Crack Paths 2009

t h K ∆ is the threshold value of stress intensity range,

n is the strain hardening exponent,

( S i i

)

Nf is the fatigue life. In equation (2)

=123,, and Sp are elastic and plastic

coefficients respectively. The work [1] contains more details about the determination of

( ) S S n I M Y Y p p n p e I I = , , , , ~ , , νfoσr the general case of

( ) S S Y Y i i I I I = θκβη,,,,, and

mixed-mode elastic-plastic fracture.

Geometrical modeling of crack trajectories

Criterion (1) was applied for the crack path prediction for the two geometric

configurations containing the single-edge and the central initial cracks of length and obliq eness a0 β0 as shown in Fi .1 (a) and (b). Crack path predi tion for the mixe

modes I and II initial crack involves replacing a bent crack with a staightline crack

approximation, as shown in Fig.1. The principal feature such modelling is determination

of the crack growth direction and definition of crack length increment in this direction.

a

P

b

Y

A

B

r

0

O

A

D

E

O

X

C

D

B

r1

∆θβ∗1

C

1

Figure 1. Crack growth trajectory approximation by fracture damage zone size,

(a) single-edge crack geometry, (b) central notched biaxially loaded crack geometry.

Crack may be assumed to grow in a number of discrete steps. After each increment

of crack growth, the crack angle changes from the original angle β0

and so does the

effective length of the crack. For the next increment of crack growth, one has to

β1. As shown in Fig. 1, O Ais the

consider the new crack length

a 1

and crack angle

initial crack length equation (1), δ

β0. c be computed. The value is then extended Let = A B be the crack growth r0

a0 and hence r

oriented at an angle

increment for the first growth step. It would correspond to the F D Zsize. Making use of

0 = δ

a0

r0

θ∗

θ0∗

=

θcos

0

0 x

∗0

∗θ0sin

along A Bwith the angle . For the single-edge crack geometry ( Fig.1,a) the first step

and

of crack growth obtained as

,

= ro 0 y

. The next step

φ =

0r

0

A C =

along B C oriented at the angle , while

∑

y 2 + ∑

θ1∗

plotting

r1

x

,

2

,∗θ+

( φ1 1 = − ∑ ∑ t a n y x ) a d

1 1 1 γ c =o s r x,

1 1 1 γ s =i n r y,

where

γ = β∆

1

1

1

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