Removal of finite deformation using strain trajectories

A technique is described for removing the effects of finite deformation, given the principal values and orientations of strain at a number of points throughout a deformed body.Using the principal orientations, strain trajectories are constructed for the deformed state. The body is divided into finite elements bounded by these trajectories. Each element is then unstrained without changing its orientation or position. This process creates artificial voids and overlaps, which are minimized by imparting rigid translations and rotations to the elements according to a least squares method.The result is the pattern of strain trajectories for the undeformed state. It is shown that the trajectories for the deformed and undeformed states may be used as reference coordinates in order to map the change in shape of any body as it passes from the deformed to the undeformed state or vice versa. The technique is tested using models of a folded layer and a shear zone. It is suggested that the technique is versatile enough to allow for errors in original strain data. Although the technique has so far been applied to two-dimensional deformations, a similar method should be usable in three dimensions.     

The estimation of finite strain from the orientation distribution of passively deformed linear markers: Eigenvalue relationships

The eigenvalues and eigenvectors of a direction cosine dispersion matrix may be used to characterise sets of orientation measurements in either two or three dimensions. General equations are presented that relate ratios of the eigenvalues of such a measured set to the finite strain ratios of the set assuming that the original orientation distribution, before a deformation, was uniform. The measured linear features in the rock are assumed to have behaved passively during a homogeneous deformation. The application of these relationships to strain estimation is discussed and tables for the three-dimensional case are given.     

基于高效随机有限元法的边坡风险评估

提出了基于子集模拟的边坡风险评估的高效随机有限元法（RFEM）,推导了基于子集模拟的边坡失效概率和失效风险的计算公式,并给出了基于高效RFEM的边坡可靠度分析和风险评估流程图。采用一个边坡算例验证了所提方法的有效性。结果表明,基于子集模拟的高效RFEM可以视为是对基于蒙特卡洛模拟的传统RFEM的改进,显著地提高了失效概率和失效风险的计算效率以及失效样本的产生能力,非常适用于分析小失效概率的可靠度问题,极大地增强了RFEM在边坡可靠度分析和风险评估中的实用性。高效RFEM将边坡的整体失效风险分解为对应不同概率水平的边坡失效风险,并量化了它们对整体风险的相对贡献度。在该方法中,边坡可靠度分析和风险评估与确定性边坡有限元分析互不耦合,极大地简化了它们的计算过程。此外,土体不排水抗剪强度的竖向空间变异性对边坡失效风险具有显著的影响。     

Evaluate the performance of an accelerated initial stiffness method in three dimensional finite element analysis

Newton’s method is a commonly used algorithm for elasto-plastic finite element analysis and has three common variations: the full Newton–Raphson method, the modified Newton–Raphson method and the initial stiffness method. The Newton–Raphson methods can converge to the solution in a small number of iterations when the system is stable; however, the methods can be quite computationally expensive in some types of problems, for example where the tangent stiffness matrix is unsymmetric or the plasticity is highly localized. The initial stiffness method is robust in those cases but requires a larger number of iterations. This prompted the formulation of many acceleration techniques in literature. In this paper, those techniques will be briefly discussed. This will be followed by the development of a modified acceleration technique for the initial stiffness method. The performance of the modified accelerated initial stiffness method will be examined in elasto-plastic analyses, using both direct and iterative matrix solvers. The results will be compared – in terms of the required number of iterations and the computation time – with an existing accelerated initial stiffness method, the non-accelerated initial stiffness method and the Newton–Raphson tangent stiffness method.     

