一种基于弹性应变能的裂隙岩体等效弹性模量评价方法

基于线弹性断裂力学裂隙面张开位移及剪切位移理论公式,考虑裂隙存在常法向和常切向刚度情况,研究了含单个裂隙岩体加载过程中由于裂隙存在而附加的弹性应变能。基于应变能等效方法并假设两种裂隙变形模型--非均匀变形模型和均匀变形模型,研究了二维非贯通裂隙岩体的等效杨氏模量和等效剪切模量解析表达式。研究结果表明,对于贯通裂隙规则分布情况,均匀变形模型得到的解析解与Amadei等的结果一致;对于非贯通裂隙正态分布情况,考虑裂隙相互作用的非均匀变形模型解明显低估裂隙岩体的等效杨氏模量和等效剪切模量,而考虑裂隙相互作用的均匀变形模型解与有限元数值解的偏差在10%以内。得到的解析表达式在一定条件下可以作为裂隙岩体等效弹性模量评价方法之一。     

Reconstruction of deformation and flow parameters from deformed vein sets

In any homogeneously deformed material, sets of material lines with a similar stretch history occupy sectors with a scale and orientation that depend on finite strain, area change rate and vorticity of the parent flow. If the size and shape of these “material line sectors” are known, they can be used to determine the sense of shear and to calculate deformation and flow parameters, even if actual stretches along material lines are unknown. In rocks which have undergone near-homogeneous deformation on a large scale, the orientation of sets of boudinaged folded and folded-boudinaged or boudinaged-folded veins can be used for such calculations. A general theoretical background to the problem is presented and a Mohr circle construction to carry out the deformation and flow analysis is introduced. Among other statements, the theory predicts that shortened or folded boudins must develop in non-coaxial monophase deformation, with considerable area change in the plane of observation, or in polyphase deformation.     

Einübertragungsnetz zur kluftstatistik

This paper describes the application and characteristics of a new net for crack statistics. The net, here called transmission net or “u-net” (“Übertragungsnetz”, “Ü-Netz”) is used in combination with a transmission table or “u-table” (“Übertragungstabelle”, “Ü-Tabelle”). Its purpose is to obtain a sphere of crack locations from a series of cracks, having been measured according to strike and inclination.

The “u-net” is composed of a grid subdivided into degrees or degree-grid (“Gradnetz”) and an equal-area-grid (“flächengleiches Netz”). Having replaced the direction σ of strike by its normal δ, the cracks are registered into the meshes of the degree-grid and are there counted. Then they are transmitted into the equal-area-grid according to the percent values of the u-table. From the pattern of frequency numbers (“Häufigkeitszahlen”) in the equal-area-grid the sphere of crack locations is obtained.

Particular specifications regulate the procedure for special measure values, i.e., those of the integral multiples of 5°, especially the angles of inclination τ = 0° and τ = 90° (see 1.4). With greater inaccuracies in measurements, one changes, by means of a given table (see Table V), to a degree-grid of 10°. With very small inaccuracies, on the other hand, the procedure may be simplified, the degree-grid becoming unnecessary (see 1.5). The meshes near the centre, being too long, may be avoided by an additional circle (“Zusatzkreis”—see 1.3).

The “u-net” was constructed in such a way that the spheres of crack locations report the real frequency distribution at all times, free from systematic errors. This is achieved by the method that all calculations follow the principle of area equality or area proportionality on the hemisphere (see 2.1). The procedure using the “u-net” can be adapted to differential accuracies of measuring. It is especially suitable for large numbers of cracks; it is simple in calculation and may easily be programmed for digital computers. Thus the “u-net” is advantageous for all applications in which a large number of cracks has to be dealt with. Such applications are very frequent in rock mechanics, in engineering geology for the purpose of foundation of large hydraulic buildings (dams, caverns), in petrography, tectonics and in geophysical investigations such as the determination of crack structures with a view to explaining micromagnetic occurrences, for instance. Furthermore, the “u-net” is applicable not only to crack statistics but also to other similar statistical methods, e.g., to the statistics of cristal axes or to geographic frequency analyses.     

