Shakedown in layered pavements under moving surface loads

Elastic–plastic deformations in pavements consisting of layers of different frictional materials are investigated. The upper bound, kinematic shakedown theorem is used to obtain estimates of the critical shakedown loads. Fully general three-dimensional deformations are considered. The influence of the loading distribution, interactions between loads, and the effect of varying the thickness, stiffness and strength of the layers are explored. Consequences of this investigation for particular existing designs of flexible pavements are investigated. In particular, it is found that the strength of the subgrade has no effect on the magnitude of the critical shakedown load.     

On the buckling analyses of embedded layers

The published buckling analyses of axially compressed layers embedded in a softer medium, or of multilayers, are based on the notion that the buckling load is P_cr; the so-called critical load which is obtained from a linear eigenvalue problem. The purpose of the present paper is to show that this assumption is generally not justified. For these cases a linearized buckling analysis is not sufficient and buckling of a layer, or of multilayers, has to be determined from a non-linear formulation.     

Strain distribution in superposed buckling folds and the problem of reorientation of early lineations

Two series of experiments were carried out with soft model-materials in order to assess the relative importance of initial homogeneous strain, external rotation and late-stage strain in reorienting early lineations during superposed buckle-folding. In the first series cylindrical buckling folds were produced in embedded planar sheets containing a “lineation”. In the second series noncylindrical folds were produced by compression of a set of cylindrical folds. The experiments indicate that the ratio of buckle shortening to layer-parallel strain is much smaller when the principal extension is parallel to the fold-axis than in the case when the principal extension is perpendicular to the fold-axis. In very competent rocks, the reorientation of old lineations is mainly by external rotation and by the associated concentric longitudinal strain. In moderately competent rocks, the orientation of early lineations always changes by initial homogeneous strain before buckling becomes significant. Because of the unlike amounts of initial strain in layers of different competences, orientations of unrolled lineations may not be parallel in disharmonically folded layers of unlike competences. Under certain conditions the early lineation may become virtually parallel to the later fold-axis. The experiments indicate that the effects of late-stage strain in buckle-folding are largely restricted to the incompetent layers of a multilayer. Hence, if orientation data of early lineations in both competent and incompetent rocks are lumped together, the pattern of orientation may become quite complex. Even for a single competent layer, the pattern of early lineations can locally become complex because of the complex nature of concentric longitudinal strain (and strain resulting from stretching of middle surface of the layer), development of conical folds, development of shear strain along hinge zones of deformed early folds and also because of the development of different orders of folds in both the first and the second deformations.     

A paradoxical situation in determining relative competence from wavelength/arclength ratios within buckle folded multilayers

By measuring the ratio of wavelength to arclength of folds in closely associated disharmonically folded competent layers, it is possible to determine which layer has undergone a greater layer-parallel strain and has a smaller competence. This method may lead to a paradoxical situation. For example when foliated quartzite and mica schist layers are folded together, the mica schist laminae show a much larger buckle shortening than the quartzite layers. On the other hand, the geometry of folds in quartzite indicates that quartzite was more competent than the mica schist. The structure can be explained by different modes of buckling, general buckling in quartzite layers and internal buckling in strongly anisotropic mica schists.     

A numerical study of flexural buckling of foliated rock slopes

The occurrence of foliated rock masses is common in mining environment. Methods employing continuum approximation in describing the deformation of such rock masses possess a clear advantage over methods where each rock layer and each inter‐layer interface (joint) is explicitly modelled. In devising such a continuum model it is imperative that moment (couple) stresses and internal rotations associated with the bending of the rock layers be properly incorporated in the model formulation. Such an approach will lead to a Cosserat‐type theory. In the present model, the behaviour of the intact rock layer is assumed to be linearly elastic and the joints are assumed to be elastic–perfectly plastic. Condition of slip at the interfaces are determined by a Mohr–Coulomb criterion with tension cut off at zero normal stress. The theory is valid for large deformations. The model is incorporated into the finite element program AFENA and validated against an analytical solution of elementary buckling problems of a layered medium under gravity loading. A design chart suitable for assessing the stability of slopes in foliated rock masses against flexural buckling failure has been developed. The design chart is easy to use and provides a quick estimate of critical loading factors for slopes in foliated rock masses. It is shown that the model based on Euler's buckling theory as proposed by Cavers (Rock Mechanics and Rock Engineering 1981; 14 :87–104) substantially overestimates the critical heights for a vertical slope and underestimates the same for sub‐vertical slopes. Copyright © 2001 John Wiley & Sons, Ltd.     

