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We model in 2-D the formation and development of half-graben faults above a low-angle normal detachment fault. The model, based on a “sequential limit analysis” accounting for mechanical equilibrium and energy dissipation, simulates the incremental deformation of a frictional, cohesive, and fluid-saturated rock wedge above the detachment. Two modes of deformation, gravitational collapse and tectonic collapse, are revealed which compare well with the results of the critical Coulomb wedge theory. We additionally show that the fault and the axial surface of the half-graben rotate as topographic subsidence increases. This progressive rotation makes some of the footwall material being sheared and entering into the hanging wall, creating a specific region called foot-to-hanging wall (FHW). The model allows introducing additional effects, such as weakening of the faults once they have slipped and sedimentation in their hanging wall. These processes are shown to control the size of the FHW region and the number of fault-bounded blocks it eventually contains. Fault weakening tends to make fault rotation more discontinuous and this results in the FHW zone containing multiple blocks of intact material separated by faults. By compensating the topographic subsidence of the half-graben, sedimentation tends to slow the fault rotation and this results in the reduction of the size of the FHW zone and of its number of fault-bounded blocks. We apply the new approach to reproduce the faults observed along a seismic line in the Southern Jeanne d'Arc Basin, Grand Banks, offshore Newfoundland. There, a single block exists in the hanging wall of the principal fault. The model explains well this situation provided that a slow sedimentation rate in the Lower Jurassic is proposed followed by an increasing rate over time as the main detachment fault was growing.

International audience; Estimating the stress in reservoirs and their surroundings prior to the production is a key issue for reservoir management planning. In this study, we propose an integrated inverse method to estimate such initial stress state. The 3D stress state is constructed with the displacement-based finite element method assuming linear isotropic elasticity and small perturbations in the current geometry of the geological structures. The Neumann boundary conditions are defined as piecewise linear functions of depth. The discontinuous functions are determined with the CMA-ES (Covariance Matrix Adaptation Evolution Strategy) optimization algorithm to fit wellbore stress data deduced from leak-off tests and breakouts. The disregard of the geological history and the simplified rheological assumptions mean that only the stress field, statically admissible and matching the wellbore data should be exploited. The spatial domain of validity of this statement is assessed by comparing the stress estimations for a synthetic folded structure of finite amplitude with a history constructed assuming a viscous response.

Two modes of extensional collapse in a cohesive and frictional wedge of arbitrary topography, finite extent, and resting on an inclined weak decollement are examined by analytical means. The first mode consists of the gravitational collapse by the action of a half-graben, rooting on the decollement and pushing seaward the frontal part of the wedge. The second mode results from the tectonics extension at the back wall with a similar half-graben kinematics and the landward sliding of the rear part of the wedge. The predictions of the maximum strength theorem, equivalent to the kinematic approach of limit analysis and based on these two collapse mechanisms, not only match exactly the solutions of the critical Coulomb wedge theory, once properly amended, but generalizes them in several aspects: wedge of finite size, composed of cohesive material and of arbitrary topography. This generalization is advantageous to progress in our understanding of many laboratory experiments and field cases. For example, it is claimed from analytical results validated by experiments that the stability transition for a cohesive, triangular wedge occurs with the activation of the maximum length of the decollement. It is shown that the details of the topography, for the particular example of the Mejillones peninsula (North Chile) is, however, responsible for the selection of a short length-scale, dynamic instability corresponding to a frontal gravitational instability. A reasonable amount of cohesion is sufficient for the pressures proposed in the literature to correspond to a stability transition and not with a dynamically unstable state.

Summary Stress estimation in reservoirs and overbuden has become a key point during the exploration and the exploitation of the oil ans gas fields. We propose in this abstract an integrated method to compute a physically admissible (i.e. satisfying the equilibrium equations) 3D stress field in whole geological models. Stress field is computed using a simple elastic behavior and it is constrained to the wellbore data using an inverse approach. Forward problem is solved using a Finite Element Analysis. The model parameters to invert are the Neumann conditions which are assumed to be piecewise linear functions along the vertical direction. The data parameters are stress observations which came from the hydraulic fracturing and the borehole breakouts. Misfit between the computed stress and the observed stress is minimized using the CMA-ES algorithm. The method is tested with a synthetic case by taking a stress field computed with the Limit Analysis method as reference. The inversion results show that the method is able to well retrieve the stress variation in the geological model.

Listric normal faults are widespread in the extension of the upper crust. Despite major advances in understanding the formation of listric faults through various experiments, the mechanical conditions that allow their formation are highly debated. In particular, Anderson's faulting theory predicts that newly formed normal faults are planar and are dipping at least at 45 ∘ , and in practice, at 60–65∘ for most rock types. Here, we develop Limit Analysis to investigate the formation of a listric fault at the onset of slip linking a deep detachment to the topographic surface. We find that listric normal faults can occur in the brittle upper crust without appealing to viscous or ductile behaviours, nor to flexural stresses. The disequilibrium-compaction fluid pressures, typically observed in many sedimentary basins, are essential for the formation of listric faults. The fluid pressure is hydrostatic down to the fluid-retention depth Z F R D and sustains a higher gradient below this depth. Parametric studies show that the surface slope is also essential for the formation of listric faults even with a gently dipping surface slope ( ≤ 4 ∘ ) , whereas a flat topographic surface leads to a simple Andersonian geometry at the onset of slip. The method is applied to two field examples in order to determine fluid overpressures that best match the fault shapes interpreted from seismic data. For the offshore Niger Delta, the simulated normal faults match the observed listric faults with a very shallow Z F R D = 0.5–0.75 km, and below Z F R D the fluid pressure has a lithostatic gradient, consistent with the observed fluid-pressure profiles. To reproduce a series of listric faults joining on one common low-angle detachment in the NW Gulf of Mexico, we demonstrate that a shallow Z F R D = 0.7–1.1 km is required, below which the fluid pressure increases to the lithostatic pressure on the detachment, in agreement with the fluid-pressure observations.

We analyze the mechanical properties needed to account for the large shallow slip during the 2011 Tohoku-Oki earthquake and the activation of landward normal faulting within the forearc. We show that the morphology and internal structure of the forearc follows closely the prediction of the critical Coulomb wedge in horizontal compression, implying a high internal pore pressure ratio (λ = 0.7 + 0.14/ – 0.48) and a low effective basal friction (μ^(eff)_b = 0.14 + 0.18/ – 0.04). We then show that the activation of the normal fault requires a lower effective basal friction beneath the outer wedge than beneath the inner wedge (μ_outer ≤ 0.015), possibly due to transient dynamic weakening associated to the seismic rupture. Forearc normal faults could be considered as evidence for very efficient dynamic weakening along the megathrust and typify megathrust with high tsunamigenic potential.

