Abstract

In this paper a numerical procedure is presented for the calculation of the openingof an internally pressurized equilibrium line crack in an infinite elastic solid. The method is based on England and Green'sintegral equations and allows a solution to be obtained for any pressure distribution along the crack length. As an example, theproblem of Christianovich and Zheltov (constant pressure along a portion of the crack length) is solved numerically and shown to be in excellent agreement with analytical results. Also, the numerical solutions toa number of other pressure functions are presented here. The paper is prepared with the idea of using its findings in the design of a hydraulic fracture.

Introduction

In order to design a hydraulic fracture one needs to know the quantitative expression yielding the opening of an internally pressurized crack in an infinite medium. The derivation of such an expression is hindered by the mathematical problemsencountered during a formal solution. Thus one makes the assumption that the medium is isotropic, homogeneous and linearly elastic) that the crack is two dimensional, i.e., no variations in its geometry in the third dimension, and that the plane of the crack is under the influence of a constant hydrostatic stress (-S) at infinity (compression negative). Under these assumptions the fracture propagates along a straight line (line crack) and the problem is reduced to the calculation of the opening of an internally pressurized two-dimensional line crack in an infinite elastic solid.

In order to cause any fracture opening, the magnitude of the-internal pressure in the crack must be sufficientlylarge to:

  • apply enough force on the crack walls to neutralize the opposite force due to the far-field stress (-S) and

  • provide the required energy to account for the change in the strain energy of the deformed body.

Crack extension starts when the tensile stresses induced at thecrack tip equal or exceed a critical value and cease when the potential energy or the system attains a minimum (the condition for stable equilibrium). The solution of the "moving" crack problem is difficult to obtain and is beyond the scope of the present work. Thus it is assumed that the internal pressure is not large enough to cause a fracture extension. This means that even if this pressure has been sufficient at some point in the past to cause an extension, enough time has passed for the crack to reach a state of stable equilibrium.

The problem of a two-dimensional line crack in an infinite elastic solid under the influence of a constant internal pressure has been solved by Sneddon.1 He found that under these conditions the crack deforms into an ellipse and derived the mathematical expressions for the opening of the fracture. For the case of variable pressure along the crack length, Sneddon and Elliot 2 have derived equations yielding the crack opening. The use of these equations requires the Maclaurin series expansion of the function describing the pressure variations along the crack length.

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