Crack tips with contained plasticity

The crack-driving force G may be estimated from energy considerations. Consider an arbitrarily shaped body containing a crack, with area A, loaded in tension by a force P applied in a direction perpendicular to the crack plane as illustrated in Fig. 2.6. For simplicity, the body is assumed to be pinned at the opposite end. Under load, the stresses in the body will be elastic, except in a small zone near the crack tip i.e., in the crack-tip plastic zone). If the zone of plastic deformation is small relative to the size of the crack and the dimensions of the body, a linear elastic analysis may be justihed as being a good approximation. The stressed body, then, may be characterized by an elastic strain energy function U that depends on the load P and the crack area A i.e., U = U(P, A)), and the elastic constants of the material. 

The inductance contained in the equivalent circuit shown in Figure 4.4.52 corresponds to an energy adsorption process, presumably due to the plastic deformation of the steel matrix in front of the crack tip. The size of the plastically deformed region, and hence the amount of energy deposited in the mafiix in front of the crack tip, is known to increase with increasing K, (Chung [1983]), and this accounts for the observed increase in L(Ki). 

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