Strain memory of 2D and 3D rigid inclusion populations in viscous flows — What is clast SPO telling us?

We model the development of shape preferred orientation (SPO) of a large population of two- and three-dimensional (2D and 3D) rigid clasts suspended in a linear viscous matrix deformed by superposed steady and continuously non-steady plane strain flows to investigate the sensitivity of clasts to changing boundary conditions during a single or superposed deformation events. Resultant clast SPOs are compared to one developed by an identical initial population that experienced a steady flow history of constant kinematic vorticity and reached an identical finite strain state, allowing examination of SPO sensitivity to deformation path. Rotation paths of individual triaxial inclusions are complex, even for steady plane strain flow histories. It has been suggested that the 3D nature of the system renders predictions based on 2D models inadequate for applied clast-based kinematic vorticity gauges. We demonstrate that for a large population of clasts, simplification to a 2D model does provide a good approximation to the SPO predicted by full 3D analysis for steady and non-steady plane strain deformation paths. Predictions of shape fabric development from 2D models are not only qualitatively similar to the more complex 3D analysis, but they display the same limitations of techniques based on clast SPO commonly used as a quantitative kinematic vorticity gauge. Our model results from steady, superposed, and non-steady flow histories with a significant pure shearing component at a wide range of finite strain resemble predictions for an identical initial population that experienced a single steady simple shearing deformation. We conclude that individual 2D and 3D clasts respond instantaneously to changes in boundary conditions, however, in aggregate, the SPO of a population of rigid inclusions does not reflect the late-stage kinematics of deformation, nor is it an indicator of the unique ‘mean’ kinematic vorticity experienced by a deformed rock volume.     

运动学涡度的理论与实践

一般剪切作用下,碎斑或顺剪切作用方向向前或逆向旋转向两特征方向或流脊(非旋转方向)靠拢。高应变条件下,二长比大于一特定值的碎斑,当其顺向或逆向旋转至以特征方向为渐近线的双曲线的稳定翼上时,便稳定下来不再旋转。二长比小于该值的碎斑将不断向前旋转。这一特定值位于该双曲线的顶点,相应的临界形态因子(B*)或/和两特征方向间夹角的余弦定义为运动学涡度(Wk)。Wk是确定一相关韧性变形带纯剪切和简单剪切组分相对大小的重要度量,是根据内旋转(涡度)与线应变速率之间的比值而定的数值度量。就变形带而言,一般剪切带的运动学涡度变化为0~1,纯剪切为零,简单剪切为1。这是一非线性尺度,纯剪切和简单剪切各占50%的运动学涡度为0.71,而不是0.5。运动学涡度可通过计算、图解(双曲线网(PHD)、刚性颗粒网(RGN)法、Passchier图解、Wallis图解)、极摩尔圆法和应力或瞬时增量应变方向获得。运动学涡度与有限应变测量相结合很可能是估算一地地壳减薄/伸展量或增厚/缩短量的最佳途径。变形过程中运动学涡度很可能变化,应根据不同时期形成的构造获得相应时期的运动学涡度。     

运动学涡度和极摩尔圆的基本原理与应用

运动学涡度是岩石递进变形中非同性的一种量度，利用其对韧性剪切带进行应变分解，确定剪切作用类型是当今构造地质学研究的较新课题。极摩尔圆同时适用共轴和非共为形，是应变分析的一种有利工具，并特别适用于从应变测量数据求取运动学涡度；从力学理论角度对运动学涡度进行了系统而简明的论述，地极摩尔圆的原理进行了推导，提出了它们在韧性剪切带应变分析中的应用。     

