The opening of a Griffith crack under internal pressure
1. The determination of the distribution of stress in the neighbourhood of a crack in an elastic body is of importance in the discussion of certain properties of the solid state. The theory of cracks in a two-dimensional elastic medium was first developed by Griffith1 who succeeded in solving the equations of elastic equilibrium in two dimensions for a space bounded by two concentric coaxial ellipses; by considering the inner ellipse to be of zero eccentricity and by assuming that the major axis of the outer ellipse was very large Griffith then derived the solution corresponding to a very thin crack in the interor of an infinite elastic solid. Because of the nature of the coordinate system employed by Griffith the expressions he derives for the components of stress in the vicinity of the crack do not lend themselves easily to computation. An alternative method of determining the distribution of stress in the neighbourhood of a Griffith crack was given recently by one of us2 making use of a complex stressfunction stated by Westergaard.3 This method suffers from the disadvantage that the Westergaard stress-function refers only "to the case in which the Griffith crack is opened under the action of a uniform internal pressure; the stress-function corresponding to a variable internal pressure does not appear to be known. In the present note we discuss the distribution of stress in the neighbourhood of a Griffith crack which is subject to an internal pressure, which may vary along the length of the crack, by considering the corresponding boundary value problem for a semi-infinite two-dimensional medium. The analysis is the exact analogue of that for the three-dimensional "circular" cracks developed in the previous paper2 except that now we employ a Fourier cosine transform method in place of the Hankel transform method used there. A method is given for determining the shape of the crack resulting from the application of a variable internal pressure to a very thin crevice in the interior of an elastic solid, and for determining the distribution of stress throughout the solid. The converse problem of determining the distribution of pressure necessary to open a crevice to a crack of prescribed shape is also considered. As an example of the use of the method the expressions for the components of stress, due to the opening of a crack under a uniform pressure, are derived and are found to be in agreement with those found in the earlier paper.2 2. We consider the distribution of stress in the interior of an infinite two-dimensional elastic medium when a very thin internal crack — c^y^c, * = 0 is opened under the action of a pressure which may be considered to vary in magnitude along the length of the crack. For simplicity we shall consider the symmetrical case in which the applied pressure is a function of |y| but the analysis may easily be extended to the