The influence of the plastic potential on plane strain failure

The influence of the shape of the plastic potential in the deviatoric plane on plane strain collapse is investigated. The most commonly employed elastic‐perfect plastic models are considered, which adopt well‐known failure criteria for defining the yield and plastic potential surfaces, namely the von Mises, the Drucker–Prager, the Tresca, the Mohr–Coulomb and the Matsuoka–Nakai criteria. Finally, the conclusions are also extended to strain hardening/softening models. For simple constitutive models based on perfect plasticity, it is shown that the value of the Lode's angle at plastic collapse in plane strain conditions strongly depends on the specific failure surface adopted for reproducing the plastic potential surface. If the value of the Lode's angle at yield coincides with the failure value prescribed by the plastic potential, the stress–strain curves exhibit the typical perfect plastic behaviour with yield coinciding with failure, otherwise the stress changes after yield and the stress‐strain curves resemble those of strain hardening/softening models. The infinite strength which is in some situations exhibited by the Drucker–Prager model in plane strain condition is investigated and explained, and it is shown that this can also affect strain hardening/softening models. Copyright © 2013 John Wiley & Sons, Ltd.     

Grain boundary plane populations in minerals: the example of wet NaCl after low strain deformation

Many physical properties of rocks are sensitive to grain size and hence to the structure of grain boundaries. Depending on their properties, such as deformation and transport behaviour, boundaries may be divided into two broad types, namely special and general grain boundaries. Electron backscattered diffraction (EBSD) is used to investigate the misorientation distributions of grain boundaries and, more recently, to determine the population of grain boundary planes. Studies on metals and ceramics suggest that the grain boundary plane, rather than the misorientation, is the key parameter when defining special and general grain boundaries. In this study, the distribution of grain boundary plane orientations has been successfully determined using EBSD for a slightly deformed, synthetic NaCl material containing 22 ppm water. Boundaries showed a preference for {100} planes, which occurred with twice the frequency of a random distribution. The grain boundary plane distributions found in NaCl were largely in agreement with studies on MgO. Grain boundaries, with a coincident site lattice (CSL) misorientation, also showed a preference for {100} planes, rather than the planes of high coincident density associated with the CSL. Three main types of boundary were identified, namely {100} twist boundaries, boundaries with {100}{hkl} planes and general {hkl}{hkl} boundaries. As the properties of these three types of boundary differ, then the transport and creep properties in wet NaCl will depend on the fraction of the different boundary types found in the grain boundary population.     

平面应变条件下土工格室加筋垫层的变形分析

将铺设在软土地基上的土工格室加筋垫层看作具有一定刚度的梁板。地基上梁板的变形计算一般以弹理论的文克尔假定为基础。针对文克尔假定中不考虑水平抗力影响的问题,提出了考虑水平抗力的双参数法,并以该法应用于土工格室加筋垫层工法的变形分析。     

An automated machine for pure shear deformation of analogue materials in plane strain

A plane strain, pure shear rig has been developed for the deformation of rock-analogue materials under tightly controlled experimental conditions. Shortening of up to 42% is possible, with constant natural strain rates and a predetermined confining side stress (σ₃). Differential stress, side stress, temperature and strain rate are recorded during the experiments, allowing rheological calibration under exactly the same conditions used in the applied studies of mechanical instabilities (folds, boudinage, etc.). The relatively large specimen size, three-piece side plate design and the extensive use of low-friction facing materials in contact with the model result in boundary conditions which closely approximate plane strain, pure shear.     

