A general deformation matrix for three-dimensions

A deformation that is obtained by any simultaneous combination of two steady-state progressive deformations: simple shearing and a coaxial progressive deformation, involving or not a volume change, can be expressed by a single transformation, or deformation matrix. In the general situation of simple shearing in a direction non-orthogonal with the principal strains of the coaxial progressive deformation, this deformation matrix is a function of the strain components and the orientation of shearing. In this example, two coordinate systems are defined: one for the coaxial progressive deformation (x_i system), where the principal and intermediate strains are two horizontal coordinate axes, and another for the simple shear (x _i ^t’ system), with any orientation in space. For steady-state progressive deformations, from the direction cosines matrix that defines the orientation of shear strains in the x_i coordinate system, an asymmetric finite-deformation matrix is derived. From this deformation matrix, the orientation and ellipticity of the strain ellipse, or the strain ellipsoid for three-dimensional deformations, can be determined. This deformation matrix also can be described as a combination of a rigid-body rotation and a stretching represented by a general coaxial progressive deformation. The kinematic vorticity number (W _k is derived for the general deformation matrix to characterize the non-coaxiality of the three-dimensional deformation. An application of the deformation matrix concept is given as an example, analyzing the changes in orientation and stretching that variously-oriented passive linear markers undergo after a general two-dimensional deformation. The influence of the kinematic vorticity number, the simple and pure shear strains, and the obliquity between the two deformation components, on the linear marker distribution after deformation is discussed.     

Using the Maximum Effective Moment Criterion to Interpret Quartz <c>‐Fabric Patterns

The Maximum Effective Moment (MEM) criterion predicts that the initial orientation of ductile shear zones and shear bands is ~55° relative to the maximum principal stress axis (σ1) and that the kinematic vorticity number (Wk) is ~0.94. These preferred orientations should be reflected in the pattern of quartz -fabrics in shear zones and shear bands. Common quartz -fabrics in plane strain can be divided into low-temperature (L) and high-temperature (H) fabrics, with each group showing three patterns. A steady flow with a constant value of Wk≈0.94 gives rise to L-1 and H-1 patterns, which are commonly characterized by a single axis girdle normal to the shear zone and a single -point maximum parallel to the shear zone.Once the conjugate set develops, L-1 and H-1 have opening angles of ~70° and ~110°, respectively. L-2 and H-2 are asymmetric patterns associated with variable deformation partitioning and vorticity values of 0< Wk<0.94. In contrast, L-3 and H-3 are symmetric patterns associated with 100% deformation partitioning and Wk=0. The opening angle in quartz -fabrics is implicitly linked to the temperature during deformation. The opening angle is ~70° at low temperature and ~110° at high temperature. However, a linear correction between the opening angle and the temperature cannot be established. During deformation partitioning, synthetic shear bands form earlier than antithetic bands and are more easily developed. This may result in opening angles of <70° for low-temperature fabrics and of >110° for high-temperature fabrics. The following criteria can be used to recognize reworked shear zones that have experienced multiple orogenic phases and changes in the stress state: 1) the initial Wk is larger or smaller than ~0.94; 2) the change in Wk is abrupt, rather than progressive; 3) inconsistent shear senses are inferred for the different phases of deformation; and 4) a negative value of Wk is found in reworked shear zones.     

Vorticity and non-coaxiality in progressive deformations

A measure of the non-coaxiality involved in progressive deformation histories is proposed in the form of the kinematical vorticity number, W_k. This number is a measure of the relative effects of rotation of material lines (relative to the instantaneous stretching axes) and of stretching of these material lines. As such, W_k, is a measure of the instantaneous degree of non-coaxiality. A detailed example is first presented in the form of a progressive simple shearing in which the shear plane rotates relative to an external coodinate system. This is followed by examples of more complicated deformation histories. Three specific types of progressive, isochoric (constant volume) deformation histories are recognized. Those for which 0 ≤ W_k < 1 correspond to deformation histories where no line that has been extended is shortened in future increments; W_k > 0 is a special case of these corresponding to a coaxial history. Histories with W_k > 0 are non-coaxial. Those histories with W_k = 1 correspond to progressive simple shearing. Those histories with 1 < W_k < ∞ are pulsating and lines that have been extended may be shortened in future increments.     

基于分裂节点法的地震同震和震后形变数值模拟及其在汶川大地震中的应用

大地震导致的同震及震后效应,对于分析不同地震之间的相互影响及区域地震危险性等有着重要的作用.文中开发了模拟地震同震及震后效应的三维黏弹性有限元程序,通过计算走滑断层震例(概念性模型)引起的同震及震后效应,并与解析/半解析解进行对比,验证了程序的可靠性.同时基于概念性模型,分析了不同介质参数对同震及震后的地表变形的影响....     