The maximum strength theorem is applied to determine the pressure conditions in accretionary wedges responsible for the activation of splay faults (SF). This theorem, based on the limit analysis developed in soil mechanics, selects among several modes of deformation the one prevailing for a given pressure distribution. It applies for frictional rock properties and for an arbitrary wedge geometry. In particular, the decollement is assumed here to be partitioned into two mechanically distinct regions, the internal region being part of the seismogenic zone. The considered modes of deformation include the activation of a thrust fold rooting on the decollement and outcropping anywhere on the topography. This mode, if close to the transition between the seismic and the aseismic region of the decollement is in competition with the SF which constitutes also a potential mode of deformation. The dominance of the various modes of deformation is presented in stability maps in the plane spanned by the pressure ratios of the two regions of the decollement (pore pressure divided by lithostatic pressure, both referenced to the sea floor). Several maps are proposed for the particular case of the Kumano transect of Nankai wedge, South-West Japan, varying the SF pressure ratio, the position of the SF root on the decollement with respect to the transition between the seismic and aseismic regions as well as the respective length of these two regions. In addition, these stability maps are suggesting various scenarios to explain the changes through time in the prevailing mode of deformation in an accretionary wedges. For the Kumano transect, three scenarios could explain the observed deactivation of the SF, one of them being the seaward migration of the decollement transition point.

SUMMARY A triangular wedge, composed of a frictional material such as sand, and accreting additional material at its front, is the classical prototype for accretionary wedges and fold-and-thrust belts. A simplified method is proposed to capture the internal deformation of this structure resulting from a large number of faulting events during compression. The method combines the application of the kinematic approach of limit analysis to predict the optimum thrust-fold and a set of geometrical rules to update the geometry accordingly, at each increment of shortening. It is shown that the structure topography remains approximately planar with a slope predicted by the critical Coulomb wedge theory. Failure by faulting occurs anywhere within the wedge at criticality, and its exact position is sensitive to topographic perturbations resulting from the deformation history. The convergence analysis in terms of the shortening increments and of the topography discretization reveals that the timing and the position of a single faulting event cannot be predicted. The convergence is achieved nevertheless in terms of the statistics of the distribution of the faulting events throughout the structure and during the entire deformation history. It is these two convergence properties that are presented to justify the claim that these compressed frictional wedges are imperfection sensitive, chaotic systems. Copyright © 2013 John Wiley & Sons, Ltd.

International audience; A technique is presented for improving the structural analysis of natural fractures development in large scale fold structures. A 3D restoration of a fold provides the external displacement loading conditions to solve, by the finite element method, the forward mechanical problem of an idealized rock material with a stress-strain relationship based on the activation of pervasive fracture sets. In this elasto-plasticity constitutive law, any activated fracture set contributes to the total plastic strain by either an opening or a sliding mode of rock failure. Inherited versus syn-folding fracture sets development can be studied using this mechanical model. The workflow of this methodology was applied to the Weber sandstone formation deformed by forced folding at Split Mountain Anticline, Utah for which the different fracture sets were created and developed successively during the Sevier and the syn-folding Laramide orogenic phases. The field observations at the top stratigraphic surface of the Weber sandstone lead to classify the fracture sets into a pre-fold WNW-ESE fracture set, and a NE-SW fracture set post-dating the former. The development and relative chronology of the fracture sets are discussed based on the geomechanical modeling results. Starting with a 3D restoration of the Split Mountain Anticline, three fold-fracture development models were generated, alternately assuming that the WNW-ESE fracture set is either present or absent prior to folding process. Depending on the initial fracture configuration, the calculated fracture patterns are markedly different, showing that assuming a WNW-ESE joint set to predate the fold best correlates with field observations. This study is a first step addressing the complex problem of identification of fold-related fracturing events using an elementary concept of rock mechanics. When tight to complementary field observations, including petrography, diagenesis and burial history, the approach can be used to better constrain fractured reservoir characterization. © 2012.

The principal concern of this paper is with the accuracy and validity of experimental sand box models of geological structures – and of thrust faults in particular – as influenced by side wall friction. Side wall friction can produce undesirable artifacts, in particular in experiments aimed at reproducing plane deformations. The importance of such artifacts is revealed in the present study by the results of experiments that are affected in opposite ways by side wall friction. Two distinct experimental configurations have been chosen for this purpose for a set-up that comprises a body of sand placed between vertical side walls on a basal plate against a back wall and a front wall, all made of glass. In configuration M the bottom plate and attached front wall move relative to the side walls towards a fixed back wall, while in configuration F the back wall moves relative to the side walls and attached bottom plate towards a fixed front wall (‘fixed’ = ‘fixed relative to the side walls’). In both configurations, thrust traces on the top surface of the sand become curved near the side walls because of wall friction. The outcome of altogether 31 experiments in the two configurations shows that the magnitude of the experimental bias depends on the ratio S L / S B of the area of contact of the sand body with the side walls ( S L ) to its area of contact with the basal surface ( S B ). The bias is the difference in thrust locations in the two configurations. It is measured on photographs taken through the side walls during the experiments. Away from the side walls, the bias is estimated by the thrust curvatures observed in top views. For S L / S B ≤ 0.1, i.e. very little contact with side walls, there is no identifiable bias. Furthermore, a 2D theoretical solution reproduces well the results. For S L / S B increasing to 0.25, curvatures increase, yielding rather different cross-sections at the side walls, but still similar cross-sections in the central part of the sand body. For S L / S B ≥ 0.3, thrust planes occupy different regions of the box for each configuration, although their curvature is decreasing. At S L / S B = 0.9, curvatures have almost disappeared but the location of the thrusts and the length of the decollement depend strongly on the experimental configuration. Departures from the 2D theoretical solution are clearly stronger with configuration M than F. Another important conclusion is that the absence of curvature of the thrust traces is not a proof of the 2D character of the experiment. Mechanical equilibrium of the total forces recorded at both front and back walls during shortening shows that the thrust locations are controlled by the difference between basal and side wall shear forces.

International audience; 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 in- herited 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 tetra- hedra in 3D and six-noded triangles in 2D. The re- sulting, discrete optimization problem is first presented emphasizing the dual formalism found most appropri- ate 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 fail- ure 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 stabil- ity conditions, respectively. The failure mode is char- acterized 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.

