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.     

3-D geomechanical restoration and paleomagnetic analysis of fault-related folds: An example from the Yanjinggou anticline,southern Sichuan Basin

We examine the development of the Yanjinggou anticline, a fault-propagation-fold in the southern Longmen Shan, through an integrated study of structural geometry, strain, and paleomagnetism. The 3-D structural and strain restoration models generated in our analysis reveal that the NE-trending Yanjinggou fold has a curved map trace that is convex to the southeast. The fold has three distinct regions characterized by different strain patterns: contraction in the core of the fold, extension in the outer arc, and a forelimb with distributed shear. To further understand the kinematics of the Yanjinggou anticline, we performed paleomagnetic analysis on 184 oriented samples collected across the structure. Anisotropy of magnetic susceptibility (AMS) measurements and stepwise thermal demagnetization were conducted. A strike test was applied to the high temperature component (HTC) in order to identify rotation around the arc. The result indicates that the Yanjinggou anticline is a progressive arc, with a minor initial curvature and a dominant secondary curvature related to vertical-axis rotation synchronous with thrusting. The primary curvature and initial development of the structure correlates with the growth of the southern Longmen Shan in Late Miocene. The secondary curvature correlates with displacement extending since Late Pleistocene toward the southeast into the central basin along the detachments that underlie the structure. Lateral gradients in displacement along this underlying detachment provide a mechanism for producing the vertical rotation of the anticline. AMS results and historical earthquake analysis imply that the fault-propagation fold, along with other NE trending structures in the southern Sichuan basin, are tectonically active and accommodate east-west crustal shortening in the basin. By integrating 3-D structural and strain restoration modeling with systematic AMS and paleomagnetic methods using statistical analysis, we closely constrain how the Yanjinggou anticline developed, and provide insights into the formation of fault-related folds with curved shapes in map view, which are common in other fold-and-thrust belts around the world.     

A new approach for calculating strain for particulate media

Discrete element modelling is a viable alternative to conventional continuum‐based analysis for analysing problems involving localized deformations of particulate media. However, to aid in the interpretation of the results, it is useful to express the results of discrete element analyses in terms of the continuum parameters of stress and strain. A number of homogenization methods have been proposed to calculate strain in discrete systems; however, two significant limitations of these methods remain. First, none of these methods incorporate particle rotation effects satisfactorily, although significant particle rotation occurs in shear bands in both physical tests and numerical simulations of granular materials. Additionally, observations of the particle displacement fields in shear bands in granular materials indicate that the displacements within the localizations are erratic. Consequently, existing linear, local interpolation approaches produce substantial variations in the strain values calculated in adjacent elements in the region of localization, hindering clear visualization of the strain localization as it evolves. A new method of domain discretization for calculating strain is proposed. This method is capable of capturing particle rotation and employs a non‐local meshfree interpolation procedure capable of smoothing the erratic displacements in strain localizations, which better defines their evolution. The proposed method is validated for problems involving both two and three dimensions. A number of methods are compared with the proposed method and pertinent insights are made. Copyright © 2003 John Wiley & Sons, Ltd.     

A non‐coaxial constitutive model for sand deformation under rotation of principal stress axes

A constitutive model for the simulation of non‐coaxiality, an aspect of anisotropic behavior of sand subjected to the rotation of the principal stress axes, is presented in this paper. Experimental studies have shown that non‐coaxiality or non‐coincidence of principal plastic strain increments with principal stress axes under loadings involving the rotation of principal stress axes may be considerable. Besides, the rotation of the principal stress axes results in dramatic effects on stiffness and dilatant behavior of sand. Therefore, the consequences of principal stress axes rotation on deformational behavior, dilatancy and soil stiffness must be taken into account in theoretical and practical problems. To this aim, the following steps are taken: (1) A general relationship for flow direction with respect to possibility of non‐coaxial flow is developed. Moreover, special circumstances linking non‐coaxiality to instantaneous interaction between loading and soil fabric are proposed. (2) Proposing novel expressions for plastic modulus and dilatancy function, the model is enforced to provide realistic simulations when sand is subjected to the rotation of the principal stress axes. Finally, with numerous examples and comparisons, the model capabilities are shown under various stress paths and drainage conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

