A calculation has been carried out of the changes in Gibbs free energy associated with the initiation of crack propagation in an isotropic, brittle solid of the Griffith type when the solid is subjected to a homogeneous, uniform but otherwise arbitrary triaxial system of stresses at infinity. The cracks have been taken to be flattened ellipsoidal cavities which are free from surface tractions. In accordance with the two-dimensional treatment of the problem and with experimental observations it is shown that minimisation of the Gibbs free energy, though necessary, is not a sufficient condition for brittle fracture initiation. There is in general more than enough energy available in the system to enable fracture to be initiated; the excess energy being particularly large in bodies fractured in compression.
In this paper we consider the thermodynamic aspects of the three dimensional treatment of Griffith's (1920Griffith's ( , 1924) ) model of the brittle fracture process which was given by Murrell and Digby (1970a, b).
In order to circumvent the problem of calculating the interatomic cohesive strength, and yet to enable an explicit value to be calculated for the tensile strength of a body containing a crack, Griffith (1920) derived a condition for crack propagation from the condition that there should be a reduction in the Gibbs free energy G of the system consisting of the body containing the crack together with the straining agent. Murrell (1964a) showed, for the case of plane strain or stress, that this condition, although a necessary one, is in general not identical with the condition that the local tensile stress close to the tip of a crack should attain a value equal to the interatomic cohesive strength T (the latter condition is of course a sufficient one).
In this paper we show that this result is true in general. In two previous papers (Murrell & Digby 1970a, b) we have derived conditions for the initiation of brittle fracture in triaxially stressed bodies containing flat ellipsoidal Griffith cracks on the basis of a critical local tensile stress criterion. (For brevity we shall denote these papers by I and I! and equations given in the papers will be distinguished by a bracketed I or II).
Three terms contribute to the change (AG) in Gibbs free energy of the system resulting from an isothermal extension ("propagation") of a crack.
These terms are: the elastic strain energy U stored in the body, the potential energy W of the external forces applied to the body, and the total surface energy 7S of the body (where ~ is the specific surface energy of the body). We consider the case in which the applied loads remain constant while the crack propagates, in which case there is an increase in elastic energy U, and in 7S, and the external forces do work on the body so that W decreases. Denote the surface of a crack by St, and the external surface of the body by $2. We consider two states of the body, the elastic quantities in the two states being distinguished by affixes I and II. In state I a given set of surface tractions is applied to Sz and another set of tractions is applied to $1, so that the body deforms as though no crack is present. In state II the same set of surface tractions is applied to S 2 but no tractions are applied to S 1.
By using the Rayleigh-Betti reciprocal theorem, Murrell (1964 (a)) derived an expression for the increase in elastic potential energy of a body produced by the presence of a traction-free crack.