The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

An extended finite element method (X-FEM) for three-dimensional crack modelling is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three-dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.

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Extended finite element method for three-dimensional crack modelling

A model for the axisymmetric growth and coalescence of small internal voids in elastoplastic solids is proposed and assessed using void cell computations. Two contributions existing in the literature have been integrated into the enhanced model. The first is the model of Gologanu-Leblond-Devaux, extending the Gurson model to void shape effects. The second is the approach of Thomason for the onset of void coalescence. Each of these has been extended heuristically to account for strain hardening. In addition, a micromechanically-based simple constitutive model for the void coalescence stage is proposed to supplement the criterion for the onset of coalescence. The fully enhanced Gurson model depends on the flow properties of the material and the dimensional ratios of the void-cell representative volume element. Phenomenological parameters such as critical porosities are not employed in the enhanced model. It incorporates the effect of void shape, relative void spacing, strain hardening, and porosity. The effect of the relative void spacing on void coalescence, which has not yet been carefully addressed in the literature. has received special attention. Using cell model computations, accurate predictions through final fracture have been obtained for a wide range of porosity, void spacing, initial void shape, strain hardening, and stress triaxiality. These predictions have been used to assess the enhanced model. (C) 2000 Elsevier Science Ltd. All rights reserved.

An extended model for void growth and coalescence

Failure of metals I : brittle and ductile fracture

Theoretical concepts of sintering were originally based upon ideas of the discrete nature of particulate media. However, the actual sintering kinetics of particulate bodies are determined not only by the properties of the particles themselves and the nature of their local interaction with each other, but also by macroscopic factors. Among them are externally applied forces, kinematic constraints (e.g. adhesion of the sample's end face and furnace surface), and inhomogeneity of properties in the volume under investigation (e.g. inhomogeneity of initial density distribution created during preliminary forming operations). Insufficient treatment of the questions enumerated above was one of the basic reasons hindering the use of sintering theory. A promising approach is connected with the use of continuum mechanics, which has been successfully applied to the analysis of compaction of porous bodies. This approach is based upon the theories of plastic and nonlinear-viscous deformation of porous bodies. Similar ideas have recently been embodied in a continuum theory of sintering. The main results of the application of this theory for the solution of certain technological problems of sintering are introduced including their thermo–mechanical aspects.

Theory of sintering: from discrete to continuum

Symmetric weak-form integral equation method for three-dimensional fracture analysis

A theory of cold rolling of thin gauge strip is presented which, within the idealizations of homogeneous deformation and a constant coefficient of Coulomb friction, rigorously models the elastic deformation of the rolls and the frictional traction at the interface. In contrast with classical theories (3) it is shown that, for gauges less than a critical value, plastic reduction takes place in two zones, at entry and exit, which are separated by a neutral zone in which the rolls are compressed fiat and there is no slip between the rolls and the strip. Roll load and torque are governed by five independent non-dimensional parameters which express the influence of gauge, reduction, friction and front and back tensions. Values of load and torque have been computed (for zero front and back tensions) for a wide range of thickness, reduction and friction and have been found to collapse approximately on to a single master curve.

Cold Rolling of Foil

T he macroscopic response of an incompressible power-law matrix containing a dispersion of aligned, spheroidal voids is investigated. Attention is restricted to dilute concentrations of voids and to axisymmetric deformation of the solid. The essential step in the analysis is the solution of a kernel problem for an isolated void, and this solution is obtained accurately and efficiently using a Ritz procedure developed for this purpose. Results for macroscopic strain-rates are presented for void shapes ranging from penny-shaped cracks to infinitely long circular cylinders and for a wide range of triaxialities and matrix hardening exponents. These results are used to assess the role of void shape on the overall response of porous solids.

Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids

The initiation of crack growth under a combination of opening and anti-plane shearing mode loading is considered in this paper. It is shown that such cracks do not grow through a continuous evolution of the crack surface. Rather, an abrupt fragmentation or segmentation of the crack front is generated. Through experimental observations and a theoretical model, we postulate a relationship between the scale of the fragmentation and the mode mix.

Criterion for initiation of cracks under mixed-mode I + III loading

We present a non-planar fracture model in a poro-elastic medium. The medium in which the fracture is embedded is governed by the standard Biot equations of linear poro-elasticity and the flow of the fluid within the fracture is governed by the lubrication equation. We establish existence and uniqueness of the linearized coupled system under weak assumptions on the data. Two discretizations of the problem are formulated: one with a continuous Galerkin method and one with a mixed method for the flow in the reservoir and in the fracture. We perform the numerical analysis of the former and we provide an algorithm and numerical experiments for the latter.

Mark E. Mear

Papers

Symmetric weak-form integral equation method for three-dimensional fracture analysis

Cold Rolling of Foil

Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids

Criterion for initiation of cracks under mixed-mode I + III loading

A lubrication fracture model in a poro-elastic medium