Impact of extreme waves on a vertical wall

This study addresses the impact of nonlinear wave evolution on the resulting wave force values on a vertical wall. To this end, the problem of interaction between non-breaking water waves and a vertical wall over constant depth is investigated. The investigation is performed using a two-dimensional wave flume model which is based on the high-order spectral method. Wave generation is simulated at the flume entrance by means of the additional potential concept. This model enables to preserve full dispersivity. Therefore, the model enables to examine the role of nonlinear evolution in the formation of extreme wave force values on a vertical wall for a wide range of water depths. The results for the force exerted on a vertical wall are presented for shallow and deep water conditions. In shallow water, extreme wave force values occur due to the formation of an undular bore. In deep water, extreme wave forces have been obtained as a result of disintegration process of incident wave groups into envelope solitons. Multiple maximum force values have been detected for each of the highest run-up peaks. This phenomenon has been introduced in shallow water conditions and is extended here for deep water conditions.     

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.     

A comparative study of the hypoplasticity and the fabric‐dependent dilatant double shearing models for granular materials

In this paper, we consider the mechanical response of granular materials and compare the predictions of a hypoplastic model with that of a recently developed dilatant double shearing model which includes the effects of fabric. We implement the constitutive relations of the dilatant double shearing model and the hypoplastic model in the finite element program ABACUS/Explicit and compare their predictions in the triaxial compression and cyclic shear loading tests. Although the origins and the constitutive relations of the double shearing model and the hypoplastic model are quite different, we find that both models are capable of capturing typical behaviours of granular materials. This is significant because while hypoplasticity is phenomenological in nature, the double shearing model is based on a kinematic hypothesis and microstructural considerations, and can easily be calibrated through standard tests. Copyright © 2006 John Wiley & Sons, Ltd.     

韧性剪切带的剪切作用类型和韧性减薄量

韧性剪切带组构的演化和剪切作用类型受到许多研究者的关注。运用极莫尔圆法、有限应变法、刚性颗粒法、石英光轴组构结合有限应变测量法、拖尾形态法、剪切带内变形脉体(岩墙)法、碎斑法等方法可以估算剪切带变形过程中的运动学涡度,进而判别剪切带中单剪切组分与纯剪切组分的相对含量。自然界的剪切带一般介于单剪与纯剪之间,运动学涡度Wk介于0~1之间,表明韧性剪切带在变形过程中发生了垂直于剪切带边界(Z轴)方向的韧性减薄。剪切带变形过程中的韧性减薄量可依据有限应变测量与运动学涡度估算求得,也可依据剪切带内的石香肠(布丁)构造求解,还可依据构建极莫尔圆求解。以华北克拉通北缘的楼子店变质核杂岩及其韧性剪切带,以及希腊西奈山的Chelmos剪切带为例,介绍估算韧性剪切带韧性减薄的方法,这种韧性减薄是对大规模岩石圈减薄的有益补充和完善。研究结果表明,定量估算与变质核杂岩相关的韧性剪切带的剪切作用类型是分析变质核杂岩形成机制的有效途径和方法。     

桐柏造山带中剪切带的位移量及年代学研究

桐柏造山带在形成和演化过程中产生了多条韧性剪切带, 本文选取其中大河-固县剪切带和殷店-马垅剪切带作为研究对象, 分析其构造变形特征, 并利用数学法和图解法, 分别计算出两条韧性剪切带的位移量; 对殷店-马垅剪切带尝试用同构造变形石英脉的ESR测年来确定剪切带的活动年龄, 其结果与前人在该地区所获得的年龄十分吻合。大河-固县剪切带具右旋剪切的运动学特征, 运动学涡度W_k明显大于0.75, 指示以简单剪切为主, 位移量为857~867 m。殷店-马垅剪切带同样为右行剪切, 其位移量为1 081~1 113 m, 石英ESR测年结果140.6~131.5 Ma, 表明桐柏造山带碰撞后的构造变形以剪切活动为主, 活动年龄在早白垩世时期。     

