Crack Paths 2012

ξ = x/Δa fixes the crack tip in the point (−1,0) while the outer boundary moves

to infinity as Δ a→ 0. W edefine the weight functions at infinity, ηj = Xj + ˜ηj, as

solution to the elasticity problem (in ξ-coordinates) in the plane with half infinite

crack ending in (−1,0), the interface is situated on the line x1 = 0, and η˜j is regular

at infinity, that is

(16)

˜ηj(ξ)∼MjkYk(ξ)+..., | ξ | → ∞ .

The 2 × 2 matrix M = ( M jk ) is symmetric resp. hermitian and negative definite. Near the crack tip, the solution u−Δa is approximated in terms of solutions of these

solutions in stretched coordinates, that is by the inner expansion of the form

uΔa

(17)

(

= uΔa(ξ) ∼ b1η1(ξ)+ b2η2(ξ) + ... ,

Δ a≪ 1.

Δa−1x)

The coefficients aj in (12) and bj in (17) depend on the distance Δa, of course.

T H EC A L C U L A TOIFOTNH E N E R RG EYL E A SR EA T E S

Exploiting the homogeneity relations of the power-law solutions Xj, Yj together

with the representation (16) of the weight functions ηj, the asymptotic representa

tion at infinity of the inner decomposition (17) can be rewritten in x coordinates. The asymptotic representation (13) of the weight functions ζj gives the asymptotics

of the inner decomposition (17) near the point (0,0). In a matching zone between

the crack tip and the outer boundary Γ both decompositions must coincide. Equal izing the coefficients in front of the power-law solutions Xj, Yj gives a system of

four equations for the coefficients aj and bj, j = 1,2. To be more specific, we put

(

(

)

K1K 2 )

a1a

b1b

K =(

a =

, b =

,

,

2 )

2

(

)

(

)

m 11 m 12

M 11 M 12

m =

,

M =

,

m 21 m 22

M 21 M 22

where Kj are the coefficients in eq. (7), mij are given in eq. (13), and Mij in eq. (16).

Case1: T w oreal eigenvalues 0 < λ1 ≤ λ2.

With

(

)

(Δa)λ1 0 0 (Δa)λ2 )

(Δa)λ1 0

M ( Δ a )=

· M ·(

0 (Δa)λ2

we obtain

(

)

1 0

(18)

a = M ( Δ a )(I·− m · M ( Δ) a ) − 1 · K , I =

.

0 1

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