Crack Paths 2012

However, the numerical simulation is stable if one appends crack tip elements of

approximately the same length aΔ. This implies in different number N Δ of cycles per

simulation step. The latter is determined from Eq. (13) as

()n a N A K Δ Δ ≈ .

(14)

If aΔ is small enough, the numerical results are convergent.

E F F E C T I VPER O P E R T I EOSFA C R A C K EA NDI S O T R O PMI CA T E R I A L

To determine the effective properties of a cracked anisotropic material one has to solve

i j σ , the best of

the problem for 5 linearly independent variants of an average stress

which are

1)

Pa, 0ijσ= ()()()11ij≠∧≠;

1 1 1 σ =

2)

2 2 1 σ = Pa, 0ijσ= ()()()22ij≠∧≠;

3)

3 2 1 σ = Pa, 0ijσ= ()()()32ij≠∧≠;

4)

Pa, 0ijσ= ()()()31ij≠∧≠;

3 1 1 σ =

5)

Pa, 0ijσ= ()()()12ij≠∧≠

.

1 2 1 σ =

Thenamatrix

(5)

1

−

1,

(1)

1,1

ª

º

...

u

u

«

»

«

»

...

2,2

2,2

(5)

«

»

(1) 3,2

3,2

«

»

c

...

(15)

u

u

=

«

»

«

»

...

3,1

3,1

(5)

(5)

«

»

(1)

(1)

...

u

u

1,2 u u

1,2

2,1

2,1

« ¬

+

+

» ¼

is the matrix of effective properties of M E Ecomposite material, which relates an

average stress vector

T 1 1 2 2 3 2 3 1 1 2 , , , , σ σ σ σ σ = ª º ¬ ¼ with an average strain

T 1 1 2 32 2 312 1 , , 2 , ,2 ε ε ε ε ε º

¼ : = c . The superscript ahead of

vector

= ª ¬

,iju in Eq. (15) denotes the variant of the applied average load. The values of ( ) , k i j u

used in Eq. (15) are determined using Eq. (7).

Proposed approach utilizes special boundary integral equations (6), which use the

doubly periodic kernels [11]. Thus, for determination of the effective properties of a

mediumwith a doubly periodic array of cracks one has to consider only the surface of

one crack, because Eqs. (6), (7) already include the periodic boundary conditions at the

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