平面应变条件下粗粒土的变形特性试验研究

使用河海大学TSW－40型真三轴仪,对粗粒土分别进行平面应变和常规三轴压缩试验,研究了粗粒土在平面应变条件下的应力-应变关系及各种应变之间的关系,包括：主应力差? 1-? 3(? 2-? 3)与大主应变?1、主应力比? 1 /? 3(? 2 /? 3)与大主应变?1、球应力p与体积应变?v、偏应力q与偏应变? s、广义应力比q/p与广义剪应变? s、小主应变? 3与大主应变?1、体积应变? v与广义剪应变? s之间的关系。试验结果表明,对于相同的? 3,平面应变试验的(? 1-? 3)(或? 1 /? 3)-ε1曲线位于常规三轴压缩试验相应曲线的上方;? 3大的(? 1-? 3)(或? 2-? 3)-? 1曲线在上方,而? 3大的? 1 /? 3(或? 2 /? 3)-? 1曲线在下方;对于相同的? 3,在相同的? 1下,平面应变条件下的小主应力方向膨胀量要比常规三轴压缩条件下的大;产生相同的? s时,? 3越大,? v越大,对于相同的? 3,平面应变条件下的? v要比常规三轴压缩条件下的大;粗粒土在平面应变和常规三轴压缩条件下加荷时,具有归一化的双曲线应力-应变关系。研究表明,采用非线性弹性模型计算粗粒土的平面应变问题时,有必要考虑弹性模量E和泊松比? 的各向异性。     

On implementation of bounding surface plasticity models with no overshooting effect in solving boundary value problems

This paper presents a new scheme that can be used to overcome the overshooting effect, one of the well known problems occurs during application of bounding surface plasticity models in numerical analysis of boundary value problems. The scheme is based on definition of clouds of loading surfaces with a specific margin of strains within which unloading does not accompanied kinematic hardening. The basic concept of the scheme is introduced and the methodology and the relevant step by step algorithm to implement this scheme are presented. This scheme has been incorporated in the UNSW bounding surface model and implemented in a finite difference code and used to simulate cyclic triaxial tests as well as complicated monotonic and dynamic boundary value problems. The satisfactory performance of the scheme is demonstrated and its efficiency is discussed thorough these simulations.     

The effect of the basal-prism mechanism switch on fabric development during plastic deformation of quartzite

Blacic described a transition from basal 〈a〉 to prism 〈c〉 dislocation glide systems as temperature increased or strain-rate decreased in sequences of experiments involving deformation of single crystals of quartz. In this paper theoretical aspects of competition between these systems during plastic deformation are discussed. It is concluded that the basal-prism mechanism switch should have important consequences for the development of crystallographic fabrics during plastic deformation of quartzite, and the Taylor-Bishop-Hill analysis is used in an attempt to predict expected fabric transitions.     

Seismic plastic deformation in the free-field

An analytical solution for the free-field seismic elastoplastic deformation of granular dry soil is developed. This solution is possible since the stress field solution from pseudostatic analysis is known at any acceleration level. A plastic potential function with two plastic material parameters is suggested to model the soil. This model can describe both the volume dilation and contraction of soil under shearing. Closed-form deformation solutions are derived for two special cases and the results from a numerical evaluation of the solution are compared with those measured from shaking table tests. From the deformation solution the shape of the horizontal plastic displacement distribution with depth is identical to that for the elastic deformation since both depend only on the distribution of the elastic shear modulus. Copyright © 1999 John Wiley & Sons, Ltd.     

Measurement of the deformation of sand in a plane strain compression experiment using incremental digital image correlation

Acta Geotechnica - This work presents the results from a series of plane strain compression experiments on medium-density sands. In these experiments, a new type of plane strain experimental...     

Semi-analytical solutions of two-dimensional plane strain consolidation in unsaturated soils subjected to the lateral semipermeable drainage boundary

The study presents semi-analytical solutions of two-dimensional plane strain consolidation problem in unsaturated soils incorporating the lateral semipermeable drainage boundary by adopting Fourier sine series and Laplace transform. The two-dimensional plane strain consolidation equations in the form of two-order partial differential equations with three variables are firstly converted to two-order partial differential equations with two variables, which are similar to those of one-dimensional consolidation problem. The four-order ordinary differential equations about excess pore-air and excess pore-water pressures are got by applying Laplace transform and the substitution method. Then, the solutions of excess pore pressures and settlement are achieved in the Laplace transform domain. Afterwards, on the basis of Crump's method, the inverse Laplace transform is conducted to obtain the analytical solutions in time domain. The comparison is conducted to verify the exactness of the obtained solutions, and the two-dimensional plane strain consolidation property with the lateral semipermeable drainage boundary is illustrated and discussed. Parametric studies are demonstrated for the excess pore pressures and normalized settlement with the change of the boundary parameters, air-water and lateral-vertical permeability coefficients, and the distance and depth. It can be found that the lateral semipermeable drainage boundary impedes the consolidation rate obviously, and when different investigated parameters are adopted, the consolidation property is similar to each other under the later permeable and semipermeable drainage boundary conditions.     