Strain rates at high temporal resolution from curved inclusion trails in garnet,Passo del Sole,Central Swiss Alps

Quantitative strain rates at outcrop scale are very difficult to obtain, but they may be estimated from crystals with curved inclusion trails by calculating rotation rates from growth rates and corresponding deflections of the internal foliation. Garnet in a quartzose pelite at Passo del Sole in the central Swiss Alps is extraordinarily valuable for calculation of strain rates during Alpine orogenesis, because the unusual zoning patterns clearly define the kinetics of its nucleation and growth. Complex concentric zoning patterns can be correlated from one crystal to another in a hand sample, based on compositional and microstructural similarities; the ubiquity of these features demonstrates that all garnet crystals nucleated at nearly the same time. Compositional bands whose radial widths are proportional to crystal size provide evidence for growth governed by the kinetics of intergranular diffusion of locally sourced nutrients. Together, these constraints increase the reliability of estimates of rates of garnet growth, and the strain‐rate calculations that depend on them. To obtain growth rates, P–T conditions during garnet crystallization were modelled in a series of pseudosections, and compositional evolution was connected to rates of garnet growth by means of an independently determined heating rate. These growth rates, combined with measured amounts of curvature of inclusion trails, indicate that the time‐averaged strain rate at Passo del Sole during Alpine metamorphism was on the order of 10^?14 s^?1. Strain rates calculated using rotational v. non‐rotational models are similar in magnitude. The constraints on crystallization kinetics also allow direct calculation of strain rates during individual stages of garnet growth, revealing short‐term increases to values on the order of 10^?13 s^?1. These higher strain rates are correlated with the growth of concentric high‐Ca or high‐Mn zones in garnet, which implies that strain softening associated with the transient passage of fluids is responsible for acceleration of deformation during these intervals.     

Numerical modeling of heterogeneous flow fields around rigid objects with special reference to particle paths, strain shadows and foliation drag

With the help of two-dimensional numerical models this paper investigates three aspects of heterogeneous deformation around rigid objects: (1) the nature of particle paths; (2) the development of strain shadow zones; and (3) the drag patterns of passive markers. In simple shear, spherical objects develop typically a concentric vortex motion, showing particle paths with an eye (double-bulge)-shaped separatrix. The separatrix has no finite dimension along the central line, parallel to the shear direction. Under a combination of pure shear and simple shear, the particle paths assume a pattern with a bow-tie shaped separatrix. With increase in the ratio of pure shear to simple shear (S_r), the separatrix around the object shrinks in size. The axial ratio of the object (R) is another important factor that controls the geometry of particle paths. When R<1.5, the loci of a particle close to the object form an elliptical shell with the long axis lying along the central line. With increase in axial ratio R, the loci form a doublet elliptical shell structure. Objects with R>3 do not show closed particle paths, but give rise to elliptical or circular spiral particle paths.

The development of strain shadow zones against equant rigid bodies depends strongly on the strain ratio S_r. When S_r=0 (simple shear), they develop opposite to the extensional faces of the object, forming a typical σ-type tail. The structure has a tendency to die out with an increase in the pure shear component of the bulk deformation (S_r). The initial angle of the long axis of the object with the shear direction (φ) and the axial ratio of the object (R) determine the development of strain shadow zones near inequant rigid objects. Objects with large R and φ between 60 and 120° form pronounced zones of low finite strain, giving rise to strain shadow structures. A geometrical classification of diverse drag patterns of passive markers around rigid objects is presented along with their conditions of formation.     

A novel analytical approach for predicting the noncylindrical pile penetration‐induced soil displacement in undrained soil by combining use of cavity expansion and strain path methods

