则程度,当柱体完全不规则相对于完全规则情况,等效模量提高大约10%左右;在其他条件不变情况下,当柱体平均边长从0.1 m增加至0.5 m时,等效模量从5.36 GPa增加至23.4 GPa;对不同加载方向的研究结果表明,柱状节理在柱轴横向平面上可以视作各向同性;当节理刚度线性递增时,相应的岩体等效模量也基本符合线性递增的规律。在平行柱体轴线平面,节理组2间距越大,柱状节理岩体的等效变形模量越大;随着错距比的增加,柱状节理岩体的等效变形模量呈现先增加后减少的趋势,当错距比为50%时,节理岩体的等效变形模量取最大值;对于节理刚度的影响,与柱轴横向平面类似的,随柱体轴线平面上节理刚度的线性增加,相应的岩体模量也基本符合线性递增的规律。该研究成果与已有的现场研究成果及三维数值模型成果相比,吻合良好。说明了文中提供的研究结论具有较好的可信度及参考价值。     

弹塑性土质边坡的ALE方法有限元分析

有限元法被广泛用于解决几何和材料非线性的问题,但标准的有限元方法难以有效解决某些材料的大变形问题和计算中的网格扭曲问题。任意拉格朗日-欧拉法（ALE法）吸取了拉格朗日和欧拉法的优点,并克服了两者的缺点,可用于解决仅用拉格朗日或欧拉有限元法所难以解决的问题。基于ALE有限元方法和弹塑性大变形基本原理,研究了岩土工程中土质边坡在自重作用下的稳定问题;计算结果不仅能直观地显示失稳时的大变形状态,并能确定较符合实际的临界滑移面形状;同时分析了含软弱夹层复杂土质边坡的稳定性。结果表明,ALE方法能有效分析土质边坡的稳定性问题,适用于岩土工程的弹塑性分析。