Steven L. Crouch

Soft, stiff and servo-controlled testing machines: a review with reference to rock failure

Abstract Testing machines have become increasingly sophisticated and versatile since they were first introduced in the early eighteenth century. Two recent advances in testing machine technology are of particular importance for laboratory studies of rock failure: the development of machines with stiff frames and the use of feedback control systems. Force has been traditionally regarded as the independent variable in materials testing but the inevitable outcome of a rock mechanics test with a constant loading rate is violent uncontrolled failure at the peak of the force—displacement curve. Rock failure can be controlled if displacement is regarded as the independent variable and a constant displacement rate can be achieved in a testing machine with a stiff frame or a feedback control system. This review contains a brief history of testing machines and a detailed discussion of the principles involved in controlling rock failure with stiff and servo-controlled testing machines. The development of stiff testing machines is followed from 1935 to the present day and the rock specimen stiffness and testing machine stiffness (factors that are neither fixed nor independent) are discussed in detail. In a closed-loop servo-controlled testing machine, the ability to control rock failure is governed by the fast response time and correct choice of feedback signal. These factors are explained and examples are given of the precise control that can now be achieved in experimental studies of rock failure that were impractical or impossible several years ago.

A complex boundary integral method for multiple circular holes in an infinite plane

Abstract A complex boundary integral equation method, combined with series expansion technique, is presented for the problem of an infinite, isotropic elastic plane containing multiple circular holes. Loading is applied at infinity or on the boundaries of the holes. The sizes and locations of the holes are arbitrary provided they do not overlap. The analysis procedure is based on the use of a complex hypersingular integral equation that expresses a direct relationship between all the boundary tractions and displacements. The unknown displacements on each circular boundary are represented by truncated complex Fourier series, and all of the integrals involved in the complex integral equation are evaluated analytically. A system of linear algebraic equations is obtained by using a Taylor series expansion, and the block Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the accuracy, versatility, and efficiency of the approach.

A Galerkin boundary integral method for multiple circular elastic inclusions with homogeneously imperfect interfaces

A Galerkin boundary integral method is presented to solve the problem of an infinite, isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with homogeneously imperfect interfaces. Problems of interest might involve thousands of inclusions with no restrictions on their locations (except that the inclusions may not overlap), sizes, and elastic properties. The tractions are assumed to be continuous across the interfaces and proportional to the corresponding displacement discontinuities. The analysis is based on a numerical solution of a complex hypersingular integral equation with the unknown tractions and displacement discontinuities at each circular boundary approximated by truncated complex Fourier series. The method allows one to calculate the stress and displacement fields everywhere in the matrix and inside the inclusions. Numerical examples are included to demonstrate the effectiveness of the approach.

Prediction of interfacial crack path: a direct boundary integral approach and experimental study

This paper presents the development of a higher-order direct boundary integral-displacement discontinuity method for crack propagation in layered elastic materials. The method is based on the dual boundary integral equations of linear elasticity which are solved by means of a quadratic boundary element formulation. The analytical solution for a point force within a bonded half-plane region is used to derive the kernel functions of the boundary integral equations. Square-root displacement-discontinuity elements are used to model the crack tips, and stress intensity factors may be computed using the numerically predicted values of the displacement discontinuity components at the midpoints of these crack-tip elements. An algorithm based on the maximum tensile-stress criterion is then developed and incorporated into the boundary element model to predict the paths of cracks propagating in layered elastic materials.In the experimental part of this study, crack profiles for straight-through-cracked, compact-tension specimens of the anodically bonded silicon/Pyrex glass system are measured by profilometry. The plane strain prediction of the crack-propagation path is compared with the experimentally measured crack profiles. Consistent with the prediction, the interfacial crack is observed to kink away from the strong, anodically-bonded interface and propagate into the more compliant glass layer. The predicted initial kink angle of 26° agrees very well with the average measured value of 28°. The measured path of the crack is also in very good agreement with the predicted path over about the first 120 microns of crack growth with increasing deviation observed beyond that.

