Crack Paths 2009

Fig. 5. Crack with wedge— shaped asperities and microstresses acting on asperities facet

1"” and stress acting on the nominal crack plane segment F0 .

Denote the local stresses acting on wedge flanks 1", by 0'5‘ and T: , so that the local

friction condition on 1", is

1: Ira: zany/‘a:

(6)

where 77” is the local friction coefflcient and y” is the friction angle. The stress 0'; and

If acting on the nominal crack segment are expressed from the equilibrium equations

for a single asperity, thus

6.5 I 65‘ S—I(C<>S7Z — rfsinyz),

1.? I I5‘ s—1(sin W + 17” 605W)

(7)

S2

82

where s1 denotes the wedgeflank length and s2 is the asperity length within the plane

F0 . The conditions on the nominal plane F0 can be expressed as follows

M3 I iu, tany/z,

If I 0': tan(7/Z + 7”), 3 I i n

(8)

These conditions can be expressed in terms of the asymptotic flelds with neglect of T

stress. Denoting by

the effective SIFs for the dilatation crack model and by

K1”, KI‘; the SIFs resulting from the stress on the wedgeasperity flanks 1"” , we obtain

K? I iKiiwnyi

iKri I Kr tanw w”),

(9)

Denote by KI ,KII the SIFs for a smooth and plane crack. Applying the superposition

principle for the external loading 6,2‘ and the wedge asperity loading 65,15, the

effective SIFs are

, 7 K10’: KI + K,‘ : fi fls1n(7~ + 7*‘q) + ‘KH‘cosQ/Z +7”)), sin z .

(10)

d

;,

cosyz

.

z

,

Z

,

‘Kll‘ 2 K11 + K n I Sin7,,(K1$1n(7+ 7 ) +‘KH‘COSU/ + 7 These relations are valid whenthe contact occurs on 1"”, thus

—KI tan(;/Z +y”)£‘KH‘ZKcIot(;/Z).

(11)

These inequalities specify the sliding domain B, Fig. 6. W h e n‘KH‘

crack opening occurs, and the

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