PSI - Issue 42

Vera Petrova et al. / Procedia Structural Integrity 42 (2022) 1145–1152 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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loads on the crack faces ( p n ). Besides, with changing the temperature, e.g. the structure is cooled on ΔT , the residual stresses ( σ n T ) are arising due to mismatch in the coefficients of thermal expansion. Thus, the full load on the n -th crack consists of p n , σ n T and σ n e , where the index “ n ” denotes that the functions are written in the local coordinate system ( x n , y n ) connected with the n -th crack and σ n e is due to E ( y ) variation (Afsar anbd Sekine (2000)):

* n p p

e      T

( n = 1, 2, …, N ),

(3)

( ) n 

( )[ ( ) 1] [ ( ) 1] ( ) E y Q y E y     

pf

pf





n

n

n

n

t

( ) n  t1 1 Q TE    It is assumed that p = Q , otherwise the additional loading parameter p/Q should be considered. E 1 and α t 1 are the (1 exp(2 )) / 2 p i  n n n n pf p i        ,

material parameters of the substrate, α t ( y ) and E ( y ) are Eq. (2). 3. Solution and determination of main fracture characteristics

The method of analytical functions of complex variables is used for formulation of the singular integral equations (see, Panasyuk et al. (1976)). The equations are solved numerically using the method of mechanical quadrature (Panasyuk et al. (1976), Erdogan and Gupta (1972)). With this method, the singular integral equations are reduced to a system of algebraic equations. The unknown derivatives of the displacements jumps on the crack lines are determined; then the stress intensity factors near the crack tips are calculated. The details of this solution can be found in Petrova and Schmauder (2020a, b, 2021). The cracks are under mixed-mode conditions due to their interaction and applied thermo-mechanical loading. That is, both stress intensity factors, Mode I and Mode II, are generally non-zero. In this case, the cracks will deviate from their initial paths. To predict the crack growth and determine the direction of crack growth, the criterion of maximum circumferential stresses (Erdogan and Sih (1963)) is applied. According to this criterion, the crack deflection angle ϕ (or the so-called fracture angle, Fig. 1c), the critical stress intensity factors, and then the critical stresses are calculated from the following relations:   2 2 2arctan 8 4 n In In IIn IIn K K K K           , (4) eq n K    3 , cos ( / 2) 3 tan( / 2) n In IIn n Ic tip K K K     or , eq n Ic tip K K  . (5) From Eq. (5), the critical loads are obtained as 0 1 Ic p K a   is the critical load for a reference single crack subjected to a load p normal to the crack line with the stress intensity factor 0 K p a   and a = max a n ( n =1, 2, …, N ), here N is the number of cracks. K Ic is defined by the an expression similar to Eq. (2) for E ( y ). In Eqs. (4), (5), and (6), dimensional and non-dimensional stress intensity factors (SIFs) are related as follows: ( ) In IIn n In IIn K iK p a k ik     . (7) The weakest crack is defined from the condition 0 min / cr cr n n P p p  ( n = 1, 2, …, N ) . (8) 4. Results and discussion Typical crack patterns resulting from thermo-mechanical loads are multiple surface cracks at the ceramic top of FGCs (Rangaraj, Kokini K. 2003). Many studies are devoted to different edge crack systems, such as periodic edge cracks (Afsar, Sekine, 2000), or three or more edge cracks (Petrova et al. 2016). In a number of works (Afsar, Sekine, 2000, Li et al. 2018), it is reported that the fracture resistance of a TBC can be increased by introducing dense vertical   , Ic n tip 3 cos ( / 2)  0 3 tan( / 2)  cr n n In k k  IIn n n p K a p a  , (6) where