Crack Paths 2012

• Determination of the equilibrium shape of a crack front penetrating into a harder

obstacle of infinite length in the direction of propagation, up to second order in

the gap of toughness between the matrix and the obstacle.

RICE’SF O R M U LFAOERA NA R B I T R A PR YL A N ACRR A C K

Consider an isotropic elastic body Ω containing a planar crack with arbitrarily shaped

contour. Assume the body and the loading to be symmetric about the crack plane. The

crack is then in a situation of pure mode I all along its front; let s and

0 ( ) K s denote a

curvilinear abscissa along this front and the local SIF, respectively.

N o wdisplace the crack front, within the crack plane, by some infinitesimal distance

()asδ perpendicularly to itself, while keeping the loading unchanged. The resulting

infinitesimal variation

1 ( ) K s δ of the local SIF is given by Rice’s first formula [4]:

[ ]

[ ] 0 ( , ) ( ) ( ) ( ) C F Z s s K s a s a s d δ δ − ∫ (1) 1

( ) ( ) K s K s

δ

δ

=

+

PV

1

1

1 ( ) ( ) , a s a s s δ = δ ∀

where the integral over the crack front (CF ) is to be understood as a Cauchy principal

[ ] 1 1 ( ), ) ( ( ) a s a s s K s δ δ δ = ∀ denotes the value of

value (PV ). In this expression,

1 ( ) K s δ for a

uniform advance of the front equal to

1 ( ) a s δ , and

1 ( , ) Z s s the F K of the cracked

geometry considered. This quantity, which is tied to Bueckner's mode I crack-face

weight function, has no dependence upon the loading other than on which portions of

Ω and ∂Ω have forces versus displacements imposed, and verifies the following

properties:

1

s s

1 2 2 1 2 ( , ) ( , ) ; ( ),2 ( ) Z s s Z s s Z s s s s π = − ∼ for 2 1 1 2

1 2 − → (2) 0.

and

2s, the infinitesimal variation

1 2 ( , ) Z s s δ of

In addition, if ()asδ vanishes at 1s

the F K at these points is given by Rice’s second formula [4], which involves two

principal values, at the points

1s and

2 s :

1 2 2 ( , ) ( , ) ( , ) ( ) . C F Z s s PV Z s s Z s s a s d s δ δ = ∫ (3) 1

A P P L I C A T I O NFRICE’SF O R M U LTAOEA SEMI-INFINITCE R A C K

Generalities

W enow consider (Figure 1) a semi-infinite tensile crack located in some infinite body

subjected to prescribed forces only. The crack front is assumed to be slightly curved, its

equation in the plane Oxz being of the form

752