Enhancement of random finite element method in reliability analysis and risk assessment of soil slopes using Subset Simulation

Random finite element method (RFEM) provides a rigorous tool to incorporate spatial variability of soil properties into reliability analysis and risk assessment of slope stability. However, it suffers from a common criticism of requiring extensive computational efforts and a lack of efficiency, particularly at small probability levels (e.g., slope failure probability P _f ?<?0.001). To address this problem, this study integrates RFEM with an advanced Monte Carlo Simulation (MCS) method called “Subset Simulation (SS)” to develop an efficient RFEM (i.e., SS-based RFEM) for reliability analysis and risk assessment of soil slopes. The proposed SS-based RFEM expresses the overall risk of slope failure as a weighed aggregation of slope failure risk at different probability levels and quantifies the relative contributions of slope failure risk at different probability levels to the overall risk of slope failure. Equations are derived for integrating SS with RFEM to evaluate the probability (P _f ) and risk (R) of slope failure. These equations are illustrated using a soil slope example. It is shown that the P _f and R are evaluated properly using the proposed approach. Compared with the original RFEM with direct MCS, the SS-based RFEM improves, significantly, the computational efficiency of evaluating P _f and R. This enhances the applications of RFEM in the reliability analysis and risk assessment of slope stability. With the aid of improved computational efficiency, a sensitivity study is also performed to explore effects of vertical spatial variability of soil properties on R. It is found that the vertical spatial variability affects the slope failure risk significantly.     

应变局部化弹黏塑性分析的内嵌局部软化带模型

针对岩土介质结构在破坏过程中局部化变形的问题,结合位移不连续的思想,提出内嵌局部软化带模型来捕捉结构中的局部化带。通过虚功原理建立了含局部化带影响的弹黏塑性的有限元计算模式,其中分叉理论作为局部化判断条件。模型将局部化带的形成视为一个黏塑性屈服流动过程,从而能够连续地描述局部化变形前后的力学性质。特点是计算量小、物理意义明确,可以方便地整合到传统有限元分析程序中。算例表明,计算模型是合理和有效的。     

The algebraic evaluation of two-dimensional finite strain rosettes

Arrays or rosettes of lines along which the extension or relative extension can be determined constitute finite strain gauge rosettes. Expressions for the principal strains and their orientation in terms of three nonparallel gauge extensions can be established in a suitable form for algebraic evaluation, thus replacing graphical methods based on Mohr construction. Three types of strain rosette problems, including one which utilizes angles, are particularly relevant to the study of deformed rocks. These, together with their relation to grid methods, are discussed and simple examples of their use given. Finally, an approach to the problem of finding best-fitting solutions to overdetermined regular strain rosettes is discussed.     

Analysis of three-dimensional, homogeneous, finite strain using ellipsoidal objects

It has been suggested (Oertel, 1971, 1972;Owens, 1974; Shimamoto and Ikeda, 1976) that some methods for analysis of finite homogeneous strain from deformed ellipsoidal objects (Ramsay, 1967; Dunnet, 1969a; Elliott, 1970; Dunnet and Siddans, 1971; Matthews et al., 1974) require sections to be cut in principal planes of the finite strain ellipsoid. A mathematical model is presented which enables the homogeneous deformation of a randomly oriented ellipsoid to be investigated. In particular the elliptical shapes that result on any three mutually perpendicular sections through the ellipsoid, in the deformed state, can be computed, together with the corresponding strain ellipses. The resulting ellipses can be unstrained in the section planes by applying the corresponding reciprocal strain ellipses. It is shown that these restored ellipses are identical with the elliptical shapes that result on planes through the original ellipsoid when the planes are parallel to the unstrained orientation of the section planes.The model is extended to investigate the finite homogeneous deformation of a suite of 100 randomly oriented ellipsoids of constant initial axial ratio. The pattern of elliptical shapes that result on any three mutually perpendicular section planes, in the deformed state, is computed. From this data the two-dimensional strain states in the section planes are estimated by a variety of methods. These are combined to recalculate the three-dimensional finite strain that was imposed on the system. It is thus possible to compare the results of the two- and three-dimensional analyses obtained by the various methods. It is found that providing all six independent combinations of the two-dimensional strain data are used to compute a best finite strain ellipsoid, the methods of Dunnet (1969a), Matthews et al. (1974) and Shimamoto and Ikeda (1976) provide accurate estimates of the three-dimensional finite strain state.It is concluded that measurement of the two-dimensional data on section planes parallel to the principal planes of the finite strain ellipsoid is not necessary and that all six independent combinations of the two-dimensional strain data should always be made and used to compute a best finite strain ellipsoid.     