The first objective is to study the influence of the burial depth and of an imperfection, in the form of a tilt in the inclination of the otherwise straight, laminated beam, on the conditions for the initiation and the development of kink folds. The beam, typical of sedimentary rocks, is layered with weak interfaces between the competent beds, promoting the onset of kinking by their slip. The results are analytical and based on the upper bound approach of the classical limit analysis, referred to as the maximum strength theorem in the absence of any discussion of plasticity. The weak interface strength is described by the Coulomb criterion, whereas the bulk material is also cohesive and frictional but with an additional closure in the compressive stress domain to depict the action of compacting deformation mechanisms. The new twist to the methodology is to extend its application to the development of the failure mode beyond the onset, assuming that the structure finite response is well described by the least upper bound solution. The second objective is to compare in terms of upper bounds three different failure mechanisms which are the compaction band, the reverse fault and the kink fold. Their respective domain of dominance is constructed in failure-mechanism maps in the space spanned by the imperfection angle and the burial depth. Compaction bands are predicted at the deepest end of the beam and the reverse fault and the kink fold at the shallower end. These depth differences are resulting from the geometrical imperfection. It is found that the kink-band mode at its onset, with compaction band dominant conditions, resembles to a slip-enhanced compaction band due to the weak interface activation and the compaction along the two parallel hinges. This hybrid mode migrates suddenly through the competent beam from the deepest towards the shallowest region and develops as a kink fold, after a negligible amount of shortening. The kink fold development, beyond the onset, occurs in two phases, the first corresponding to the rotation of the kink band and the second, to its widening. The associated least upper bound is first decreasing during the development and then increasing, the minimum being controlled more by the increase in the potential energy of the system than by the most favorable orientation of the frictional weak interfaces. It is finally found that the continuous activation of slip over the weak interfaces and the widening of the kink band prevent the rotation of the kink towards the large angles which are necessary to induce its locking. It is proposed that the introduction of damage along the hinges could palliate these two effects and prompt the locking observed experimentally and in the field.

The structure of interest occupies initially a rectangular-shaped domain and is composed of a laminated material with weak interfaces defining the horizontal bedding. It is subjected to a compressive lateral force parallel to the bedding. The postulated fold kinematics relies on the existence of a kink band with two parallel sharp boundaries, or hinges, separating two rigid blocks. One block moves upward, the other sideways, by distances defining the fold amplitude and the horizontal shortening, respectively. Internal work is assumed to be done only along the hinges and the weak interfaces. The orientations of the hinges (hinge dip) and of the beds within the kink band (kink dip) are optimised for every fold amplitude to provide the least upper bound on the compressive force according to the maximum strength theorem. It is shown that the fold onset requires the introduction of a compaction mechanism for the hinges. The kink initiates as a sub-vertical compaction band of finite thickness equal to the bed thickness times the sine of the friction angle over the bedding. During the first phase of the fold development, the kink band rotates with the property that the sum of the kink dip and the hinge dip is always complementary to the friction angle over the bedding. Each bed along the hinges sees the activation of two deformation mechanisms: compaction and opening. The boundary between the regions over each bed where they are activated migrates from the bottom of the bed—pure compaction at the onset—towards a position at 90% of the bed thickness—development dominantly controlled by opening. The second phase of the fold development is marked by a thickening of the kink band with minor evolution of the dips. This two-phase development leads to a sharp decrease of the compressive load from the onset, a minimum as the two dips are approximately equal and then a moderate increase in applied load. In conclusion, it is noted that such combination of postulated fold kinematics and the application of the maximum strength theorem to optimise the structure could be generalised to folding in the presence of ramps, providing a useful tool to comprehend the mechanics of fold-and-thrust belts and of accretionary wedges.

A linear stability criterion for strain-rate sensitive solids and structures is proposed and validated with the help of two versions of Shanleys column, the first with two discrete supports and the second with a continuous distribution of supports. Linear stability transition is defined by the change in sign of the second derivative with respect to time of the columns angular position evaluated at the onset of perturbation. This criterion pertains to the initiation of instabilities but is not expected to provide information on their long term development. Two parameters influence linear stability: the dimensionless number T, defined as the ratio of the relaxation time of the viscous support to the characteristic loading time, and the perturbation size. It is found that the critical load of principal equilibria, defined for a straight column and a zero value of T, is the classical reduced modulus load, in agreement with existing stability criteria for rate-independent models based on maximum dissipation. For arbitrary values of T, two critical loads are identified at the linear stability transition. The first is named the rate-dependent tangent modulus load and is valid for perturbations sufficiently small to prevent initial unloading. That load coincides with the classical tangent modulus load for T tending to zero and is, surprisingly, a decreasing function of that dimensionless number. The second critical load is termed the rate-dependent reduced modulus load, and is applicable to columns that are partly unloaded at the onset of perturbation. This critical load approaches the classical reduced modulus load in the limit of T tending to zero, is a decreasing function of T, and depends on the imperfection size. Similar results are found for the second model, with a new insight on the role of the unloading zone extent in determining the critical load in the singular limit of T equal to zero. The proposed stability criterion is validated by comparing its predictions with the outcome of nonlinear perturbation analyses and of imperfection sensitivity studies. It is shown, in particular, that an imperfection evolves according to our stability predictions as long as the relative difference between the irreversible displacements of the supports can be disregarded. A generalization of the proposed linear stability criterion to viscoplastic continua is finally sketched.

The motivation of the study is to demonstrate that geometrical constructions of fault‐related folds can be mechanically balanced according to the theory of limit analysis assuming that materials are cohesive and frictional. The advantage of the proposed methodology is to remain simple in terms of geometry and to introduce explicitly the various discontinuities such as the ramps which compose the geological structure. The main outcome of such mechanically balanced cross section construction is the availability of the stress field history which is fundamental to estimate the fracture pattern. These fracture predictions could be compared and benchmarked with data from outcrops. The proposed methodology is presented with the help of the field example of the Nerthe duplex, South‐Est France, with two horses, each well described with the classical fault‐bend fold geometry. The results highlight the role of erosion in the tectonic history of the folds and try to answer why early, inherited, fracture networks have not been reactivated during folding

The present study deals with the morphological alterations resulting from the pressure solution of fluid-filled tubular pores embedded in a stressed solid matrix. Building upon previous work concerning transformation kinetics and the kinetically unstable nature of cylindrical solid-fluid phase boundaries subjected to nonhydrostatic stress, we investigate the response of such a pore to circumferential and longitudinal perturbations in its shape by means of a linear stability analysis. We obtain criteria for the onset of instability, determine the wavenumber of the disturbance with fastest growth, and demonstrate that with increasing pore pressure the dominant instability changes from a circumferential to a longitudinal mode. This last feature could distinguish interface migration that is limited by the dissolution/precipitation kinetics from that controlled by diffusion through the pore fluid. We finally obtain estimates of the number and spacing of the inclusions formed when a cylindrical pore breaks up following unstable circumferential or longitudinal growth.

