Calculation of the Crack Mouth Opening Displacement

As an important application of the foregoing general result, let us calculate the expression for the crack mouth opening displacement (CMOD), which we denote as wm (Fig – 3.5.2). Aside from the actual load P, we also consider a virtual loading Pm consisting in a pair of forces at the crack mouth working

Подпись: „ D


Figure 3.5.2 Crack mouth opening wm, applied load P, and crack mouth load PM.

through u>m. In the previous expressions we now have n — 2, and we set P] г= P, Pi = Pm, U = u, Ui = wm, Сц (a) = C(a), and Cn{a) = Cm {a) so that we write the displacements as

и — C(a) Р + См{а) Pm (3.5.21)

wm = См(а) Р + Смм(а) Рм (3.5.22)

We also set k (a) = k(a) and ^(o) = км(а) for the shape factors corresponding to forces P and Pm – Noting that См о — Ct2o = 0 (because when the crack length is zero, the crack opening is also zero), the cross-compliance for the CMOD, Cm {a-), follows from (3.5.18):

2 ^ ~

Cm (a) = — j k(a)kM(a)dn (3.5.23)

Thus, according to (3.5.22), the crack mouth opening displacement when the structure is loaded by P alone is

p fa ~ – і і

wm = ЬР’^м(а) ’ ^r(«)=2 J к{а’)км(а)(1а’ (3.5.24)

Again, this can be expressed in terms of cr^ instead of P; the result is


wm – ~-Dvm(&) » =2 / к(аг)км(а,)(1а/ (3.5.25)

& Jo

where we notice again that the expression is identical to the previous one except that the hat is removed for k(a) (but not for км(а)).

Подпись: kM (a) ■ Подпись: 2.594 ■/тю Подпись: (3.5.26)

Example 3.5.2 Consider again the plate of Fig. 3.5.1. When a pair of loads Pm is applied to the crack mouth (as shown in Fig. 3.5.2), the corresponding stress intensity factor is expressed as Ki — 2.594Pm/bi/тга (Ouchterlony 1975; alsoTada, Paris and Irwin 1985). This can be rewritten as Ki — {Pm/bfD)2.594/^/Wa. Therefore, the shape function км(а) is

Calculation of the Crack Mouth Opening Displacement Подпись: (3.5.27)

Substituting this and (3.5.8) into (3.5.25) we get the CMOD:

which is the expression found, for example, in Tada, Paris and Irwin (1985).