Rock-soil slope stability analysis by two-phase random media and finite elements

To investigate the strong random nature of the geometric interfaces between soil and rock, a rock-soil slope is considered as a two-phase random medium. A nonlinear translation of a Gaussian field is utilized to simulate the two-phase random media, such that the soil(or rock) volume fraction and the inclination of the soil layer can be examined. The finite element method with random media incorporated as the material properties is used to determine the factor of safety of the rock-soil slope. Monte-Carlo simulations are used to estimate the statistical characteristics of the factor of safety. The failure mode of the rock-soil slope is examined by observing the maximum principal plastic strain at incipient slope failure. It is found that the critical surface of a rock-soil slope is fairly irregular, and it significantly differs from that of a pure soil slope. The factor of safety is sensitive to the soil volume faction, but it is predictable. The average factor of safety could be well predicted by the weighted harmonic average between the strength of soil and rock; the prediction model is practical and simple. Parametric studies on the inclination of the soil layer demonstrate that the most instable scenario occurs when the slope angle is consistent with the inclination of the soil layer.     

An improved accelerated initial stress procedure for elasto-plastic finite element analysis

The most flexible and generally applicable methods for elasto-plastic analysis are those based on an incremental-iterative form of the initial stress approach, but such methods often exhibit slow convergence. The acceleration procedure known as the alpha-constant stiffness method is reconsidered and some modifications are proposed. The principal difference in the present approach lies in the use of a single acceleration parameter, rather than a diagonal matrix of acceleration coefficients. The new scheme shows a significant improvement in numerical stability and converges three times faster than the standard initial stress method. Some practical aspects associated with the method are discussed and a number of applications are presented.     

Three-dimensional slope reliability and risk assessment using auxiliary random finite element method

This paper aims to propose an auxiliary random finite element method (ARFEM) for efficient three-dimensional (3-D) slope reliability analysis and risk assessment considering spatial variability of soil properties. The ARFEM mainly consists of two steps: (1) preliminary analysis using a relatively coarse finite-element model and Subset Simulation, and (2) target analysis using a detailed finite-element model and response conditioning method. The 3-D spatial variability of soil properties is explicitly modeled using the expansion optimal linear estimation approach. A 3-D soil slope example is presented to demonstrate the validity of ARFEM. Finally, a sensitivity study is carried out to explore the effect of horizontal spatial variability. The results indicate that the proposed ARFEM not only provides reasonably accurate estimates of slope failure probability and risk, but also significantly reduces the computational effort at small probability levels. 3-D slope probabilistic analysis (including both 3-D slope stability analysis and 3-D spatial variability modeling) can reflect slope failure mechanism more realistically in terms of the shape, location and length of slip surface. Horizontal spatial variability can significantly influence the failure mode, reliability and risk of 3-D slopes, especially for long slopes with relatively strong horizontal spatial variability. These effects can be properly incorporated into 3-D slope reliability analysis and risk assessment using ARFEM.     

Strain analysis using deformed pebbles: The influence of initial pebble shape

Several methods exist for the determination of the finite strain ellipsoid from deformed pebble shapes. These methods are critically evaluated and others are proposed on the basis of calculations which predict both the sectional and three-dimensional shape of pebbles in simple deformed simulated conglomerates. In many cases it is found preferable to use an average pebble shape to estimate the tectonic strain and that the harmonic mean of the ratios of axial lengths yields an average pebble shape which is closest to the strain ellipsoid shape.     

A computer program for the determination of finite strain using fry method

Fry method enables rapid estimate of finite strain from deformed aggregates such as clastic grains, fossil colonies, oolitic or pisolitic aggregates, prophyroblastic minerals or phenocrysts. It has an advantage over the other methods of finite strain analysis in its very quality of enabling rapid estimation with a reasonable degree of accuracy. Details of the software to prepare a plot using Fry method are outlined. This program has an advantage over other computer based programs on the world wide web in its aesthetic getup, small size, user friendliness and a help file.     

