A new method to calculate the equivalent Mohr–Coulomb friction angle for cohesive and frictional materials

In this note, a new method to calculate the equivalent Mohr–Coulomb friction angle ?′_mc for cohesive and frictional materials is presented. This method makes a transformation from the failure surface for cohesive materials to the failure surface for cohesionless materials and obtains ?′_mc as well as the principal stress ratio σ′₁/σ′₃ for cohesionless materials in the transformed space first, then obtains ?′_mc for cohesive materials by linking σ′₁/σ′₃ in the transformed space and in the original space. In the application example, an analytical solution of the invariant stress ratio L is derived from the failure function in the transformed space. The influence of the intermediate effective principal stress σ′₂ is also demonstrated using the already calculated ?′_mc. Copyright © 2010 John Wiley & Sons, Ltd.     

Failure in accretionary wedges with the maximum strength theorem: numerical algorithm and 2D validation

The objective is to capture the 3D spatial variation in the failure mode occurring in accretionary wedges and their analog experiments in the laboratory from the sole knowledge of the material strength and the structure geometry. The proposed methodology relies on the maximum strength theorem which is inherited from the kinematic approach of the classical limit analysis. It selects the optimum virtual velocity field which minimizes the tectonic force. These velocity fields are constructed by interpolation thanks to the spatial discretization conducted with ten-noded tetrahedra in 3D and six-noded triangles in 2D. The resulting, discrete optimization problem is first presented emphasizing the dual formalism found most appropriate in the presence of nonlinear strength criteria, such as the Drucker–Prager criterion used in all reported examples. The numerical scheme is first applied to a perfectly triangular 2D wedge. It is known that failure occurs to the back for topographic slope smaller than and to the front for slope larger than a critical slope, defining subcritical and supercritical slope stability conditions, respectively. The failure mode is characterized by the activation of a ramp, its conjugate back thrust, and the partial or complete activation of the décollement. It is shown that the critical slope is captured precisely by the proposed numerical scheme, the ramp, and the back thrust corresponding to regions of localized virtual strain. The influence of the back-wall friction on this critical slope is explored. It is found that the failure mechanism reduces to a thrust rooting at the base of the back wall and the absence of back thrust, for small enough values of the friction angle. This influence is well explained by the Mohr construction and further validated with experimental results with sand, considered as an analog material. 3D applications of the same methodology are presented in a companion paper.     

Interaction of surface erosion and sequential thrust progression: Implications on exhumation processes

This paper investigates the evolution of thrust wedges with concomitant surface erosion, and its bearing on the exhumation processes in orogenic belts. We performed sandbox experiments, simulating syn-orogenic erosion on forelandward sloping surfaces (∼4°). Experiments show that the erosion process has a significant control on the progression of frontal thrusts. In case of no-erosion condition, wedges with high basal friction develop frontal thrusts with strongly increasing spacing. In contrast, for the same basal friction the thrusts show uniform spacing as the wedge development involves concomitant surface erosion. On the other hand, the erosion promotes reactivation of hinterland thrusts in wedges with low basal friction. We show that erosion-assisted thrust reactivation is the principal mechanism for exhumation of deeper level materials in orogens. Efficiency of this mechanism is largely controlled by basal friction. The exhumation of deeper level materials is limited, and occurs within a narrow, sub-vertical zone in the extreme hinterland when the basal friction is high (μ_b = 0.46). In contrast, the process is quite effective in wedges with low basal friction (μ_b =0.36), resulting in exhumation along gently dipping foreland-vergent thrusts as well as along thrusts, subsequently rotated into steep attitude. The zone of exhumation also shifts in the foreland direction in the course of horizontal movement. Consequently, deeper level materials cover a large area of the elevated part of the wedge.     

Role of lateral thickness variations on the development of oblique structures at the Western end of the South Pyrenean Central Unit

The Montsec unit is one of the most important detached South-verging nappes within the South Pyrenean Central Unit (SPCU, Southern Pyrenees). A N–S cross-section of its Western sector, based on seismic reflection profiles, shows a hangingwall ramp geometry in Mesozoic strata, overlain by a syntectonic series of Lower Eocene sediments with growth geometry. The geometry of growth strata constrains the age of its movement between the Paleocene and the Middle Eocene. The geometry of the Western, oblique ramp of the South Pyrenean Central Unit is defined by a series of N–S folds, in some cases associated with underlying West-verging thrusts, as indicated by seismic reflection profiles and field data. In this paper, we propose that the geometry of the thrust wedge of Mesozoic units, progressively thinning from East to West, strongly contributed to constrain the location and geometry of the Western termination of the Montsec thrust. The hypothesis proposed is checked by a series of experimental wedges developed in a sandpack with lateral and three-dimensional thickness variations. Oblique structures form as thrusting progresses at the tip of the sand wedge.     