Benchmark Results for the Problem of Interaction Between a Crack and a Circular Inclusion

This paper is a reply to the challenge by Helsing and Jonsson (2002, ASME J. Appl. Mech., 69, pp. 88-90) for other investigators to confirm or disprove their new numerical results for the stress intensity factors for a crack in the neighborhood of a circular inclusion. We examined the same problem as Helsing and Jonsson using two different approaches-a Galerkin boundary integral method (Wang et al., 2001, in Rock Mechanics in the National Interest, pp. 1453-1460) (Mogilevskaya and Crouch, 2001, Int. J. Numer Meth. Eng., 52, pp. 1069-1106) and a complex variables boundary element method (Mogilevskaya, 1996, Comput. Mech., 18, pp. 127-138). Our results agree with Helsing and Jonssons in all cases considered.

Elastodynamic boundary element method for piecewise homogeneous media

A general higher-order formulation for the time domain elastodynamic direct boundary element method is presented for computing the transient displacements and stresses in multiply connected two-dimensional solids. The displacement and traction interpolation functions are linear in time and quadratic in space. All integrations are analytical, and are expressed in terms of twelve basic recurring integrals. Causality is ensured by integrating only over the dynamically active parts of each element, and the algorithm presented is time-marching and implicit. The use of analytical integrations allows both unbounded and bounded domain problems to be solved without having to introduce special enclosing elements. All of these improved features allow for a formulation that is very efficient and accurate. The stability and accuracy of the elastodynamic boundary element algorithm is demonstrated by solving several example problems and comparing the results with available analytical and numerical solutions.

THE POST-FAILURE BEHAVIOR OF NORITE IN TRIAXIAL COMPRESSION

Abstract Results of a comprehensive series of quasi-static triaxial compression tests on a South African norite tested at confining stresses of up to 8,000 p.s.i. are presented. The tests were performed in a stiff testing machine so that the post-failure behavior could be observed, and lateral volumetric strains were continuously recorded using the “constant confining stress” technique. A comparison of the results of these tests with some data previously presented for a Witwatersrand quartzite reveals that there are significant differences between the deformational behavior of norite and quartzite, particularly in the manner in which the lateral expansion at the maximum axial stresses varies with confining stress. For norite, the amount of lateral expansion in excess of that due to linear, elastic compression was found to steadily increase about threefold between confining stresses of 50 and 6,000 p.s.i., and thereafter remain approximately constant. The “excess lateral expansion” for quartzite is about four times higher at a confining stress of 2,500 p.s.i. than at 50 and 5,000 p.s.i. The rate of lateral expansion at the maximum axial stress decreases with increased confinement in a similar manner for both rocks, although at any particular confining stress the absolute rate is higher for quartzite than for norite.

Computational Modeling of Viscoelastic Porous Materials

This paper is concerned with the development of a computational basis for modeling time‐dependent effects in a finite or an infinite viscoelastic medium containing multiple circular cavities. A two‐dimensional model for a suitably oriented plane section through a porous polymeric material is adopted. The solution of the problem is based on the use of the correspondence principle. The governing equation for this problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements at the boundaries of the holes. The method allows to accurately calculate the viscoelastic fields anywhere within the material. The effective viscoelastic properties of an equivalent homogeneous material can then be found directly from the corresponding constitutive equations for the average field values.

Transient Heat Conduction in a Porous Medium

This paper is concerned with modeling time‐dependent effects due to the diffusion processes in a medium containing multiple circular (in two dimensions) or spherical (in three dimensions) cavities (pores). The cavities may have different sizes provided that they do not overlap. The application of interest is for transient heat conduction in a porous material, and the aim is to devise a method that is capable of accurately computing the temperature and heat flux at any point and any time, without the need to consider a series of discrete time steps, as in conventional numerical solution procedures involving finite elements and finite differences. The approach is based on the use of the analytical solution to a corresponding problem of a single cavity in an infinite domain and superposition. Application of the analytical Laplace transform and its inversion results in a semi‐analytical solution for the case of multiple cavities in the form of a truncated Fourier series (in two dimensions) or a series of surf...