Crack Paths 2012

12”? (y) = [gugp (x,y)zr;>(x)dr(x)

- [313;]: (X,y)Au3(X)dr(X)

+ 1;“ (y),

2

l m... 5Ati°(y)=nj (y)[e, + L$D;,f(x,y)Zt£(x)dF(x)— +

[35,13 (X,y)Au;3(X)dr(X)],

(6)

where the doubly periodic kernels Kdp =[U;p,Tlj“p,D;,§,S;f]

are explicitly defined in

Ref. [11], functions [17° and E; define the external load [11].

n 362

F 0 :

.. -

'\ ff fr-z\ 5;" m r‘.

I“

u’.

z" e o e

I’.

m I...

I’

O

34"

m"...

F1

b

b

N...‘

Figure 1. A doubly periodic set of curved cracks

Average strains are as follows [11]

_ _ “(2) (1)

(1) (2) _ (2) (1)

1101“ ‘I

< u i ’ 1 > _ A u i wxz /(0)X1 (0X2

60x1 60x2 ) + u i l j( H i m ) ,

(7)

_ Am (1) hom : - Au, 60%/((0,160,2 —(0x1 (0x2 )+u,,j (Hkm), <1) <2) (2) <1)

horn where u,’j (o'km) are gradients of displacements in the homogeneousmediumcaused

by the far-field load O'km; 00(1) =[will),o)g)]T

and 00(2) =[o)§l2),o)i22)]T

are the period

vectors (see Fig. 1). Cyclic constants are equal to [11]

Atty‘) = ujjjm (5km ) (of) + Any‘), A6,“) = 0,

A

*

t

(8)

Auirl) : ‘Lg [UH (X)Zi;) (x) —7], (x) Au? (xfldlix).

F A T I G UCER A CGKR O W STIHM U L A T I O N

Boundary integral equations (6) can be solved numerically using the direct boundary

element method. Since the singularity of corresponding kernels of the periodic and

doubly periodic BIEs is the same as that of the nonperiodic BIEs, for the numerical

evaluation of the weakly, strongly and hypersingular integrals one can utilize

1107