Contribution à l'étude de la vitesse critique d'érosion des sols cohésifs

The phenomena of erosion and sedimentation in rivers are treated qualitatively and quantitatively. An analytical solution of the bedload critical velocity for erosion and sedimentation is proposed, depending on material property, geometry and on flow characteristics. The critical velocity of erosion of the river bed, as defined by Hjulström in 1935, can be explained by a cohesive behaviour of the grains. Whereas the cohesive force is dominated by an 1/r² (r is the radius of grains) force when grains are small, it can be modelled by the cohesion C and the friction angle φ for larger grains. To cite this article: J. Gargani, C. R. Geoscience 336 (2004).     

Structural consequences of cohesion in gravitational instabilities triggered by fluid overpressure: Analytical derivation and experimental testing

The critical taper theory of Coulomb wedges has been classically applied to compressive regimes (accretionary prisms/fold-and-thrust belts), and more recently to gravitational instabilities. Following the initial hypothesis of the theory, we provide an alternative expression of the exact solution for a non-cohesive wedge by considering the balance of forces applied to the external surfaces. Then, we use this approach to derive a solution for the case of cohesive wedges. We show that cohesion has conspicuous structural effects, including a minimum length required for sliding and the formation of listric faults. The stabilizing effect of cohesion is accentuated in the foremost thin domain of the wedge, defining a required Minimum Failure Length (MFL), and producing sliding of a rigid mass above the detachment. This MFL decreases with less cohesion, a smaller coefficient of internal friction, larger fluid overpressure ratio, and steeper upper and basal surfaces for the wedge. Listricity of the normal faults depends on the fluid overpressure magnitude within the wedge. For moderate fluid overpressure, normal faults are curved close to the surface, and become straight at depth. In contrast, where fluid overpressure exceeds a critical value corresponding to the fluid pressure required to destabilize the surface of a noncohesive wedge, the state of stress changes and rotates at depth. The faults are straight close to the surface and listric at depth, becoming parallel to the upper surface if the wedge is thick enough. We tested some of these structural effects of a cohesive wedge on gravitational instabilities using analogue models where cohesive material was subjected to pore-fluid pressure. The shape of the faults obtained in the models is consistent with the predictions of the theory.     

Topography as a major factor in the development of arcuate thrust belts: insights from sandbox experiments

We have used sandbox experiments to investigate and to illustrate the effects of topography upon the development of arcuate thrust belts. In experiments where a sand pack shortened and thickened in front of an advancing rectilinear piston, the geometry of the developing thrust wedge was highly sensitive to variations in surface topography. In the absence of erosion and sedimentation, the surface slope tended to become uniform, as predicted by the theory of critical taper. Under these conditions, the wedge propagated by sequential accretion of new thrust slices. In contrast, where erosion or sedimentation caused the topographic profile to become irregular, thrusts developed out of sequence. For example, erosion throughout a hinterland caused underlying thrusts to remain active and inhibited the development of new thrusts in the foreland. Where initial topography was irregular in plan view, accreting thrusts tended to be arcuate. They were convex towards the foreland, around an initially high area; concave towards the foreland, around an initially low area. Initial plateaux tended to behave rigidly, while arcuate thrust slices accreted to them. Thrust motions were radial with respect to each plateau. Within transfer zones to each side, fault blocks rotated about vertical axes and thrust motions were oblique-slip. At late stages of deformation, the surface slope of the thrust wedge tended towards a uniform value. Initial mountains of conical shape (representing volcanoes) also escaped deformation, except at depth, where they detached. Arcuate thrust slices accreted to front and back. Where a developing thrust wedge was subject to local incision, accreting thrust slices dipped towards surrounding areas of high topography, forming Vs across valleys.Arcuate structural patterns are to be found around the three highest plateaux on Earth (Tibet, Pamirs and Altiplano) and around the Tromen volcanic ridge in the Neuquén Basin of northern Patagonia. We infer that these areas behaved in quasi-rigid fashion, protected as they were by their high topography.     

