Boudinage structure: some new interpretations based on elastic-plastic finite element simulations

An elastic-plastic finite element method, based on the Prandtl-Reuss equations of plastic flow and involving equivalent stresses and strains, is used to study boudinage structure. Our choice of data for the simulations was guided by published stress-strain curves for marble (matrix) and quartzite (boudin), the essential parameters being yield stress and rock ‘hardness’ (defined by the slope of the stress-strain curve). All models assume an initial fracture and slight separation and therefore only simulate post-fracture behaviour. The simulations suggest that boudin shape is determined by boudin hardness; maximum stresses are concentrated in the corners which therefore shows the most shape modification. Matrix hardness determines the amount of boudin separation. Direct comparison with natural examples is restricted to boudins suffering no significant pre-fracture plastic deformation (i.e. rectangular- and barrel-shaped boudins), although other types are likely to have the characteristics of barrel and pinch-and-swell styles. The simulations do not consider the nature and timing of boudin-defining fractures but these are important in determining the style of boudinage which ultimately develops. Some mechanical problems associated with the infilling of inter-boudin gaps by ductile rock matrix are discussed and two models proposed. The first, based on yielding fracture mechanics, is used to explain boudins with wedge-shaped (or otherwise nonmatching) ends. The second, a hydraulic model, is proposed to account for gaps between rectangular boudins that are filled by ductile rock matrix.     

A stress-transfer model for the development of extension fracture boudinage

The term boudinage is used to describe a wide variety of extensional structures in deformed rocks. This paper is mainly concerned with boudinage resulting from through-layer extension fractures followed by separation of the layer segments, thus forming boudins with more or less rectangular cross-sections. In principle, this process is similar to the break up of fibres in fibre-reinforced composite materials extended parallel to the fibre direction. Both processes are controlled by the transfer of stress from the matrix to the fibre (or layer) and a mathematical model for fibre-matrix stress transfer (the ‘fibre-loading’ model) is well established. We have used this as a basis for developing a stress transfer model for boudinage. The only difference in the basic mathematical formulation results from geometric differences between the two systems; the geometric expressions in the fibre-loading model have, therefore, been rederived for the layer-matrix case.Stress-transfer theory predicts that the tensile stress in a layer segment rises from a minimum at the end of a segment to a maximum at the centre. This behaviour, which is clearly shown by finite-element models of boudinage structure, suggests that extension fracture boudinage develops by successive ‘mid-point’ fracturing. According to stress-transfer theory, the process will continue until a layer is reduced to segments (boudins) all of which are shorter than some critical length (for which the tensile fracture strength of the layer is equal to the tensile stress at the mid-point). In practice, successive fracturing will be influenced by two other factors: (1) in nature the controlling material properties (tensile fracture strength, elastic moduli) will not be single-valued but will have a distribution reflecting local variations in lithology and microstructure and (2) major pre-deformation flaws may be present in a layer which will control the ‘starting length’ of layer segments. These factors are incorporated with the stress-transfer theory into a statistical (Monte Carlo) model for extension fracture boudinage which results in a prediction of boudin aspect ratios. The predicted distribution compares very closely with the observed distribution of 91 quartzite boudins within Lower Carboniferous slates at Tintagel, Cornwall.The stress-transfer model implies that boudin-defining fractures occur sequentially so that inter-boudin gap lengths will be unequal. Strain estimates based on boudinage structure will vary according to which part and how much of a layer is sampled. A much improved strain estimate is possible based on sequentially closing the inter-boudin gaps. The stress-transfer theory also leads to the possibility of estimating palaeostress from boudinage structure and is the only model available which predicts an aspect ratio distribution of boudins formed by extension fracture. Other than our own, we know of no published data on boudin aspect ratio distributions. Hence, further elaboration of the model is not possible until more field data is available. We hope that our work will encourage the systematic measurement of boudinage as well as the development of alternative models.     

