PSI - Issue 3

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

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

stress intensity factors are obtained as follows

σ y ( ξ )

1 √ π s

s

s + m ξ s − m ξ

d ξ

(18)

K I = −

− s

where s = ( a 2 − a 1 ) / 2, x = ξ + ( a 1 + a 2 ) / 2, m = − 1 for crack tip at x = a 1 and m = 1 for crack tip at x = a 2 . Replacing σ y ( x ) from (14), the previous integral is conveniently split into two parts

(1) I, n + K

(2) I, n

K I, n = K

(19)

where n equals a 1 or a 2 . With t = − ( a 1 + a 2 ) / 2 and k ( m , ξ ) = s + m ξ s − m ξ

, stress intensity factors are

σ o √ π s

t

k ( m , ξ )d ξ = − σ o √ π s − t + ξ ( ξ − t )(2 − t + ξ )

K (1)

(20)

I, n = −

− s

σ o √ π s

s

K (2)

k ( m , ξ )d ξ

(21)

I, n =

t

By applying the transformation u = ξ/ s , ˆ t = t / s and ˆ = / s to (20) and (21), we obtain

σ o √ π

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

1

K I, n = − σ o √ π +

k ( m , u )d u

(22)

ˆ t

and using condition (16), we obtain the following expression, for n = a 1

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

= 0 ⇒

1

1 − u 1 + u

(1) I, a 1

(2) I, a 1

d u = π

(23)

K I, a 1 = K

+ K

ˆ t

Equation (23) provides a relation between a 1 and a 2 for any given ˆ , and it can be rewritten with the following recursion which yields a single root d i = 1 / ˆ = s / d i = 1 π 1 ˆ t 1 + ( u − ˆ t ) d i − 1 ( ( u − ˆ t )(2 + ( u − ˆ t ) d i − 1 ) 1 − u 1 + u d u 2 , with d 0 = 0 (24)