Yves M. Leroy

A finite element method for localized failure analysis

A method is proposed which aims at enhancing the performance of general classes of elements in problems involving strain localization. The method exploits information concerning the process of localization which is readily available at the element level. A bifurcation analysis is used to determine the geometry of the localized deformation modes. When the onset of localization is detected, suitably defined shape functions are added to the element interpolation which closely reproduce the localized modes. The extra degrees of freedom representing the amplitudes of these modes are eliminated by static condensation. The proposed methodology can be applied to 2-D and 3-D problems involving arbitrary rate-independent material behavior. Numerical examples demonstrate the ability of the method to resolve the geometry of localized failure modes to the highest resolution allowed by the mesh.

Mechanical constraints on the chronology of fracture activation in folded Devonian sandstone of the western Moroccan Anti-Atlas

The three-dimensional meter-scale fracture networks, observed on exposed folds between the towns of Tata and Akka, western Moroccan Anti-Atlas, consist mostly of planar discontinuities, which are sub-perpendicular to the bedding and partitioned in three main sets. The chronology of their activation is proposed in five stages since the Hercynian orogeny. Stage 1 predates folding and involves the horizontal compression of the Emsian sandstone. It involves fracture set I, composed of systematic joints parallel to the direction of compression. Stages 2 ‐ 4 correspond to the folding and are marked in the outer-arc by the activation of fracture set II, composed mainly of joints parallel to the fold axial plane. Stage 5 is a regional shear event during which sets I and III, separated by an angle close to 608, are activated in a conjugate manner. To throw light on the recurrent difficulty in discriminating between activation of inherited and new fractures, an elasto-plastic model is used to construct a stress path in the pervasively fractured medium idealized as a continuum. Each fracture set obeys the Mohr‐ Coulomb criterion truncated in tension to describe both sliding and opening activations. Finite-element simulations of a simple buckling event accounting for the field fracture sets are presented. It is shown that set I cannot be generated by folding and thus does belong to stage 1. Set II is activated at a later stage of folding than expected from the field interpretation. Set III cannot be activated during stage 2, confirming its role in stage 5. The advantages and limitations of the proposed modeling are finally discussed. q 2002 Elsevier Science Ltd. All rights reserved.

Mechanics of low-angle extensional shear zones at the brittle-ductile transition

corresponding to low values of the velocity ratio Ve/Vs. The 2-D solution (for Ve/Vs =1 0 � 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. INDEX TERMS: 8109 Tectonophysics: Continental tectonics—extensional (0905); 8159 Tectonophysics: Rheology—crust and lithosphere; 8164 Tectonophysics: Stresses—crust and lithosphere; 7218 Seismology: Lithosphere and upper mantle; 3230 Mathematical Geophysics: Numerical solutions; KEYWORDS: continental extension, strain localization, detachment fault, rheology, reaction softening, numerical model

Stability of steady states in shear zones

Abstract 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.

Solid–fluid phase transformation within grain boundaries during compaction by pressure solution

Abstract 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.

Spatial patterns and size effects in shear zones : a hyperelastic model with higher-order gradients

Abstract 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.

Stability of a frictional material layer resting on a viscous half-space

(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.

Geometrical evolution of stressed and curved solid-fluid phase boundaries: 2. Stability of cylindrical pores

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.

Geometrical evolution of stressed and curved solid‐fluid phase boundaries: 1. Transformation kinetics

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 pores 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.

Chapter 3 ▸ – Sandstone Compaction by Intergranular Pressure Solution

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.