The principal objective of this work is to make a systematic study of a generalization of the Griffith theory in three dimensional fracture mechanics from mathematical viewpoint. We consider the situation where an elastic body containing a crack, in its non-deformed state, occupies a domain in R of the form Ω = G — Σ. Here we consider the crack as a discontinuity in the material in the form of a surface Σ9 and we assume that G is a domain in R 3 with local Lipschitz property and Σ is a two dimensional manifold with boundary contained in G. This body is in a state of equilibrium under the influence of a load 3? consisting of a body force in Ω and a surface force on the boundary dG of G. By I(j£? Σ) we denote the potential energy of the elastic body containing the crack Σ under the load 3f. The generalization of the Griffith theory can be expressed in terms of the concept of energy release rate as follows (cf. Palamiswamy and Knauss [19]). The crack extension process is considered to occur in a quasistatic manner, so that when we refer to time we use it as a parameter which indicates the sequence of events. We denote by Σ(t) the surface obtained from Σ by extending it in the length of time ί(^0). Of course Σ(t)aΣ(t) if t<t, and Σ = Σ(0) = Γ\t^0Σ(i). During crack extension let the load & be independent of t. If the crack extends from Σ to Z(ί), the potential energy released by the increment Σ(f) — Σ is given by