Extrusion vs. accretion at the frictional–viscous décollement transition in experimental thrust wedges: the role of convergence velocity

In many cases, thrust wedges accreted at shallow crustal levels show an across‐strike rheological variability along the basal décollement, notably from brittle to ductile behaviour. In this paper, we illustrate the results of sandbox analogue modelling research devoted to studying the influence of convergence velocity on wedge architecture when laterally juxtaposed frictional and viscous materials occur along the basal décollement of accreting thrust wedges. Our results show that slow convergence favours a near symmetrical distribution of thrust vergence within wedge sectors accreted above viscous décollement material, whereas fast convergence favours vergence asymmetry. In particular, at fast convergence rates the hinterlandward extrusion of viscous décollement material at the toe of the frictional wedge is favoured and contributed to accommodate a significant amount of the total contraction. Terra Nova, 18, 241–247, 2006     

Experimental investigation of shear strength of sands with inherent fabric anisotropy

Loading direction-dependent strength of sand has been traditionally characterized in the principal stress space as a direct extension of the Mohr–Coulomb criterion. A recent study found that it is more appropriate to define anisotropic strength of sand on failure/shear planes, but this proposition has only been demonstrated with discrete element method (DEM) simulations. The present study experimentally investigates anisotropic shear strength of sands in this new framework. Three granular materials with distinct grain characteristics ranging from smooth and rounded particles to flaky and angular particles are tested with the bedding plane inclination angle ψ _b varying over the full range of 0°–180°. The main objective is to study how the peak friction angle ? _p of sand is affected by the ψ _b angle and how the ψ _b–? _p relationship evolves with the change of characteristics of constituent sand particles. We find that the general trend of ψ _b–? _p curves for real sands resembles what was predicted by DEM in a previous study, whereas rich anisotropic strength behavior is revealed by the laboratory data. The effects of normal stress and initial density, as well as shear dilation behavior at different shear directions, are also studied.     

Numerical solution of active earth pressures on rigid retaining walls built near rock faces

For retaining walls built in mountainous regions, narrow backfill spaces are often encountered. The space to fully develop the active wedge is restricted for walls with a limited backfill space. This paper presents a numerical study on the behaviour of active earth pressures behind a rigid retaining wall with limited backfill space of various geometries. The active earth pressure for a wall built with limited backfill space is considerably less than that of the Coulomb solution, and the location of the resultant of active earth pressures is noticeably higher than one-third of the wall height. The coefficient of active earth pressures is as low as 0.5–0.6 times the Coulomb solution and the h/H value reaches up to 0.4–0.37 if aspect ratio of the fill space is in the range from 0.1 to 0.2. A clear trend between the ratio of the coefficient of active earth pressures at constrained fill conditions over the Coulomb K_a value and the aspect ratio of the fill-space geometry is obtained.     

Inverse method applied to a sand wedge: Estimation of friction parameters and uncertainty analysis

We examine the fundamental mechanism of frontal accretion in a sand wedge from the occurrence of a forward thrust ramp evolving into a fault-bend fold to the jump of deformation to a new frontal ramp ahead. We use inverse problem theory to extract quantitative information on friction parameters from the systematic comparison of experimental observations and theoretical predictions. The observables are locations, dips and lifetime of thrust ramps, hinge and associated compression. The experimental values (observed data) are cast into statistical models describing the error bars. The theory of limit analysis provides calculated data, requiring five parameters : material density, friction coefficient of the décollement plane, friction coefficient of the bulk material, and the variation of friction with slip on the ramp as well as the distance for this variation. The misfit between observed data and calculated data is determined for all physically admissible values of the parameters. Values yielding a small misfit are interpreted as highly probable. The mean misfit per observable is within their error bars and therefore application of the theory reproduces the observables. Bulk and décollement friction coefficient values with high probability are compared to independent measurements. The inversion also reveals systematic discrepancies: the frictional weakening on the ramps is overestimated, while the force is underestimated, the calculated thrust sheet is longer than observed and the calculated jump to a second ramp occurs earlier than observed. These conclusions allow us to identify necessary improvements for the experimental set-up and of the theoretical assumptions.     