Kinematic analysis using AMS data from a deformed granitoid

This study highlights the usefulness of anisotropy of magnetic susceptibility data from a deformed granitoid in deciphering its kinematic evolution vis-à-vis shear zone. Data are presented from the Chakradharpur Granitoid (CKPG) that lies to the north of the northerly dipping, ENE–WSW striking Singhbhum Shear Zone (SSZ; eastern India). Whilst the foliation recorded in the field in some parts of the granitoid is parallel to the SSZ, the magnetic foliation is N54°E/90° (mean orientation). It is suggested that the magnetic fabric provides a window into an evolutionary stage prior to the final shearing/thrusting event, the evidence of which is preserved on the mesoscopic scale. It is envisaged that during the initial stages of deformation there was simple shear along the evolving SSZ that resulted in sinistral strike-slip movement; the vorticity axis at this stage was steeply plunging and sense of rotation was anticlockwise. Space was generated in a direction ∼N25°E (perpendicular to maximum-Instantaneous Stretching Axis) into which CKPG emplaced synchronously with regional deformation and evolving SSZ. With continued deformation, there was thrusting along the SSZ. The vorticity axis flipped to a sub-horizontal orientation, thus leading to the development of down-dip stretching lineations and sheath folds within the SSZ. However, at the same time, the vorticity axis responsible for fabric evolution within the syntectonically crystallizing/cooling CKPG was steeply plunging with clockwise rotation. The magnetic foliation (mean orientation N54°E/90°) developed during the final stage of syntectonic crystallization. However, deformation in the region and thrusting along the SSZ continued even after the CKPG had fully crystallized and solidified, which led to the development of the ENE–WSW striking mesoscopic foliation that is parallel with the SSZ. We propose that the angle between the magnetic foliation and the SSZ/foliation recorded in the field, enables to decipher the kinematic vorticity number of flow responsible for fabric evolution of the CKPG. It is concluded that transpression was an important mechanism, and during regional deformation, whilst the SSZ developed structures by dominantly simple shear, the CKPG underwent dominantly pure shear.     

An elastoplastic constitutive model for rockfills incorporating energy dissipation of nonlinear friction and particle breakage

A new elastoplastic model is developed for rockfills within the general critical state framework incorporating the state parameter. Two state functions are proposed to characterize the evolution of volume dilation and strain softening of rockfills, and a modified breakage index based on the concept of Hardin's relative breakage is defined to describe the progressive crushing of rockfills. The nonassociated plastic flow rule is derived from a state dependent dilatancy equation, and it incorporates energy dissipation due to intrinsic nonlinear friction and particle breakage upon shearing. Thus, their couple effect on the plastic deviatoric and volumetric deformation of rockfills is taken into account in the current model. The numerical analyses are carried out for a series of drained triaxial tests on the modeled rockfills at various consolidation pressures and stress paths. The volume dilation/contraction and strain softening/hardening of rockfills are accurately predicted by the proposed model, and the particle breakage and nonlinear critical state shear strength of rockfills are also well captured. The research findings indicate that the current model is applicable to represent the complex stress–strain–volume change behavior of rockfills in general. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparison of linear and nonlinear shallow wave water equations applied to tsunami waves over the China Sea

This paper discusses the applications of linear and nonlinear shallow water wave equations in practical tsunami simulations. We verify which hydrodynamic theory would be most appropriate for different ocean depths. The linear and nonlinear shallow water wave equations in describing tsunami wave propagation are compared for the China Sea. There is a critical zone between 400 and 500 m depth for employing linear and nonlinear models. Furthermore, the bottom frictional term exerts a noticeable influence on the propagation of the nonlinear waves in shallow water. We also apply different models based on these characteristics for forecasting potential seismogenic tsunamis along the Chinese coast. Our results indicate that tsunami waves can be modeled with linear theory with enough accuracy in South China Sea, but the nonlinear terms should not be neglected in the eastern China Sea region.     

