Removal of finite deformation using strain trajectories

A technique is described for removing the effects of finite deformation, given the principal values and orientations of strain at a number of points throughout a deformed body.Using the principal orientations, strain trajectories are constructed for the deformed state. The body is divided into finite elements bounded by these trajectories. Each element is then unstrained without changing its orientation or position. This process creates artificial voids and overlaps, which are minimized by imparting rigid translations and rotations to the elements according to a least squares method.The result is the pattern of strain trajectories for the undeformed state. It is shown that the trajectories for the deformed and undeformed states may be used as reference coordinates in order to map the change in shape of any body as it passes from the deformed to the undeformed state or vice versa. The technique is tested using models of a folded layer and a shear zone. It is suggested that the technique is versatile enough to allow for errors in original strain data. Although the technique has so far been applied to two-dimensional deformations, a similar method should be usable in three dimensions.     

Compatibility of two-dimensional strains and rotations along strain trajectories

A study of the compatibility of strains and rotations is necessary for a proper understanding of deformation patterns. Principles of compatibility also underlie all methods for removing the effects of deformation and finding the pretectonic shape of deformed regions. The object of this paper is to derive compatibility equations in a direct way and in a simple form which can be interpreted geometrically. This is done using sets of orthogonal curvilinear coordinates parallel to strain trajectories. As a result, all shear components vanish and only principal strains appear in the equations. Such advantages are obtained at the expense of the extra complexity of curvilinear coordinates compared with Cartesian ones. Orthogonal curvilinear coordinates are described first, with particular reference to curvature. Compatibility of strains and rotations is then discussed by comparing curvatures of trajectories in the undeformed and deformed states. Results are compared with previous theoretical work and numerical methods.     

Tectonic inheritance and kinematic strain localization as trigger for the formation of the Helvetic nappes,Switzerland

The Helvetic nappes in Switzerland consist of sediments, which have been sheared off and thrust over the crystalline basement of the European passive continental margin during Alpine orogeny. Their basal shear zones usually root above the external crystalline massifs. However, the mechanisms that initiated the shear zones and the associated nappe formation are still debated. We perform two-dimensional numerical simulations of the shearing of linear viscous fluids above a linear viscous fluid with considerably higher viscosity (quasi-undeformable). The boundary between the fluid, mimicking the sediments, and the quasi-undeformable fluid, mimicking the basement, exhibits geometrical perturbations, mimicking half-grabens. These geometrical perturbations can trigger significant strain localization and the formation of shear zones within the linear viscous fluid although no rheological softening mechanism is active. This kinematic, ductile strain localization is caused by the half-grabens and the viscosity ratio between basement and sediments. The viscosity ratio has a strong control on the kinematics of strain localization, whereas the depth of the half-grabens has a weak control. For sediment viscosities in the order of 10²¹ Pas and typical half-graben geometries of 5 km depth and 25 km width the localization generates (a) low-angle shear zones at the basement-sediment interface, but also entirely within the sediments, (b) horizontal transport >10 km associated with the shear zones, (c) shear zones with thickness in the order of 100 m, (d) an ordered stacking of model nappes and (e) shear zones that root above the basement. The results suggest that tectonic inheritance in the form of half-grabens and associated kinematic strain localization could have been the triggering mechanism for Helvetic nappe formation, and not rheological softening mechanisms, which might, however, have subsequently further intensified shear localization significantly.     

Dust fallout characteristics within global dust storm major trajectories

Satellite images from 2000 to 2010 were used to identify major dust trajectories within seven major deserts in the world. The fallen dust from these trajectories were collected and analyzed. The fallen dust rates tend to be lower downwind. The average samples of the world's fallen dust are negatively skewed trimodal with dominancy of silt size fractions (61 %). Most of the fallen dust within major dust trajectories is fining downwind. The fallen dust from the eastern zones (Taklimakan, Gobi, and Australian deserts) are characterized by higher percentages of feldspars and clay minerals, low carbonate content, and grain surface area in comparison to the Western zones (Sahara and Arabian deserts). The Western Sahara Desert dust is differentiated by the highest depositional rates and average quartz percentage (66 %). The dust samples in this zone contain low carbonates and grain surface area compared to dust from the Eastern Sahara Desert. The dust samples within Northern Arabia and Ethiopian-south Arabia zones show the highest average of carbonate amounts and grain surface area.     