This contribution is concerned with the fundamental thermodynamic aspects of solid-fluid phase transformations in stressed rocks, specifically in the context of pressure solution. We concentrate in particular on the formulation of a kinetic law governing the migration of stressed and curved solid-fluid phase boundaries, an objective that is achieved by using the methods of the thermodynamics of irreversible processes. We then apply our result to the study of the geometrical evolution of a fluid-filled cylindrical pore embedded in an isotropic, linear elastic solid that is subject to a hydrostatic remote stress, assuming that the interface kinetics controls the phase boundary migration and allowing for the effects of capillarity. On the basis of this investigation, we obtain an analytical expression for the pore's growth and show that phase equilibrium along the cylindrical solid-fluid phase boundary is possible only when the pore pressure exceeds a critical value. The phase equilibrium is found to be kinetically unstable: when subjected to a small perturbation of its radius, the pore will either grow or shrink. The nature of this instability is further explored in the companion paper.

(R~~ir~tl4 MUITII 1993) AESTRACI THE STABILITY OF il geological two-layer system composed of a frictional material layer of finite thickness. called the overburden, resting on a viscous half-space of lower density is investigated. The salient features of this study are a realistic description of the stiffness of the overburden and its state of (in sitzl) prestress. and the USC of the viscosity of the substratum to define a characteristic time for the stability analysis. A general variational formulation for the linearized. non-selfadjoint stability problem is prcsentcd, followed by asymptotic analyses for the cases of large and small perturbation wavelengths and by an analytical solution in the absence of gravity. Results obtained by a finite-element method are compared with the analytical and asymptotic predictions: they permit the detection of various modes of instability : interfaceand beam-type modes in the compressive range of deformation. and neck-type modes in the tensile ransc. It is found that the system’s stability is not only governed by geometry and density contrast. as expected from the conclusions of earlier studies on viscous and viscoelastic models, but is also scnsitivc to the state of in .viru stress. A complete parametric study reveals that the overburden material cohesion and workhardening properties have more influence on stability than the friction angle. Furthermore, it is found that critical stresses at neutral stability predicted by deformation theory. which is an appropriate model for studying the initiation of faulting in rocks. are smaller in magnitude than those obtained by the corresponding flow theory with a smooth yield surface. Implications of this work for the interpretation of various laboratory analogue model experiments pertaining to geological two-layer systems arc also discussed.

This introduction to the finite-element method is offered to Earth Scientists with an interest in numerical methods, continuum mechanics and the theory of plasticity. No previous exposure to this material is required. Starting with the theorem of minimum potential energy for a one dimensional poro-elastic column, the finite-element method is introduced as the minimization in the subspace of the set of admissible displacements constructed by local, linear interpolation. The method is then extended to two dimensions to introduce the simplest element, the three-noded triangle. These two examples are also used to introduce the concept of assembly for static problems. The numerical quadrature, and in particular Gauss’ and Lobatto’s quadrature rules are then presented before constructing the family of Lagrange elements. The finite-element method is then explored in the context of non-linear material response and, in particular, for plasticity, first under the small-strain approximation and then for finite transformations. Examples are presented showing the good performance of the so-called spectral elements in problems involving near incompressible plastic flow, strain localization and buckling. Two exercises and a problem are finally proposed, the latter providing an introduction to these spectral elements which have become popular in earthquake modeling.

The onset of localization of the deformation is analysed from the results of a two-dimensional bifurcation analysis for a shear zone of infinite extent and of finite thickness. The material is assumed to be incompressible and hyperelastic and, to account for micro-structural effects, to have an energy density that is a function of second-order deformation gradients. With such a model, the onset of strain localization is associated with a bifurcation but not with a change of type of the governing partial differential equations. It is demonstrated that both layer thickness and the exact nature of the boundary conditions affect that bifurcation. As expected, the incremental displacement field that characterizes the first loss of uniqueness has a spatial variation across but not along the shear zone. However, and contrary to a common belief, it is shown that the displacement field could vary in these two directions, for a range of values of the material properties. Consequences of this wavy pattern for the onset of failure during localization are discussed.

The existence and stability of stationary shear flows are studied for materials modelled as non-Newtonian temperature-sensitive fluids. Two temperature dependences are considered: the Arrhenius law and a first-order approximation referred to as the exponential law. Mechanical boundary conditions are introduced as a linear combination of stress and velocity. These mixed boundary conditions, known to be pertinent to geological problems, are proved to prevail during the main part of high-strain-rate tests in a torsional Kolsky bar. Analytical results are obtained for the steady states and the conditions of neutral stability for the exponential law. Approximate solutions, based on the Galerkin method, are then presented; these compare favourably with the exact results. This approximate method is aimed at the study of the Arrhenius law, for which no exact solutions are available. Finally, as an application, it is shown that the shear bands formed on thin-walled specimens tested in a torsional Kolsky bar do not tend towards stable steady states.

[1] The objective is to demonstrate that the equilibrium element method (EEM) provides the stress distribution in geometrical models of folds, relevant to fold-and-thrust belts as well as accretionary wedges. The core of the method, inherited from limit analysis, is the search for an optimum stress field that (1) is in equilibrium, (2) remains everywhere below or equal to the maximum strength of the rock, and (3) balances the largest possible applied tectonic force. This force and the associated stress field are interpreted as those at the onset of rupture. The method makes no appeal to the rock rheology nor to its elasticity, except for its maximum strength described here with the Coulomb criterion. The stress fields are discretized by elements covering the whole domain and allowing for discontinuities. The example chosen to illustrate the potential of the EEM and to validate our implementation is the thrusting of a rectangular sheet over a flat and weak decollement. The EEM reproduces the solution proposed by Hafner (1951) on the basis of linear elasticity, as long as the strength limit is not reached in the bulk of the domain. The EEM shows in addition that failure in the bulk prevents the activation of the decollement. The EEM is then applied to two fault-bend folds, with known ramp and flat decollement, with and without relief buildup. It is shown that the transition from the flat to the ramp hanging walls occurs through a narrow fan defining the back thrust. The predicted dip of this back thrust decreases with increase in the ramp friction angle, the relief buildup, as well as the ramp curvature. A sharp increase and then a sharp decrease in the magnitudes of the equivalent shear stress and of the mean stress are observed as one moves from the lower flat, through the back thrust up the ramp. If the ramp friction angle is too large, or the relief too important, the EEM predicts the initiation of a new thrust rooting at the back wall, instead of activating the proposed ramp. The application to detect the incipient thrust system within the toe of Nankai's accretionary wedge, southeast coast of Japan, is proposed in the auxiliary material.

The analysis of a Shanley column is first presented, in order to illustrate the role or rate dependence in the uniqueness of and stability conditions for the solution of the discrete system. Stability conditions for the solution of boundary-value problems for rate-dependent solids are then considered, and a weak formulation of the linear stability criterion is presented. This formulation is well-suited for obtaining solutions to both the instability modes and the corresponding critical loads in systems with a finite number of degrees of freedom, typically obtained by discretization of the continuum. As an application, a stability analysis by the finite-element method of a von Mises solid under plane strain tensile loading is presented. The influence of both the rate sensitivity and the wavelength of the mode of perturbation on the subsequent localization of the deformation in shear bands is discussed.