Pleistocene glaciolacustrine breccias of seismic origin in an active graben (central Poland)

Varved layers in the partly consolidated glaciolacustrine Early Saalian deposits exposed in the Bełchatów mine (central Poland), show specific types of deformations. Two types dominate: half-varve deformations constituting the uppermost, clayey part of a single varve, and multi-varve deformations affecting a few varves. The latter are found either as clast-supported breccia beds with varve fragments in a quasi-horizontal position within a clayey matrix, or as deformed beds resulting from subsequent plastic deformation and folding. The deformed beds are interpreted as seismites derived from earthquakes related to the deep graben structure. The magnitudes of the earthquakes have determined the type of deformations. The seismites are thought to have originated intraformationally.     

砂土高压蠕变微观机理分析

砂土层在地面沉降过程中呈现一定的蠕变特性。文章采用一维压缩试验对4 MPa 下砂土的变形特征进行了研究,并 采用改进的试验装置对砂土的微观结构进行了提取。通过对8 组不同级配砂土样的蠕变特性和微观结构分析,得到如下结 论：粒径单一的砂土在蠕变过程中更容易出现突变,砂粒受力后更容易发生破碎,产生滑移错动;砂土蠕变的蠕变量与砂 粒级配密切相关,砂土的粒径越单一且粒径越大时,砂土的蠕变量也越大;采用表征砂土颗粒形状复杂程度的形状系数, 从微观上定量分析了砂土在蠕变过程中发生破碎的情况和蠕变量大小,且形状系数越小,表明该种砂土形状越复杂,在蠕 变过程中砂土的破碎性越高,蠕变量越大。     

Bifurcations of thick-walled hollow cylinders of geomaterials under axisymmetric compression

This paper derives analytically the condition for the onset of diffuse mode bifurcations in thick-walled hollow cylinders with internal radius a, external radius b and length L under axial compression and confining pressure. The thick-walled cylindrical specimens are made of geomaterial characterized by Rudnicki's constitutive model, and the method of solution for the governing equations is the velocity potential approach employed by Chau. Numerical results show that thick-walled cylinders are stronger than thin-walled cylinders against diffuse mode bifurcations, including both buckling, axisymmetric and non-axisymmetric deformations. In contrast to the conclusion for solid cylinders (Chau), no buckling solution is found for γ = mπa / L smaller than about 0·7 under compression for a fixed and finite value of a / b (i.e. no buckling for long and slender hollow cylinders with small a / L and fixed b / a). When 0·7 < γ < 0·9, buckling is the expected first bifurcation; whereas, when γ > 0·9, bulging or barrelling is anticipated. The exact value of γ that excludes buckling and separates the buckling and barrelling phenomena depends on the current values of the constitutive parameters of the solid. Hollow cylinders with higher degree of anisotropy, disobeying normality flow rule, and subjected to confining pressure are more conducive to bifurcations than cylinders made of materials with isotropy, obeying normality, and subjected to no confining pressure. In addition, diffuse mode bifurcations are found possible in the pre-peak regime of the stress–strain curve. © 1998 John Wiley & Sons, Ltd.     

Strain analysis and fold shape in a limestone layer and implications for layer rheology

Folds are developed in thin limestone layers within slates of the McKay Formation exposed to the east of the Rocky Mountain Trench, British Columbia, Canada. They possess geometrical characteristics expected of development by buckling. Strain in the profile plane of a selected fold is similar to that predicted by tangential longitudinal strain, except that magnitudes are too low for the observed curvature. This is attributed to inhomogeneity of strain on the scale of measurement, largely because of pressure solution. Material removed by pressure solution from the inner arc of the fold appears to form veins perpendicular to the hinge, a direction of tectonic stretching. Bedding-parallel stylolites developed diagenetically prior to tectonism.Layer-parallel shortening during the initiation of buckling was less than 20%, and probably less than 10%. The mean arclength/thickness ratio is 6.5 and 7.1, with a dispersion of 0.48 and 0.37 for local and regional populations of 29 and 212 folds, respectively. Application of buckling theory to this data suggests that folding followed a non-linear flow law. The viscosity contrast between limestone and slate would be higher and the power law exponent lower, if initial irregularities in the layers were in the form of a constant amplitude spectrum rather than one of white roughness. The data do not allow a choice of initial amplitude spectrum to be made, nor do they closely constrain estimates of n the power law exponent and viscosity contrast.Deformation in the limestone layers was accommodated by intracrystalline flow (twin gliding), pressure solution, and extensional veining (the last two linked by diffusive mass transfer). The first two dominated deformation in the profile plane of the fold and the last, in association with fracturing, allowed for extension parallel to the hinge. Experimental and theoretical considerations suggest that deformation by a combination of these processes should be non-linear. The non-linear flow law deduced from buckling analysis is consistent with expectations based on observations of active deformation mechanisms.     

