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E.Z. Lajtai

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Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 34th U.S. Symposium on Rock Mechanics (USRMS), June 28–30, 1993

Paper Number: ARMA-93-0705

Abstract

ABSTRACT INTRODUCTION How tensile fractures form and then propagate in response to the crack-parallel compression has been puzzling the rock mechanics community for decades. The direct application of engineering fracture mechanics to the problem appears to be invalid, because its zero-width mathematical crack model is unresponsive to the normal stress which is coaxial with the crack direction. A Finite-Width Elliptical Crack (FIWEC) model, which retains the crack width as an additional parameter, is proposed to make crack propagation sensitive to the compressive stress acting along the crack path. ASYMPTOTIC STRESS FIELD AROUND THE FIWEC CRACK The analysis starts with the stress functions for an elliptical hole [1]. The elliptical coordinates are replaced with their polar equivalents using the focal point as origin. The terms that include the radial distance from the focal point are expanded into power series. The asymptotic variation of the stresses at the crack tip are then obtained by neglecting the higher order terms. The elastic asymptotic stress field around the FIWEC crack tip consists of two singular terms including the 3/2 and the 1/2 powers of the distance from the focal point (r): [Equation available in full paper] (1) Here the external loading, P, acts at an angle â to the crack axis; e is the crack aspect ratio of minor axis (b) to major axis (a) and c is the focal length of the ellipse (Figure 1). The mathematical expressions for the coefficients of fij3, fij1, and fij0 are quite complex.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 23rd U.S Symposium on Rock Mechanics (USRMS), August 25–27, 1982

Paper Number: ARMA-82-496

Abstract

INTRODUCTION ABSTRACT The probability distribution of fracture data from a rock mechanics test contains information about the effectiveness of the testing procedure and/or about the nature of the operating failure mechanism. The tensile strength of Lac du Bonnet granite forms a bimodal probability distribution when determined through line loading as in the Brazilian test. The same data obtained from three point bends tests are unimodal. The bimodality can in part be removed by distributing the line load over a finite area. Moisture has the anticipated effect in lowering both the average compressive and the average tensile strength of Lac du Bonnet granite. The reduction in strength however is not uniform within the probability distribution; the change is the largest at low and the smallest at high strength. Conversely, long-term loading is more damaging to the high strength members of the distribution. The large scatter of failure times in static fatigue experiments can in part be removed or even utilized when the probability distribution of instantaneous strength is taken into account. Rock mechanics professionals are all aware of the probabilistic nature of geotechnical measurements and events. They routinely make use of statistical techniques in reducing data and often make statistical inferences about the means based on the assumption of a normal or "student t" distribution. In the process of analyzing fracture data relating to the Lac du Bonnet granite, the comparison of the distributions of fracture data has been yielding more useful information than statistical tests of population means. CRACK GROWTH AND FAILURE IN COMPRESSION THE BRAZILIAN TEST The measurement of tensile strength by diametral compression of rock core slices has always been somewhat contraversial. The reason for this is that fracture may not start at the centre, i.e. under the influence of the tensile stress concentration, but at the platen rock contact where high compressive stress concentrations may develop. In the testing of Lac du Bonnet granite, the use of a flat, ungrooved platten (concentrated line loading) results in a probability distribution that is clearly bimodal (Figure 1); the Weibull two-parameter cumulative distribution function (Weibull, 1951) gives a relatively poor fit at the low-strength tail. The source of the bimodal distribution is not the rock itself, because an alternate test, the three point bend test, gives a unimodal form (Figure 1). The use of the recommended finite area loading (Mellor and Hawks, 1971) seems to produce the desired result (Figure 2). The biomodality has not completely disappeared, but the unimodal Weibull fit is much better as quantified by the higher value of the coefficient of codetermination (r2). The growth of microcracks in a uniaxial compression test is usually demonstrated by tracking the course of the volumetric strain. Alternatively, the intensity of cracking at a certain value of compression may be characterized by the tensile strength of the test specimen with tension applied across diametral planes as in the Brazilian test. Although a large number of specimens are required for meaningful results, there are two advantages.