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

2362

13

By introducing Eqs. (24) into (23), and recalling Eqs. (A.2), the mode II crack closure force in the x -direction and relative displacement in the z -direction are obtained:

k xx k zz − k 2 xz k zz

1 c xx c zx c xx

Q II

∆ u x =

∆ u x ,

x =

(25)

k xz k zz

∆ u II

∆ u x = −

∆ u x .

z =

In the second crack closure step, corresponding to mode I (Fig. 7b), the remainders of the crack closure forces, Eqs. (16), are applied:

Q I Q I

II x , II z .

x = Q x − Q z = Q z − Q

(26)

By substituting Eqs. (24) and (25) into (26), and recalling Eqs. (8) and (16), we obtain:

k xz k zz

Q I

( k xz ∆ u x + k zz ∆ u z ) ,

x =

(27)

Q I z = k xz ∆ u x + k zz ∆ u z .

By substituting Eqs. (27) into (22), and simplifying, the mode I crack-tip relative displacements are deduced:

∆ u I

x = 0 ,

(28)

k xz k zz

∆ u I

∆ u x + ∆ u z .

z =

The amounts of work done by the mode I and II systems of crack closure forces respectively are

1 2

Q I

I z ,

z ∆ u

∆ W I =

(29)

1 2

Q II

II x .

∆ W II =

x ∆ u

Correspondingly, the modal contributions to the energy release rate are

∆ W I B ∆ a

G I = G II =

,

(30)

∆ W II B ∆ a

.