反倾层状岩体斜坡弯曲-拉裂两种失稳破坏之判据探讨

对反倾层状岩体斜坡弯曲-拉裂的失稳破坏判据,已有研究分别基于两种力学模型进行推导,即竖直压杆弹性屈曲稳定和平直梁弯折破坏模型,但对层间错动阻力及挠度产生附加弯矩等因素未加以考虑,不尽合理。在反倾斜坡岩层受力分析基础上,建立考虑了板侧层间错动阻力的下端嵌固、上端自由的斜置等厚弹性悬臂板梁模型,统一地通过瑞利-里兹能量方法,推导了弹性屈曲临界条件和嵌固端弯折破坏临界条件。实例计算及讨论表明,弹性屈曲判据适用于陡立岩层;而中-陡反倾岩层应主要为弯折破坏,但层间的力学性质对弯折临界判据值具有较大影响。     

Orogen-transverse tectonic window in the Eastern Himalayan fold belt: A superposed buckling model

The Eastern Lesser Himalayan fold-thrust belt is punctuated by a row of orogen-transverse domal tectonic windows. To evaluate their origin, a variety of thrust-stack models have been proposed, assuming that the crustal shortening occurred dominantly by brittle deformations. However, the Rangit Window (RW) in the Darjeeling-Sikkim Himalaya (DSH) shows unequivocal structural imprints of ductile deformations of multiple episodes. Based on new structural maps, coupled with outcrop-scale field observations, we recognize at least four major episodes of folding in the litho-tectonic units of DSH. The last episode has produced regionally orogen-transverse upright folds (F₄), the interference of which with the third-generation (F₃) orogen-parallel folds has shaped the large-scale structural patterns in DSH. We propose a new genetic model for the RW, invoking the mechanics of superposed buckling in the mechanically stratified litho-tectonic systems. We substantiate this superposed buckling model with results obtained from analogue experiments. The model explains contrasting F₃–F₄ interferences in the Lesser Himalayan Sequence (LHS). The lower-order (terrain-scale) folds have undergone superposed buckling in Mode 1, producing large-scale domes and basins, whereas the RW occurs as a relatively higher-order dome nested in the first-order Tista Dome. The Gondwana and the Proterozoic rocks within the RW underwent superposed buckling in Modes 3 and 4, leading to Type 2 fold interferences, as evident from their structural patterns.     

Buckling analysis of 3D layered structures using a Cosserat continuum approach

This paper concerns the buckling analysis of three-dimensional (3D) layered continua using the Cosserat approach. The finite element method (FEM) is used to implement the adopted Cosserat formulation. As a result, the interfaces between the layers need not be explicitly modelled. Instead, the internal characteristic length, i.e., the layer thickness and also the interaction conditions between the layers are incorporated into the governing equations of the solution. This paper introduces a new 3D geometric stiffness matrix based on the principle of virtual work. The proposed geometric stiffness matrix is applied to the linear buckling analysis of a number of benchmark problems with various geometries, boundary conditions, and interaction conditions between the layers. In all cases, the FEM Cosserat solution exhibits a high level of consistency with the analytical solution.     

