John W. Rudnicki

Conditions for the localization of deformation in pressure-sensitive dilatant materials

SUMMARY Tms PAPER investigates the hypothesis that localization of deformation into a shear band may be considered a result of an instability in the constitutive description of homogeneous deformation. General conditions for a bifurcation, corresponding to the localization of deformation into a planar band, are derived. Although the analysis is general and applications to other localization phenomena are noted, the constitutive relations which are examined in application of the criterion for IocaIization are intended to model the behavior of brittle rock masses under compressive principal stresses. These relations are strongly pressure-sensitive since inelasticity results from frictional sliding on an array of fissures; the resulting inelastic response is dilatant, owing to uplift in sliding at asperities and to local tensile cracking from fissure tips. The appropriate constitutive descriptions involve non-normality of plastic strain increments to the yield hyper-surface. Also, it is argued that the subsequent yield surfaces will develop a vertex-like structure. Both of these features are shown to be destabilizing and to strongly influence the resulting predictions for localization by comparison to predictions based on classical plasticity idealizations, involving normality and smooth yield surfaces. These results seem widely applicable to discussions of the inception of rupture as a ~nstitutive instability. ZONES of localized deformation, in the form of narrow shear bands, are a common feature of brittle rock masses that have failed under compressive principal stresses, both in laboratory experiments (e.g. BRACE, 1964, WAWERSIK and FAIRHURST, 1970, and WAWERSIK and BRACE, 1971) and, naturally, as earth faults. It is possible that such behavior can be explained only by modelling in detail the processes of growth and interaction of the many individual fissures that ultimately join together in forming the rubble-like macroscopic surface of rupture. However, we investigate here an alternative hypothesis: That ~ocali~~~io~ can be u~l~ers~oo~ as an instability in the macroscopic constitutive description of inelastic deformation of the material. Specifically, instability is understood in the sense that the constitutive relations may allow the homogeneous deformation of an initially uniform material to lead to a bifurcation point, at which non-uniform deformation can be incipient in a planar band under conditions of continuing equihbrium and continuing homogeneous deformation outside the zone of localization. An explanation of this kind has been proposed by BERG (1970) for the inception of rupture in ductile metals owing to the nucleation and progressive growth of microscopic voids, and RICE (1973) has given a formulation for localization in

A NOTE ON SOME FEATURES OF THE THEORY OF LOCALIZATION OF DEFORMATION

Abstract This note examines two aspects of the theory which treats localization of deformation as a bifurcation from homogeneous deformation. The results are obtained for solids modelled as elastic-plastic and having smooth yield and plastic potential surfaces, but it is not required that inelastic strain increments be normal to the yield surface. First, it is demonstrated that discontinuous bifurcations, for which elastic unloading occurs outside the zone of incipient localization, first become possible at the point of continuous bifurcation, for which further plastic deformation is assumed to occur both inside and outside of the zone of localization. Second, we investigate an apparent paradox which arises in the rigid plastic limit of an elastic-plastic localization calculation if normality does not apply. This is resolved by consideration of the relative amounts of the bifurcation mode corresponding to elastic and to plastic deformation and it is demonstrated that even for very small amounts of elasticity, bifurcation modes which are inadmissible in the rigid plastic case become possible.

Conditions for compaction bands in porous rock

Reexamination of the results of Rudnicki and Rice for shear localization reveals that solutions for compaction bands are possible in a range of parameters typical of porous rock. Compaction bands are narrow planar zones of localized compressive deformation perpendicular to the maximum compressive stress, which have been observed in high-porosity rocks in the laboratory and field. Solutions for compaction bands, as an alternative to homogenous deformation, are possible when the inelastic volume deformation is compactive and is associated with stress states on a yield surface “cap.” The cap implies that the shear stress required for further inelastic deformation decreases with increasing compressive mean stress. While the expressions for the critical hardening modulus for compaction and shear bands differ, in both cases, deviations from normality promote band formation. Inelastic compaction deformation associated with mean stress (suggested by Aydin and Johnson) promotes localization by decreasing the magnitude of the critical hardening modulus. Axisymmetric compression is the most favorable deviatoric stress state for formation of compaction bands. Predictions for compaction bands suggest that they could form on the “shelf” typically observed in axisymmetric compression stress strain curves of porous rock at high confining stress. Either shear or compaction bands may occur depending on the stress path and confining stress. If the increase in local density and decrease in grain size associated with compaction band formation result in strengthening rather than weakening of the band material, formation of a compaction band may not preclude later formation of a shear band.