挡土墙库仑土压力的遗传算法求解分析

在对破裂面上滑动土体静力极限平衡分析的基础上,建立了基于优化方法求解无黏性土、黏性土库仑土压力的自变量取值区间和目标函数模型,并采用遗传进化方法进行了实例求解分析。研究结果表明,遗传算法在计算挡土墙库仑主动土压力的过程中,收敛速度快、用时短,并具有较高的计算精度。算例1中5组无黏性土挡土墙的主动土压力的计算结果与经典库仑解析解非常接近,平均误差为1.748 %,平均进化代数为15代。算例2中8组黏性土挡土墙的主动土压力计算结果与文献的解答非常吻合,平均误差仅为0.017 %,平均进化代数为17.125代。遗传算法具有良好的适应性和强大的搜索性能,非常适合求解岩土工程优化问题。     

Application of statistical learning algorithms to ultimate bearing capacity of shallow foundation on cohesionless soil

This study employs two statistical learning algorithms (Support Vector Machine (SVM) and Relevance Vector Machine (RVM)) for the determination of ultimate bearing capacity (q_u) of shallow foundation on cohesionless soil. SVM is firmly based on the theory of statistical learning, uses regression technique by introducing varepsilon‐insensitive loss function. RVM is based on a Bayesian formulation of a linear model with an appropriate prior that results in a sparse representation. It also gives variance of predicted data. The inputs of models are width of footing (B), depth of footing (D), footing geometry (L/B), unit weight of sand (γ) and angle of shearing resistance (?). Equations have been developed for the determination of q_u of shallow foundation on cohesionless soil based on the SVM and RVM models. Sensitivity analysis has also been carried out to determine the effect of each input parameter. This study shows that the developed SVM and RVM are robust models for the prediction of q_u of shallow foundation on cohesionless soil. Copyright © 2010 John Wiley & Sons, Ltd.     

Sensitivity of shear zones in orogenic wedges to surface processes and strain softening

We investigate the internal deformation of orogenic wedges growing by frontal accretion with a two-dimensional numerical model. Our models are limited to crustal deformation and assume a horizontal detachment as observed for various natural orogens (e.g. Alaska and Costa Rica). The model wedges develop as a result of convergence of a brittle sediment layer in front of a strong backstop. We find that our reference model develops in-sequence forward-thrusts which propagate upward from the basal detachment. For this reference model we investigate the sensitivity of shear zone activity to surface processes and strain softening. Model results show that diffusive or slope dependent erosion enhances material transport across the wedge and slows down forward propagation of the deformation front. Frictional strain softening focuses deformation into narrow shear zones and enhances displacement along them. This has also been postulated for natural thrusts such as the Glarus thrust in the Swiss Alps and the Moine thrust in the Scottish Caledonides. A second series of models investigates the effects of regularly spaced weak inclusions within the sediment layer which simulate remnants of previous deformation phases. These inclusions facilitate and focus internal deformation, influence the thrust dip and thrust vergence and enable thrust reactivation in the internal part of the wedge. Our results show that inactive thrusts in the internal part of the wedges may be reactivated in models with diffusive surface processes, strain softening or weak inclusions. Thrust reactivation occurs as models seek to maintain their critical taper angle. First order characteristics of our numerical models agree well with natural orogenic wedges and results from other numerical and analogue models.     

Bearing capacity of strip and circular footings in sand using finite elements

Design of shallow foundations relies on bearing capacity values calculated using procedures that are based in part on solutions obtained using the method of characteristics, which assumes a soil following an associated flow rule. In this paper, we use the finite element method to determine the vertical bearing capacity of strip and circular footings resting on a sand layer. Analyses were performed using an elastic–perfectly plastic Mohr–Coulomb constitutive model. To investigate the effect of dilatancy angle on the footing bearing capacity, two series of analyses were performed, one using an associated flow rule and one using a non-associated flow rule. The study focuses on the values of the bearing capacity factors N_q and N_γ and of the shape factors s_q and s_γ for circular footings. Relationships for these factors that are valid for realistic pairs of friction angle and dilatancy angle values are also proposed.     