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

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

The solvus in the system MnCO3-CaCO3

The solvus on the Mn-rich side in the system MnCO₃-CaCO₃ was studied experimentally at 10 kb and lower pressures. The critical point lies at 540 °C and Mn_0.68Ca_0.32CO₃. Reaction rates of demising are extremely sluggish. X-ray calibration curves for (Mn, Ca)CO₃ solid solutions are given. Depicting microprobe analyses of about 300 carbonates from manganese calc silicate marbles affected by different degrees of metamorphism confirm the experimentally derived solvus.Margules parameters were derived from the experimental solvus, giving for spinodal unmixing W _a=20.1+0.032·T and W _b=18.3-0.0033·T, and for binodal unmixing W _a=–22.9+0.035 ·T and W _b=3.36+0.0147·T (All W, in kJ/-mol)A physical interpretation of the Margules parameters based on deformations of the ionic radii in the solid solution is presented. The Margules parameters obtained by this model agree fairly well with the experimental data.     

秦岭沙沟街韧性剪切带的岩石磁学、磁组构和运动学涡度分析

秦岭商丹构造带内发育的晚三叠世沙沟街韧性剪切带蕴含大量地质信息,很好地记录了秦岭印支期碰撞造山过程。为了探究该剪切带的运动学特征及其动力学背景,在野外观测、显微构造分析的基础上,对其中发育的糜棱岩进行了磁组构和运动学涡度研究。岩石磁学和磁组构分析结果显示:样品的平均磁化率Km值总体较高,载磁矿物主要为磁铁矿等铁磁性矿物;磁化率各向异性度PJ值较大,表明构造变形较为强烈;形态参数T值多大于0,反映磁化率椭球体以扁球体为主;磁线、面理优势产状与野外观测到的矿物线、面理较为一致。结合磁组构、边界断层以及C面理产状,认为沙沟街韧性剪切带具有左行走滑挤压的运动学特征。运动学涡度Wk值及其分布特征表明,沙沟街剪切带中纯剪切作用所占的比重总体大于简单剪切作用,并且剪切带的核部应位于北界断层附近。综合分析认为,沙沟街韧性剪切带的运动学特征反映了总体斜向汇聚背景下的局部走滑挤压,与商丹带西段发育的同期韧性剪切带具有完全反向的运动学指向,这可能与碰撞导致的侧向挤出构造有关     

Investigations on the mechanical significance of crenulation cleavage

Measurements of total, incremental and progressive strains associated with the development of small scale crenulation cleavage in some low-grade metamorphic rocks from Australia and Switzerland are applied to a discussion of the mechanical significance of the cleavage.Limits are placed on the amount of incremental and total slip or simple shear possible along the cleavage by the observation that the XY principal plane trace of bulk total crenulation strain coincides within 4° of the crenulation cleavage trace in all cases where this strain has been measured or estimated. The measurements are made on eight specimens using deformed porphyroblasts, crystal fibres in pressure-shadows around pyrite and flattened folds and include deformations with coaxial and non-coaxial histories.Further measurements derived from pressure-shadow fibres (eight specimens) show that the style and orientation of incremental deformation are essentially independent of the crenulation cleavage, except for a limit (43°) to the obliquity of the principal incremental extension axis during a given cleavage episode. The only special deformation related to the cleavage is the coaxial one. An indication of passive cleavage behaviour at high strain is shown by the progressive strain history of one specimen. Evidence for passive rotation of a transected axial plane is shown by another. A model is proposed to account for these observations, especially the conditions necessary for initiation and continued development of a new cleavage fabric.Some further applications of existing strain measurement techniques are described: of the R_f/Ø_f method to heterogeneously superposed tectonic strains and of an improved procedure of t′_α/α flattening analysis.     

Determination of stress directions from plagioclase fabrics in high grade deformed rocks (Além Paraíba shear zone, Ribeira fold belt, southeastern Brazil)

A new method to determine stress directions using the preferential orientation of plagioclase mechanical twins has been applied to high-temperature mylonitic rocks from the Além Paraíba shear zone, Ribeira fold belt, southeastern Brazil. We have measured the lattice-preferred orientation of plagioclase grains and calculated the orientation of the stress axes possible for the observed twin orientations. The maximum compressive stress direction (σ₁), determined for all studied samples, is a function of the mechanical twin orientations of a number of distinct plagioclase populations. The σ₁ direction is generally subperpendicular to the (010) plane. The statistical treatment for most of the plagioclase grains examined for each sample shows that σ₁ is almost perpendicular to the foliation plane, suggesting a significant coaxial component in the deformation process of these rocks.     

