Page 259 - The Combined Finite-Discrete Element Method

242 TRANSITION FROM CONTINUA TO DISCONTINUA

where the maximum crack opening displacement at x = 0isgivenby
4a
δ 0 = σ (7.16)
E

The lower value estimate of the plastic zone length for a short crack can be obtained by
substituting ∆ = a, δ o = δ t and σ = f t into (7.16).

E
= δ t (7.17)
4f t
In a similar way, for a long crack, Westergaard’s asymptotic solution for the crack
opening near the crack tip (x = a) is obtained from equation (7.15) by substituting
x . x
2
1 − = 2 1 − (7.18)
a a
i.e.

4a x
δ . = σ 2 1 − (7.19)
x=a
E a
For a plane stress mode I crack, the strain energy release rate is given in terms of stress
intensity factor K 1 (Irwin’s formula):

K 1 2 √
G f = 2γ = and K 1 = σ πa (7.20)
E

from which it follows that

G f E
σ = (7.21)
πa
which, when substituted into (7.19), gives

G f (a − x)
δ . = 32 (7.22)
x=a
πE
The lower value estimate of the plastic zone length for a long crack can therefore be
estimated by substituting

. . .
= (a − x); G f = δ t f t and δ = δ t (7.23)

into (7.22), which then yields
. πEδ t
= (7.24)
32f t
The meaning of the plastic zone length is best illustrated by the problem shown in
Figure 7.11. For a very coarse mesh, the stress field close to the crack tip is almost