The superposition of compaction and plane strain

The coaxial and non-coaxial superposition of tectonic (biaxial or plane) strain on compactional (uniaxial) strain is examined and total finite strains and deformation paths calculated. Both prolate and oblate strains may result, depending on the relative orientations of the two strains. The implications of superposed strain are discussed in relation to naturally occurring strains in rocks, and various patterns of finite strain and deformation fields recognized.     

The influence of matrix rheology and vorticity on fabric development of populations of rigid objects during plane strain deformation

The influence of vorticity and rheology of matrix material on the development of shape-preferred orientation (SPO) of populations of rigid objects was experimentally studied. Experiments in plane strain monoclinic flow were performed to model the fabric development of two populations of rectangular rigid objects with object aspect ratios (R_ob) 2 and 3. The density of the rigid object populations was 14% of the total area. Objects were dispersed in a Newtonian and a non-Newtonian, power law matrix material with a power law exponent n of 1.2. The kinematic vorticity number (W_n) of the plane strain monoclinic flow was 1, 0.8 and 0.6 with finite simple shear strain of 4.6, 3.0 and 0.9, respectively. In experiments with R_ob=3, the SPO is strongly influenced by W_n and the material properties of the matrix. Deformation of a power law matrix material and low W_n resulted in a stronger SPO than deformation of a linear viscous matrix and high W_n. Strain localization coupled with particle interaction plays a significant role in the development of a shape-preferred orientation. High strain simple shear zones separate trains of rigid objects that are surrounded by low strain zones with W_n lower than the bulk W_n. In fabrics involving populations of objects with R_ob=2, rheology of the matrix materials does not systematically influence the intensity of the SPO.     

亚洲中部天山—贝加尔陆内变形终止边界

亚洲中部,沿天山的西端、阿尔泰山的西北端到贝加尔地区,是一条显著的NE向边界(天山—贝加尔边界),在它的两侧,地形、构造、地震活动和其他地球物理特征都有显著的差异。沿这条边界不存在同方向的大断层,同时,一系列NW向大型走滑断层与这条边界以大角度相交,这些断层的西北段在边界西北侧附近终止。总体上,这条边界西北侧是稳定的哈萨克斯坦地台和西伯利亚地台,东南侧主要是构造活动的天山、阿尔泰山及蒙古西部,因此,这条NE向构造边界是大陆内部的变形终止边界。中亚地区的陆块拼合过程在古生代末已结束,天山、阿尔泰山等造山带与北边的哈萨克斯坦地台和西伯利亚地台焊接为一体,经历中生代的固化。新生代以来,受印度-欧亚板块碰撞的影响,天山、阿尔泰山等造山带在NS向挤压力作用下发生变形,包括近东西向逆断层和NW向走滑断层的活动以及分散的地震。这些变形向北扩展时,被刚性、结构完整的哈萨克斯坦地台和西伯利亚地台截断终止,因而造成NE向天山—贝加尔边界。亚洲大陆内部块体的地壳强度和变形特征有明显的继承性,新生代以来,原来的地台(克拉通)继续保持结构完整和稳定性,几乎没有内部变形,而天山、阿尔泰山等古老的造山带则再次经历强烈变形。这不仅是由于印度板块继续向北推挤的影响,而且与这些地区自身的地壳和上地幔结构有关,同时NE向天山—贝加尔边界的西北侧哈萨克斯坦地台和西伯利亚地台对变形扩散的阻挡也起了重要作用。     

