PSI - Issue 23

Vera Petrova et al. / Procedia Structural Integrity 23 (2019) 407–412 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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 The non-homogeneity of the functionally graded material is revealed in the form of the corresponding inhomogeneous stress distributions on the surfaces of the cracks, see Afsar and Sekine (2000), Petrova and Sadowski (2014). In this case, the properties of the FGM should vary slightly with the depth of the layer. The edges of the cracks are free of tractions, and the FGC and the substrate are perfectly bonded, i.e. ideal thermal and mechanical conditions are fulfilled outside of an interface crack, namely, the tractions and displacements are equal at the interface. For the formulation of the problem the method of complex variables is used, and the method of superposition. Due to the superposition principle the common problem is decomposed into sub-problems, each of which contains one crack. Besides, the loads at infinity are reduced to the corresponding loads on the crack faces. The solution of each sub-problem can be obtained in explicit form (and can be analyzed separately) or in an integral form, and this integral form is used for constructing the equations of the complete problem. The methods for constructing integral equations for systems of cracks in a homogeneous material can be found in Panasyuk et al. (1976). The thermal and mechanical properties of an FGC are continuous functions of the thickness coordinate y . As in the previous works, e.g. Petrova and Schmauder (2017), El-Borgi et al. (2006), the exponential form of these properties is used: 1 ( ) exp( ( )) n f y f y h    , 0 h y    , (1) { , , } t f k E   , 1 1 1 1 { , , } t f k E   , { , , } n      , (2) In Eq. (2) k is thermal conductivity, α t – thermal expansion coefficient and E – the Young’s modulus with non homogeneity parameters δ, ε and ω, respectively. f 1 are thermal and mechanical properties of the homogeneous substrate. The Poisson’s ratio is assumed to be constant and is equal to the value of the homogeneous substrate. The previous studies show that the effect of Poiss on’s ratio on the stress intensity factors is negligibly small. The values of the dimensionless graded parameters ζ nh ( h is the thickness of the FGC) are obtained from Eq. (1) as follows 2 1 ln( / ) n h f f   , 2 0 ( ) y f f y   1 ( ) y h f f y   An example of a ceramic/metal FGC/H material is (ZrO 2 /Ti-6Al-4V)/Ti-6Al-4V with f 2 / f 1 ={2/6.7, 10/8.6, 200/114} and, respectively, with the inhomogeneity parameters { δh, εh, ωh } ={ – 1.2, 0.15, 0.56} (see Shackelford and Alexander (2001) for material properties, and Petrova and Schmauder (2018) for the inhomogeneity parameters). The variation of non-dimensional properties 1 / exp( ( / 1)) n f f h y h    with dimensionless coordinate y/h was presented in Petrova and Schmauder (2018), Fig. 2a. It was demonstrated that the coefficient of thermal expansion and the Young's modulus increase towards the upper part of the FGC/H structure, while the coefficient of thermal conductivity decreases.

3. Singular integral equations for the problem and solution

As it was mentioned in Section 2, assuming that the inhomogeneity of the FGM is revealed in the form of the corresponding inhomogeneous stress distributions on the surfaces of cracks, the solution of the boundary value problem of elasticity is reduced to the solution of the system of singular integral equations (Afsar and Sekine (2000), Petrova and Sadowski (2014)):

a

( ) g t dt  n

k N a  

n  a

[ ( ) ( , ) g t R t x g t S t x dt  ( ) ( , )] nk k  nk k 

n p x

( )







, | x | < a n , n = 1,2, …, N .

(3)

t x 

k

1



a





n

k

k n 









n g x 

[ ] [ ] u i v 

( ) 2 

n

n

i

x

( 1) 

 