Crack Paths 2009

f:Hv,

(4)

where H is the generalized stress intensity factor (it can generally be complex), v,- is a

complex eigenvector corresponding to the eigenvalue 5, where 1 — 6:1—(6’+i6”)

represents the exponent of the stress singularity at the notch tip. Eigenvector v,- and

eigenvalue 5 are the solution of the eigenvalue problem leading from the prescribed

notch boundary and compatibility conditions. The expression <55 > is a diagonal matrix

for which : diagpfs,

Then the relation between the polar coordinates (136‘) and the coordinates (R,-,‘~P,) in

the plane of Re(z,-), Im(z,-):

Rf : (cos6 + p,’ sin 6)2 + (,ul'sin

6)2

O

for a I 0

\P arccot((cos6+,u,'

sin a) / p,” sin a)

for a e (0, n)

(5)

'

arccot((cos6+,u,.'

sin a) / p,” sin a) - n for 9 E (W0)

_”

for 6 : —a

Fromthe relations (2) and (4) we get the potentials:

(Mr, 6) : 2 HRe{—(/1i’ +i,uln)(vil +iv1”)r5'R16’e’m"eiwnhmmn’el+5

W1) +

(6)

_ (#2!+ iiuzn x v ;+ ivzn)rs'Rj'ees’qlezi(5"]n r+5”1nR2+6'\I/2)}

%(r, 6) : 2 HRe{(v1’ +iv1")r5'RfS 'ei‘wleimmylml“w” +

(7)

, , , _._(VZ _HVZ ) r a R j e6Y2e1(5]nr+6]nR2+6\IJ2)} _ , . , i . . .

Finally, the stress components in polar coordinates are obtained via the relations (3).

Further we assume the existence of more than one singular term. In the following the

subscript k denotes the pertinence to the specific stress singularity exponent l-é‘k:

a r r: Z ( n l t r l

+ n 2 t r 2 ())6’k6 r 6: Z ( m 1 t r 1 (m62kt)r+2 )()’6 k6 6 6I Z ( m l t e l + m z t e( z6 k) )

In most practical cases, as well as in the cases studied in the paper, there are two

singular terms corresponding to two stress singularity exponents l-61 and l-62. Note

that for the final determination of the stress field in the bi-material notch vicinity the

generalized stress intensity factors have to be estimated by means of numerical

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