Crack Paths 2009

Here, we use the sum convention and g - u I g,-u,-.

The last equation follows from

CLAPEYRON’tsheorem. Let uh be the solution to problem (2) in the solid Q;,, where

the crack has propagated along a small shoot T;, of (small) length h. For simplicity,

w e supposethat the shoot is a linear polygon, starting from the tip of the crack E0

at an angle 0 € (—1r,7r):

T;,(0) :I {x : 0 5 x1 5 hcos(0),x2 I 161 tan(0)}

The change in the potential energy produced by crack propagation can be calculated

using CLAPEYRON’tsheorem:

A U U h — U — l / ( u h — u ) - g d s l g /uh-o(")(u)ds. (7)

2

2 i

F

rifle

)

At the new crack tip xtip I h(cos(0),sin(0))T, the displacement field uh has an

expansion similar to (4):

1mg) I uhmip) t KllfitliplCg)

+ K§Ut2i’p1(y)+

--->

lyl —> 0,

(8)

where y denote local GARTESIAN(crack) coordinates at xtip directing along Thw).

are related to the homogeneouselasticity problem (5) with H O O K Ematrix

A(xtip) I + 6(xtip)H(0), where A, H are the H O O K Etensors A, B, rotated to

crack coordinates y (see e.g. [6] for more details). K ? are the SIFs at the new

tip. T o evaluate formula (7), w e replace uh and u by their asymptoticexpansions

at the new crack tip xtip and at 1'0 respectively. Using the jumprelations (6) and

expansions (4) and (8), short calculations lead to

2 AU ~ wild.- / 1/2U£">ds

E1 M0)

2

~-

~-

h h — r

I 0(9) 2 K§_,K§ (§’1(A;0)h.