Localized failure of fibre‐reinforced elastic–plastic materials subjected to plane strain loading

We consider discontinuous bifurcations as the indicator of a localized failure for a class of composites that are characterized by elastic fibres reinforcing an elastic–plastic matrix. A macroscopic tangent stiffness tensor for the fibre‐reinforced composite is developed by consistently homogenizing the contribution of fibres in a spherical representative volume element. Analytical solutions are derived for the critical hardening modulus and corresponding bifurcation directions for the case of plane strain loading. Properties of the solutions are further illustrated on the example of the non‐associated Drucker–Prager model at onset of yielding. Results show that presence of fibres decreases the critical hardening modulus, thus inhibiting the onset of strain localization. The rate of decrease in the critical hardening modulus is the highest for pure shear, followed by uniaxial tension, uniaxial compression, biaxial tension and biaxial compression. The main fibre parameters that control the onset of strain localization are their volumetric content and their stiffness modulus whereby very stiff fibres can produce the most significant decrease in the critical hardening modulus, especially for the state of biaxial tension. The critical hardening modulus for the non‐associated Drucker–Prager model exhibits a full range of localization modes including compaction bands, dilation bands, and transition in the form of shear bands regardless of the presence of fibres. Presence of fibres affects bifurcation directions, except in the case when Poisson's ratio of the matrix is equal to 0.25. The results demonstrate stabilizing effects of fibres by which they provide the control against the onset of strain localization. Copyright © 2006 John Wiley & Sons, Ltd.     

An edge-based strain smoothing particle finite element method for large deformation problems in geotechnical engineering

To solve large deformation geotechnical problems, a novel strain-smoothed particle finite element method (SPFEM) is proposed that incorporates a simple and effective edge-based strain smoothing method within the framework of original PFEM. Compared with the original PFEM, the proposed novel SPFEM can solve the volumetric locking problem like previously developed node-based smoothed PFEM when lower-order triangular element is used. Compared with the node-based smoothed PFEM known as “overly soft” or underestimation property, the proposed SPFEM offers super-convergent and very accurate solutions due to the implementation of edge-based strain smoothing method. To guarantee the computational stability, the proposed SPFEM uses an explicit time integration scheme and adopts an adaptive updating time step. Performance of the proposed SPFEM for geotechnical problems is first examined by four benchmark numerical examples: (a) bar vibrations, (b) large settlement of strip footing, (c) collapse of aluminium bars column, and (d) failure of a homogeneous soil slope. Finally, the progressive failure of slope of sensitive clay is simulated using the proposed SPFEM to show its outstanding performance in solving large deformation geotechnical problems. All results demonstrate that the novel SPFEM is a powerful and easily extensible numerical method for analysing large deformation problems in geotechnical engineering.     

平面应变状态下岩石剪切带网络数值模拟研究

研究了围压、端面约束、试件高度及界面特性对剪切带网络形成的影响。首先对影响剪切带网络形成的因素（包括试件高度、围压及温度、试件端部轴向应变大小的计算方法的差异）进行了分析。之后采用FLAC 3D对平面剪切带网络进行了数值模拟,其中摩擦角及内聚力为应变软化。探讨了倾向于出现剪切带网络的若干条件。若端面缚束较强且存在一定的侧压力,试件中部出现多重剪切带。若端面缚束较弱,试件端部易于形成剪切带网络,且随试件高度的增加剪切带条数增加。界面法向和切向刚度越大,剪切带穿越断层并按其固有方向延伸能力越强,剪切带网络格局越明显。获得的数值结果可在实验中找到佐证,并且可以用来解释地震中的一些剪切应变局部化现象。     

Biotic characteristics of the Sinian-Cambrian boundary beds in China and the boundary problems