Since development of cavity expansion theory and strain path method, almost all the conventional analyses of pile penetration problem have been based on circular cross section penetrometer. However, noncylindrical pile (with noncircular cross section) is also required in geotechnical engineering such as rectangular cross‐sectional pile, X‐sectional cast‐in‐place concrete pile, H‐shaped steel pile, prefabricated vertical drains, and flat dilatometer. This paper presents a novel and general analytical approach for capturing the soil deformation mechanism around the pile with arbitrary cross section. The penetration problem is simulated by a new 2‐dimensional (radial and circumferential) cavity expansion model. Based on the theoretical framework of strain path method, the kinematics (velocity field) of the noncylindrical cavity expansion is reduced to solve the Laplace equation with arbitrary velocity boundary conditions by using the conformal mapping technique. Then, solutions for the strain and displacement, which could consider the large deformation effect, are obtained by the integration of the strain rate and velocity along the streamline. The analytical solution is validated by comparing the degenerate solution of this study with conventional circular (cylindrical) cavity expansion theory. Subsequently, typical numerical examples for the deformation mechanism of elliptical and rectangular cavity expansion are presented to prove the advantage of the proposed new solution particularly in capturing the noncylindrical symmetric displacement field. A brief application of the proposed new analytical solution to the interpretation of the smear effect of prefabricated vertical drain installation confirms its useful in geotechnical engineering.     

Fibre reinforced sands: from experiments to modelling and beyond

Based on hypotheses derived directly from experimental observations of the triaxial behaviour, a constitutive model for fibre reinforced sands is built in this paper. Both the sand matrix and the fibres obey their own constitutive law, whereas their contributions are superimposed using a volumetric homogenization procedure. The Severn‐Trent sand model, which combines well‐known concepts such as critical state theory, Mohr‐Coulomb like strength criterion, bounding surface plasticity and kinematic hardening, is adopted for the sand matrix. Although the fibres are treated as discrete forces with defined orientation, an equivalent continuum stress for the fibre phase is derived to allow the superposition of effects of sand and fibres. The fibres are considered as purely tensile elements following a linear elastic constitutive rule. The strain in the fibres is expressed as a fraction of the strain in the reinforced sample so that imperfect bonding is assumed at the sand‐fibre interface. Only those fibres oriented within the tensile strain domain of the sample can mobilize tensile stress—the orientation of fibres is one of the key ingredients to capture the anisotropic behaviour of fibre reinforced soil that is observed for triaxial compression and extension loading. A further mechanism of partition of the volume of voids between the fibres and the sand matrix is introduced and shown to be fundamental for the simulation of the volumetric behaviour of fibre‐reinforced soils. Copyright © 2012 John Wiley & Sons, Ltd.     

模型相似材料内部应变光纤量测应变传递

应用光纤传感技术测量模型相似材料内部应变,必须解决光纤与相似材料耦合的问题。传统的研究中普遍存在应变传递理论分析结果无法通过试验检验的问题。设计实现了一种基于光纤-胶体-相似材料3层应变传递结构的光纤Bragg光栅（FBG）相似材料一维传感器。通过对该传感器应变传递特性和标定方法的研究探索解决光纤与相似材料耦合性的问题。该传感器采用无涂覆层的光纤Bragg光栅作为传感元件,将光纤埋入相似材料应变试件内部直接测量轴向中心应变。通过应变传递理论分析、数值模拟以及全新设计的标定试验研究3种方法研究了该相似材料光栅传感器的应变传递特性。平均应变传递率结果显示3种方法所得到的数值非常接近,表明该传感器的光纤、胶体、相似材料具有良好的耦合性以及应变测量的准确性。该方法为解决模型相似材料内部三维应变测量提供了可行方案。     

A virtual experiment technique on the elementary behaviour of granular materials with discrete element method

Multi‐scale investigations aided by the discrete element method (DEM) play a vital role for current state‐of‐the‐art research on the elementary behaviour of granular materials. Similar to laboratory tests, there are three important aspects to be considered carefully, which are the proper stress/strain definition and measurement, the application of target loading paths and the designed experiment setup, to be addressed in the present paper. Considering the volume sensitive characteristics of granular materials, in the proposed technique, the deformation of the tested specimen is controlled and measured by deformation gradient tensor involving both the undeformed configuration and the current configuration. Definitions of Biot strain and Cauchy stress are adopted. The expressions of them in terms of contact forces and particle displacements, respectively, are derived. The boundary of the tested specimen consists of rigid massless planar units. It is suggested that the representative element uses a convex polyhedral (polygonal) shape to minimize possible boundary arching effects. General loading paths are described by directly specifying the changes in the stress/strain invariants or directions. Loading can be applied in the strain‐controlled mode by specifying the translations and rotations of the boundary units, or in the stress‐controlled mode by using a servo‐control mechanism, or in the combination of the two methods to realize mixed boundary conditions. Taking the simulation results as the natural consequences originated from a complex system, virtual experiments provide particle‐scale information database to conduct multi‐scale investigations for better understanding in granular material behaviours and possible development of the constitutive theories provided the qualitative similarity between the simulation results from virtual experiments and observations on real material behaviour. Copyright © 2011 John Wiley & Sons, Ltd.     

