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

Cracking and coalescence behavior in a rectangular rock-like specimen containing two parallel (stepped and coplanar) pre-existing open flaws under uniaxial compression load has been numerically studied by a parallel bonded-particle model, which is a type of bonded-particle model. Crack initiation and propagation from two flaws replicate most of the phenomena observed in prior physical experiments, such as the type (tensile/shear) and the initiation stress of the first crack, as well as the coalescence pattern. Eight crack coalescence categories representing different crack types and trajectories are identified. New coalescence categories namely “New 1” and “New 2”, which are first observed in the present simulation, are incorporated into categories 3 and 4, and category 5 previously proposed by the MIT Rock Mechanics Research Group, respectively. The flaw inclination angle (β), the ligament length (L) (spacing between two flaws) and the bridging angle (α) (inclination of a line linking up the inner flaw tips, between two flaws) have different effects on the coalescence patterns, coalescence stresses (before, at or post the peak stress) as well as peak strength of specimens. Some insights on the coalescence processes, such as the initiation of cracks in the intact part of specimens at a distance away from the flaw tips, and coalescence due to the development and linkage of a number of steeply inclined to vertical macro-tensile cracks are revealed by the present numerical study.

Crack Initiation, Propagation and Coalescence in Rock-Like Material Containing Two Flaws: a Numerical Study Based on Bonded-Particle Model Approach

A novel meshless numerical method, called general particle dynamics (GPD), is proposed to simulate samples of rock-like brittle heterogeneous material containing four preexisting flaws under uniaxial compressive loads. Numerical simulations are conducted to investigate the initiation, growth, and coalescence of cracks using a GPD code. An elasto-brittle damage model based on an extension of the Hoek–Brown strength criterion is applied to reflect crack initiation, growth, and coalescence and the macrofailure of the rock-like material. The preexisting flaws are simulated by empty particles. The particle is killed when its stresses satisfy the Hoek–Brown strength criterion, and the growth path of cracks is captured through the sequence of such damaged particles. A statistical approach is applied to model the heterogeneity of the rock-like material. It is found from the numerical results that samples containing four preexisting flaws may produce five types of cracks at or near the tips of preexisting flaws including wing, coplanar or quasi-coplanar secondary, oblique secondary, out-of-plane tensile, and out-of-plane shear cracks. Four coalescence modes are observed from the numerical results: tensile (T), compression (C), shear (S), and mixed tension/shear (TS). A higher load is required to induce crack coalescence in the shear mode (S) than the tensile (T) or mixed (TS) mode. It is concluded from the numerical results that crack coalescence occurs following the weakest coalescence path among all possible paths between any two flaws. The numerical results are in good agreement with reported experimental observations.

Numerical Simulation of Crack Growth and Coalescence in Rock-Like Materials Containing Multiple Pre-existing Flaws

SUMMARY
Aiming to solve, in a unified way, continuous and discontinuous problems in geotechnical engineering, the numerical manifold method introduces two covers, namely, the mathematical cover and the physical cover. In order to reach the goal, some issues in the simulation of crack propagation have to be solved, among which are the four issues to be treated in this study: (1) to reduce the rank deficiency induced by high degree polynomials as local approximation, a new variational principle is formulated, which suppresses the gradient-dependent DOFs; (2) to evaluate the integrals with singularity of 1/r, a new numerical quadrature scheme is developed, which is simpler but more efficient than the existing Duffy transformation; (3) to analyze kinked cracks, a sign convention for argument in the polar system at the crack tip is specified, which leads to more accurate results in a simpler way than the existing mapping technique; and (4) to demonstrate the mesh independency of numerical manifold method in handling strong singularity, a mesh deployment scheme is advised, which can reproduce all singular locations of the crack with regard to the mesh. Corresponding to the four issues, typical examples are given to demonstrate the effectiveness of the proposed schemes. Copyright © 2013 John Wiley & Sons, Ltd.

New strategies for some issues of numerical manifold method in simulation of crack propagation

Mechanical anisotropy of aligned fiber reinforced composite for extrusion-based 3D printing

Frictional crack initiation and propagation analysis using the numerical manifold method

Experimental investigation of thermal effects on dynamic behavior of granite

An investigation of thermal effects on micro-properties of granite by X-ray CT technique

Micro-mechanical modeling of the macro-mechanical response and fracture behavior of rock using the numerical manifold method

Strength and deformation behaviors of rockfill materials, key factors for determining the stability of dams, pertain strongly to the grain crushing characteristics. In this study, single-particle crushing tests were carried out on rockfill materials with nominal particle diameters of 2.5 mm, 5 mm and 10 mm to investigate the particle size effect on the single-particle strength and the relationship between the characteristic stress and probability of non-failure. Test data were found to be described by the Weibull distribution with the Weibull modulus of 3.24. Assemblies with uniform nominal grains were then subjected to one-dimensional compression tests at eight levels of vertical stress with a maximum of 100 MPa. The yield stress in one-dimensional compression tests increased with decreasing the particle size, which could be estimated from the single-particle crushing tests. The void ratio-vertical stress curve could be predicted by an exponential function. The particle size distribution curve increased obviously with applied stresses less than 16 MPa and gradually reached the ultimate fractal grading. The relative breakage index became constant with stress up to 64 MPa and was obtained from the ultimate grading at the fractal dimension ( α = 2.7 ). A hyperbolical function was also found useful for describing the relationship between the relative breakage index and input work during one-dimensional compression tests.

Zhijun Wu

Papers

Frictional crack initiation and propagation analysis using the numerical manifold method

Experimental investigation of thermal effects on dynamic behavior of granite

An investigation of thermal effects on micro-properties of granite by X-ray CT technique

Micro-mechanical modeling of the macro-mechanical response and fracture behavior of rock using the numerical manifold method

Effects of particle size on crushing and deformation behaviors of rockfill materials