Fluid mass sources and point forces in linear elastic diffusive solids

Abstract Solutions for the point force and line load suddenly applied in a linear elstic fluid-infiltrated porous solid are rederived in a manner that emphasizes the relation to solutions for the homogeneous diffusion equation. Specifically, the stress, displacement, and pore pressure due to instantaneous and continuous injection of fluid mass are obtained. These are differentiated to yield solutions for fluid mass dipoles. Finally, the desired solutions are obtained by adding a constant multiple of the continuous dipole solution to the pure elasticity solution based on the undrained (short-time) moduli.

A microcrack-based continuous damage model for brittle geomaterials

A new microcrack-based continuous damage model is developed to describe the behavior of brittle geomaterials under compression dominated stress fields. The induced damage is represented by a second rank tensor, which reflects density and orientation of microcracks. The damage evolution law is related to the propagation condition of microcracks. Based on micromechanical analyses of sliding wing cracks, the actual microcrack distributions are replaced by an equivalent set of cracks subjected to a macroscopic local tensile stress. The principles of the linear fracture mechanics are used to develop a suitable macroscopic propagation criterion. The onset of microcrack coalescence leading to localization phenomenon and softening behavior is included by using a critical crack length. The constitutive equations are developed by considering that microcrack growth induces an added material flexibility. The effective elastic compliance of damaged material is obtained from the definition of a particular Gibbs free energy function. Irreversible damage-related strains due to residual opening of microcracks after unloading are also taken into account. The resulting constitutive equations can be arranged to reveal the physical meaning of each model parameter and to determine its value from standard laboratory tests. An explicit expression for the macroscopic effective constitutive tensor (compliance or stiffness) makes it possible, in principal, to determine the critical damage intensity at which the localization condition is satisfied. The proposed model is applied to two typical brittle rocks (a French granite and Tennessee marble). Comparison between test data and numerical simulations show that the proposed model is able to describe main features of mechanical behaviors observed in brittle geomaterials under compressive stresses.

Stability and localization of rapid shear in fluid‐saturated fault gouge: 2. Localized zone width and strength evolution

Field and laboratory observations indicate that at seismic slip rates most shearing is confined to a very narrow zone, just a few tens to hundreds of microns wide, and sometimes as small as a few microns. Rice et al. (2014) analyzed the stability of uniform shear in a fluid-saturated gouge material. They considered two distinct mechanisms to limit localization to a finite thickness zone, rate-strengthening friction, and dilatancy. In this paper we use numerical simulations to extend beyond the linearized perturbation context in Rice et al. (2014), and study the behavior after the loss of stability. Neglecting dilatancy we find that straining localizes to a width that is almost independent of the gouge layer width, suggesting that the localized zone width is set by the physical properties of the gouge material. Choosing parameters thought to be representative of a crustal depth of 7 km, this predicts that deformation should be confined to a zone between 4 and 44 μm wide. Next, considering dilatancy alone we again find a localized zone thickness that is independent of gouge layer thickness. For dilatancy alone we predict localized zone thicknesses between 1 and 2 μm wide for a depth of 7 km. Finally, we study the impact of localization on the shear strength and temperature evolution of the gouge material. Strain rate localization focuses frictional heating into a narrower zone, leading to a much faster temperature rise than that predicted when localization is not accounted for. Since the dynamic weakening mechanism considered here is thermally driven, this leads to accelerated dynamic weakening.

