PSI - Issue 3

P. Ståhle et al. / Procedia Structural Integrity 3 (2017) 468–476

474

P. Ståhle et al. / Structural Integrity Procedia 00 (2017) 000–000

7

The recursion cycles are assumed to proceed until a converged result is obtained after N cycles. The obtained value d N is denoted d . The stress intensity factor at x = a 2 is given by

ˆ + ( u − ˆ t ) ( u − ˆ t )(2 ˆ + u − ˆ t )

= σ o √ π

d u − 1

1 π

1

1 + u 1 − u

(1) I, a 2

(2) I, a 2

K I, a 2 = K

+ K

(25)

ˆ t

Normalizing with K R , from (13), the dimensionless stress intensity factor is

1 + ( u − ˆ t ) d ( u − ˆ t )(2 + ( u − ˆ t ) d )

1 π

1

K I, a 2 K R

1 + u 1 − u

d u − √ d

(26)

=

ˆ t

3.3. A wedge with a parabolic shape

Equation (1) can be written as

h ( x ) = h o 2 −

− x

− x

(27)

Accordingly, it can be noticed that near the wedge tip, i.e. for ( − x ) / → 0, the ellipse degenerates to a parabola (Fig. 4), namely h ( x ) = h o 2 √ − x

thus, equation (24) for a parabolic wedge is rewritten as

1 ˆ t

1 4 π 2

d u

2

1 − u u − ˆ t

d =

(28)

with the limit solution ˆ t → 1 as d → 0 . The stress intensity factor K I , a 2 is obtained as the limit for → ∞ in equation (25), and in dimensionless form is

1 π

1

K I, a 2 K R

d u ( u − ˆ t )(1 − u )

(29)

=

ˆ t

However, the contact conditions require that (23) is fulfilled for both the elliptic and parabolic wedge.