Transmission of Elastic P-waves across Single Fractures with a Nonlinear Normal Deformational Behavior

Summary This paper presents a theoretical study on normally incident elastic P-wave transmission across single dry fractures with a nonlinear normal deformational behavior. The effects of nonlinear fracture normal behavior on P-wave transmission are examined without the mixture of fracture shear behavior. The linear displacement discontinuity model for wave propagation across fractures is extended to a nonlinear model – the hyperbolic elastic model (BB model). Numeric solutions of magnitudes of transmission (|T _non|) and reflection (|R _non|) coefficients, for normally incident P-wave transmission across the nonlinear deformable fractures, are obtained and related to the closure behavior of fractures. Parametric studies are conducted to acquire an insight into the effects of the nonlinear fracture normal deformation on P-wave transmission, in terms of initial normal stiffness and the ratio of current maximum closure to maximum allowable closure of the fractures, as well as the incident wave amplitude and frequency. Comparisons between the linear and nonlinear models are presented. It is shown that, |T _lin| and |R _lin| for the linear model are special solutions of |T _non| and |R _non| for the nonlinear model, when the incident wave amplitude is so low that the current maximum closure of fracture incurred during the wave transmission is much smaller, relative to the maximum allowable closure. In addition, the nonlinear fracture behavior gives rise to a phenomenon of higher harmonics during the wave transmission across the fracture. The higher harmonics contribute to the increase of |T _non| from |T _lin|.     

膨胀土卸荷蠕变特性及其非线性蠕变模型

鉴于膨胀土滑坡往往表现为长期性、渐进性等与时间相关的特性,利用GDS应力路径三轴仪对膨胀土进行了三轴卸荷蠕变试验。试验结果表明：当偏应力较小时,膨胀土的蠕变曲线仅出现瞬时变形和衰减蠕变;当偏应力达到一定值时,其蠕变曲线也呈现衰减蠕变、稳态蠕变和加速蠕变3个阶段,但其加速蠕变阶段的特征与一般岩土体不同,其蠕变速度近乎常数。膨胀土的应力?应变等时曲线显示,膨胀土卸荷蠕变具有非线性特征,且其非线性程度与蠕变时间和应力水平相关,蠕变时间越长、应力水平越高,其非线性程度越高。基于非线性流变力学理论,提出了一种非线性四元件蠕变模型,将标准线性体与一个非线性黏壶串联,该模型可描述等围压三轴压缩应力状态下膨胀土轴向应变随时间的演变规律。根据膨胀土卸荷蠕变试验结果,采用曲线拟合法对三维非线性模型的参数进行反演识别。拟合曲线和试验曲线对比显示,两者吻合良好,说明该模型可以很好地描述膨胀土的蠕变特性。此外,基于该蠕变模型获取了膨胀土的临界破坏应力,其与常规剪切破坏应力的比值随着固结压力的减小而减小,表明越接近坡面的土层越容易发生蠕变破坏。     

Sensitivity of the tropical nonlinear stationary Kelvin and Rossby waves to the vertical structure of heating

Idealized experiments using linear (LM) and nonlinear (NM) multilevel global spectral models have been carried out to investigate and understand the impact of nonlinearities on the stationary wave response in the tropical atmosphere and its sensitivity to the vertical profile of heating. It is found that nonlinearities exert a dominant influence on the low-latitude stationary Kelvin and Rossby waves particularly in the vicinity of the forcing region. Our study shows that nonlinear effects on the upper tropospheric response produce prominent eastward displacement of the anticyclonic vorticity and horizontal shifts of the maximum equilibrium divergence relative to the prescribed heating. These changes due to nonlinear terms are found to be quite sensitive to the vertical structure of diabatic heating. The strongest nonlinear effects are found to occur when the vertical level under consideration is strongly forced from below. Detailed vorticity budget calculations indicate that stronger nonlinear contributions from stretching and horizontal advection of relative vorticity favour the generation of upper tropospheric anticyclonic circulation and its eastward displacement. Larger vertical advection and twisting terms appear to oppose the generation of upper tropospheric anticyclonic vorticity. It is found that the nonlinear terms which affect the vorticity generation in the upper levels are crucially controlled by the vertical profile of heating. The mid-tropospheric response due to deep convective heating in the NM is characterized by anomalous equatorial westerlies in the low-latitude Rossby regime and exhibits prominent ageostrophic motions. Such nonlinear effects appear probably because of a vertical shift of the low level circulation anomalies in the NM. In the case of shallow convective heating the occurrence of anomalous zonal flows and ageostrophic motions in the low latitude regions of the NM takes place near the level of the maximum heating. Our study shows that large heating amplitudes and small vertical gradient of heating at a given vertical level together favour generation of anomalous zonal flows and ageostrophic motions in the near equatorial regions. These anomalous basic flows in the low-latitudes have implications on the propagation of transients from the tropics to midlatitudes. Non-linear effects on the lower tropospheric stationary waves are prominently seen in the case of strong low level heating which produces a large strengthening of the lower tropospheric cyclonic anomalies that exhibit distinct eastward shifts in the NM relative to the LM.     

