Calculation of the Volume of the Crack

A further interesting application of the general relation (3.5.18) is the determination of the volume of the opened crack. To this end, we must consider a loading which is work-conjugate to volume, i. e., such that the work is expressible as the product of the conjugate generalized force with the variation of crack volume (IV. Such a loading is a uniform internal pressure p over the crack faces; then p and V are conjugate variables and can be used directly in the energetic expressions. It is now necessary to adjust (3.5.18), because the dimensions of the variables are different. This is fairly easy and is left to the reader as an exercise. Here we execute a simple trick to be able to use (3.5.18) as it is; we define a generalized. force Py from p, and its associated displacement uv ос V so that PyUy pV and the dimensions be those of force and length:

*» у

Pv — pbD and uy — —- (3.5.28)


(Note that the crack length a must not appear in the definition of Pv and uy.) Working now as in our calculation of the CMOD, wc have n = 2, and set P P, P2 rz Py, u t= u. m = uy, Си (a) = C(a), and £42(0) == Су (a). So we may write the displacements as

и — С (a) P + Су (a) Py (3.5.29)

uv — Су (a) P + Су у (a) Py (3.5.30)

We also set кі (a) = k(a) for the shape factor corresponding to force P, and /^(a) s= ky(a) for the one corresponding to the internal pressure ■— in which the stress intensity factor must be written in the form К і = (Pv/bfD)ky (a). Thus, the cross-compliance for uy, Су (a), follows from (3.5.18) with Су о = C120 = 0 (because when the crack length is zero, the crack volume is also zero):

2 [at ®

Cy(a) – — j k{a)ky{a)da The crack volume follows from (3.5.30) and (3.5.28):



V = bDuy = bDCyP = – vy{a)




vy(a)r-~-2 J k(a’)ky(a’)da’


If we write V in terms of cr/v instead of P, we get

V — bD2vy(a) , vy(a) =2 f k(a,)ky(a’)da/ E‘ * Jo


where k(a’) is the shape factor defined in (2.3.11) and ky(a’) is defined so that the stress intensity factor created by a uniform pressure inside the crack is written as Iij = pV~Dky(a).

Example 3.5.3 Consider again the plate of Fig. 3.5.1. When a uniform pressure p is applied to the crack faces, the superposition sketched in Fig. 3.5.3 shows that the stress intensity factor is identical to that corresponding to a remote uniaxial stress cr = p. The corresponding stress intensity factor is К і — .25pJva, and so the shape function ky (a) is

kv(a) = 1.1215 фїа (3.5.35)

Substituting this and (3.5.8) in (3.5.34), we get the crack volume:

V – ^bD22 [ 1.25 87га’ da’ = bD2a2 = —1.25877ba2 (3.5.36)

E Jo Ь b

where we wrote Da — a. Q

Figure 3.5.3 Stress intensity factors for internal pressure p and remote uniaxial stress p are identical.

Calculation of the Volume of the Crack

Figure 3.5.4 Virtual loading used in the computation of the crack opening profile.

For an internal crack of length 2a, the results are similar, except that the shape factors at both tips of the crack appear explicitly, as in previous sections. Then the expression for vz(a) in (3.5.34) must be replaced by

«l/(a)=2 f [k+(a’)ky(a’) + k+(a’)ky(a’)l da’ (3.5.37)