中低应变率下的岩石损伤本构模型研究

深部岩石的破碎与钻探中,岩石主要处在中低应变率的载荷作用下,而目前的研究还沿用中高应变率推导的动态损伤本构关系。从工程背景和试验角度分析了中低应变率下的岩石动力学特性,提出Maxwell体、Bingham体和损伤体的并联模型,借助拉普拉斯变换,引入基于岩石孔隙、裂隙劣化的损伤变量,导出了中低应变率下的岩石动态损伤本构模型。利用该模型对泥质砂岩和花岗岩在应变率为87 s-1、382 s-1、673 s-1时的应力-应变试验数据进行拟合,并将拟合结果与实测数值、原文献拟合曲线对比,结果表明该理论模型适用性较好。分析拟合参数可知,该动态损伤本构模型能够主动适应中低应变率两类情况,本构曲线和损伤变量的变化符合实际情况。     

Scale dependence of mineral dissolution rates within single pores and fractures

The possibility that gradients in concentration may develop within single pores and fractures, potentially giving rise to scale-dependent mineral dissolution rates, was investigated with experimentally validated reactive transport modeling. Three important subsurface mineral phases that dissolve at widely different rates, calcite, plagioclase, and iron hydroxide, were considered. Two models for analyzing mineral dissolution kinetics within a single pore were developed: (1) a Poiseuille Flow model that applies laboratory-measured dissolution kinetics at the pore or fracture wall and couples this to a rigorous treatment of both advective and diffusive transport within the pore, and (2) a Well-Mixed Reactor model that assumes complete mixing within the pore, while maintaining the same reactive surface area, average flow rate, geometry, and multicomponent chemistry as the Poiseuille Flow model. For the case of a single fracture, a 1D Plug Flow Reactor model was also considered to quantify the effects of longitudinal versus transverse mixing. Excellent agreement was obtained between results from the Poiseuille Flow model and microfluidic laboratory experiments in which pH 4 and 5 solutions were flowed through a single 500 μm diameter by 4000 μm long cylindrical pore in calcite. The numerical modeling and time scale analysis indicated that rate discrepancies arise primarily where concentration gradients develop under two necessary conditions: (1) comparable rates of reaction and advective transport, and (2) incomplete mixing via molecular diffusion. For plagioclase and iron hydroxide, the scaling effects are negligible at the single pore and fracture scale because of their slow rates. In the case of calcite, where dissolution rates are rapid, scaling effects can develop at high flow rates from 0.1 to 1000 cm/s and for fracture lengths less than 1 cm. Under more normal flow conditions where flow is usually slower than 0.001 cm/s, however, mixing via molecular diffusion is effective in homogenizing the concentration field, thus eliminating any discrepancies between the Poiseuille Flow and the Well-Mixed Reactor model. The analysis suggests that concentration gradients are unlikely to develop within single pores and fractures under typical geological/hydrologic conditions, implying that the discrepancy between laboratory and field rates must be attributed to other factors.     

Neoproterozoic nappes and superposed folding of the Itremo Group, west-central Madagascar

The Precambrian geology of west-central Madagascar is reviewed and re-interpreted in light of new field observations, Landsat Thematic Mapper image analysis, and U–Pb geochronology. The bedrock of the area consists of: (1) late Archean (to Paleoproterozoic) migmatite gneiss and schist; (2) Mesoproterozoic stratified rocks (Itremo, Amborompotsy, and Malakialina Groups) perhaps deposited unconformably on the older metamorphic rocks (1, above); (3) Proterozoic ( 1000 Ma–720 Ma) plutonic rocks emplaced into both units above (1 and 2), and; (4) latest Neoproterozoic to middle Cambrian ( 570–520 Ma) granitoids emplaced as regionally discordant and weakly foliated plutons throughout the regions.

