PSI - Issue 28

Vera Petrova et al. / Procedia Structural Integrity 28 (2020) 608–618 Author name / Structural Integrity Procedia 00 (2019) 000–000

615

8

( In IIn K iK p a k ik     . ) In IIn

(19)

By substituting (19) into condition (16), the critical loads are obtained

( )

Ic K y

1

p







cr n

3 cos ( / 2) 

3 tan( / 2) 

In k k 

a



n

IIn

n

n

or

p

0 exp( (1 / 

 ( / ) sin )) 3 tan( / 2) n n   IIn n

h y h x h  

a

cr n

,

(20)



n



3 cos ( / 2) 

p

In k k 

a

0

n

n

where p 0 is defined by Eq. (18) and K Ic by Eqs. (4) and (6). The fracture angle ϕ n is shown in Fig. 3. First, the angle of the crack propagation (fracture angle) Eq. (15) is obtained using the results of the calculated stress intensity factors, Eqs. (13) and (14). Next, the local fracture stability is evaluated by Eq. (16). Then, the critical loads are obtained near the crack tips, Eq. (20). Finally, the weakest crack or crack tip is defined from the condition 0 min / cr cr n n P p p  ( n = 1, 2, …, N ) . (21) 4. Results: stress intensity factors and critical loads for a system of interacting cracks As illustrative examples, consider the geometries shown in Fig. 4. Three edge cracks with inclination angles β n = β ( n = 1, 2, 3) and three internal cracks with β = 0 in a week zone. The inclination angles for internal cracks are fixed and equal to zero, i.e. the cracks are parallel to the FGC boundary. Fig. 4a shows the geometry with different crack sizes ( a n = 2 a 1 ), and Fig. 4b – for the same crack sizes. These crack patterns are observed in experiments, which were reported in the literature, e.g. see Rangaraj and Kokini (2003). The dimensionless stress intensity factors (SIFs) are denoted as k I,II = K I,II / K 0 , where K 0 is the SIF for a single crack, Eq. (17). The SIF for a single edge crack normal to the surface of the layer is equal to K I = 1.58 p (π a ) 1/2 ( a is the half-length of edge crack). This definition for SIFs is more convenient for the considered mixed system of internal and edge cracks than the commonly used one for the SIF for an edge crack: K I = 1.12 p (2π a ) 1/2 , where 2 a is the full length of the edge crack.

a

b

Fig. 4. FGC/H structure with a system of three edge cracks and three internal cracks, (a) for a n = 2 a 1 and (b) for the equal sizes of cracks.

The inhomogeneity parameters for the thermal expansion coefficient, Young’s modulus and fracture toughness are the following: ε h = –0.5, ω h = –1.5, γ h = –2.3. Other parameters (geometrical) are h/a = 4, d/a = (2, 4, 6), a = max a k ,