Theory of chocolate tablet boudinage

Two-dimensional boudinage in a flattening type of bulk deformation, with equal layer-parallel extensions in all directions, leads to the development of roundish or polygonal outlines of boudins in plan-view. As combined experimental and theoretical studies show, chocolate tablet boudinage with two sets of mutually perpendicular boudin axes may form in different ways. (1) Unequal layer-parallel extension in the matrix results in one set of extension fractures forming perpendicular to the greatest principal stress in the matrix. Once these long narrow boudins are formed, the greatest principal stress in the brittle layer becomes approximately parallel to the long axis of the boudin. As a result a second set of fractures forms normal to the first set. (2) In lineated rocks the anisotropy of tensile strength leads to the sequential formation of two sets of extension fractures, parallel and perpendicular to the lineation. Depending on the orientation of the lineation the boudin axes may or may not be parallel to the principal stresses in the matrix. (3) Boudins with rectangular plan-view may also form when two successive events of unidirectional boudinage are superposed on one another. Irrespective of the direction of principal extensional strain rate in the matrix, the second generation extension fractures are likely to form approximately perpendicular to the first generation boudin axes.     

鄂东南铁山不对称骨节状石香肠构造基质层中的应变测量与分析

鄂东南铁山是亚洲首个发现骨节状石香肠构造的地区。以该区不对称骨节状石香肠构造为研究对象,利用惯量椭圆法对其基质层进行有限应变测量,获得真应变差、运动学涡度等相关参数及有限应变椭圆长轴展布方位的分布图。对所获数据资料研究表明:该不对称骨节状石香肠构造基质层的应变受其能干层控制,有限应变值与其离能干层的距离趋于负相关,且与其矿物颗粒粒径呈负相关,在平行于石香肠构造伸展方向上基质层有限应变分布不均;石香肠体附近与之相近规模的变形构造可使其相应基质中的应变分布紊乱;该不对称骨节状石香肠构造是由早期平行于层面简单剪切叠加晚期平行层面伸长、垂直层面压缩的纯剪切作用形成。与不对称鱼嘴状石香肠构造对比研究表明,两者基质层中简单剪切与纯剪切的分布均分别与其相对增厚与减薄区段对应,而两者形态的不同主要与石香肠体不连续处充填物的能干性不同有关,再次表明两者均是较好的岩石流变学标志。     

A back-of-the-envelope approach to boudinage mechanics

Smith has shown from linear hydrodynamic stability that pinch-and-swell boudinage is possible only in a non-Newtonian material. By assuming uniaxial stress and strain and large viscosity contrast between the competent layer and the incompetent matrix, Smith's result can be derived in a very simple way. The one-dimensional force balance also predicts growth rates, although not dominant wavelengths. However, we question the ability of even linear stability analysis to accurately predict boudin widths.     

P-T path development derived from shearband boudin microstructure

This work focuses on the development of a regional P-T-path from the Malpica-Lamego Ductile Shear Zone, NW Portugal, based on the microstructures of shearband boudins evolved during progressive simple shear. The combination of microstructural analysis, fluid inclusion studies, crystallographic preffered orientation and fractal geometry analyses, allows to link several stages in the internal evolution of the boudin to regional P-T conditions. The boudinage process is initiated under differential stress after the original layer achieved sufficient viscosity contrast relative to the surrounding matrix. Two main transformations occur simultaneously: i) change in the external shape with continuous evolution from tabular rigid body to sigmoidal asymmetric morphology (shearband boudin) and ii) localized dynamic recrystallization in the sharp-tips of the structure (acute edge of shearband boudin), and along the boudin's margin and grain boundaries. Smaller recrystallized grains, particularly in the sharp-tip domains, accommodate most of the external strain, and larger relict grains are preserved in the centre. Dynamic recrystallization under constant strain rates and strain partitioning inside the boudins is indicated by fractal geometry based on grain boundary and grain area analysis. Progressive deformation leads to the generation of structural and textural heterogeneous domains inside the boudins, and is recorded by quartz c-axis orientation analysis and fluid inclusion studies. The last deformation episode shows the final formation of the blunt-tip domain and internal secondary shear planes. The regional P-T path begins with the crystallization of andalusite after an internal shearband boudin dilation event and ends with quartz dynamic recrystallization on boudin tips. The main deformation stage (310/315 Ma) led to reactivation of internal secondary shear zones with sillimanite crystallization.     

Data pertinent to the phenomenon of boudinage at Bastogne in the Ardennes

Data are presented pertinent to the genesis of the boudinage structure at Bastogne. It are the geometrical relations between the structural elements, the state of strain of the boudined layer and the relations between the dimensions of individual boudins. These data are consistent with a hypothesis of boudin formation by layer-parallel elongation, preceding the folding of the layers.     

