PSI - Issue 13

Lapin V.N. et al. / Procedia Structural Integrity 13 (2018) 1171–1176 Lapin V.N. and Cherny S.G. / Structural Integrity Procedia 00 (2018) 000–000

1173

3

maximum principal stress σ , also called special principal stress. It is a principal stress where the radial components of the stress tensor are neglected. It can be calculated as

1 2

σ θ + σ z 2

σ 1 ( θ ) =

2 + 4 τ 2

(3)

( σ θ − σ z )

θ z .

+

where σ θ , σ z , τ θ z are functions of SIF. Kinking angle θ ∗ can be found as the one that provides the maximum of σ 1 , and twisting angle is calculated using the explicit formula

arctan

2 τ θ z ( θ 0 ) σ θ ( θ 0 ) − σ z ( θ 0 ) .

1 2

(4)

ψ 0 =

2.2. Global type criterion of Lazarus and Leblond

Starting from the idea that the twisting of the crack front is caused by the presence of mode III, a criterion linked to the disappearance of mode III has to be derived. In Lazarus et al. (2008) two criteria are described: MVG that maximizes the mean value of total energy release rate

1 − ν 2

2 II +

E

1 + ν E

K 2

K 2

(5)

G =

I + K

III

with respect to kinking angle θ ( l ) and MVK that maximizes the mean value of stress intensity factor K I ( l , δ ) along the front with respect to θ ( l ) . Here l = x 3 / d is dimensionless front coordinate normalized on half length of the front d (see Fig. 1). It is supposed in Lazarus et al. (2008) that the fracture increment δ c is uniform at all points of the front and small enough to ally analytical formulas for SIF that characterize stress state after the fracture propagation on δ c . The fracture front is treated as straight lite that is given by the increment δ c and the angle between the specimens face and front line θ m . For the case of rectangular parallelepiped containing an initial inclined fracture (Fig. 1) it is possible to write the system of equation for θ m and δ c that gives the fracture front form (Lazarus et al. (2008)). The criterion proposed in Cherny at al. (2016) uses the zero condition for SIF modes II and III to obtain the kinking and twisting angles As long as the twisting angle can be calculated using the derivative of the kinking angle with respect to the local front coordinate l one can rewrite the condition as K II ( θ ( l )) = 0 , K III ( θ ( l )) = 0 . (6) It is impossible to fulfill both conditions in (6) at each point of the crack front because two conditions should be satisfied and just one parameters θ can be varied. Therefore we have connected both modes K II and K III with weight β into a single function and considered this function integrally along the whole crack front on every new time step and require that the function reach the minimum 2.3. Implicit criterion of fracture propagation

F ( θ ( l )) , F ( θ ( l )) = C

II ( θ ( l )) + β K 2

(1 − β ) K 2

F ( θ ∗ ( l )) = min θ ( l )

(7)

III ( θ ( l )) dl .

Optimization problem (7) is solved iteratively at each step of the fracture propagation as it has been described in Cherny at al. (2016). Parameter β allows to consider different propagation criteria. For example, β = 0 gives the implicit formulation of maximal tangential stress (2DMTS) criterion and β = 0 . 5 gives strain energy density criterion Sih (1974). In Cherny at al. (2016) there was no recommendations of β was made. No comparison with any experiments was performed also. The next section is devoted to eliminate this lag.