An evaluation of models for the kinematic evolution of thrust and fold belts: structural analysis of a transverse fault zone in the Front Ranges of the Canadian Rockies north of Banff,Alberta

The prevailing ‘piggyback’ conceptual model for the kinematics of ‘thin-skinned’ thrust and fold belts maintains that the main faults develop sequentially from the hinterland to the foreland, and from the top to the bottom of the accretionary wedge. Moreover, it presumes that when younger thrust faults originate, overlying older thrust faults become inactive and are carried forward passively. This appears to contradict the prevailing mechanical model for the evolution of ‘thin-skinned’ thrust and fold belts, the critical Coulomb wedge model, which requires that lateral growth of the wedge must be accompanied by vertical thickening of the wedge. Crosscutting relationships along a transverse fault zone in the Front Ranges of the Canadian Rockies north of Banff, Alberta, and patterns of overprinting of thrust-related folding on pre-existing thrust sheets, demonstrate substantial overlap in the times of displacement on four major thrust faults in this part of the Front Ranges. The presumption that displacement on one major thrust fault ends when displacement on a younger underlying thrust begins is a fallacy. There is no contradiction between the ‘piggyback’ conceptual kinematic model and the critical Coulomb wedge mechanical model for the evolution of ‘thin-skinned’ foreland thrust and fold belts. The main faults do originate sequentially from the hinterland to the foreland, and from the top to the bottom of the evolving wedge; but displacement occurs simultaneously on several major faults.     

Strontium and oxygen isotopic variations in Mesozoic and Tertiary plutons of central Idaho

Regional variations in initial ⁸⁷Sr/⁸⁶Sr ratios (r _i) of Mesozoic plutons in central Idaho locate the edge of Precambrian continental crust at the boundary between the late Paleozoic-Mesozoic accreted terranes and Precambrian sialic crust in western Idaho. The r _i values increase abruptly but continuously from less than 0.704 in the accreted terranes to greater than 0.708 across a narrow, 5 to 15 km zone, characterized by elongate, lens-shaped, highly deformed plutons and schistose metasedimentary and metavolcanic units. The chemical and petrologic character of the plutons changes concomitantly from ocean-arc-type, diorite-tonalite-trondhjemite units to a weakly peraluminous, calcic to calcalkalic tonalite-granodiorite-granite suite (the Idaho batholith). Plutons in both suites yield Late Cretaceous ages, but Permian through Early Cretaceous bodies are confined to the accreted terranes and early Tertiary intrusions are restricted to areas underlain by Precambrian crust. The two major terranes were juxtaposed between 75 and 130 m.y. ago, probably between 80 and 95 m.y. Oxygen and strontium isotopic ratios and Rb and Sr concentrations of the plutonic rocks document a significant upper-crustal contribution to the magmas that intrude Precambrian crust. Magmas intruding the arc terranes were derived from the upper mantle/subducted oceanic lithosphere and may have been modified by anatexis of earlier island-arc volcanic and sedimentary units. Plutons near the edge of Precambrian sialic crust represent simple mixtures of the Precambrian wall-rocks with melts derived from the upper mantle or subducted oceanic lithosphere with r _i of 0.7035. Rb/Sr varies linearly with r _i, producing “pseudoisochrons” with apparent “ages” close to the age of the wall rocks. Measured δ ¹⁸O values of the wall rocks are less than those required for the assimilated end-member by Sr-O covariation in the plutons, however, indicating that wall-rock δ ¹⁸O was reduced significantly by exchange with circulating fluids. Metasedimentary rocks of the Belt Supergroup are similarly affected near the batholith, documenting a systematic depletion in ¹⁸O as much as 50 km from the margin of the batholith. Plutons of the Bitterroot lobe of the Idaho batholith are remote from the accreted terranes and represent mixtures of Precambrian wall-rocks with melts dominated by continental lower crust (r _i>0.708) rather than mantle. “Pseudoisochrons” resulting from these data are actually mixing lines that yield apparent “ages” less than the true age of the wall rocks and meaningless “r_i”. Assimilation/ fractional-crystallization models permit only insignificant amounts of crystal fractionation during anatexis and mixing for the majority of plutons of the region.     

