PSI - Issue 28

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

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(a)

(b)

Fig. 8: Mode II closure of an open crack in compression: (a) first sub-step; (b) second sub-step.

For the second sub-step, we have

= k xx ∆ u x − ∆ u = − P .

II , a x ,

II , b x II , b x

II , b z

Q II , b x Q II , b z

= k xx ∆ u

+ k xz ∆ u

(35)

II , b z

= k zx ∆ u

+ k zz ∆ u

Hence, by recalling previous expressions and simplifying,

k xx k xz

Q II , b x

( k xz ∆ u x + k zz ∆ u z ) ,

=

k zz k xz

(36)

∆ u II , b x

= ∆ u x +

∆ u z ,

P

= − k zx ∆ u x − k zz ∆ u z = − F z .

The work done by Q II , a

II , b x in the above described sub-steps is to be ascribed entirely to mode II (the contact

x and Q

force does not produce any work as ∆ u II , b z

= 0). By using Eqs. (34) and (36), and simplifying, we obtain

II , b x ∆ u

II , b x =

1 2

1 2 ( k xx ∆ u x + k xz ∆ u z ) ∆ u x .

Q II , a x

II , a x

+ Q

+ ∆ u

(37)

∆ W II =

At the end of the second sub-step, the crack turns out to be completely closed. Thus, the second crack closure step, corresponding to mode I, is not necessary for an open crack in compression. Hence, a null mode I crack closure work is considered for this case, ∆ W I = 0.