The effects of Neoproterozoic orogenic processes are widespread throughout the region and our observations and isotopic measurements provide important constraints on the tectonic history of the region: (i) Archean gneisses and Mesoproterozoic stratified rocks are the crystalline basement and platformal sedimentary cover, respectively, of a continental fragment of undetermined tectonic affinity (East or West Gondwanan, or neither). (ii) This continental fragment (both basement and cover) was extensively invaded by subduction-related plutons in the period from  1000 Ma to  720 Ma that were emplaced prior to the onset of regional metamorphism and deformation. (iii) Continental collision related to Gondwana's amalgamation began after  720 Ma and before  570 Ma. Collision related deformation and metamorphism continued throughout the rest of the Neoproterozoic with thermal effects that lasted until  520 Ma. The oldest structures produced during continental collision were km-scale fold- and thrust-nappes with east or southeast-directed vergence (present-day direction). They resulted in the inversion and repetition of Archean and Proterozoic rocks throughout the region. During this early phase of convergence warm rocks were thrust over cool rocks thereby producing the present distribution of regional metamorphic isograds. The vergence of the nappes and the distribution of metamorphic rocks are consistent with their formation within a zone of west or northwest-dipping continental convergence (present-day direction). (iv) Later upright folding of the nappes (and related folds and thrusts) produced km-scale interference fold patterns. The geometry and orientation of these younger upright folds is consistent with E–W horizontal shortening (present-day direction) within a sinistral transpressive regime. We relate this final phase of deformation to motion along the Ranotsara and related shear zones of south Madagascar, and to the initial phases of lower crustal exhumation and extensional tectonics within greater Gondwana.     

A numerical scheme for calculating stresses and strain rates in glaciers

The stress and strain-rate fields in a glacier or other type of rock with known rheological properties can be calculated by using a numerical solution technique. Calculations are based on force-balance equations and the constitutive relation for polar ice, and do not involve any mathematical approximations. The geometry is prescribed. Basal velocities also are prescribed, either by specifying their values or by relying on a sliding relation. All other quantities are determined numerically.     

Metamorphic and structural diachroneity in the Finnmarkian nappes of north Norway

Abstract Two periods of garnet growth (Gt₁ and Gt₂) have been found in the Finnmarkian nappes of north Norway. In the Kolvik Nappe (the lowest nappe) Gt₁ has preserved an S₂ syntectonic spiral inclusion fabric; in the Olderfjord Nappe an earlier S₁ fabric and an interkinematic inter-D₁–D₂ fabric have been preserved in Gt₁ whilst only the S₁ fabric has been found in Gt₁ in the Brennsvik Nappe (the highest nappe). In each nappe Gt₂ overgrew a penetrative fabric (S₂) wrapped around Gt₁. In the Kolvik Nappe inclusion fabrics may be continuous from Gt₁ into Gt₂ but in the higher nappes there is a distinct break. Gt₂ may have been partially syntectonic with D₃ in the Brennsvik Nappe. Chemically Gt₁ in the Kolvik Nappe and in parts of the Olderfjord and Brennsvik Nappes has antithetic Fe-Mn zoning. In all nappes X_Ca and X_Mg are weakly zoned in Gt₁; X_Mg increases outwards and is greater in the higher nappes in Gt₁ suggesting higher nucleation temperatures. In the Olderfjord and Brennsvik Nappes Gt₂ is marked by increasing X_Ca, probably due to changing garnet-plagioclase equilibria, although the Fe/Mg ratio remains constant. X_Mg is higher in Gt₂ than Gt₁. Basement rocks within the nappe pile have an early pre-Finnmarkian growth (Gt₁) and a later Finnmarkian growth (Gt_H) correlated with Gt₂ on the basis of chemical zoning patterns. The diachroneity of Gt₁ is ascribed to progressively earlier (compared to the structural development) cessation of overstepping of garnet-forming reactions before peak metamorphism in the higher nappes, resulting in earlier structural events being preserved.     

