PSI - Issue 7

7

G. Härkegård / Procedia Structural Integrity 7 (2017) 343–350 F / Structural Integrity Procedia 00 (2017) 000–000 G. Härkegård 0

349

Introducing the axis ratio (

) a c α = , an exact solution for the (maximum) stress-intensity factor at the ends of the

, 0 x a z = ± = , is given by

minor axis,

( ) ( ) ( ) a a E k α σ σ α = π π

(A.1)

,

Ia K F =

where E ( k ) denotes the complete elliptic integral of the second kind and k its ‘modulus’:

(A.2)

π 2

( )

∫

2 2 1 sin d , k θ θ −

E k

=

0

2

2 1 . α

k

= −

An approximate expression for the elliptic integral, valid for the entire range of the modulus, 0 1 k ≤ ≤ , may be obtained by fitting 2 1 E − to a power function of α , cf. Newman & Raju (1986) and Dowling (2012). Thus, regression analysis using the ten axis ratios α = 0.1, 0.2..., 1.0 yields

(A.3)

( ) α

( ) α F ⇒ ≈ +

1.653

1.653

1 1.467

1 1 1.467

E

.

α

α

≈ +

The deviation of the approximate eq. (A.3) from the exact eq. (A.2) is < 0.1% for the entire range of axis ratios. A.2. Solution for the stress-intensity factor of a crack of a given area Consider now axis ratios ranging from α = 0 (through-crack) to α = 1 (circular crack, a c r = = ) all having the constant area

(A.4)

2 π π area ac r = = ,

as demonstrated by the two cracks in Fig. A.1. Thus, the axis ratio may be expressed as

(A.5)

( ) 2 a c a r α = = ,

and the ratio between the stress-intensity factor of the elliptical crack and that of the circular crack becomes ( ) ( ) ( ) ( ) 1 4 1 4 I 1.653 I π 2.467 1 1 1.467 1 π a r F a F K K F F r α σ α α α α σ = = ≈ + . a r K K against a c in Fig. A.2 has a rather flat maximum = 1.093 at a c = 0.483. In fact, the stress intensity factor of an embedded elliptical crack with a given area varies less than 10% in the range a c = 0.2–1. (A.6) The graph of I I