PSI - Issue 28

Paolo S. Valvo et al. / Procedia Structural Integrity 28 (2020) 2350–2369 P.S. Valvo / Structural Integrity Procedia 00 (2020) 000–000

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7

(a)

(b)

Fig. 3: Virtual crack closure technique: (a) initial crack; (b) crack propagation.

3.2. Crack-tip flexibility coe ffi cients

The magnitude of the crack-tip forces may be evaluated by introducing suitable tie constraints into the model, e.g. springs with “infinite” (i.e. numerically large) sti ff ness between nodes C − and C + . As an alternative, the crack tip forces can be evaluated as follows. We define the crack-tip flexibility coe ffi cients as the relative displacements occurring between the crack tip nodes, C − and C + , when the body is subject to unit force loads at the same nodes [Jerram (1970)]. In particular, we denote with c xx and c zx the flexibility coe ffi cients corresponding to the relative displacement in the x - and z -directions, respectively, produced by unit force loads in the x -direction (Fig. 4a). Besides, we denote with c xz and c zz the flexibility coe ffi cients corresponding to the relative displacement in the x - and z directions, respectively, produced by unit force loads in the z -direction (Fig. 4b). By virtue of Betti’s theorem, it is c zx = c xz . The above definition can be used also for practical calculation of the flexibility coe ffi cients by carrying out two separate auxiliary analyses on the finite element mesh with the propagated crack [Valvo (2012)]. Because of linearity, the relative displacements produced by general (not unit) crack-tip forces are:

∆ u x = c xx F x + c xz F z , ∆ u z = c zx F x + c zz F z .

(7)

Inversion of Eqs. (7) furnishes:

F x = k xx ∆ u x + k xz ∆ u z , F z = k zx ∆ u x + k zz ∆ u z ,

(8)

where k xx , k xz = k zx , and k zz are sti ff ness coe ffi cients, whose expressions are given by Eqs. (A.1) in the Appendix.