Crack Paths 2006

The function f(i) in Eq. (2), which is the same for all three separation modes,

determines the shape of the single-mode cohesive law, whereas gi(j) introduces the

modecoupling, particularly between normal and shear separation. Using this approach,

the shape of the single-mode cohesive law and the mixed-mode behaviour can be

chosen independently. For three-dimensional simulations, it is reasonable to calculate

the shear separation in one resultant tangential direction, only, since the separation

behaviour should not depend on the finite element orientation. Only for shell structures,

where all three separations denote different failure modes, the shear coupling is defined

by

(3)

g1Gj g G k g G l (i I,II,III ; k II,III,I;l III,I,II)

The functions f() and g() are defined as follows:

2

­

§

·

§

·

G

G

°

G G1

2

© ¨

¹ ¸

© ¨

¹ ¸

G

G

°

1

1

°

(4)

G1 G G2 ,

f(G)

1

®

°

3

§

·

G G 2 §©¨ ·

G G 2

3

°

G2 G G 0

2

1

¹¸ 2

© ¨

¹ ¸

°

G

G

G

G

¯

0

2

0

2

containing two additional shape parameters,

1 and

2, and

(5)

g ( G ) 2 G / G 0 3 3 G / G 0 2 1

The shape of these functions is shown in Figure 3. The single-mode cohesive law, eq.

(4), is similar to the trapezoidal shape proposed by Tvergaard [15], but the present one

is differentiable, and the mixture rule is different.

f(G) 1

1 g(G)

1

2 initial stiffness unloading/reloading path

T*/T0

(mm)

0

(mm)

0

0

Figure 3. Functions of the traction separation law used in Eq. (2).