土工格栅加筋砂土的变形与破坏机理解析

利用可考虑局部破坏的非线性弹塑性有限元，对无加筋和加筋砂土的平面应变压缩试验结果进行了从小变形到破坏的全过程数值解析。加筋砂土试验体用土工格栅分6层和11层进行加筋加固。将等价二维有限元解析所得到的解析结果与试验得到的实测值进行了较为伞面的比较，结果表明：合理的二维非线性弹塑性有限元解析，不仅可以较为精确的模拟加筋砂土的平均心力-应变特性，而且还可以全面地调查试验体的局部应力-应变分布以及剪切破坏发生状况，从而，对加筋砂土的变彤破坏以及加筋材的加州机理有一个更加全面合理的认识。     

Evolution of metamorphic fluids in shear zones: The record from the emeralds of Habachtal, Tauern Window, Austria

Fluid inclusions in emeralds from the Habachtal, Central Tauern Window, have been studied by microthermometry. Results allow a detailed reconstruction of trapping history and evolution of the metamorphic fluids during the Middle Alpine Tauernkristallisation metamorphic event and some of the subsequent cooling period. Five different types of fluid inclusions, corresponding to at least five trapping periods, have been distinguished. In general, the earliest primary (type 1) inclusions, which occur as negative crystals or thin long tubes, are represented by low salinity ( < 10 wt. % NaCl equivalent) aqueous fluids with or without CO₂ with up to X_CO2 ≈ 0.04. Later primary type 2 inclusions are distinguished by different morphologies and distribution patterns. Lower salinity CO₂-free brines and CO₂-bearing denser inclusions with higher CO₂ contents (up to X_CO2 ≈ 0.11) are characteristic of this stage. The type 2 inclusions may also occur as pseudosecondary arrays. The effects of necking have been studied, and found to be considerable in the type 1 primary inclusions. This mechanism has occasionally resulted in the appearance of almost pure CO₂ fluids. The possibility of fluid immiscibility has been examined, and rejected, for the apparent “coexistence” of primary brine and CO₂-bearing inclusions. Instead, mixing of fluids which fluctuated between two different compositions is proposed. The fluctuation was probably due to the sequence of hydration reactions during the Tauernkristallisation. Maximum trapping pressures (3.6 kbar) obtained for stage 1 of the Tauernkristallisation are thought to represent a situation where sublithostatic fluid pressures exested in shear zones during the crystallisation period of many of the emerald cores and coexisting biotite and actinolite. Maximum fluid pressures of 7 kbar were obtained from the type 2 inclusions. This is similar to pressure estimates obtained from mineral equilibria. At least four phases of deformation are indicated by the trapping history. A pressure-temperature-time path for the Tauernkristallisation and the subsequent cooling/uplift period has been constructed for the Habachtal area, using the maximum pressure estimates obtained in this work together with previously existing data. In the cooling period, fluid pressures lower than the lithostatic load again prevailed. This difference, about 1–2 kbar, was probably due to late stage fracturing and/or the development of an open system. At least two more phases of minor deformation and three more stages of entrapment have been defined for this period. During this time, fluids gradually evolved towards more CO₂-poor, and less saline compositions. The present work shows that the possibility of fluctuations in fluid pressures must be considered seriously when attempting to define the PT cooling path from fluid inclusions in metamorphic rocks, especially those in shear zones. Postulations of retrograde PT paths based on fluid inclusions alone may result in pressure estimates which are too low.     

Slipping strip analysis of reinforced earth

Reinforced earth in plane strain is idealized as a homogeneous material with the strips attached to the elastic soil matirx by a conceptual shear zone. A ‘no-slip’ finite element model is derived by assigning a large shear modulus to the shear zone. Relaxation of this modulus using a tangential stiffness algorithm in conjunction with a Mohr–Coulomb strip-slip criterion allows slipping to be simulated. The finite element formulation is validated and the finite element discretization assessed by comparisons against exact solutions for a simple test problem. An idealized reinforced earth wall example is used to demonstrate the feasibility of the method and to answer the question: ‘is slipping significant?’ The method is shown to be potentially useful, and slipping is shown to be significant.     

Determination of strain from stretched belemnites

Belemnites are valuable strain markers and can be used to determine the elongation that has been suffered by the matrix in which they are situated. Experiments simulating the development of stretched belemnites have been performed to verify their deformation history. The analysis of the experimental results is used in the interpretation of field examples collected from Leytron, Valais, Switzerland. The amount of strain suffered by Leytron belemnites has been evaluated using the “least square” method and the result has been compared with that obtained by the “average elongation” method. The relationship between the final orientation of the whole specimen (θ′_b) and that of the fragments (θ′_f) demonstrates that the Leytron belemnites have suffered an irrotational type of deformation.     