Development of garnet porphyroblasts by multiple nucleation,coalescence and boundary misorientation‐driven rotations

Two types of garnet porphyroblast occur in the Schneeberg Complex of the Italian Alps. Type 1 porphyroblasts form ellipsoidal pods with a centre consisting of unstrained quartz, decussate mica and small garnet grains, and a margin containing large garnet grains. Orientation contrast imaging using the scanning electron microscope shows that the larger marginal garnet grains comprise a number of orientation subdomains. Individual garnet grains without subdomains are small (< 50 µm), faceted and idioblastic, and have simple zoning profiles with Ca‐rich cores and Ca‐poor rims. Subdomains of larger garnet grains are similar in size to the individual, small garnet grains. Type 2 porphyroblasts comprise only ellipsoidal garnet, with small subdomains in the centre and larger subdomains at the margin. Each subdomain has its own Ca high, Ca dropping towards subdomain boundaries. Garnet grains, with or without subdomains, all have the same Ca‐poor composition at rims in contact with other minerals. The compositional zonation patterns are best explained by simultaneous, multiple nucleation, followed by growth and amalgamation of individual garnet grains. The range of individual garnet and garnet subdomain sizes can be explained by a faster growth rate at the porphyroblast margin than in the centre. The difference between Type 1 and Type 2 porphyroblasts is probably related to the growth rate differential across the porphyroblast. Electron backscatter diffraction shows that small, individual garnet grains are randomly oriented. Large marginal garnet grains and subdomain‐bearing garnet grains have a strong preferred orientation, clustering around a single garnet orientation. Misorientations across subdomain boundaries are small and misorientation axes are randomly oriented with respect to crystallographic orientations. The only explanation that fits the observational data is that individual garnet grains rotated towards coincident orientations once they came into contact with each other. This process was driven by the reduction of subdomain boundary energy associated with misorientation loss. Rotation of garnet grains was accommodated by diffusion in the subdomain boundary and diffusional creep and rigid body rotation of other minerals (quartz and mica) around the garnet. An analytical model, in which the kinetics of garnet rotation are controlled by the rheology of surrounding quartz, suggests that, at the conditions of metamorphism, the rotation required to give a strong preferred orientation can occur on a similar time‐scale to that of porphyroblast growth.     

Comparison of vibrocompaction methods by numerical simulations

Soils can be best compacted by repeated shearing. The strain amplitude plays an important role for the maximum compaction that can be reached. Experimental evidence emphasizes a vital impact of simultaneous multidirectional shear loading on the rate and magnitude of soil compaction. Two different vibrocompaction methods were analysed by the numerical simulations in the light of these findings. In an elastic finite element (FE) analysis, strain paths were determined. A strain amplitude‐dependent stiffness at small strains was introduced by multiple runs of the FE calculation to reach an appropriate stiffness for particular distances from the vibrator. Subsequently, the obtained strain paths were used to control single element simulations using hypoplasticity with intergranular strains. The calculated compaction profiles show three zones known from practical evidence: a limited compaction close to the vibrator, a zone of maximum compaction and a non‐densified zone remote from the vibrator. The deep vibrator produces a faster compaction than the top vibrator, especially in the more distant zone. The more efficient work of the deep vibrator can be attributed to a more general multidirectional shearing. Copyright © 2009 John Wiley & Sons, Ltd.     