There is general agreement that the first appearance of various small shelly fossils should be used to help delineate the beginning of the Cambrian Period. The history of research on this stratigraphic topic in China is briefly outlined and leads on to a discussion of the boundary problem in the light of information available both inside and outside China.The sections across the Precambrian—Cambrian (Sinian—Cambrian) boundary are described with notes on fossils in: (1) Tianzhushan section, Yichang, Hubei; (2) Maidiping section, Emli, Sichuan; (3) Wangjiawan section, Jinning County, Yunnan; and (4) Kunyang section, Yunnan.This is followed by a consideration of the biotic characteristics of Sinian—Cambrian Period. The history of research on this stratigraphic topic in China is briefly outlined and leads on to a discussion of the boundary problem in the light of information available both inside and outside China. This is followed by a consideration of the biotic characteristics of Sinian—Cambrian boundary beds, with respect to stratigraphy as well as organic evolution. The Precambrian—Cambrian boundary in China is estimated to have an age of 615 ± 20 Ma.It is commented that the above sections are qualified to be important candidates for the selection of the Sinian—Cambrian boundary stratotype and are also important working reference sections for the international Precambrian—Cambrian boundary stratotype.     

Successive development of plane noncylindrical folds in progressive deformation

In the history of superposed deformations of the iron formations at the western border of the Kolar Gold Field in S India, an important event was the successive growth of broadly coaxial plane noncylindrical folds in course of a progressive deformation concomitant with development of ductile mesoscopic shear zones. The noncylindrical folds were initiated as active folds by the creation of a buckling instability at successive stages on newly developed foliation surfaces. The nucleation of noncylindrical folds and the subsequent axial-plane folding of the tightened mature folds are explained by the mechanical inhomogeneity of the rocks and the heterogeneous character of strain. The correlation between increasing tightness and increasing noncylindricity of the folds indicates that the initial curvatures of hinge lines were accentuated by an extension parallel to the subhorizontal stretching lineation. From the patterns of deformed lineations over folds of varying tightnesses, it is concluded that the passive accentuation of hinge-line curvatures was mostly achieved when the folds had already become isoclinal or very tight.     

The simulation of fabric development during plastic deformation and its application to quartzite: the influence of deformation history

The effect of deformation history on the development of crystallographic preferred orientation in quartzities has been simulated using a computer program based on the Taylor-Bishop-Hill analysis. Model quartzities with different combinations of glide systems have been subjected to various coaxial and non-coaxial deformation histories. It is possible to obtain information from the fabrics that develop during simple histories; for example, the location of the axis of extension is generally associated with a pole free area on a c-axis plot, and progressive axial shortening, plane strain and axial shortening produce characteristic fabrics. In progressive simple shear the fabric skeleton becomes asymmetric relative to the sense of shear and a-axes preferentially align in the flow plane parallel to the flow direction. However, this example illustrates that the fabric orientation and characteristics are controlled by the kinematic framework and bear only an indirect relationship to the finite strain accumulated to that point in the history.The imprint of the closing stages of deformation limits to some degree the use of crystallographic fabrics as a tool for structural geologists, but in favourable circumstances data can be obtained concerning characteristics of the deformation history, on the scale of the hand-specimen, for the last part of this history.     

From discrete to continuum modelling of boundary value problems in geomechanics: An integrated FEM-DEM approach

Double-scale numerical methods constitute an effective tool for simultaneously representing the complex nature of geomaterials and treating real-scale engineering problems such as a tunnel excavation or a pressuremetre at a reasonable numerical cost. This paper presents an approach coupling discrete elements (DEM) at the microscale with finite elements (FEM) at the macroscale. In this approach, a DEM-based numerical constitutive law is embedded into a standard FEM formulation. In this regard, an exhaustive discussion is presented on how a 2D/3D granular assembly can be used to generate, step by step along the overall computation process, a consistent Numerically Homogenised Law. The paper also focuses on some recent developments including a comprehensive discussion of the efficiency of Newton-like operators, the introduction of a regularisation technique at the macroscale by means of a second gradient framework, and the development of parallelisation techniques to alleviate the computational cost of the proposed approach. Some real-scale problems taking into account the material spatial variability are illustrated, proving the numerical efficiency of the proposed approach and the benefit of a particle-based strategy.