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.     

Vorticity and strain analysis using Mohr diagrams

Fabric elements in naturally deformed rocks are usually of a highly variable nature, and measurements contain a high degree of uncertainty. Calculation of general deformation parameters such as finite strain, volume change or the vorticity number of the flow can be difficult with such data. We present an application of the Mohr diagram for stretch which can be used with poorly constrained data on stretch and rotation of lines to construct the best fit to the position gradient tensor; this tensor describes all deformation parameters. The method has been tested on a slate specimen, yielding a kinematic vorticity number of 0.8 ± 0.1.     

极限分析有限元法讲座—— Ⅰ岩土工程极限分析有限元法

经典岩土工程极限分析方法一般采用解析方法，有些还要对滑动面作假设，且不适用于非均质材料，尤其是强度不均的岩石工程，从而使极限分析法的应用受到限制。随着计算技术的发展，极限分析有限元法应运而生，它能通过强度降低或者荷载增加直接算得岩土工程的安全系数和滑动面，十分贴近工程设计。为此，探讨了极限分析有限元法及其在边坡、地基、隧道稳定性计算中的应用，算例表明了此法的可行性，拓宽了该方法的应用范围。随着计算机技术与计算力学的发展，岩土工程极限分析有限元法正在成为一门新的学问，而且有着良好的发展前景。     

边坡稳定的有限元塑性极限分析

系统地介绍了结构塑性极限分析的原理和方法。借助有限单元法和线性规划,运用塑性极限分析的下限法,可求解岩土边坡的极限承载力和安全系数。算例表明了该方法的正确性并具有较高的精度,同时,分析了该方法的困难所在。     

The strain history of a dalradian slide: Using pebbles with low fluctuations in axis orientation

The strain associated with the Horn Head Slide, a major tectonic break in the Dalradian rocks of NW Ireland, is recorded by pebbles in an adjacent quartzite horizon. Mean X/Y ratios of the deformed pebble shapes in excess of 8.0 are seen closest to the slide and the field of three-dimensional shapes lies along the K = 1 line. The usual methods of separating initial shape ratio (R_i) and tectonic strain ratio (R_s) from the deformed shape ratio (R_f) of ellipsoidal markers are based on measurements of variation in fluctuation (e.g., the (R_f/φ technique). However, due to the high X/Y strains in this situation and since the pebbles initially lay parallel to bedding and to a principal plane of the finite strain ellipsoid, fluctuation is generally very low. Thus, except for the least deformed X/Y data, the R_f/φ technique is inapplicable and other methods are used. For X/Y data with mean (R_f > 4.0: R_s is calculated as the harmonic mean of R_f; maximum R_i values only are obtained from the range of R_f data. For all Y/Z and X/Z data: R_i is calculated from the logarithmic range (ω_log) of R_f; R_s is simply obtained from the geometric mean of R_f modified by R_i. It is concluded from this that a varying prolate tectonic strain (K - 1.5) reaching X/Y values in excess of 8.00 was coaxially superimposed on an initial oblate shape fabric to produce the present field of deformed pebbles in the quartzite near the slide.     

The analysis of finite elasto-plastic consolidation

A theoretical formulation and a numerical solution method are proposed for the problem of the time dependent consolidation of an elasto-plastic soil subject to finite deformations. The soil is assumed to be a two-phase material with a skeleton which may yield according to a general yield criterion with plastic flow governed by a general flow law, and whose pore fluid flows according to Darcy's Law. Governing equations are cast in a rate form and constitutive laws are expressed in a frame indifferent manner. The method of analysis is illustrated by several examples of practical interest for both a soil with an elastic skeleton and a soil with an elasto-plastic skeleton which obeys a Morh–Coulomb yield criterion and a non-associated flow law.