International audience; The main objective is to determine the three stages of the life of a thrust in an accretionary wedge which are the onset of thrusting along its ramp, the development with the construction of the relief, and the arrest because of the onset of another thrusting event. A simple kinematics is proposed for the geometry of the developing thrust fold based on rigid regions separated by velocity discontinuities along which work is dissipated according to the Coulomb criterion. The evolution of the thrust fold satisfies mechanical equilibrium and is optimized at every time of the three stages to provide the least upper bound in tectonic force according to the maximum strength theorem. The development of the thrust or its arrest because of the initiation of another thrust is decided by selecting the event which leads to the least upper bound in tectonic force. The approach is first validated by proving that the critical slope angle a c for the classical triangular wedge is properly captured. It is shown that a perturbation, in the form of an extra relief in this perfectly triangular geometry, leads to the onset of thrusting with the ramp or the back thrust outcropping either at the back or to the front of the perturbation, respectively, for a range of slope dip close to the critical angle a c. The study of normal thrust sequences (from the rear to the front in the wedge toe) reveals that weakening of the ramp, accounted for by changing its friction angle from an initial to a smaller final value, is necessary for each thrust to have a finite life span. This life span is longer with a larger relief buildup for more pronounced weakening. Decreasing the décollement friction angle results in an increase in the number of thrusts in the sequence, each thrust creating milder relief. The normal sequence is ended with the first out of sequence thrust which occurs earlier for smaller weakening over the ramp. The proposed methodology is partly used to construct an inverse method proposed to assess the likeliness for the transfer of activity from the active to the incipient thrust in a section of Nankai's accretionary wedge. The inverse method provides the initial friction angle over the incipient ramp and the final friction angle over the fully active ramp, from the geometry of the corresponding thrusts, and the topography. It is shown that the friction angle over the incipient ramp is most likely to be larger then the one over the active ramp, justifying a key hypothesis needed to predict discrete sequences of thrusting.

Finite elements with embedded shocks are used to investigate transient strain localization phenomena in frictional solids. In particular, we seek to elucidate the effect of rate sensitivity and inertia on the development of shear bands in solids subjected to impulsive loading. As in the static case, our results show that shear banding may induce severe softening of the specimen even as the material steadily hardens. As expected, rate sensitivity retards the onset of structural softening and tends to stabilize the post-peak response. It is verified that the static solution is indeed recovered in the inviscid limit. Under dynamic conditions, shear bands are observed to propagate discontinuously, arresting and resuming propagation repeatedly before linking up with the boundary of the specimen. The direction of the band is equally unsteady. In addition, multiple shear banding, with the development of secondary and even tertiary bands, appears to be a prevalent mechanism at sufficiently high impact velocities.

SUMMARY The very low water content of the granulitic unit of Holsnoy, in the Bergen Arcs, Norway, caused its partial metastable preservation throughout Caledonian burial end exhumation, leading to the observed mixture of completely eclogitized and uneclogitized rocks. The eclogitization of the granulite-facies protolith led to a density increase ∼10%, but also to a large rheological weakening that resulted in the localization of all deep ductile deformation in the eclogite fraction. We address the consequences on subduction dynamics of the concomitant evolution of large-scale density and rheology during progressive eclogitization of crust by comparing the behaviour of buried crust with various properties in the channel-flow model. Large-scale buoyancy and density are integrated in a single adimensional parameter, the ‘exhumation number’ α, which describes the capacity of the crust to exhume. For a crust whose eclogitization causes a very large viscosity decrease (‘burial resistant’ end-member), α goes through a maximum in the partially eclogitized zone, that is, there is a given proportion of eclogite for which the crust is strongly weakened but still significantly lighter than surrounding mantle. For this partially eclogitized crust with α max, the maximum possible downward flow is very low, this zone acts as a bottleneck and all incoming crust in excess is forced backwards and starts exhuming. This return flow zone is bounded downwards in the partially eclogitized zone by α max, while its upper limit propagates upwards at a constant rate. The curve α(eclogite fraction) controls the maximum proportion of eclogite of the crust that can be exhumed. We also demonstrate that large exhumation rates reached by certain UHP units are incompatible with any stationary flow regime, whatever the nature of subducted crust. The additional presence of a weak layer on top of the channel favours burial for narrow channels and exhumation for large ones, but does not qualitatively change the dynamics of the subduction channel. This study shows that deep crustal circulation in subduction zones and exhumation from large depths are controlled by the evolution of large-scale crustal properties with progressive eclogitization.

The northwestern part of Holsnøy island, in the Bergen Arcs, Norway, consists of a granulite-facies protolith partially transformed at depth in eclogite (700 °C, > 19 kbars) and amphibolite (650 °C, 8-10 kbars) facies during the Caledonian orogenesis. Eclogitized zones are mainly planar objects (fractures with parallel reaction bands and cm-to-100 m-scale shear zones). Eclogitic zones are distributed in two sets of orientations and the associated deformation can be described as 'bookshelf tectonics'. The major shear zones strike around N120 and dip to the North, and show consistent top-to-the-NE shear sense throughout the area. In the large-scale kinematic frame of Caledonian NW-dipping slab, eclogitic shear zones are interpreted as the way to detach crustal units from the subducting slab and to prevent their further sinking. As the retrograde amphibolitic deformation pattern is similar to the eclogitic one, the detached crustal units started their way up along these eclogitic shear zones. Radiometric ages of eclogitic and amphibolitic metamorphism and their comparison with the chronology of Caledonian orogenesis show that the deformation recorded on Holsnøy occurred in a convergent context. The mechanism we propose can thus account for the first steps of exhumation during collision.

Direct observation of peak pressure deformation in exhumed subduction channels is difficult because little evidence of this deformation survives later syn-exhumation deformation. Most ultrahigh-pressure parageneses are found in continental derived metamorphic rocks making continental subduction the best context to observe peak pressure deformation. Whereas many studies have enlightened the main driving parameters of exhumation such as buoyancy forces, low viscosity in the subduction channel, overburden removal by erosion and normal faulting, a basic question is seldom considered: why is a tectonic unit disconnected from the descending lithosphere and why does it start its way towards the surface? This event, seminal to exhumation processes, must involve some deformation and decoupling of the exhumed slice from the descending slab at peak pressure conditions or close to it. Our field observations in the Bergen arc show that Caledonian eclogitization and later amphibolitization of a granulitic terrane was achieved with a consistent component of simple shear compatible with the sense of the Caledonian subduction. Thus, the sequence of deformation preserved in the Bergen Arc documents the decoupling of subducted crustal material from the descending slab at the onset of exhumation. This observation suggests that deformation in the subduction channel is largely controlled by kinematic boundary conditions, i.e. underthrusting of the subducting slab. In this context of simple shear, metamorphic reactions assisted by fracturating, fluid infiltration and ductile deformation lower the resistance of rocks and allow the localisation of shear zones and the decoupling of buoyant tectonic units from the subducting slab. These tectonic units can then be incorporated into the channel circulation and start their upward travel.