The role of initial perturbations in the development of folds in a rock-analogue

A series of experiments is described in which layered specimens were shortened parallel to the layering. The specimens comprise two salt (NaCl) layers sandwiched between three layers of salt–mica synthetic schist. All specimens were prepared and deformed under the same conditions, except for the amount of shortening, which was varied. The resulting fold-shapes are variable, even where the amounts of shortening are the same. In one specimen, folds are believed to have developed essentially by buckling with very little concomitant bulk homogeneous shortening perpendicular to the axial-plane. The specimen lacks an axial-plane foliation. Other folds are believed to have experienced varying amounts of bulk homogeneous shortening before and during buckling, and all have axial-plane foliations that have developed by grain-scale transposition of the original bedding-parallel mica foliation. The difference in the behaviour of the various specimens is explained in terms of initial perturbations. These irregularities take the form of initial deflections in the almost planar bedding, variation in the degree of preferred orientation of the mica grains, and local compositional variation within individual salt–mica layers.     

敦煌莫高窟北区岩体变异变形及修复对策

敦煌莫高窟北区岩体长期以来在特殊的地形地貌、气象水文、岩土性质、地质构造条件下,受人工开挖、重力、风蚀、雨蚀和洪水冲刷等共同作用,岩体发生风化、卸荷等变异以及崩塌、坍塌和危岩、开裂与裂隙(构造裂隙、卸荷裂隙)等多种变形,严重影响岩体的稳定和洞窟的安全。针对上述变异、变形破坏情况,采用PS材料加粉煤灰浆液灌浆封闭裂隙和岩面喷浆加固相结合的修复方法,有效地防止了岩体变异的进一步扩展和表面风化的加剧,减少了岩体变形和表面风化程度,使石窟得以保护和修复。该方法对黄土及干旱地区石窟保护具有良好的借鉴作用。     

羌塘地块东南部上三叠统巴贡组软沉积变形特征及其意义

羌塘地块东南部巴青、索县一带,在上三叠统巴贡组内发育大量未固结状态下发生的软沉积变形。其形态特征主要表现为枕状、球状、链状以及卷曲状,在平面较大范围内均有发现,纵向上密集分布,100余米长的扎德改剖面中发育有7层明显的软沉积变形构造。经过详细地变形特征描述与比对,认为这些软沉积变形主要是与地震振动相关的震积岩,部分变形清晰地显示出受伸展背景的控制。震积岩是软沉积物对古构造活动的记录,研究区内巴贡组震积岩的密集发育表明巴贡组沉积期区域构造活动强烈,对比晚三叠世早期地层中震积岩的发育状况,发现构造活动在诺利中晚期更加强烈,与羌塘地块中晚三叠世火山岩的年代呼应。     

大陆板内构造变形及其动力学机制

典型大陆板内变形发生在克拉通化的大陆岩石圈内部,距离同变形期活动板块构造边界数百至2000km以上。收缩变形主要表现为区域尺度的盆地构造反转、结晶基底与上覆盖层共同卷入变形的厚皮式逆冲构造,具有变形局部化特征。因为流变学分层特征不同,大陆板内变形可以发生在中上部地壳、整个地壳乃至岩石圈尺度上,表现为不同波长的地壳或岩石圈尺度纵弯弯曲。大陆岩石圈板块内部物质组成与结构的不均一性、流体活动、热作用、克拉通内盆地巨厚沉积产生的覆盖效应、地壳加厚等导致的岩石圈强度的局部降低等,是导致大陆板内变形以及应变局部化的原因。构造活化是大陆板内变形的重要方式。板块俯冲或碰撞远程效应被认为是大陆板内变形的主导动力学模型,但是放射性元素积累导致的岩石圈强度热弱化,或大陆冰川消退触发板内应力状态变化等导致大陆板内变形的动力学模型也应该引起关注。     

The development of folds in bedded chert and related rocks in the Malaguide Complex, southern Spain

Folds and folding mechanism in a chert sequence and related rocks of the Maláguide Complex (the uppermost tectonic unit of the Betic Zone) have been investigated. The geometric study shows that folds that developed in the chert sequence are usually angular in shape and asymmetric. Chevron and conjugate folds are common.Folding in bedded chert is explained in terms of a suggested model:

1. (1) Development of folds by kink and conjugate kinking.

2. (2) As the shortening increases, the interlimb angles decrease; in the kink folds this is caused by a reduction of the angle between the layers within the kink and the kink boundaries. There seems to be a relationship between this angle and the asymmetrical thinning-out in the limbs of many folds: the smaller is the angle between the kink boundary and the layers within the kink band, the larger is the reduction of the thickness in these layers.

Single limestone layers embedded in slate deform very probably by a buckling mechanism, implying tangential longitudinal strain and an additional flattening.     

