Fracture mechanics: A unified view for Griffith-Irwin-Orowan cracks

Abstract

A unified picture of fracture mechanics is proposed in this paper for small scale yielding conditions (a small plastic zone at a crack tip) for two dimensional cracks. For a Griffith crack in a brittle material two equivalent fracture criteria that give the same result can be used. One is an energy argument that uses the true surface energy γ of a solid. The other is based on the intensity of the stress singularity at a crack tip. If the stress singularity is intense enough the stresses at the crack tip can break the solid apart and create surfaces of energy γ. The generalization of the Griffith crack theory to plastic small scale yielding until recently has used only the energy argument. In the generalization the true surface energy is replaced with an effective surface energy that presumably has its origin in plastic work. To have a unified picture of fracture mechanics the intensity of the stress singularity at the actual tip of the crack (rather than the attainment of some stress or strain value within the interior of the plastic zone) should also be used as a fracture criterion and shown to be equivalent to the energy criterion. This demonstration is given in this paper. It is proposed that the stress-strain curve to use in small scale fracture theory is one that is elastic to a yield stress σ o , is plastic up to a stress of the order of the theoretical strength σ T ( σ T ≈ μ 10 , where μ is the shear modulus), and then is elastic again to infinite stress and strain. Thus the crack tip is in elastic material. (A similar crack tip model has been used by R. Thomson.) It is shown how to find the true stress intensity factor K t using a proof of the J-integral theorem that is valid for a growing crack. Because of plastic blunting the value of K t is smaller than the stress intensity factor K of a brittle crack. ( K = σ a(πa) 1 2 for a crack of length 2 a in an infinite plate of infinite width and thickness subjected to an applied stress σ a .) The fracture equation is found by setting K t = K cb where K cb is the critical value of K for a crack in a perfectly brittle solid.