On wave–current interaction by the Green–Naghdi equations in shallow water

This work is on the use of the Green–Naghdi (GN) nonlinear wave equations for simulating wave–current interaction in shallow water. The stream-function wave theory is used at the wave-maker boundary to generate nonlinear incident waves to consider the wave–current interaction. The nonlinear GN equations are solved in the time domain by use of the finite-difference method. The model is evaluated with data from three experimental studies. A strong opposing current over a submerged bar is investigated in the first test case. In the second test case, the interaction of waves with a uniform current over flat bottom is considered. In the third case, wave–current interaction over a variable bathymetry with the following and opposing currents is studied. The numerical results obtained by the GN equations are compared with the experimental data and results based on the Boussinesq equations. A good agreement is obtained for the three experimental studies considered for a wide range of wave and current conditions.     

3D structural model and kinematic interpretation of the Panixer Pass Transverse Zone (Infrahelvetic Complex,eastern Switzerland)

The Panixer Pass Transverse Zone in the eastern Swiss Alps is oriented perpendicular to most alpine structures in the area. Its main element is the SSE-trending Crena-Martin Fold, a downward facing fold with Permian Verrucano in its core, which is cut by the Glarus Thrust. Hence Verrucano can be found below the Glarus Thrust in the Infrahelvetic Complex. Across the Panixer Pass Transverse Zone the structural buildup of the Infrahelvetic Complex changes considerably. Multiple published theories of the structural evolution are not satisfying particularly because traditional 2D geological cross-sections are insufficient to understand the 3D complexity. The main result and product of our study is a 3D structural model of the Panixer Pass Transverse Zone providing insight into its geometry. As modeling input, we produced a lithostratigraphic map and collected structural orientation data. The 3D structural model honors the observed surface geology and the expected 3D subsurface geometry. Our field data indicates that the shearing and transport direction was continuously NNW-directed, except for a phase of north-directed shearing during the early movement along the Glarus Thrust (late Calanda Phase) and related foliation development in the Helvetic Nappes. The Panixer Pass Transverse Zone developed prior to the penetrative foliation during a thrust-dominated deformation phase (Cavistrau Phase), for which we created a kinematic block model. According to this model, the Crena-Martin Fold is the result of multiple lateral ramps and related lateral fault-bend folds that all developed in a similar positon. In particular, we do not propose ENE-WSW-directed shortening to form the Crena-Martin Fold. The latter was finally cut at low angle by a dextral strike-slip fault to create the final geometry of the Panixer Pass Transverse Zone. Our kinematic model reproduces the main features of the 3D structural model and embeds well into previously proposed sequences of deformation phases.     

