Consider the symmetry requirements on the deformed crack surface. Symmetry requires that x = 0 at в = n/2, and у = 0 at в = 0. The constraint imposed by continuity is that the elastic and have the same value at the interface (a = af, в = вУ) between two domains as a consequence, the points on the elastic segment of the deformed crack are found to have coordinates
For an elastic solid, E = En and hence
The crack opening parameter is therefore related to Young’s modulus E and far-fleld stress S. In fact 2S
tan af =———- .
1 E – S
However, this equation is not available if the behavior is elastic plastic.
To satisfy the continuity of stress across the elastic-plastic interface, we assume the same value for crack opening parameter on both elastic and plastic segments. Hence, the deformed position of the points on the plastic segment is
— (sinha/ + 2 cosh a/) sin ft.
The second of the above equation yield 2S
Therefore, the value
En + S
En — 3S of the crack opening parameter depends on the modulus En of the material at the tip and the far-fleld stress S.
For further generalization, divide the non-linear stress-strain curve into N segments. At the same time, divide the crack surface also in N domains such that points in k obey the stress-strain rule of the segment k, and в = вk is the interface between domains k — 1 and k.
Therefore, the deformed position of this interface is
= cS(sinh af + 2 cosh a/) I
i = 1 k
For the purpose of book-keeping, вИ and во in this equation corresponds, respectively, to the tip (в = 0) and the crown (в = n/2) and Ei is the modulus of the material in the ith segment. It is of course possible to convert the summed terms into integral forms by letting the number of segments become infinitely large such that, in the limit, each domain shrinks to a point and
where Et is the tangent modulus. The right-hand side can be integrated provided a relationship between the tangent modulus and в can be established. But such a relation does exist. Recall the first of the two expressions for stress field in (5). Since the crack surface is free of traction, stress normal to it must vanish. Thus, in plane stress, ayy + axx = at is the only non-zero stress, and Equation (5)3 can be rearranged in the form
This equation relates в on the crack surface to the tangential stress at which in turn is related to the tangential modulus via the stress strain law. In that case, the stress-strain law need not be linearized. However, it must be emphasized that the purpose of linearization was to make the stress function biharmonic and allow the use of analytic function f and g in the analysis. It is of course possible to assume a form for stress and strain fields on some other ground in such a manner that the stress field is an approximate solution of the equations of equilibrium. This is the course adopted by Hutchinson (1968) and Rice and Rosenberg (1968). In our case, we have opted in favor of approximating the stress-strain law but, at the same time, choosing a stress field that satisfies the equations of equilibrium.