鄂尔多斯盆地富县探区三叠系延长组震积岩及其地质意义

通过对26口井岩心观察,结合区域构造背景,在鄂尔多斯盆地富县探区三叠系延长组中识别出典型的震积岩。该区震积岩代表性沉积构造有液化砂脉、液化卷曲变形、环状层理、负载构造、球枕构造、微断层、微裂缝、震裂岩与震碎角砾岩,其中液化砂脉和液化卷曲变形是地震引起的主要软沉积变形构造。震积岩垂向序列划分为两类:一种以脆性变形为主,自下向上依次为下伏未震层,扰动层,微断层、微裂缝及震裂岩层段,液化卷曲变形层段,震碎角砾岩层段,液化砂脉层段及上覆未震层;另一种以软沉积变形为主,自下至上依次为下伏未震层,环状层段,扰动层,球枕构造及负载构造层段,液化砂脉及液化卷曲变形层,枕状层及上覆未震层。富县探区延长组中的震积岩说明在该盆地晚三叠世存在一个强地震事件活跃期;震积岩中的微裂缝、微断层、震裂岩及液化砂脉可改善储集层物性,同时震碎角砾岩和网状砂脉也是良好的储集层。     

Buckling stability of a viscous single-layer system,oblique to the principal compression

A new theory is developed for single-layer buckling, where the layer is not parallel to the principal stresses. The model chosen consists of a single layer with Newtonian viscosity η embedded in an infinite matrix of viscosity η₁. The layer lies at an angle θ to the bulk principal compressive stress in the embedding medium. It is deformed in equal-area plane strain, with the direction of no strain and the third principal bulk stress, parallel to the layer; hence the obliqueness to the principal stresses is only in two dimensions. It is shown that stress refraction is a necessary condition for this system, and an expression is derived for its value in terms of η, η₁ and θ. Buckling stability equations are completely developed which satisfy the Navier-Stokes equilibrium equations for the buckling layer, and the condition of stress continuity at the layer-embedding medium interface. The dominant wavelength of the buckles is shown to be independent of θ, but the stress required increases with θ.The results of this work have an important bearing on natural folds, since there is no evidence that rock layers are initially parallel to the stresses which fold them, an assumption made in former buckling theories. It is suggested that refraction of stresses and the resulting incremental strains gives rise to the finite structure of cleavage refraction so common in deformed rocks, and that the progressive development of folds in layers oblique to the principal bulk stresses gives rise to asymmetry.     

Layer shortening and fold-shape development in the buckling of single layers

The progressive development of folds by buckling in single isolated viscous layers compressed parallel to the layering and embedded in a less viscous host is examined in several ways; by use of experiments, an analogue model to simulate simultaneous buckling and flattening and by an application of finite-element analysis.The appearance of folds with a characteristic wavelength in an initially flat layer occurs in the experiments for viscosity ratios (μ_layer/μ_host = μ₁/μ₂) of between 11 and 100; progressive fold development after the initial folds have appeared is similar in the experiments and in the finite-element models. Except for the finite-element model for μ₁/μ₂ = 1,000 layer-parallel shortening occurs in the early stages of folding and a stage is reached where little further changes in arc length occur. The amount of layer-parallel shortening increases with decreasing viscosity contrast, and becomes relatively unimportant after the folds have attained limb dips of about 15°–25°.Thickness variations with dip are only significant here for the finite-element model with μ₁/μ₂ = 10, and in experiments for μ₁/μ₂ = 5 where the layer is initially in the form of a moderate-amplitude sine wave. The variations range from a parallel to a near-similar fold geometry, and in general depend on the viscosity contrast, the degree of shortening and the initial wavelength/thickness ratio. They are very similar to the variations predicted by the analogue model of combined buckling and flattening. The difference between the thickness/dip variations in a fold produced by buckling at low viscosity contrast and one produced by flattening a parallel fold is marked at high limb dips and very slight at low limb dips.Many natural folds in isolated rock layers or veins show thickness/dip relationships expected for a flattened parallel fold, and some show relationships expected for buckling at low viscosity contrasts. Studies of the wavelength/thickness ratios in natural folds have suggested that competence contrast is often low. Many folds in isolated rock layers or veins whose geometry may vary between parallel and almost similar, and may be indistinguishable from those of flattened parallel folds, have probably developed by a process of buckling at low viscosity contrasts.