A new algorithm for three-dimensional rate dependent crystal plasticity is presented which involves first, the explicit update of the dependent variables based on a high-order accurate scheme and second, the derivation of the consistent operator by the lineariziation of this update algorithm, in order to warrant a quadratic rate of convergence during the iterative equilibrium search. A detailed and complete description of the governing equations amenable to numerical implementation is provided using the convected coordinate formalism. The algorithm has been implemented for an update based on the second-order accurate Runge–Kutta integration scheme in a displacement based finite-element code. The methodology for extending the algorithm to the fourth-order accurate Runge–Kutta scheme is discussed. The algorithm is illustrated for one-dimensional loading of otherwise traction free single crystals loaded along specific crystallographic directions and for a 3D simulation of a tension test for a face centered cubic single crystal oriented for symmetric double slip up to the onset of shear localization. These first numerical simulations are consistent with the analytical solutions. For the 3D case, they provide an insight on the development of plasticity through the sample, with an initial first phase leading to an homogeneous deformation, followed by a second phase where the deformation is more and more limited to the central sections of the sample and ultimately becomes localized, without the need for any initial defect or perturbation.

Present address : Institut des Sciences de la Terre d'Orléans UMR 7327, Université d'Orléans, CNRS/INSU, BGRM; International audience; Low-angle midcrustal ductile shear zones and the related microseismic activity recorded below regions of active extension are seen here as two consequences of strain localization. The feldspar-to-mica reaction which occurs once feldspar grains are fractured is the destabilizing mechanism selected to explain the strain localization. The model problem considered to substantiate these claims is solved by numerical means and combines the simple shear due to the rigid gliding of the upper crust (at the velocity of V s ) and the stretch resulting from the extension of the whole crust (at the velocity V e ). The rheological model accounts for dislocation creep of quartz, feldspar, and mica, the feldspar-to-mica reaction, and its prerequisite, which is the feldspar fracturing detected by the Mohr-Coulomb criterion. The one-dimensional (1-D) solution, which constrains shear bands to be horizontal, shows the depth partitioning in deformation mode between the simple shear of the low-viscosity deep crust and the stretching of the highly viscous midcrust. Strain localization occurs during rapid increase of the shearing velocity V s , corresponding to low values of the velocity ratio V e /V s . The 2-D solution (for V e /V s = 10−3) reveals the development of a periodic system of extensional shear bands, dipping at 30° toward the shearing direction at a depth of 12 to 14 km. Shear bands are formed after less than half a million years at the base of the reaction zone defined by the region where feldspar-to-mica reaction is completed. Shear bands do not propagate to greater depths because the pressure prevents the feldspar from fracturing and thus the reaction to occur. The periodic system of shear bands defines a midcrustal flat weakened zone within which the equivalent shear stress is enhanced by at least a factor of three at the shear band tips. Brittle fracture could thus occur within the midcrustal flat weakened zone, explaining therefore the microseismicity monitored at these depths in regions of active extension.

International audience; This chapter discusses sandstone compaction by intergranular pressure solution and presents the theoretical means for exploring problems of stress-sensitive dissolution and coupled macroscopic deformation and transport. The simple one-dimensional compaction model discussed in this chapter represents only a first exploratory study of the behavior of the proposed model. Despite carrying out a numerical study for only one-layer thickness and a single set of initial and boundary conditions, the results obtained for this case have yielded some basic insights into the behavior of the compaction model that can also point the way for future studies. When precipitation was inhibited, the exceedingly slow diffusive transport over 5,000 m of layer thickness was found to lead to the build-up of high (possibly unrealistic) supersaturations, approximately 1.5 times the equilibrium concentration, which slowed down intergranular dissolution to extremely low rates. However, because this low-rate limit in compaction behavior is likely to be attained only in the absence of any significant advective influx of undersaturated fluid through the base of the layer, the role of the prevailing hydrological regime in compacting sedimentary layers is immediately apparent. This observation suggests that in a sedimentary column that is open only at its top, macroscopic advection will be activated only if there exist local sinks—the free faces—on the grain scale, where the material at a nearby grain boundary can be deposited.

A thermomechanical model for a shape memory alloy (SMA) wire under uniaxial loading is implemented in a finite element framework, and its results are compared with new experimental data. The constitutive model is a one-dimensional continuum model of an SMA element, including two internal field variables, strain gradient effects, possible unstable mechanical behavior, and the relevant thermomechanical couplings resulting from latent heat effects. The model is calibrated to recent experiments of typical commercially available polycrystalline NiTi wire. The shape memory effect and pseudoelastic behaviors are demonstrated numerically as a function of applied loading rate and environmental parameters, and the results are found to be quite similar to experimental data. The model is then used to simulate a simple SMA actuator device, and the model proves to be a useful tool to assess the performance.

[1] Back thrusts and hinges are two types of transition between rigid sections of hanging walls observed in fold-and-thrust belts. Back thrusts are typical of frictional and homogeneous solids and hinges of creeping and layered materials. Our objective is to study the orientation of these transitions for the special case of a lower flat-ramp transition in a fault-bend fold with the following general two-step methodology. In the first step, the forces acting on the transition are determined using equilibrium of each rigid section. In the second step, the optimal dip of the transition is obtained by minimizing the total dissipation of the structure. The three sources of dissipation, of comparable magnitude, are at the transition, on the flat, and on the ramp. For frictional material flows, the back thrusts are velocity discontinuities with optimal dips always less than half the complementary ramp angle, leading to hanging wall thickening. The optimal dip agrees well with the results of physical analogue and numerical experiments. For creeping and layered materials, it is shown that a destabilizing deformation mechanism, selected to be flexural slip, is necessary for the strain to localize and the existence of hinges to be justified. Activation of flexural slip reduces dissipation at the transition and affects the optimal transition dip. The two-step methodology proposed here could be seen as a first attempt in producing mechanically balanced cross sections accounting for material rheology. This approach should complement the now classical kinematic models of folding.