辽西地区台里韧性剪切带中长英质脉体的变形演化机制研究

华北克拉通构造演化及破坏机制一直是地学界的热点话题,中生代华北克拉通东部主要受扬子板块的俯冲使区域内岩浆-变质事件频繁,形成一系列近EW向构造,包括位于辽西地区的一条中生代NEE向台里韧性剪切带。该韧性剪切带中发育有S-C组构、旋转碎斑、石香肠、长英质脉体等丰富的构造现象。通过野外观测和室内综合分析,识别出其中长英质脉体的演化序列为"分异脉→香肠化→透镜化→碎斑化→细粒化→丝带化"六个变形阶段。对各变形阶段样品切制垂直面理平行线理的薄片,并对薄片基质层进行二维有限应变测量分析与统计,包括真应变差(ε_1-ε_2)等值线图、岩层厚度比(S)等值线图、运动学涡度(W_k)等值线图。其中W_k在演化前期主要小于0.71,演化中期主要大于0.71,演化后期主要小于0.71,表明长英质脉体演化过程前期主要为纯剪切作用,中期为简单剪切作用,后期为纯剪切作用,同时W_k为正值和S小于1反映研究区内韧性剪切带在伸展减薄的动力学背景下形成,这对了解华北克拉通东部中生代构造演化提供了新的约束。     

Numerical investigations of shear localization in a micro‐polar hypoplastic material

In this paper a micro‐polar continuum approach is proposed to model the essential properties of cohesionless granular materials like sand. The model takes into account the influence of particle rotations, the mean grain size, the void ratio, the stresses and couple stresses. The constitutive equations for the stresses and couple stresses are incrementally non‐linear and based on the concept of hypoplasticity. For plane strain problems the implementation of the model in a finite element program is described. Numerical studies of the evolution of micro‐polar effects within a granular strip under plane shearing are presented. It is shown that the location and evolution of shear localization is strongly influenced by the initial state and the micro‐polar boundary conditions. For large shearing the state quantities tend towards a stationary state for which a certain coupling between the norm of the stress deviator and the norm of the couple stress tensor can be derived. Copyright © 2003 John Wiley & Sons, Ltd.     

Application of kinematic vorticity and gold mineralization for the wall rock alterations of shear zone at Dungash gold mining,Central Eastern Desert,Egypt

The use of porphyroclasts rotating in a flowing matrix to estimate mean kinematic vorticity number (Wm) is important for quantifying the relative contributions of pure and simple shear in wall rocks alterations of shear zone at Dungash gold mine. Furthermore, it shows the relationship between the gold mineralization and deformation and also detects the orientation of rigid objects during progressive deformation. The Dungash gold mine area is situated in an EW-trending quartz vein along a shear zone in metavolcanic and metasedimentary host rocks in the Eastern Desert of Egypt. These rocks are associated with the major geologic structures which are attributed to various deformational stages of the Neoproterozoic basement rocks. We conclude that finite strain in the deformed rocks is of the same order of magnitude for all units of metavolcano-sedimentary rocks. The kinematic vorticity number for the metavolcanic and metasedimentary samples in the Dungash area range from 0.80 to 0.92, and together with the strain data suggest deviations from simple shear. It is concluded that nappe stacking occurred early during the underthrusting event probably by brittle imbrication and that ductile strain was superimposed on the nappe structure during thrusting. Furthermore, we conclude that disseminated mineralization, chloritization, carbonatization and silicification of the wall rocks are associated with fluids migrating along shearing, fracturing and foliation of the metamorphosed wall rocks.     

堆石料等应力比路径四模量增量非线性模型

已有研究表明,土石坝内堆石料在施工填筑期的应力路径可近似为主应力比为常数的路径。根据大型三轴仪上进行的两种应力路径排水试验,即等向压缩的固结试验和等应力比值的剪切试验,提出了一个四模量增量非线性模型。模型除给出了堆石料体积模量K和剪切模量G的表达式外,对剪应力产生的体积应变和平均主应力产生的剪切应变也分别采用了交叉模量J1、J2表达。通过与试验数据的对比表明,四模量增量非线性模型能够较好地预测堆石料在等应力比路径下的应力-应变关系。