Folding of boudinaged layers

The boudinage structure in the Väddö area in Sweden developed during an early isoclinal folding. During a second deformation which deformed the axial surfaces and the limbs of the isoclinal folds, the shapes of boudins were variously modified. The resulting structures include folding of the individual boudins into half-waves, a tile-like piling up of straight or gently folded boudins and arrangement of straight boudins in a herring-bone pattern. Similar structures have been produced experimentally in soft models containing ‘boudins’ of different competencies. The relative importance of the different physical parameters which contribute towards the development of the various structures is examined. In particular, the experiments showed that when the competence contrast between the boudin and the host material was rather small, the shape of the boudins was modified by a second deformation into trapezoidal forms often with flame-shaped projections at the edges. From an analysis of the experimental and natural structures, a set of criteria has been formulated which can be used to identify structures formed by superposed deformations and to give an approximate idea of the competence contrast between the boudin and the host rock during the second deformation.     

Boudinage control on the emplacement of lodes of the Kalgoorlie goldfield

Literature on the Kalgoorlie goldfield is reviewed and boudinage is shown to be an important aspect of the structure of the field, the lode distribution to be coincident with the principal necks, and the configuration of the lodes to match the characteristic fracture patterns of classical boudin necks. Boudinage is therefore interpreted to be an important control on the emplacement of the mineralization. Boudinage and the concomitant introduction of mineralization is related to flattening of the Boomerang Anticline, which is consistent with the general consensus that mineralization is emplaced as hydrothermal veins during late‐stage deformation. This interpretation is proposed as a simpler alternative to other more complex shear‐related models and may be useful in exploration for deposits of similar type.     

Granite emplacement by crustal boudinage: example of the Calmayo and El Hongo plutons (Córdoba,Argentina)

This study deals with the structure and emplacement of the Calmayo and El Hongo trondhjemite plutons (Famatinian belt of Córdoba, Argentina). It provides structural data from the granites and the country rocks and a study of the magnetic fabric in the plutons. New U/Pb geochronological data yield intrusion ages of 512.1 ± 3.4 Ma and 500.6 ± 4.5 Ma for the Calmayo and El Hongo plutons respectively. The El Hongo massif and the southern part of the Calmayo trondhjemite preserve magmatic structures, whereas the northern domain of Calmayo shows the imprint of solid‐state deformation. The main foliation in the country rocks outlines a boudin‐like pattern at the map scale and the granites are located along boudin necks, suggesting that the emplacement of these trondhjemite plutons was linked to large‐scale boudinage of the country rocks.     

塔中北斜坡走滑断裂断距对碳酸盐岩油气藏的影响

塔中北斜坡奥陶系碳酸盐岩的储层改造和油气分布,都受到走滑断裂的影响。由于区域内较大规模的走滑断裂均表现为北东向,水平断距较小,使得利用地震资料直接计算断距时存在着较大的困难。实际上,走滑断裂还普遍表现出北西向拉张特征,并沿断裂走向形成了宽度不同的拉分区域。断裂多沿北东向平直延伸,表明这些拉分区域的宽度和高度主要是受局部应力作用而形成的。这些拉分区域的宽度和高度在三维地震资料中较容易识别。根据断裂的拉分特征,提出通过计算拉分区域的宽度和高度间接识别走滑断裂的断距的方法。断距计算结果符合走滑断裂发育的地质规律,断距较大处是走滑断裂活动性较强的区域,同时也是储层改造作用强烈和油气分布较好的区域。     

岩石变形破坏全过程的变形场和能量演化研究

以白光数字散斑相关方法作为试验观测手段,通过单轴压缩试验对一种红砂岩变形破坏全过程的变形场和能量演化特征进行研究。利用CCD(charge-coupled device)相机记录试验加载全过程的试件表面散斑场图像,采用数字散斑相关方法对试验图像进行分析,计算得到了岩石变形破坏过程中试件表面变形场、变形局部化区域的位移错动量值和试件表面变形能密度量值,并对变形局部化带的位移演化和变形能量演化特征进行分析。从试验结果分析得出：岩石试件变形局部化带的位移演化在加载峰值前基本上保持线性演化规律,而其位移错动加速与加载峰值点相对应,岩石试件加载峰值后阶段的承载力变化主要受变形局部化带的位移演化影响;岩石试件在加载过程中的能量释放和能量积累规律与局部化带的演化有关,体现出局部能量释放和整体能量释放两种形式。     