The role of non‐coaxiality in the simulation of strain localization based on classical and Cosserat continua

It is normally accepted that materials inside the shear band undergo severe rotation of the principal stress direction, which causes non‐coaxiality between the principal stress and principal plastic strain rate. However, classical plasticity flow theory implicitly assumes that the principal stress and the principal plastic strain rate are coaxial; thus, it may not correctly predict the onset of the shear band. In addition, classical continuum does not contain any internal length scales; as a result, it cannot provide a reasonable shear band thickness. In this study, the original vertex non‐coaxial plastic model based on the classical continuum is extended to the Cosserat continuum. The corresponding codes are implemented via the interface of the user defined element subroutine in ABAQUS. Through a simple shear test, the effectiveness of the user's codes is verified. Through a uniaxial compression test, the influence of non‐coaxiality on the onset, the orientation, and the thickness of the shear band is investigated. Results show that the onset of the shear localization is delayed, and the thickness of the shear band is widened when the non‐coaxial degree increases, while the orientation of the shear band is little affected by the non‐coaxial degree. In addition, it is found that the non‐coaxiality can weaken the micro‐polar effect to some extent; nonetheless, the Cosserat non‐coaxial model still has its advantage over the classical non‐coaxial model in capturing the pre‐bifurcation as well as the post‐bifurcation behaviors of strain localization. Copyright © 2016 John Wiley & Sons, Ltd.     

Diffusion-creep modelling of fibrous pressure-shadows

Grain-scale deformation of a bimineralic composite by normal-stress-driven grain-boundary diffusive mass-transfer has been modelled using diffusion-creep software that, for the first time, allows different diffusion coefficients for interphase boundaries and grain-boundaries as well as for different components. By selecting suitable coefficients we have obtained a fully coupled mechanical and mass-transfer numerical solution for the formation of fibrous pressure-shadows or strain fringes at the interface between a rigid object and a deforming rock matrix of different compositions. The model is two-dimensional and run in pure-shear velocity mode to a stretch ratio of 1.4. The first results are reported, which show dissolution and face-controlled antitaxial growth of a soluble, polycrystalline, calcite matrix at the interface with a circular and relatively insoluble pyrite object.The ratio of CaCO₃ diffusion along calcite–pyrite versus calcite–calcite boundaries strongly influences the extent of pressure-shadow growth and the pattern of strains in the matrix. A low ratio produces little growth, significant perturbation, and deflection of total-strain axes around the pyrite. An intermediate ratio produces appreciable growth, little perturbation, and near-parallel total-strain axes. A high ratio produces larger pressure-shadows, and total-strain axes that bend in towards the pyrite as if it were a weak object. Sliding and rotation of matrix grains are also affected.These results differ fundamentally from purely physical models of ductile flow around a rigid object and demonstrate the influence of interface diffusional properties on the resulting behaviour. In particular, assumptions such as perturbation of the flow by the object, deflection of external foliation around pre-deformational porphyroblasts and relationships of pressure-shadow strain to bulk strain may need to be re-examined in the light of this finding.     

Modelling of fibre–cohesive soil mixtures

A new constitutive model for fibre-reinforced cohesive soil is proposed. The model combines a Cam-Clay like bounding surface model with an elastic–plastic one-dimensional fibrous element model. A “smearing procedure”, which can consider any spatial distribution of fibre orientation, is employed to transform discrete tensile forces developed in the fibres into stresses for the composite material. The fibre stress contribution is bounded by both degradation of soil–fibre bonding due to pull-out mechanism and tensile strength of the fibres. Eventual occurrence of fibre breakage is also considered. The model performances are analysed for both consolidation and shearing loading modes, and qualitative comparison is performed with experimental data available in the literature. For consolidation loading, tensile stresses are not developed in the fibres and thus the fibre effect is rather limited. For drained shear loading, addition of fibres can result in a consistent shear strength increase. The beneficial effect of fibres seems to be controlled by two parameters: the fibre tensile stiffness and the fibre/soil strain ratio that accounts for any possible slippage or shear deformation at the fibre/soil matrix interface. For undrained shear loading, the strengthening effect of the fibres appears to be counteracted by the increase in pore water pressure, induced by the additional confining contribution of the fibres. In agreement with published experimental data, the model suggests also that the moisture content is a key factor governing fibre effectiveness for undrained shearing. Finally, analysis of the model predicted critical states for fibre-reinforced cohesive soil is provided.     