[1] Folding of sedimentary layers is often accommodated by the opening or sliding of inherited and new discontinuities which are assumed here to be diffuse so that a continuum description applies at the fold scale. The rock rheology is then described with an elastoplasticity model for which the permanent deformation is of simple shear (sliding) or dilation (opening) with respect to specific orientations of the new or inherited diffuse discontinuities. To illustrate the relation between folding and activation of diffuse discontinuities, a three-dimensional layer under compression in the two horizontal directions and sustaining the overburden lithostatic pressure is studied. Cylindrical buckling occurs either before (elastic) or after the diffuse discontinuities have been activated. If buckling is elastic, inherited vertical discontinuities, striking obliquely to the fold geometrical axes, are activated in a sliding mode in the outer arc, leading to a rotation of the principal stress directions. Opening is then detected across new vertical planes striking obliquely to the fold axis. The activation of inherited or new vertical discontinuities can be suppressed if sliding takes place along weak bedding interfaces. Alternatively, early and homogeneous layer-parallel shortening, marked by a reverse fault mode, drastically reduces the critical buckling load compared to the Euler load and modifies the final geometry of buckling which is then more of a circular dome shape. The switching in buckling mode results in the fold limbs in a change from the early reverse fault to a strike-slip fault sliding and to opening across diffuse planes oriented consistently with the final circular structure.

[1] Low-angle extensional shear zones, which often characterize the brittle–ductile transition of the continental crust, are seen here to result from strain localization. The potentially destabilizing deformation mechanism is assumed to be the progressive transformation of fractured coarse feldspar grains into white mica as observed in the East Tenda Shear Zone, Alpine Corsica. The coupling between microfracturing and feldspar-to-mica reaction is coeval with strain localization that occurred in that field case at a depth close to 15 km. This reaction is proposed as the main destabilizing factor responsible for the onset of localization, with feldspar having a stationary dislocation creep flow stress larger than mica. To test this hypothesis, a rheological model is constructed based on the field observations for a mixture of three phases—mica, quartz and feldspar—deforming at a common strain rate. The phase concentrations change with time according to the feldspar-to-mica reaction, which takes place only if feldspar grains are fractured, a condition detected with the Mohr–Coulomb criterion. The tendency for the strain to localize is assessed by numerical means for the structure composed of an upper crust gliding rigidly over the lower crust, which sustains an overall simple shear. The onset of strain localization is defined by an increase of at least two orders of magnitude in strain rate over part of the lower crust. The upper crust gliding velocity has to be increased by at least a factor of 5 for localization to occur. The time lapse for this velocity change determines the depth of the shear zone (15–17 km). The kinetics of the metamorphic reaction and the final amount of white mica control its width (1–4 km). The time of the shear zone formation is less than half a million years.

The model problem proposed to study the buckling of stratified geological layers consists of a linearly elastic plate, capable of accommodating finite rotations, resting over an inviscid and buoyant fluid. The Lyapunov–Schmidt–Koiter decomposition is applied to construct the bifurcated equilibrium solutions. The asymptotic analysis of the post-buckling reveals a decrease in the magnitude of the lateral compressive force during an overall shortening of the stratified structure. Buckling and post-buckling are not influenced by the presence of a vertical stress gradient in the elastic plate.

The fine structural motifs of sialic acids, a frequent terminal monosaccharide of glycans, seem to contain essential biological properties. To identify such subtle structural differences, a reliable method was developed for the qualitative and quantitative identification of sialic acids present in different tissues and fluids. This method involved, after liberation of sialic acids by mild acid hydrolysis, their methyl esterification using diazomethane in the presence of methanol and the formation of volatile derivatives using heptafluorobutyric anhydride. The derivatives were analyzed by gas chromatography coupled to mass spectrometry in the electron impact mode. This technique allowed the separation and identification of a large variety of sialic acids, including different O-acylated forms of N-acetyl and N-glycolyl neuraminic acids and of 3-deoxy-D-glycero-D-galacto-nonulosonic acid (Kdn). This method allowed also identifying 8-O-methylated and 8-O-sulfated derivatives, de-N-acetylated neuraminic acid, and 1,7-sialic acid lactones. Compounds present in very complex mixtures could be identified through their fragmentation patterns. Because of the stability of the heptafluorobutyrate derivatives, this method presents important improvements compared to the previous techniques, because it can be frequently applied on very small amounts of crude samples. This methodology will support progress in the field of the biology of sialic acids.

The overall compaction of porous rocks due to intergranular pressure solution (IPS) results from the dissolution of minerals within contact regions and the diffusive transport through the grain boundary of the dissolved species towards the fluid-filled pore space. The grain boundary structure can be imagined to be composed of dry contact zones, thin fluid films and fluid-filled cavities. The connectiveness and tortuosity of this structure determine the effective diffusivity of grain contacts and thus the potential of porous rock to compact by the action of IPS. The evolution in time of the grain-boundary structure, and thus of the effective diffusivity, is discussed here with the help of two 2D initial- and boundary-value problems which are solved by analytical and numerical means. The evolution of the solid–fluid interfaces within the grain boundary is governed by a phase transformation between the non-hydrostatically stressed elastic solid and the trapped fluid assumed in mechanical equilibrium. The characteristic time is provided by a linear kinetic law. The evolution of the structure away from a state of thermodynamic equilibrium during a loading normal to the grain boundary is found to occur in two steps. The first one consists of a diffuse morphology evolution in time and results in an enhancement of any initial stress concentration. The second step is characterized by a rapid and localized dissolution in the region of stress concentration. The latency period prior to localization is governed by the magnitude of the non-hydrostatic remote stress as well as the microstructural geometric factor responsible for the initial stress concentration at the solid–fluid interface. The localized dissolution is shown to provide a mechanism for the fluid to penetrate a previously dry contact region by marginal dissolution and thus to create a fluid film. However, the newly formed thin fluid layer is found to be unstable pointing to a possible repeated reorganization or dynamic evolution of the grain boundary internal structure during the action of IPS.

Localized shear zones along low-angle normal faults have been identified in regions of extension at the brittle–ductile transition of the continental crust. The possibility of the strain localizing at a depth of 10 km is interpreted here as a consequence of an increase in the equivalent shear stress applied to the flow of the lower crust. This enhancement of the flow stress is seen as a prerequisite for the triggering of brittle deformation mechanisms leading to strain localization. The lower crust rheology used to examine this stress increase is strain-rate, temperature and grain-size dependent, due to the coupling of dislocation and diffusion creep. The model structure proposed consists of a top layer, the upper crust, gliding rigidly above a bottom layer, the lower crust, which deforms in simple shear. During a short time interval (1400 years), the equivalent shear stress is found to increase by a factor of up to 3 (67 MPa for anorthite and 17 MPa for quartz). For anorthite, this stress could explain the activation of a Mohr-Coulomb failure with a friction coefficient of 0.2, which is reasonable at the depth of 10 km. Dislocation creep is activated during a rapid change in the prescribed velocity, whereas diffusion creep dominates if the velocity is held constant, highlighting the importance of grain-size sensitivity for lower crustal rheology.