The Alpine geodynamic evolution of Penninic nappes in the eastern Central Alps

The eastern Central Alps consist of several Pennine nappes with different tectonometamorphic histories. The tectonically uppermost units (oceanic Avers Bündnerschiefer, continental Suretta and Tambo nappes, oceanic Vals Bündnerschiefer) show Cretaceous/early Tertiary W-directed thrusting with associated blueschist facies metamorphism related to subduction of the Pennine units beneath the Austroalpine continental crust. This event caused eclogite facies metamorphism in the underlying continental Adula nappe. The gross effect was crustal thickening. The tectonically lower, continental Simano nappe is devoid of any imprint from this event. In the course of continent-continent collision, high- T metamorphism and N-directed movements occurred. Both affected the whole nappe pile more or less continuously from amphibolite to greenschist facies conditions. Crustal thinning commenced during the regional temperature peak. A final phase is related to differential uplift under retrograde P–T conditions. Further thinning of the crust was accommodated by E- to NE-directed extensional deformation.     

不同应变率下砂岩动态强度准则的试验研究

分别利用RMT－150C和多功能分离式霍普金森压杆（SHPB）试验系统对砂岩进行了不同应变率下的单轴压缩、三轴压缩和拉伸试验。结果表明,在围压为0或围压一定的情况下,应变率低于102 s-1时,抗压强度的增加与应变率的量级呈正比;在应变率高于102 s-1时,岩石的抗压强度增加趋势与应变率的1/3次方呈正比。拉伸强度也存在类似规律。针对101～102 s-1应变率范围内缺少岩石动态强度准则这一现状,根据上述试验规律,结合Mohr-Coulomb准则、Hoek-Brown准则和Griffith准则的原理,给出了不同应变率范围内动态Mohr-Coulomb准则和动态Hoek-Brown准则的具体表达形式。研究结果表明,在低应变率情况下,动态Mohr-Coulomb准则和动态Hoek-Brown准则均适用,但Griffith准则判别结果误差很大。在高应变率情况下,动态Mohr-Coulomb准则比较适用,Griffith准则仅适用于评估高应变率单轴抗压强度和抗拉强度之间的关系。     

Influence of different strain rates on hydro-mechanical behaviour of reconstituted unsaturated soil

The hydro-mechanical behaviour of a reconstituted unsaturated soil under different suctions and strain rates was studied through various rate-controlled unsaturated/undrained triaxial tests. The fully saturated reconstituted specimens were desaturated to four different initial suctions (s₀?=?0, 100 kPa, 200 kPa and 300 kPa) and then triaxially sheared (conventional triaxial compression) at three different strain rates in undrained conditions (\(\dot{\varepsilon }_{1} = 0.001\) h^?1, 0.01 h^?1, and 0.1 h^?1). The observed hydro-mechanical behaviour during shearing including the volumetric strain, deviatoric stress, degree of saturation and suction is presented and discussed in this paper. The results indicate that when the strain rate rises at the given initial suctions (or pore water pressures), the maximum deviatoric stress (q_max), critical net stress ratio (M) and critical state suction (s_c) increase but the degree of saturation (S_rc) and volumetric strain at the critical state (ε^c_v ) reduce. The critical effective stress ratio (M′) is not dependent on the strain rate for saturated and unsaturated samples. The critical state lines for unsaturated soils with the constant strain rates are parallel with each other in the e???lnp′ space.

     

Three-dimensional carbon dioxide-induced strain distribution within a confined bituminous coal

Sequestration of carbon dioxide in unmineable coal seams is an option to reduce carbon dioxide emissions. It is well known that the interaction of carbon dioxide with unconfined coal induces swelling. This paper contributes three-dimensional strain distribution in confined coal at microstructural level using high-resolution X-ray computerized tomography data and image analysis. Swelling and compression/compaction of regions in the coal matrix occurs with CO₂ uptake. Normal strain varies between ? 1.15% and 0.93%, ? 3.11% and 0.94%, ? 0.43% and 0.30% along x, y and z axes respectively. Volumetric strain varies between ? 4.25% and 1.25%. The positive strains reported are consistent with typical range for unconstrained swelling. However, the average volumetric strains value (? 0.34%) reflect overall volume reduction. Overall swelling is apparently influenced by the confining stresses. The magnitudes of normal strains are heterogeneous and anisotropic. The swelling vs. compression/compaction observed after CO₂ uptake is localized and likely lithotype dependant.     

Refinement of the Hardening Soil model within the small strain range

Acta Geotechnica - The popularity of the elasto-plastic Hardening Soil (HS) model is based on simple parameter identification from standard testing and empirical formulas. The HS model is...     