Geometry of the fault zone of the 2003 Ms = 7.5 Chuya earthquake and associated stress fields, Gorny Altai

The co-seismic deformations produced during the September 27, 2003 Chuya earthquake (Ms = 7.5) that affected the Gorny Altai, Russia, are described and discussed along a 30 km long segment. The co-seismic deformations have manifested themselves both in unconsolidated sediments as R- and R′-shears, extension fractures and contraction structures, and in bedrock as the reactivation of preexisting schistosity zones and individual fractures, as well as development of new ruptures and coarse crushing zones. It has been established that the pattern of earthquake ruptures represents a typical fault zone trending NW–SE with a width reaching 4–5 km and a dextral strike–slip kinematics. The initial stress field that produced the whole structural pattern of co-seismic deformations during the Chuya earthquake, is associated with a transcurrent regime with a NNW–SSE, almost N–S, trending of compressional stress axis (σ₁), and a ENE–WSW, almost E–W, trending of tensional stress axis (σ₃). The state of stress in the newly-formed fault zone is relatively uniform. The local stress variations are expressed in insignificant deviation of σ₁ from N–S to NW–SE or NE–SW, in short-term fluctuations of relative stress values in keeping their spatial orientations, or in a local increase of the plunge angle of the σ₁. The geometry of the fault zone associated with the Chuya earthquake has been compared with the mechanical model of fracturing in large continental fault zones with dextral strike–slip kinematics. It is apparent that the observed fracture pattern corresponds to the late disjunctive stage of faulting when the master fault is not fully developed but its segments are already clearly defined. It has been shown that fracturing in widely different rocks follows the common laws of the deformation of solid bodies, even close to the Earth surface, and with high rates of movements.     

Vitrinite anisotropy under differential stress and high confining pressure and temperature: preliminary observations

The orientation of the optical indicating surface of vitrinite in reflected light has been determined following deformation at 350 and 500°C, confining pressures of 500 and 800 MPa and a strain rate of 10⁻⁵ s⁻¹. High temperature and large strain have facilitated reorientation of the indicating surface, increase in anisotropy (bireflectance) and an increase in maximum vitrinite reflectance. In a specimen deformed at 500°C and 23% axial strain the maximum vitrinite reflectance has been reoriented more than 70° from close to parallel to σ₁ in the undeformed state to perpendicular to σ₁ following deformation. Orientation of the optical indicating surface of some of the deformed specimens suggests the orientation of the maximum reflectance is a composite product of the original orientation of the indicating surface and an orientation produced during deformation.     

Strain and vorticity analysis using small-scale faults and associated drag folds

Small-scale faults with associated drag folds in brittle-ductile rocks can retain detailed information on the kinematics and amount of deformation the host rock experienced. Measured fault orientation (α), drag angle (β) and the ratio of the thickness of deflected layers at the fault (L) and further away (T) can be compared with α, β and L/T values that are calculated with a simple analytical model. Using graphs or a numerical best-fit routine, one can then determine the kinematic vorticity number and initial fault orientation that best fits the data. The proposed method was successfully tested on both analogue experiments and numerical simulations with BASIL. Using this method, a kinematic vorticity number of one (dextral simple shear) and a minimum finite strain of 2.5–3.8 was obtained for a population of antithetic faults with associated drag folds in a case study area at Mas Rabassers de Dalt on Cap de Creus in the Variscan of the easternmost Pyrenees, Spain.     

Plastic deformation of orthoenstatite and the ortho- to high-pressure clinoenstatite transition: a metadynamics simulation study

Atomic-scale mechanisms of plastic deformation in orthoenstatite, MgSiO₃ are studied by computer simulation methods. The combined use of metadynamics and molecular dynamics allows a direct observation of the structural changes during the creation of stacking faults in the (100) plane. A sequence of slip deformations in two different (100) planes at P = 15 GPa and T = 1,000 K reveals a probable transformation mechanism for the ortho- to high-pressure clinopyroxene transition. Each of the observed slips consists of at least four partial deformations crossing high-energy intermediate structures. In agreement with experimental studies, both (100)[010] and (100)[001] slip systems are activated in the deformation process. The observation of a dominant (100)[001] single slip system in pyroxenes may be related to the fact that high-energy intermediate dislocations with (100)[010] component are not stable on geological or experimental timescales.     

