Crack Paths 2012

of the arc length 5. To calculate the change of potential energy for small th(s) << 1,

we use the m e t h o dof matchedasymptoticexpansions[10]. T h eprinciple idea is the

following. For a small elongation th(s), the displacement field ut at time 15 will not

differ too muchfrom the displacement field u I no at time t I O in some distance

to the crack front T‘(t), hence w e approximateut by an outer expansion

“(%) ~ H01‘) + €1(a1;$) + €2(a2;w) + €3(a3;w) + - -

where the functions aj I aj(t, s) have to be determined. Near the crack front, the

influence of the propagated crack on the solution ut will be more significant and

to detect this at arc length 5, we change local coordinates to 5 I fly’. Sending

t —> O the outer boundary moves to infinity and very close to the crack front we

approximatethe displacementfield by an inner expansion

ut(t_1y', s) I ut(§, s) N t1/2w1(§, s) + . ...

For any arc length 5 the functions wl(-, s) are solutions of the homogeneouselasticity

problem in the plane with a semi-infinite kinked crack (compare to (4)):

zowg,S)w1(g, s) : 0, 5e 9;, Jl/O(V£, s)w1(g, s) : 0, 5e 091;,

where Q20 :I R2 \ (E00 U Th(i9(s))) is an unbounded domain with a semi-infinite

crack E00 :I {5 :

51 g 0,52 I O} and crack shoot Th(i9(s)) :I {5 : O < 51 g

[1(3) COS(’Z9(S)), 52 I 51 tan (19(5))

This is a pure two-dimensional problem depend

ing on s. Scaling g :I [7715, we arrive at a problem in a domainwith a kink of fixed

length one. Here, we can use the results in [1, 2]: There exist solutions with singular

asymptoticdecompositionat infinity:

3

W 8I UJ-O,1(£)

+ ZMi,j(Q9(5)§h)l/i?1(£)

+ - - -7

lél —> +<>O-

(8)

1:1

and shown in [2, 10] there holds

Mil-w; h) : #11” Z (M) We - flown-Ms) ds) : 111/Wow

:I:

Inner and outer expansionapproximatethe samesolution ut only in different regions

and must coincide for small |y’| and large

Both approximations have asymp

totic decompositionsfor T‘ —> O and —> +00, respectively, in terms of power-law

solutions. Rewriting the decomposition (8) in local coordinates, we find

ltl/2w1(€;8)

I EKj,1(S)UJ-O,1(y')

+15 (71(8) 2 Kj,1(5)Mi '(Q9(5))Vi(,)1(y/)>

+ - - -

j I l

i,jI1

551