The linear stability criterion, proposed for structural models in an earlier paper, is now extended for a general class of elastic–viscoplastic continua. The time-dependent trajectories, whose stability is under investigation, are functions of two characteristic times: the relaxation time of the viscous solid and the rate of the applied loading, with their ratio denoted by T . It is assumed that the loading conditions of the trajectory are not modified by the perturbation. The criterion predicts that a solid is initially unstable if there exists a unit norm perturbation in the velocity field whose time derivative is positive. This condition is equivalent to finding a positive eigenvalue in the self-adjoint part of the operator relating the initial first and second rate of the displacement perturbations. If the dominant eigenvalue is obtained from the non self-adjoint operator, the change in sign of its real part is a sufficient condition for instability. For solids with an associated flow rule, it is shown that the exclusion of instability in a trajectory, in the limit of vanishing T , coincides with stability of the corresponding rate-independent solid in the sense of Hill. The theory is applied to the analysis of a finitely strained rectangular block under uniaxial tension and compression, for different elastic–viscoplastic solids of the von Mises and Drucker–Prager type.

The initiation of two modes of instability, viz. folding and faulting, is investigated theoretically for an elasto-plastic, frictional- cohesive layer which is underlain by a viscous substratum. The destabilizing factors that are allowed to come into play are the tectonic stress, buoyancy forces resulting from a gravitationally unstable density stratification, and the redistribution of material at the top surface by erosion and deposition processes. The bending stiffness of the overburden, a non-linear function of stress, has a stabilizing influence. A variational formulation of the stability problem allows one to detect the onset of global modes of instability, such as folds and surface modes for compression as well as neck-type modes for extension. Predictions of these global modes remain valid as long as the local condition of strong ellipticity is satisfied. Failure of this condition marks the onset of discontinuities in the velocity field, which are characteristic of localized faulting. The sensitivity of these predictions to the assumed behaviour of the overburden and substratum materials is explored for a prototype representative of a dip section through the Campos salt basin on the Brazilian continental margin. These results illustrate the importance of a proper selection of analogue materials for the design of physical laboratory models. This point is underscored by employing a deformation theory of plasticity which could be seen to reproduce in a simple manner the accommodation of bulk deformation by slip along a population of pervasive small faults in sedimentary rocks as well as in analogue materials, such as sands, used in the laboratory. A historical account of the use of deformation theories of plasticity in stability analyses and a derivation of the relevant incremental moduli are also given in this paper.

International audience; This article is concerned with the development and application of a simple continuum theory for rocks that may contain both randomly as well as preferentially oriented plane discontinuity surfaces. The theory stipulates that displacement discontinuities are independently activated on these surfaces as soon as an appropriate yield criterion is fulfilled; these displacement jumps account for the irreversible, ‘plastic’ part of the bulk deformation. In stress space, the critical conditions for the activation of discontinuous slip or opening displacements define an overall yield envelope that could be initially anisotropic, reflecting for example a weakness of certain orientations due to pre-existing joint sets. For the yield conditions studied in this paper, essentially a Coulomb-type friction law and a simple fracture opening condition, the inferred stress-strain response under typical triaxial loading conditions reveals the sensitivity of the two discontinuous deformation modes to the confining pressure. The incipient growth of a geological fold in such a material is modelled as a problem of plate bending. The slip- and opening-modes of deformation are found to be activated typically in the fold intrados and extrados, respectively. Under certain conditions, both modes will be activated simultaneously at the same locality and contribute to the total deformation. Field observations on a well exposed sandstone anticline are reported here, which support this conclusion. The present plasticity model for discontinua can clearly be explored in more detail for realistic distributions of faults and joints taken from field observations. It could also be improved in various ways in its description of the underlying deformation mechanisms. Apart from its interest as a mechanical constitutive model, it can also serve as a point of departure for studies of stress-sensitive, anisotropic permeability distributions in fractured formations.

The linear stability criterion proposed generalizes to continua earlier results obtained for the viscoplastic Shanley column. A sufficient condition for instability is marked by the change of sign of the real part of the dominant eigenvalue for the system of equations constructed from the linearized version of the first and second derivatives with respect to time of the principle of virtual work. In particular, it is shown that the conditions for first instability are those for the first bifurcation of a strain-rate-independent solid obtained in the limit of inviscid plastic flow.

This study deals with the stability of a stratified structure composed of a cohesive and frictional overburden, a viscous substratum, and a rigid basement. That structure should be seen as a prototype for various salt tectonics and lithospheric plates stability analyses. The destabilizing factors are the density contrast, the tectonic compressive stress, and the possible erosion and deposition at the top surface. The overburden stiffness, a nonlinear function of in situ stress, has a stabilizing role. Two solutions are extracted from the variational formulation of the stability problem previously proposed [Leroy and Triantafyllidis, 1996]: the first is analytical and is obtained by disregarding gravity, and the second is numerical and is based on the finite element method. The latter is used to assess the validity of the previously presented asymptotic solution. It is shown that the asymptotic solution is accurate even for values of the small parameter, defined as the perturbation wavenumber times the overburden thickness, as large as 0.4. Furthermore, the possibility for the cohesive material in the overburden to accommodate part of the deformation by slip along a population of small pervasive faults is accounted for by the introduction of a deformation theory of plasticity. Stability predictions based on this theory indicate that structural modes, such as folding, and localized faulting modes are triggered for similar stress magnitudes. The parametric study presented includes the previously undetected influences of the stress gradient with depth and of the work hardening properties of the competent overburden. The role of erosion and deposition in destabilizing shallow overburdens, regardless of the magnitude of the tectonic stress, is also established. The stability predictions are then applied to a folded section through the Campos basin, offshore Brazil, revealing that the deformation theory of plasticity is necessary to explain the buckling that occurred during the Albian.

This contribution is concerned with the stability of a frictional, cohesive material layer, called the overburden, resting on a viscous, incompressible layer of lower density referred to as the substratum. The viscous layer is either perfectly bonded or free to slip with no friction on a rigid basement. The in situ stress is assigned a realistic gradient in the overburden and is assumed to be purely hydrostatic in the substratum. A general variational formulation of the linearized stability problem for the stratified system is obtained in which the viscous response of the substratum provides the characteristic time. To a given state of stress and perturbation corresponds a rate of growth or decay which is calculated by an asymptotic method for small overburden thickness compared to the perturbation wavelength. It is found that the substratum's thickness influences the rate of growth of the instability if its product by the perturbation wavenumber is smaller than 3 or 4, depending on the type of boundary condition at the basement. However, these conditions and the substratum thickness have no influence on neutral stability. Furthermore, the conditions for stability depend on the tectonic stress distribution in the overburden and not solely on the density contrast, as is the case for viscoelastic systems. It is also shown that the classical flow theory of plasticity adopted for the overburden, which successfully captures the onset of folding, fails to predict initiation of faulting for similar stress conditions.