Variation in fold geometry in the Yuso basin, northern Spain: implications for the deformation regime

First generation structures in greywackes of the Yuso Group from the Cantabrian Mountains of northern Spain show a distinct variation in geometry with depth in a regional synclinal structure (Curavacas and Lechada synclines); they are easily distinguished from other deformation events. In the structurally uppermost level we find ‘flap folds’. Flap folds are recumbent structures with the inverted limb preserved. Below this level ‘cascade folds’ are found. These structures have a vergence opposite to that of parasitic folds. The nomenclature adopted is from Harrison and Falcon. Characteristically, these structures have shallowly dipping axial surfaces, in agreement with the shallow dip of the axial plane (regional) cleavage. In the lowermost structural level, upright parasitic folds with a steep cleavage are present. The variation in fold geometry is accompanied by a general steepening of the regional cleavage with increasing depth. In the absence of overprinting relationships the F₁ fold geometries are included in a single deformation event.The steepening of the cleavage with depth reflects the change in orientation of the maximum shortening direction from sub-vertical in the upper part of the syncline to sub-horizontal in the lower part. With increasing depth the deformation regime during F₁ changed from bending to buckling. The deformation regime on the regional scale, however, is associated with basement subsidence and passive formation of the regional synclinal structure. Furthermore, the absence of a distinct microfabric for the different F₁ folds indicates that on a small scale a similar deformation regime was present. We conclude, therefore, that the scale at which we study a structure only reflects the deformation regime at that particular scale. Consequently, the overall deformation regime cannot be determined from single outcrops or microstructural analysis alone.     

Grain size as an indication of shear strain in brittle fault zones

Shear strain γ in brittle fault zones is related to final and initial grain size parameters, d_f and d_i, respectively, by an expression of the form: where μ_k the coefficient of kinetic friction of crushed rock and σ_n the normal stress across the zone at the time of faulting.Technological literature suggests that ƒ(d_f, d_i) may be given by 10 , where W_i is a material constant. The resulting relationship between shear strain and grain size seems to be compatible with existing experimental data.     

Quantitative finite strain and kinematic flow analyses along the Zagros transpression zone, Iran

The Ghouri area in southwest Iran exposes a cross section through the Zagros orogenic belt. The area provides an opportunity to investigate quantitative finite strain (Rs), kinematic vorticity number (W_k), proportions of pure shear and simple shear components, sense of shear indicators, steeply plunging lineations, and other moderate to steeply plunging stretching lineations in a transpressional zone. Based on a classical strain analysis of deformed microfossils with oblate strain ellipsoid shape, the Zagros orogenic belt is classified as a pure-shear dominated zone of transpression, but asymmetry of shear-sense indicators suggests that a significant component of simple shear was involved along the deformation zone boundaries. The long axes of the microfossils and stretched pebbles of a deformed conglomerate were used to indicate the stretching direction in this zone. The stretching lineations have a steep to moderate plunge but a constant strain magnitude. Characteristics of dextral inclined transpressional kinematics in the Zagros continental collision zone were quantified and indicate an estimated k-value < 1, an angle between the maximum horizontal axis of the instantaneous strain ellipsoid and the zone boundary (θ = 32°), asymmetrical dextral shear-sense indicators, and an angle of relative plate motion (α = 25°).     

Preferred orientation in experimentally deformed limestone

Over sixty syntectonic deformation experiments in uniaxial compression have been done on fine-grained limestones in the stability fields of calcite I, calcite II and aragonite. X-ray techniques and spherical harmonic analysis of the data were used to determine preferred orientation quantitatively, and inverse pole-figures were derived for these axially symmetric specimens. They display in most cases strong preferred orientation which varies as a function of the experimental conditions, mainly temperature and pressure. At temperatures below 350° C recrystallization is lacking and flattened grains indicate that translation, twin gliding and kinking have been the dominant deformation mechanisms. The inverse pole-figure shows a maximum at c with a shoulder towards or a second maximum at e. This is in agreement with preferred orientation observed in experimentally deformed Yule marble and can be explained as the product of dominant twin gliding on e and translation gliding on r (Turner et al., 1956). At high temperatures (900–1000° C) strong grain growth (from 4 to 50 microns) indicates that the fabric recrystallized. Grains are equidimensional and clear with a marble-like texture. The inverse pole-figure shows a single maximum at r, and c-axes are oriented in a small circle around the axis of compression, ₁. Such a pattern of preferred orientation would be expected on thermodynamic grounds assuming that recrystallized grains will be oriented in such a way that the strain energy is a maximum (e.g. MacDonald, 1960). Decrease in confining pressure caused a decrease of the maximum at c and the formation of a secondary maximum at highangle positive rhombs in the inverse pole-figure. This can be interpreted as r translation dominating over e twinning. In all deformation experiments an equilibrium in preferred orientation was reached after 20 percent shortening. The strength of preferred orientation decreased with increasing temperature. Aragonite was produced within its hydrostatic stability field at temperatures above 500° C. Close to the phase boundary, coarse-grained textures showed preferred orientation with poles to (010) parallel to ₁. At higher pressures the fabric is fine-grained and [001] is aligned parallel to ₁. Evidence is given that the phase change from calcite to aragonite in these deformation experiments is a diffusive and not a martensitic transformation.Publication No. 1043, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California.     