Terrestrial Sequestration of CO2 – An Assessment of Research Needs

Publisher Summary This chapter provides a brief review of major characteristics of reservoir structures and lithologies serving as a guide to reservoir selection for CO 2 disposal. The chapter focuses on existing experience and uncertainties in reservoir characterization and response to CO 2 injection and long-term containment of sequestration sites. Special issues germane to CO 2 disposal arise in the assessment of depleted reservoirs, whose properties are known to have changed during single or repeated pore-pressure drawdown and fluid redistribution. Oil and gas reservoirs and aquifers share some common geometric elements. Generally, both are tabular bodies in which the fluid flow is constrained by upper and lower less-permeable lithologies. Primary aspects of CO 2 sequestration in geologic formations include the geohydrologic characterization, injection behavior, and long-term containment of supercritical CO 2 for storage in aquifers and reservoirs. The efficiency of a CO 2 enhanced oil-recovery flood depends strongly on the equilibrium phase behavior of mixtures of CO 2 with the oil.

Chapter 5 ▸ - Localization: Shear Bands and Compaction Bands

The localized deformation is a ubiquitous phenomenon in geomaterials. It occurs over a vast range of size scales, from the microscale level of grains to faults extending over hundreds of kilometers. It occurs in a variety of forms: a concentration or coalescence of cracks; a distinct, planar frictional surface; a gouge zone of finely comminuted material; or simply a region of higher shear strain. In geomaterials, the severe shearing in regions of localized deformation may be accompanied by dilatancy (inelastic volume increase) or compaction (inelastic volume decrease) and by chemical alteration. Localization can even occur purely by compaction without any evident shear. If the material is fluid-saturated, as is frequently the case, inelastic volume changes can induce the flow of fluid or changes in pore pressure that affect the response. Localization occurs under a variety of conditions. Although most often associated with the formation of shear bands or faults under nominally brittle conditions, localization can also occur by cataclastic flow of rocks at higher mean stresses and by ductile shearing at temperatures and pressures typical of depths of 10 km to 15 km in the earths crust.

Mechanics of dip‐slip faulting in an elastic half‐space

The mechanics of dip-slip faulting is examined using a model of a crack in a plane strain elastic half-space loaded by remote stresses: vertical stresses are equal to the overburden and horizontal stresses are the sum of a constant (c) times the overburden and a depth-independent term σtect, which may be compressive or tensile. Opening is allowed in response to local tension, and slip on closed regions is governed by Mohr-Coulomb conditions. Because regions of opening and slip, in general, must be determined as part of the solution, relative displacements and stress intensity factors are calculated by iterative solution of integral equations. If the fault is not too shallow (h/L > 0.3, where h is the depth of the upper end of the slip zone and L is the length along dip) and the Coulomb stress (the shear stress minus the friction coefficient ƒ times the effective normal stress) is of one sign over most of the fault, simple approximations suffice. Use of these approximations and the criterion that the energy release rate equals a critical value demonstrates that updip propagation is unstable if the slip zone length exceeds a minimum value. For typical values of ƒ dip angle, and c, the slip zone will be locked below a certain depth. In general, the depth at which the Coulomb stress changes sign underestimates the locking depth, but the magnitude of the difference decreases as the depth of the sign change approaches the lower end of the slip zone. For shallow slip zones, h/L < 0.3, slip induces changes in both shear and normal stresses on the slip surface. For reverse (normal) slip, this coupling reduces (increases) the compressive normal stress and amplifies (diminishes) slip near the upper end of the slip zone surface. For extensional loading (σtect < 0), opening can occur for shallow embedded zones and, more prominently, for surface breaking zones. Although the shear stress drop is nonuniform in this model, calculations reveal that unless fault opening occurs, the surface displacement is well-approximated by that due to a uniform shear stress drop equal to the average.

Connecting microstructural attributes and permeability from 3D tomographic images of in situ shear-enhanced compaction bands using multiscale computations

Tomographic images taken inside and outside a compaction band in a field specimen of Aztec sandstone are analyzed by using numerical methods such as graph theory, level sets, and hybrid lattice Boltzmann/finite element techniques. The results reveal approximately an order of magnitude permeability reduction within the compaction band. This is less than the several orders of magnitude reduction measured from hydraulic experiments on compaction bands formed in laboratory experiments and about one order of magnitude less than inferences from two-dimensional images of Aztec sandstone. Geometrical analysis concludes that the elimination of connected pore space and increased tortuosities due to the porosity decrease are the major factors contributing to the permeability reduction. In addition, the multiscale flow simulations also indicate that permeability is fairly isotropic inside and outside the compaction band.