Crack Paths 2009

Ξ+0 and Ξ−0 we denote the upper and lower sides of the crack, considered to be tension

free. Thecoordinate system is chosen in such a way, that the crack tip x0 is at the ori

gin. Thestrain tensor (in Cartesiancoordinates) εij(u;x) = 12 (∂iuj(x) + ∂jui(x)),

i,j = 1,2, is related to the stress tensor by Hooke’slaw:

2 ∑

aklij(x)εkl(u;x),

i,j = 1,2.

σij(u;x) =

k,l=1

a(x) = akl

ij (x) is a symmetric rank-4 tensor, i.e. in an anisotropic material there are

6 different elastic moduli. For the strain tensor, we also use the vector notation

)

ε11(u;x),ε

22(u;x), √

⊤

ε(u;x) := (

2ε12(u;x)

.

Then, for the stress tensor (in vector notation) the relation holds

( σ11(u;x),σ 2 2 ( u ; x ) , √

⊤

σ(u;x) = A(x) · ε(u;x) =

2σ12(u;x) )

with the matrix function

a11(x) a21(x) √





2a31(x)

a21(x) a22(x) √





(3)

A(x) =

√2a31(x) √

2a32(x)

2a32(x) 2a33(x)

symmetric and positive definite in every point x ∈ Ω, containing the elastic moduli.

A(x) is called the mathematical H o o k etensor or H o o k ematrix (Voigtnotation). The factor √2 ensures, that strains (and stresses) have the same Euclideanorm,

in vector and in tensor notation.

A model for functionally graded materials.

Let us assume, that the specimen under consideration is composed of a F G M .With

this notion we relate the following: The H o o k ematrix depends (continuously)

on the space coordinates. Our main interest is to detect the influence of such an

inhomogeneity on the fracture process. But if the material properties depend on six

(different!) functions, there is no real chance to see which of them cause an effect on

the crack path. Therefore, we simplify the problem and introduce just one (scalar)

function in the material properties:

A(x) := A + δ(x)B

where δ is a smooth and bounded function. A, B ∈ R3×3 are (constant) symmetric

matrices of type (3). W ealways suppose that the matrix function A(x) is symmetric

and positive definite in every point x ∈ R2. The function δ can be understood as a

measure of the level of the gradation.

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