A constitutive thermomechanical model for saturated clays

The structural deformation in clays results from microscopic phenomena involving the mechanical contact-stress change, the physico-chemical variation of repulsive forces in expansive clays, and thermal dilatancy of macropores. These textural strains are associated to three plastic mechanisms represented by respectively the yield surfacesf_Tm, f_R-A andf_T. Under a thermal cycle, the sizes of interlamellar spaces between clay platelets are not modified, hence the temperature cycle is expected to have no effect on repulsive forces and thus the second mechanism is not affected by temperature changes.

This paper suggests a formulation of a model of thermo-elasto-plastic behaviour of non-expansive saturated clays characterised by two plastic mechanisms. The mechanical yield surfacef_Tm of the contact-stress mechanism is based on a modified cam-clay model; the thermal softening yield surfacef_T is a plane separating two thermal domains. In normally consolidated conditions, the resulting response to an increase of temperature is compressive. However, in highly overconsolidated conditions, a small irreversible dilative volumetric strain is observed when the temperature is above a threshold value. In intermediate conditions, the material starts with an expansion and tends to a compression.

The constitutive model combines thermo-mechanical hardening, predominant in normally consolidated states (NCS) and absent in overconsolidated states (OCS) where the thermal softening occurs. The characterisation of the model requires information about rheological parameters obtained from oedometric and triaxial paths. Lastly, some numerical simulations of thermo-mechanical tests onremoulded Boom, ‘Bassin Parisien’ andPontida clays are presented, which show satisfactory agreement between experiments and model predictions.     

Strain localization as a key to reconciling experimentally derived flow-law data with dynamic models of continental collision

Published strength profiles predict strength discontinuities within and/or at the base of continental crust during compression. We use finite element models to investigate the effect of strength discontinuities on continental collision dynamics. The style of deformation in model crust during continued subduction of underlying mantle lithosphere is controlled by: (1) experimental flow-law data; (2) the crustal geotherm; (3) strain localization by erosion; (4) strain-softening and other localization effects. In the absence of erosion and other factors causing strain localization, numerical models with typical geothermal gradients and frictional/ductile rheologies predict diffuse crustal deformation with whole-scale detachment of crust from mantle lithosphere. This prediction is at odds with earlier model studies that only considered frictional crustal rheologies and showed asymmetric, focused crustal deformation. Without localization, model deformation is not consistent with that observed in small collisional orogens such as the Swiss Alps. This suggests that strain localization by a combination of erosion and rheological effects such as strain softening must play a major role in focusing deformation, and that strength profiles derived under constant strain rates and uniform material properties cannot be used to infer crustal strength during collision dynamics.     

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.     

偶应力对层状岩体结构面边界层效应的影响

在压剪载荷作用下,层状岩体结构面附近的剪切变形梯度非常显著。基于偶应力理论,研究层状岩体结构面边界层效应。利用有限单元方法进行数值计算,并将偶应力理论的计算结果与经典弹性理论的结果进行比较。采用Serendipity单元（S单元）,导出S单元内的插值函数及位移插值公式。利用ANSYS进行前处理,并基于Matlab编制经典理论和偶应力理论相应的计算和后处理程序。结果表明：偶应力对层状岩体结构面剪应变有显著影响,尺度效应明显。相对于经典弹性理论,在偶应力理论中,应变量减小,并在边界层附近出现过渡区;剪应变突变现象得以改善,但剪应力不再连续。特征长度影响过渡区域的大小,但第二剪切模量、泊松比、弹性模量不改变过渡区域的大小。     

基于梯度塑性本构理论的岩样侧向变形分析(Ⅰ):基本理论及本构参数对侧向变形的影响

研究了岩样在单轴压缩条件下轴向应力．侧向或环向变形的全程曲线特征。基于考虑峰值剪切强度后微小结构之间相互影响和作用的梯度塑性理论，得到了由于剪切局部化而引起的侧向塑性变形。利用虎克定律描述了试件的弹性变形，得到了轴向应力．侧向变形全程曲线的解析解。在软化阶段，试件中部侧向变形及对靠近试件上端或下端部位的侧向变形并不相同。与轴向应力．应变曲线可能出现的回跳现象类似。试件中部轴向应力．侧向应变曲线也可能出现回跳现象。在应变软化阶段，与应力．侧向应变曲线相比，应力．环向应变曲线不容易发生回跳现象。若在试件内部出现多条剪切带，则应该以等效剪切带宽度替代本文中的剪切带宽度。随着剪切带倾角、内部长度参数的降低、剪切模量的增加及弹性模量的降低，轴向应力．侧向应变曲线越陡；甚至能出现弹性回跳。     