Factors affecting the kinematic interpretation of asymmetric boudinage in shear zones

Based on work along a major mylonite zone in the northern Appalachians and scale model studies, a new mechanism for the origin of asymmetric boudins in shear zones is proposed. Along the Honey Hill Fault in southern Connecticut, granitic sills intruded into calc-silicate gneisses and schists were oblique to the boundaries of the mylonite zone and experienced the following sequence in the production of sigmoidal boudins: (1) as the calc-silicate schists experienced mylonitization and flow, the more competent, coarse-grained granitic sills deformed by extensional fracturing and quartz veining; (2) continued extension of granitic sills was accommodated by ‘normal’ shear on early-formed quartz veins; (3) continuing extension of sills and ductile modification of the corners of boudins resulted in granitic ‘fish’ with tails which stream from the top of the boudin in the ‘down-dip’ direction and from the bottom in the ‘up-dip’ direction. Based on a variety of kinematic indicators, the sense of asymmetry of the tails is identical to that expected for recrystallization tails on sheared augen (σ structure). Models composed of silicone putty and Plasticine were created to investigate the effect of pre-shearing geometry on boudin evolution, and were deformed in a simple shear device. The models reproduce the kinematics deduced from field relations and suggest that one of the primary factors in controlling the amount of extension is the angle which early veins make with the shear zone boundary. Varying the angular relationships in the models suggests that other pre-shearing geometric factors may affect the geometry of boudins formed in this way. Boudins formed through this mechanism appear very similar to Hanmer's type II asymmetric boudins. Because the pre-shearing geometry can exert a control on boudin asymmetry, caution should be used when attempting to deduce shear sense or shear strain values from asymmetric boudinage.     

Distributions of cumulative displacement and seismic slip on a single normal fault surface

Displacement profiles (normalized displacement vs normalized distance from the point of maximum displacement) have been plotted for 34 horizontal radii from 25 normal faults with maximum displacements ranging from 1.0 to 37.5 m. The composite displacement profile for these faults, when corrected for systematic inaccuracies of the data, is significantly different from the theoretical slip profile for a single seismic slip event. The integration of slip displacement profiles of many slip events on a growing fault shows that a steady-state displacement profile will be established. This theoretical displacement profile is similar to the composite profile derived from the fault data. Analysis of displacement data from 488 fault traces, which do not necessarily pass through the point of maximum displacement of their respective faults, shows that although displacement measurements are strongly influenced by ductile drag the theoretical distribution can still be identified in the data. Although the slip distribution on a fault during a single slip event, or during a period of stable sliding, is not simply related to the distribution of cumulative displacement on the fault, a knowledge of both characteristics places firm constraints on fault growth models.     

日本MW9.0级地震同震位移场和应力场的有限元模拟

根据同震位移GPS观测数据, 利用有限元法反演了2011年3月11日本MW9.0级地震的断层滑移模式。在此基础上, 计算了日本MW9.0级地震引起的同震位移场和应力场, 给出了位移和应力的分布, 分析了他们的变化规律并与实测结果进行了对比。计算结果表明: 日本MW9.0级地震的静态断层滑移量最大可达25 m。地震引起断层上盘向东位移, 最大位移在震中附近, 可达24.25 m, 日本东北地区向东位移最大可达6 m。震后地表隆起, 隆起幅度可达5.6 m, 隆起的最高点也在震中附近。日本东北地区东海岸附近有一下沉带, 下沉量可达0.8 m。同震地表位移的计算值与GPS测量结果基本一致。地震引起应力变化, 导致震后应力下降。应力变化是不均匀的, 在震中附近约为9.9 MPa, 在深处可达32 MPa, 在日本东北地区地表应力变化小于4.4 MPa。地震引起的应力变化主要是水平应力, 垂直应力基本不变。     

Tension fractures and boudinage oblique to the maximum extension direction: an analogy with lüders' bands

A study has been made of the orientation of boudin necklines and extension fractures with respect to the maximum extension direction X determined by infilling fibre growth. Several localities, within the Variscan belt of Western Europe, the Pyrenees and the Alps, have been investigated in detail. It has been generally accepted that boudinage and extension fracturing occur perpendicular to X in the rock at the time of their formation; however, we have shown that is not the case: boudin necklines and extension fractures occur at an angle between 45°–90° to X, with the most frequent orientations between 65°–70° and 80°–85°. Angles of exactly 90° are rare.In order to explain · this obliquity, an analogy is made with the phenomenon of Lüders' bands (localised regions of plastic deformation) developed in thin metal plates under tensile testing. Because the thickness of a layer is negligible compared with its lateral extent, we will only be concerned here with the case of thin metal plates.     