The Maximum Effective Moment Criterion （MEMC） and Its Implications in Structural Geology

1 Introduction A high-level generalization of structures in the earth crust has been given by Ramsay (1980): low-angle thrusts in the brittle upper crust and high-angle reverse shear zones in the ductile middle-lower crust are formed in contractional regimes; high-angle normal faults in the brittle upper crust and low-angle normal shear zones in the ductile middle- lower crust are formed in extensional regimes. The formation of low-angle thrusts and high-angle normal faults in brittle domains …     

A thermodynamic model of Fe–Cr spinels

A new thermodynamic model for multi-component spinel solid solutions has been developed which takes into account thermodynamic consequences of cation mixing in spinel sublattices. It has been applied to the evaluation of thermodynamic functions of cation mixing and thermodynamic properties of Fe₃O₄–FeCr₂O₄ spinels using intracrystalline cation distribution in magnetite, lattice parameters and activity-composition relations of magnetite–chromite solid solutions. According to the model, cation distribution in binary spinels, (Fe_1-x²⁺ Fe_x³⁺)[Fe_x²⁺Fe_2-2y-x³⁺Cr_2y]O₄, and their thermodynamic properties depend strongly on Fe²⁺–Cr³⁺ cation mixing. Mixing of Fe²⁺–Fe³⁺ and Fe³⁺–Cr³⁺ can be accepted as ideal. If Fe²⁺, Fe³⁺ and Cr are denoted as 1, 3 and 4 respectively, the equation of cation distribution is –RT ln(x²/((1–x)(2–2y–x)))= G₁₃^* + (1–2x)W₁₃+y(W₁₄–W₁₃–W₃₄) where G₁₃^* is the difference between the Gibbs energy of inverse and normal magnetite, W_ij is a Margules parameter of cation mixing and G₁₃^*, J/mol =–23,000+13.4 T, W₁₄=36 kJ/mol, W₁₃=W₃₄=0. The positive nonconfigurational Gibbs energy of mixing is the main reason for changing activity–composition relations with temperature. According to the model, the solvus in Fe₃O₄–FeCr₂O₄ spinel has a critical temperature close to 500°C, which is consistent with mineralogical data.     

On the optically biaxial character and heterogeneity of anthracites

The results of the study of optical properties of 13 anthracites from different parts of the world are presented in this paper. Measurements of reflectance values were made on non-oriented vitrinite grains for a minimum of 300 points per sample. The reconstruction of Reflectance Indicating Surfaces (RIS) were made by Kilby's method [Kilby, W.E., 1988. Recognition of vitrinite with non-uniaxial negative reflectance characteristics. Int. J. Coal Geol. 9, 267–285; Kilby, W.E., 1991. Vitrinite reflectance measurement — some technique enhancements and relationships. Int. J. Coal Geol. 19, 201–218]. It was found that the use of Kilby's method for strongly anisotropic materials like anthracites did not give unambiguous results. Some improvement in Kilby's method, consisting of the division of the cumulative cross-plot into several elemental components, is suggested. Each elemental cross-plot corresponds to a textural class of anthracite, which is characterized by the values of RIS main axes R_MAX^(k), R_INT^(k) and R_MIN^(k) (k=1,2,…n; n — number of classes). The global texture of anthracite is characterized as a RIS with main axes calculated as the weighted means of , and for each class of this anthracite.The division of cumulative Kilby's cross-plot on elemental components makes possible the calculation of new coefficients H_t and H₁₀ characterizing the heterogeneity of the structure and texture of anthracites. The results of our study show that all anthracites have biaxial negative textures, but their heterogeneity varies in a wide range of H_t and H₁₀ coefficients depending upon the individual coal basin.