Intrinsic versus extrinsic variability of analogue sand-box experiments – Insights from statistical analysis of repeated accretionary sand wedge experiments

Analogue models are not perfectly reproducible even under controlled boundary conditions which make their interpretation and application not always straight forward. As any scientific experiment they include some random component which can be influenced both by intrinsic (inherent processes) and extrinsic (boundary conditions, material properties) sources. In order to help in the assessment of analogue model results, we discriminate and quantify the intrinsic versus extrinsic variability of results from “sandbox” models of accretionary wedges that were repeated in a controlled environment. The extrinsic source of variability, i.e. the parameter varied is the nature of the décollement (material, friction and thickness). Experiment observables include geometric properties of the faults (lifetime, spacing, dip) as well as wedge geometry (height, slope, length).For each variable we calculated the coefficient of variance (CV) and quantified the variability as a symmetric distribution (Normal, Laplacian) or asymmetric distribution (Gamma) using a Chi squared test (χ²). Observables like fault dip/back thrust dip (CV = 0.6–0.7/0.2–0.6) are less variable and decrease in magnitude with decreasing basal friction. Variables that are time dependent like fault lifetime (CV = 0.19–0.56) and fault spacing (CV = 0.12 – 0.36) have a higher CV consequently affecting the variability of wedge slope (CV = 0.12–0.33). These observables also increase in magnitude with increasing basal friction. As the mechanical complexity of the evolving wedge increases over time so does the CV and asymmetry of the distribution. In addition, we confirm the repeatability of experiments using an ANOVA test. Through the statistical analysis of results from repeated experiments we present a tool to quantify variability and an alternative method to gaining better insights into the dynamic mechanics of deformation in analogue sand wedges.     

不同变形速率下前陆褶皱冲断带的发育特征——基于砂箱物理模拟实验研究

构造模拟实验是研究和模拟自然界地质构造现象、变形特征、成因机制和动力学过程的一种物理实验方法。本文基于砂箱构造物理模型高、中、低速7组不同单向挤压速度的构造模拟实验,揭示不同变形速率下砂箱物质变形的几何学、运动学及其演化特征,探讨不同变形速度(尤其是不同量级速度)对前陆褶皱冲断构造变形的重要性。高速单向挤压变形过程中(0.4~0.1mm/s),砂箱模型中石英砂体后缘构造加积强烈,砂体变形主要以楔形体向前扩展变形为主,其构造样式主要表现为前展式叠瓦冲断构造。中速挤压变形过程中(0.05~0.005mm/s),砂体后缘加积相对较弱,构造变形样式主要为砂箱楔形体前缘先形成(前展式为主)逆冲断层,当达到临界楔形体后,反冲断层发育并与前展式逆冲断层构成冲起构造,构造样式上表现为叠瓦构造与冲起构造的无序叠加。低速挤压变形过程中(0.002mm/s),砂箱物质构造样式以典型冲起带为主。由于砂箱模拟过程的时效性,即如何在最有效时间内获得最全面而复杂的演变过程,通过本次系列实验,我们建议将此有效挤压速度设定为0.05~0.005mm/s范围内,可以先后清晰而全面的获得高速和低速挤压下砂体的变形过程和构造样式。     

Benchmarking numerical models of brittle thrust wedges

We report quantitative results from three brittle thrust wedge experiments, comparing numerical results directly with each other and with corresponding analogue results. We first test whether the participating codes reproduce predictions from analytical critical taper theory. Eleven codes pass the stable wedge test, showing negligible internal deformation and maintaining the initial surface slope upon horizontal translation over a frictional interface. Eight codes participated in the unstable wedge test that examines the evolution of a wedge by thrust formation from a subcritical state to the critical taper geometry. The critical taper is recovered, but the models show two deformation modes characterised by either mainly forward dipping thrusts or a series of thrust pop-ups. We speculate that the two modes are caused by differences in effective basal boundary friction related to different algorithms for modelling boundary friction. The third experiment examines stacking of forward thrusts that are translated upward along a backward thrust. The results of the seven codes that run this experiment show variability in deformation style, number of thrusts, thrust dip angles and surface slope. Overall, our experiments show that numerical models run with different numerical techniques can successfully simulate laboratory brittle thrust wedge models at the cm-scale. In more detail, however, we find that it is challenging to reproduce sandbox-type setups numerically, because of frictional boundary conditions and velocity discontinuities. We recommend that future numerical-analogue comparisons use simple boundary conditions and that the numerical Earth Science community defines a plasticity test to resolve the variability in model shear zones.