Theoretical and natural strain patterns in ductile simple shear zones

A simple empirical model representing the variation of shear strain throughout a simple shear zone allows us to determine the evolution of finite strain as well as the progressive shape changes of passive markers. Theoretical strain patterns (intensity and orientation of finite strain trajectories, deformed shapes of initially planar, equidimensional and non-equidimensional passive markers) compare remarkably well with patterns observed in natural and experimental zones of ductile simple shear (intensity and orientation of schistosity, shape changes of markers, foliation developed by deformation of markers).The deformed shapes of initially equidimensional and non-equidimensional passive markers is controlled by a coefficient P, the product of

1. (1) the ratio between marker size and shear zone thickness

2. (2) the shear gradient across the zone.

For small values of P (approximately P < 2), the original markers change nearly into ellipses, while large values of P lead to “ retort” shaped markers.This theoretical study also allows us to predict, throughout a simple shear zone, various relationships between the principal finite strain trajectory, planar passive markers and foliations developed by deformation of initially equidimensional passive markers.     

Crack damage stress as a composite function of porosity and elastic matrix stiffness in dolomites and limestones

Twenty-five uniaxial compression tests were performed to determine stress at onset of dilation, referred to herein as “the crack damage stress,” in heterogeneous dolomites and limestones. A simplified model for crack damage stress (σ_cd) is developed here using porosity, elastic modulus, Poisson's ratio and three empirical coefficients. The model shows that when porosity decreases and elastic modulus increases, σ_cd rapidly increases and approaches its maximum value. On the other hand, when porosity increases and elastic modulus decreases, σ_cd rapidly decreases and approaches its minimum value. The proposed model is validated for six heterogeneous limestone and dolomite formations which are widely distributed in Israel.     

Localization of deformations in elastic-plastic solids

Localization of deformation in elastic-plastic solids subject to plane strain deformation are investigated numerically. It is shown that the localization may be captured accurately in finite element models by employing (1) the elastic-plastic material stiffness to form the global stiffness, (2) in the case of symmetrical configurations, an imperfection in the form of a weak element, and (3) in the case of incompressible materials, a reduced selective integration scheme which alleviates mesh ‘locking’. Accuracy of the technique is demonstrated by applying it to analyse the classical punch and slope stability problems. Its versatility is illustrated by applying it to analyse finite deformation problems and shear bands formations in associative and non-associative elastic-plastic solids.     

Three-dimensional strain analysis in physical models of geological structures

A method is outlined for calculating three-dimensional finite strain in physical models of geological structures containing passive strain markers. This method makes it possible to determine the three-dimensional strain pattern in models of structures that lack any of the types of symmetry (such as that imparted by cylindrical folding) that simplified calculations in previous work. The strain markers in the new method are in the shape of stubby rectangular prisms or cubes. These form a three-dimensional grid or array occupying each of the active layers in a model (e.g., for a simple two-layer gravitationally unstable system, one grid for the overburden layer and one for the buoyant layer). Each of the grids can be described by positions of three families of “strain marker surfaces”, which are contacts between layers of strain markers.

After deformation, the model is serial-sectioned horizontally and the traces of the strain marker surfaces on the sections are digitized. The strain state is calculated at each of several hundred points arranged in a three-dimensional “output grid” extending throughout the mechanically active part of the model. An interpolation procedure is used to estimate the spacing and orientation of the strain marker surfaces in the vicinity of each of the output grid points. The following quantities are determined for each of the three families of strain marker surfaces:

1. (1) the local horizontal orientation of the strain marker surfaces;

2. (2) the local spacing of the surfaces; and

3. (3) the local inclination of the surfaces, calculated from their change in position from the serial section above, to the serial section below, the output grid point.

This information is used to generate a parallelepiped representing the strain marker geometry in the neighbourhood of the output grid point. The edges of the parallelepiped are equivalent to the coefficients of the strain matrix, from which the three principal strain magnitudes and orientations are readily derived.     