砂土的物态本构模型

基于砂土的压缩回胀性、剪切非线性及剪缩剪胀性的系统分析和包括循环荷载、主应力轴旋转及应力路径偏转等复杂应力条件下的复杂变形反应,得到了三类应力-应变基本关系。在剪缩剪胀应力-应变关系中,引入了由偏应变分量确定的应变路径长度变量,揭示了应力主轴旋转、应力路径偏转引起的剪缩剪胀性。将这些基本关系与循环荷载下砂土的物态变化相联系,建立了砂土的物态动本构关系。     

被动椭圆形标志物应变分析的计算机模拟

本文采用计算机数学模拟的方法更深入地探讨了椭圆形标志物应变分析技术(R_f/Φ去应变法、Shimamoto等提出的代数法、调和平均值法和算术平均值法)的适用条件。一共形成了十一组有代表性的应变前数组,四种变形路径:纯剪、单剪、纯剪加单剪和体变。当应变前数组在(-90°,90°)内均匀分布或随机分布时,无论变形路径是哪一种(共轴的还是非共轴的),上述的所有方法都是适用有,且以代数法最佳。当应变前数组具有优选方位(单峰分布)且变形路径为共轴递进变形时,在一定条件下,R_f/Φ法能用来估计应变。经分析发现当应变足够大时,调和平均值法是一种简单快速的应变计算法。     

A simple model to predict the effect of volume fraction,diameter, and length of fibres on strength of fibre reinforced brittle matrix composites

A simplified model is presented to predict the strength variations of brittle matrix composites, reinforced by steel fibres, with the variations of fibre parameters—length, diameter and volume fraction. This model predicts that its tensile and flexural strength increase non‐linearly with the fibre volume fraction. It also predicts that similar non‐linear behaviour should be observed with the reduction of the fibre diameter when other parameters are kept constant. The experimental results support both these theoretical predictions. It is also explained why an increase in the fibre length does not always significantly increase the fracture toughness. The objective of this paper is not to explain and understand in great detail the science of all phenomena responsible for the strength increase of fibre reinforced brittle matrix composites, but to provide a simple engineering explanation as to why its strength increases with the fibre addition, and how this increase can be quantitatively related to the variations in fibre parameters—fibre volume fraction, fibre length and diameter. These simplifying steps are needed to provide a tool that the practicing engineers can use to predict the brittle matrix strength variation with the fibre parameters. In the area of geomechanics, the results presented here can be used to assess and predict the behaviour of fibre‐reinforced earth. Copyright © 2000 John Wiley & Sons, Ltd.     

Growth and urban differentiation on the urban periphery: A case study from Szeged, Hungary

In late nineteenth century Hungarian cities there were strong contrasts between the inner ‘urban’ core and the rural periphery in terms of housing conditions, accessibility and lifestyle. Rural-urban fringe zones in the Great Plain were quite distinctive because the Turkish period was followed by an outflow of population which resulted in scattered settlement around the urban centres. The fringe could be regarded simply as a zone of transition, but the differences between urban centres and their immediate surroundings were gradually reduced through the industrialisation of the larger towns in the communist period. A study of Szeged examines the expansion of the fringe zone and its gradual urbanisation through the growth of housing as people have moved out from the centre to live among the rural population which increasingly turned from agriculture to commuting. Villages on either side of Szeged's administrative boundary accommodated more migrants from the city during the 1980s and experienced a gradual positive change in the quality of life as people escaped from the decaying residential blocs in the centre. During the years since 1989 there has been further migration, coupled with a growth of small businesses and a big improvement in services (shopping, running water, sewage and telecommunications). Further change may be expected in line with the forces of counterurbanisation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Displacement‐controlled method and its applications to material non‐linearity

For the analysis of non‐linear problems, the displacement‐controlled method (DCM) has a more extensive application scope and more powerful abilities than the load‐controlled method (LCM). However, difficulties of the DCM's procedure not amenable to most finite element implementations of the conventional LCM have restricted its applications in geomechanics. By means of Sherman–Morrison's theorem, the solution of DCM is improved. The improved procedure is characterized by high efficiency, good numerical stability and a programme structure similar to LCM. Two aspects of applications of DCM are illustrated. The first application is to compute the response of a structure under a given load level like the conventional finite element analysis. The second application is to trace the equilibrium path of a structure under a given load distribution type. A simple but effective algorithm is presented for automatically adjusting the step length in tracing the equilibrium path. Examples illustrate that the proposed procedures are suited for modelling complicated non‐linear problems in geomechanics. Copyright © 2005 John Wiley & Sons, Ltd.