## Integral

13.1 Genesis

7 Eshelby (Eshelby 1974) has defined a number of contour integrals that are path independent by virtue of the theorem of energy conservation. The two-dimensional form of one of these integrals can be written as:

Jo where w is the strain energy density; r is a closed contour followed counter-clockwise, as shown in Fig. 13.1; t is the traction vector on a plane defined by the outward drawn normal n and t = an; u the displacement vector, and dr is the element of the arc along the path r.

Figure 13.1: j Integral Definition Around a Crack

8 Whereas Eshelby had defined a number of similar path independent contour integrals, he had not assigned them with a particular physical meaning.

### 13.2 Path Independence

9 Before we establish the connection between Eshelby's expression for J, and the energy release rate J, we need to show that the former is indeed equal to zero for a closed path.

and assuming r to be defined counterclockwise, then dx = -nydr, and dy njalj where nx, ny and nj are direction cosines. Substituting nxdr and ti =

Invoking Green's theorem vinidr = / vi idU, we obtain

duj dx dxj V tj dx dw d dxdy

11 Substituting the strain energy density, Eq. 13.2, the first term in the square bracket becomes dw _ dn^d£ij_ _ d£ij_ dx deij dx 13 dx

Substituting dw dx ~

On the other hand, we have

14 Hence, dw d ( dul dx J dxj V dx which is identical to the second term of Eq. 13.6.

15 Thus the integrand of Eq. 13.3 vanishes and J = 0 for any closed contour.

16 Having shown that indeed J = 0, we will now exploit this to proove that around a crack, J is non-zero and is independent of the path.

17 With reference to Fig. 13.2 if we consider the closed path r = r + r2 + r3 + r4 in which r and r3 are arbitrarily chosen contours. Obviously J = 0 over r in order to satisfy compatibility conditions, provided that the stresses and displacement gradients are continuous. Along paths r2 and r4, the traction vector ti = 0 and also dy = 0. Consequently, the contributions to J from r and r4 vanish. Taking into account the difference sense of integration along paths r and r3 we arrive at the conclusion that the values of J integrated over paths r and r3 are identical. Because these two paths were arbitrarily chosen, the path independence of J is assured.

### 13.3 Nonlinear Elastic Energy Release Rate

18 Let us now establish the connection between the two previous interpretations of J, the one mathematically defined by Eshelby, and the one associated with the energy release rate. We shall proof that when J is applied along a contour around a crack tip, it represents the change in potential energy for a virtual crack extension da. Two slightly different derivations are presented.

### 13.3.1 Virtual Crack Extension

19 Considering a two-dimensional crack surrounded by a line r which encompasses an area Under quasi-static conditions, and in the absence of body forces, the potential energy is given y

Figure 13.2: Closed Contour for Proof of J Path Independence

20 For a virtual crack extension, the change in potential energy is

21 We have dcomposed the contour path into two parts, the first one with prescribed displacement (ru) and a second one with prescribed traction (rt). Since ru is zero along the path, we maintain a closed contour integral along rt.

22 Furthermore, the second term inside the square bracket will be zero along rt because the traction is constant during crack growth.

23 For a crack extension, the coordinate axis also moves, Fig. 13.3.