Crack Paths 2012

First page Table of contents Previous page 567 Next page Last page

In curvilinearcoordinatesH O O K E ’lsaw reads:

aijw; y) I aim; whs’twlalgjwflt

I alt-radi;

w)Tsi(i/)ltTgj(?/)l

Ialtgmdfi;y)lgm(y)lplgn(y)lqlgi(y)lalgj(y)li

., (i/fimdflw)

I: lily-m

Rewriting the divergence of the stress tensor in curvilinear coordinates, the equilib

rium equations (1) near the crack front read

I 07 a: I € T:

I 0: a: I € E i :

with the components E'JHJ- :I 533” + T2,?” + Fg-qdlq for i I 1, 2, 3, see [5] for more

details on curvilinear coordinates.

A s y m p t o t iecx p a n s i oant the crackfront.

From nowadays classical results it is known, that also in three dimensions the dis

placementfield has an asymptoticexpansionof square-root type at the crack front: H N r1/2(go;s), where (r cos(go),rsin(

plane at arc length 5 (see e.g. [4, 6, 7] and the literature cited there). If we rewrite

the elasticity equations (3) in operator notation,

3 %W W W )I 0. w I 9(a) Q T. JI/(y. Vilma) I 0. w I 6(a) Q 5i. (4)

w e can expandthe elasticity operator into a series:

{eava/aavaI}Z{rk_2£k(r.8a.r8...8.8.),t’“—W’r(ta,o,,i~a,$,a,)}.

k I O

The first operator LO(
volve derivatives of the arc length and is a homogeneoussecond-order operator with

constant coefiicients, namely the elastic moduli transformed to curvilinear coordi

nates at arc length 3 at the crack front. Exploiting (4), longer calculations show,

that the asymptoticdecompositionof the displacementfield reads

3

to) I t1” 2 K-.i~~<1>2,.+~~

t3” Z(K.-,i<1>;,.(1n.a)

+ 8sK-71(s)i1(ln(r),

90) + Kj,3(5)@3,3(§0)> + O (702%

where Kj’1(s) are the classical stress intensity factors (SIFs) and Kj,3(s) higher

order coefiicients. The functions r1/2;71 are solutions of the pure two-dimensional

homogeneousproblem

£0(go, d w r d ns)rA(g0) I r A L O Q o fi W As)<,l>(go) I 0,

go € (—7T,7T),

(5)

549

~~Made with FlippingBook Ebook Creator~~