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

2367

18

Fig. 10: Crack closure forces for an interpenetrated crack in compression.

For an interpenetrated crack ( ∆ u z < 0) in tension ( F z ≥ 0), a di ff erent derivation is required. The crack is again closed in two steps. In the first step, corresponding to mode II, a crack closure force in the x -direction is applied to close the gap in the same direction, ∆ u II x = ∆ u x − ∆ u c x . The initial contact pressure force, P , decreases gradually until it vanishes at some point. Afterwards, the crack faces open with null contact force between the crack-tip nodes, C − and C + . The final value of the crack closure force in the x -direction, Q II x , is the same given for an open crack in tension by Eqs. (25). The mode II crack closure work is

k xx k zz − k 2 xz k xx k zz

1 2

1 2

Q II

II x =

( k xx ∆ u x + k xz ∆ u z ) ∆ u x

x ∆ u

(46)

∆ W II =

In the second step, corresponding to mode I, the remainders of the crack closure forces are applied. These are equal to the mode I crack closure forces applied for an open crack in tension, Eqs. (27). Also, the associated crack-tip relative displacements are the same as those given by Eqs. (28). As a consequence, the mode I contribution to the energy release rate has the same expression of the first of Eqs. (31). To sum up, the final expressions of the modal contributions to the energy release rate for an interpenetrated crack in tension are the following:

1 k zz

1 2 B ∆ a 1 2 B ∆ a

2 ,

( k xz ∆ u x + k zz ∆ u z )

G I = G II =

(47)

k xx k zz − k 2 xz k xx k zz

( k xx ∆ u x + k xz ∆ u z ) ∆ u x .

5. Conclusions

It is now well known that the standard VCCT may yield physically inconsistent, negative values of the modal contributions to the energy release rate. This shortcoming comes out, in particular, when analysing problems of bodies with highly asymmetric cracks [Valvo (2012)]. In the present work, the physically consistent revised VCCT proposed by Valvo (2015) has been extended by introducing contact constraints to prevent interpenetration of the crack faces that may be predicted by the linearly elastic solution. In particular, local contact has been considered between the crack-tip nodes in the finite element