PSI - Issue 2_A

Haydar Dirik et al. / Procedia Structural Integrity 2 (2016) 3073–3080 Haydar Dirik and Tuncay Yalçinkaya / Structural Integrity Procedia 00 (2016) 000–000

3075

3

by Gallagher (1974) by modifying the residual SIF used in this model with an empirical material constant. This generalized model can only deal with OL e ff ect. MGW model, used in this study takes into account the e ff ect of UL as well (see Brussat (1997)). The MGW model extends the Generalized Willenborg model by taking into account the reduction of retardation e ff ects due to UL. Generelized Willenborg model considers the e ff ect of crack growth retardation by using an e ff ective SIF concept which is given as K e f f i = K i − K R in which K i is the typical SIF for the i th cycle and K R is the residual SIF represented as

oL max 1 −

a i − a oL

W R = K

K R = K

K max , i

(2)

r poL −

where a i current crack size, a oL crack size at the occurrence of the OL, r poL yield zone produced by the OL, K oL max maximum SIF of the OL and K max , i maximum SIF for the current cycle. Gallagher and Hughes (1974) introduced an empirical material constant into the calculation of residual SIF. They suggested that K R = Φ K W R where Φ is given by

K max , th K max , i

1 −

(3)

Φ =

S oL − 1

where K max , th is the threshold SIF level associated with zero FCG rates and S oL is the OL (shut-o ff ) ratio required to cause crack arrest for the given material. In this model retardation e ff ect is obtained by the change in the e ff ective stress ratio

K e f f K e f f

K min , i − K R K max , i − K R

min , i

(4)

R e f f =

=

max , i

Thus, for the i th load cycle, the crack growth increment ∆ a

i is given by

da dN =

∆ a i =

f ( ∆ K , R e f f )

(5)

MGW retardation model (see Brussat (1997)) uses a factor Φ = 2 . 523 Φ 0 / (1 + 3 . 5(0 . 25 − R U )) U < 0 . 25 and Φ = 1 when R U ≥ 0. In here R U is the ratio of current UL stress to OL stress and Φ 0 is a material dependent parameter typically ranges from 0 . 2 to 0 . 8. In this study it is used as 0 . 4 as suggested by the NASGRO material database for 7075 T6. 0 . 6 when R

2.2. Extended finite element method (XFEM)

The extended finite element method (XFEM) is based on a local enrichment of approximation space. The enrich ment is realized through the partition of unity concept. Let all nodes be represented by the set Γ , the nodes around the crack tips and faces are represented by the sub set Γ t and Γ c respectively. Then the displacement approximation for crack modeling in the XFEM has the form (see Belytschko and Black (1999)), u x f em = i ∈ Γ N i ( x )( u i ) + i ∈ Γ c N i ( x ) H ( x )( a i ) + i ∈ Γ t    N i ( x ) 4 α = 1 F α ( x ) b α i    (6)