Aspects of the Clay/Electrolyte/Water System with Special Reference to the Geotechnical Properties of Clays

The solid phase of the clay/electrolyte/water system is composed of platy clay particles (of variable thickness, δ, and variable specific surface, S) and of crystal phase (hygroscopic) water, wherefrom δ and S may be estimated. Diffuse layer water between parallel particles and other structural elements at the mutual distance 2d, W_d = d S ρ_w, and macropore water form the liquid phase. Particles are arranged in domains and and clusters, these in aggregates in an “aggregate lattice” with vacancies (macropores) and further superstructures are formed. Contact bonds may develop and long-range forces exist between structural elements, which govern the clay geotechnical behaviour. Cohesion is usually due to van der Waals attraction, friction component–to contact bonds and particle delamination, swelling and suction–to diffuse layer repulsion. Most phenomena may be explained in terms of the equilibrium condition between attraction and repulsion. Heat dissipation must be considered in mechanical processes. The angle of effective stress env envelope is correlated with particle thickness, thus therefrom clay strength may be estimated.     

Mechanical and Physical Characterization of Tabriz Marls, Iran

Three types of marls can be found in the Tabriz area (Iran): yellow, green, and gray/black marls. In the present paper, strength and deformation characteristics of Tabriz marls and their stress–strain behavior are investigated by various in situ and laboratory tests. In order to study the deformation behavior of these marls, various experiments such as the pressuremeter test, plate loading test (PLT), seismic wave velocity test, uniaxial compression test, standard penetration test (SPT), and direct shear test were carried out. Ranges of strain at the elastic and failure points were determined. Young’s and shear modulus were obtained. Test results showed that the strength characteristics increase with depth. The value of deformation modulus determined by the pressuremeter test was in good agreement with those obtained from the PLT. This implies that pressuremeter is a suitable in situ test for characterizing the deformation modulus of marl. Deformation modulus obtained from pressuremeter and plate loading tests were approximately 4–5 times the results of uniaxial compressive test and the deformation modulus obtained from seismic data was about 30–50 times the static deformation modulus. Stress–strain curves showed that the maximum value of strain at the elastic and failure points and the minimum value of strength and deformation modulus are corresponding to the yellow marls while the minimum value of strain and the maximum value of strength and deformation modulus are corresponding to the gray/black marls. Some empirical relationships between different characteristics of Tabriz marls were also derived.     

Experimental deformation of a synthetic dunite at high temperature and pressure. II. Transmission electron microscopy

We have performed detailed transmission electron microscope on most of the deformed synthetic dunite specimens prepared in the study by Zeuch and Green (1984). We have identified three basic types of sub-boundaries, simple tilt walls in (100) and (001). composed by b = [100] and b = [001] edge dislocations, respectively, and twist boundaries in (010) composed of b = [100] and b = [001] screws. We have also observed more complex, asymmetric lilt boundaries in (100) and (001). Like the (010) twist boundaries, these asymmetric tilt walls are common only at the highest temperatures and lowest strain rates. Subgrain development is extensive at the higher temperatures and lower strain rates, and subgrains are composed of the above-mentioned three types of sub-boundaries; edge components in (100) and (001) ire “knitted” to screw components in (010) as described by Kirby and Wegner (1978) for naturally deformed olivine. In many areas of the samples which we studied, subgrain development is not observed, but parallel arrays of tilt boundaries of one type or the other are present. At higher temperatures and lower strain rates. “(100) organization” (Durham et al., 1977) is common; this structure consists of parallel arrays of (100) tilt boundaries with b = [100] screws connecting the sub-boundaries. At lower temperatures we have observed an analogous arrangement of (001) sub-boundaries and b = [001] screws, which we refer to as “(001) organization”. Under all experimental conditions, dislocations with b = [100] and b = [001] are present in approximately equal numbers. However, the two types of dislocations also have distinctly different geometries under all test conditions. We suggest that the transition from slip parallel to [001] to slip parallel to [100] with increasing temperature, which has been reported in earlier studies may also depend upon water content. The substructures which we observe are virtually identical to those seen in many naturally deformed peridolites. and we conclude that the mechanisms involved in both natural and laboratory deformation of olivine polycrystals are similar. On the other hand, the substructures reported here are very different from those observed in experimentally deformed olivine single crystals. It seems likely that these substructural differences reflect fundamental differences in the behavior oh single crystals and polycrystals. which are in turn reflected in different measured creep strengths.