连拱隧道施工对洞口仰坡影响的三维数值分析

以江西某高速公路一浅埋偏压连拱隧道为例,采用Marc有限元程序对其出口段进行了动态施工的三维数值模拟。从施工过程中的应力集中、塑性区的分布形态和发展规律、仰坡轴向地表位移的分布特征等方面,系统研究了偏压连拱隧道施工对强风化岩体洞口仰坡的影响作用机制。结果表明:在模拟施工过程中塑性区和应力集中区只出现在洞体附近,可能的仰坡失稳往往是隧道施工失稳的累进性响应;洞口仰坡的轴向水平位移前缘大、后缘小,垂直位移后缘大、前缘小;隧道上方和近山脊一侧仰坡位移比近山谷一侧大,仰坡失稳往往从隧道上方和埋深较大一侧开始。     

直角域中凸起和孔洞对SH波的散射与地震动

按照波函数展开法和镜像方法,对直角域中半圆形凸起和圆形孔洞对SH波的散射进行了分析,得到其稳态解。对含孔洞和凸起的直角域做分区,等效为一个含孔洞与凹陷的直角域和一个圆域的契合,其在分界面上满足位移和应力的连续性条件,即契合条件,分别构造两个区域内的位移波函数,按照孔洞边界柱面上的应力自由和契合条件定解波函数展开式的系数。按Fourier级数展开法,得到定解条件的线性代数方程组,截断求解,进而得到问题的解析解。数值算例给出圆形孔洞边沿动应力和地表位移幅值的分布情况,得到直角域自由边界、凸起、孔洞对散射和地震动的影响。     

Numerical modelling of clast rotation during soft-sediment deformation: a case study in Miocene delta deposits

A numerical model for a rotated clast in a sedimentary matrix is presented, quantifying the deformation in associated soft-sediment deformation structures. All the structures occur in a southwards prograding deltaic sequence within the Miocene Ingering Formation, deposited at the northern margin of the Fohnsdorf Basin (Eastern Alps, Austria). Debris flow and pelitic strata contain boudins, pinch-and-swell structures, ptygmatic folds, rotated top-to-S reverse faults and rigid clasts, developed under different stress conditions within the same layers. The deformation around a 24×10 cm trapezoid-shaped rigid clast, resembling the δ-clast geometry in metamorphic rocks, has been modelled using a 2D finite element modelling software. Under the chosen initial and boundary conditions the rotational behaviour of the clast mainly depends on the proportions of pure and simple shear; best fitting results were attained with a dominantly pure shear deformation (~65–85%), with stretching parallel and shortening normal to the bedding. In this specific model set-up, the initial sedimentary thickness is reduced by 30%, explained by stretching due to sediment creeping and compaction. The high amount of pure shear deformation proposed is compatible with the observed layer-parallel boudinage and pinch-and-swell structures. Rotated faults and ptygmatic folds were caused by the minor component of bedding-parallel simple shear.     

Influence functions for stress and displacement discontinuity elements in an anisotropic medium

Influence functions, that permit us to determine stresses and displacements at an arbitrary point in an infinite, homogeneous, linear elastic, anisotropic medium due to different three-dimensional (3-D) stress or displacement discontinuities distributed on infinite, flat, band-type elements, are presented. Any straight-line segment on the band, which is perpendicular to its infinite side, has the same distribution of the discontinuities. Along with the functions, their Taylor series approximations are also provided. The last can be useful to analyse stresses and displacements at points distant from the elements. The functions allow us to avoid procedures of numerical integration in the Indirect Boundary Element Method and/or the Displacement Discontinuity Method computer codes that are able to solve complete plane-strain problems with 3-D boundary conditions for an elastic, anisotropic medium. © 1997 by John Wiley & Sons, Ltd.