Interpretation of porphyroblast inclusion trails: limitations imposed by growth kinetics and strain rates

Porphyroblast inclusion trails provide important information about the tectonometamorphic evolution of a metamorphic rock. However, there remains considerable controversy over whether porphyroblasts rotate during bulk non-coaxial deformation.
With reference to an area of the Scandinavian Caledonides and utilizing existing data from theoretical and experimental modelling, this study demonstrates that both 'straight' and 'S-shaped' inclusion trails are consistent with an interpretation in terms of syndeformational porphyroblast growth in a regime approximating to Newtonian simple shear. At crustal strain rates of 10^-14 s^-1 and porphyroblast growth times of 0.1–1.0 Ma, it is shown that a maximum of 5-9 angular rotation would occur during growth. At faster strain rates of 10^-12 s^-1 (e.g. those in a shear zone) porphyroblast angular rotations of 90 are shown to occur in 0.1–0.25 Ma (i.e. times comparable with or faster than porphyroblastesis). In view of this, 'S-shaped' inclusion trails are to be expected in porphyroblasts growing in active shear zones or other situations of high shear strain, whereas 'straight' inclusion trails can be interpreted as static overgrowth of an existing fabric or as syndeformational porphyroblastesis at low strain rates.     

Changes in strain trajectories in three different types of plate tectonic boundary deduced from earthquake focal mechanisms

Principal strain orientations (minimum horizontal compression—ex and maximum horizontal compression—ey) were established at three different types of plate tectonic boundary: two transform faults, an oceanic ridge located on the Southeast Indian Ridge and a trench located close to the South Sandwich Archipelago. To establish the strain patterns in each zone, 104 earthquake focal mechanisms (centroid-moment tensor solutions for earthquakes with mb≥4; Harvard seismology data, CMT) were examined by fault population analysis. Despite the existence of only one tectonic process that controlled deformation in these zones (divergence, convergence or passive displacement), and only one main strain tensor, several coeval strain ellipsoids were found. These differed from the main strain tensor in the location of the principal strains. In general, permutations were observed between the principal strains, i.e., interchanges between the location of the principal strain axes maintaining the strain ellipsoids in the same 3D orientation. Only in some cases were changes in the ellipsoid orientation associated with major structures.     

Present-day strain rates at the northern end of the ethiopian rift valley

Five successive geodimeter surveys during a five-year period yield hints as to the manner and rate of crustal extension in the Ethiopian rift valley. The northern geodimeter network traverses an en-echelon offset in the Wonji fault belt at latitude 8°30′N; this belt comprises the youngest volcanism and faulting of the rift floor. The northern network surveys reveal progressive rift extension at mean rates of 3–5 mm/year and strain rates of 6–16 · 10⁻⁷/yr, essentially confined within the Wonji fault belt segments. Small longitudinal motions of persistent dextral sense have occurred in the intervening zone between the offset segments. It is too early to say whether these deformations are local, regional or plate-tectonic phenomena, but the present aseismicity of the rift suggests the buildup of regional strain.     

不同围压及应变速率下页岩变形及破损特性试验研究

为揭示围压及应变速率对页岩力学特性的影响规律,对志留统龙马溪组页岩试样开展了不同围压及不同应变率下的三轴压缩力学试验研究。结果表明,围压和应变率对页岩的弹性模量、峰值强度及破裂形态等均具有显著影响,弹性模量和峰值强度均随围压的升高而增加,峰值强度增加的幅度明显大于弹性模量,峰值强度呈线性增加趋势,低围压时应变率从低到高,弹性模量和峰值强度都呈逐渐升高的趋势,两者与应变率对数的关系可用二次多项式描述;随着围压增大,页岩的应变率效应逐渐减弱,在较高高围压(50 MPa)下峰值强度和弹性模量随应变速率增加而增加现象均变得极不显著。对试验后岩样的破坏模式进行分析可知,页岩在低围压高应变率状态下主要是劈裂–剪切破坏,随着围压的增加和应变率的减小,试样的破坏由脆性劈裂–剪切破坏向单一剪切破坏转变,再逐渐向延性破坏过渡。研究结果对于合理确立页岩力学参数及设计压裂方案具有较好的参考。