PSI - Issue 54

Sjoerd T. Hengeveld et al. / Procedia Structural Integrity 54 (2024) 34–43 S.T. Hengeveld et al. / Structural Integrity Procedia 00 (2023) 000–000

37

4

with:

m

mt − U mb ,

n

nt − U nb

∆ U

= U

∆ U

= U

(4)

1 2

3 π 2

t 12 = 6 π − 20 , t 21 =

t 22 = 1

t 11 = 6 −

,

,

where G is the energy release rate, U are the nodal displacements and F are the nodal forces. The superscripts correspond to the nodes in Figure 4a. All values are evaluated in a rotated coordinate system x ′ and y ′ which is aligned with the crack in the current load increment (see Figure 4b). This alignment is based on the two elements in front of the crack tip. For every time step the energy release rates ( G ) are calculated, and subsequently these can be transformed to SIFs using Equation 5. K I ( x c ) = √ . G I ( x c ) E ef f K II ( x c ) = sgn( ∆ U x ′ ( x c ))  G II ( x c ) E ef f (5) inwhich E ef f = E for plane stress and E ef f = E 1 − ν 2 for plane strain.

y ′

y’

y ∗

m t

n c t

j

m b

x ∗

x’

n b

θ

x ′

(a)

(b)

Fig. 4: (a) Schematic overview of mesh used in the quadratic VCCT algorithm (b) Coordinate system for determination of crack growth direction

The Hourlier and Pineau (1982) criterion assumes that the crack grows in the direction in which the SIFs of an infinitely small crack increment gives the highest FCGR. The SIFs of the (kinked) crack in the rotated coordinate system are calculated using the first order approximation of the stress field given by Cotterell and Rice (1980), see Equation 6.    K ∗ I ( θ, x c ) K ∗ II ( θ, x c )    = 1 4    3cos θ 2 + cos 3 θ 2 , − 3  sin θ 2 + sin 3 θ 2  sin θ 2 + 3sin 3 θ 2 , cos θ 2 + 3cos 3 θ 2       K I ( x c ) K II ( x c )    (6) inwhich θ is the angle of the infinitely small (kinked) crack around the crack tip, see Figure 4b. To calculate the crack growth rate using these SIFs the equivalent stress range is calculated with: ∆ K ∗ I ( θ ) = max x c  ⟨ K ∗ I ( θ, x c ) ⟩  − min  ⟨ K ∗ I ( θ, x c ) ⟩  ∆ K ∗ II ( θ ) = max x c  K ∗ II ( θ, x c )  − min  K ∗ II ( θ, x c )  ∆ K ∗ eq ( θ ) =  ∆ K ∗ I ( θ ) 2 + c II ∆ K ∗ II ( θ ) 2  0 . 5 (7) where ⟨•⟩ are Macaulay brackets, ⟨•⟩ = ( • + |•| ) / 2. The material constant c II = 0 . 772 and is based on experimental fatigue crack growth data obtained for 900A (R260) rail steel, see Dubourg and Lamacq (2002). Finally the FCGR can be calculated using for example the equation developed by Paris et al. (1961), see Equation 8. da dN ( θ ) = C  ∆ K ∗ eq ( θ )  m (8)