PSI - Issue 28

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

611

4

2.3. Mechanical and physical properties for FGCs and inhomogeneity parameters In functionally graded materials, the composition of the material gradually changes mainly in one direction. In particular, in our problem, the composition varies with the y-coordinate from the ceramic in the upper part of the FGC to the metal in the substrate. Consequently, the thermal and mechanical properties of an FGC also varies continuously with the thickness coordinate y . As in the previous works, e.g. Petrova and Schmauder (2017) and (2020), an exponential form of these properties is used:

0 h y    .

( ) y y h      , exp( ( ))

,

(1)

( ) E y E 

exp( (

))

y h  

1

1

t

t

Here, α t is the coefficient of thermal expansion, and E is Young’s modulus with non-homogeneity parameters ε and ω , respectively. E 1 and α t1 stand for the thermal and mechanical properties of the homogeneous substrate. Poisson’s ratio is assumed to be constant, see, Eischen (1987), and is equal to the value of the homogeneous substrate. The values of the dimensionless inhomogeneity parameters ε h and ω h ( h is the thickness of the FGC) are obtained from Eq. (1) as follows

2 1 ln( / ) t t h     ,

2 t y y     , 0 ( ) t

1 t y h y     , ( ) t

(2)

2 1 ln( / ) h E E   ,

2 0 ( ) y E E y   ,

1 ( ) y h E E y   .

(3)

Since the exponential law was chosen for the material parameters α t and E , it is reasonable to use the exponential law also for the fracture toughness. But the comment in Jin and Batra (1996) that the fracture toughness can be overestimated in this case of the model have to be taken into account. Accordingly, it should be borne in mind that the fracture toughness values thus determined provide an upper bound for the possible values of the fracture toughness, Jin and Batra (1996). Thus the fracture toughness of a functionally graded material can be written as follows:

,

(4)

( ) Ic Ic K y K 

exp( (

))

y h  

1

where γ is the inhomogeneous parameter of the fracture toughness. The nondimensional value γ h is obtained as

2 Ic y K K y   , 0 ( ) Ic

1 Ic y h K K y   . ( ) Ic

2 Ic Ic h K K   , 1 ln( / )

(5)

In the local coordinate system ( x n , y n ) connected with the n -th crack, the function K Icn is written as

0 h y x   

( ) K x K  Icn n

exp( (

sin )) 

.

(6)

1

Ic

n

n

n

An example of a (ceramic/metal)/metal FGC/H material is (PSZ/steel)/steel with α t 2 = 9 - 12.2 (ꞏ10 -6 K -1 ), α t 1 = 15(ꞏ10 -6 K -1 ), E 2 = 48-22, E 1 = 207 (GPa) for the temperature range 20°C - 1110°C (Zhou and Hashida (2001)), and fracture toughness K Ic 2 = 7-10, K Ic 1 = 50 (MPaꞏm 1/2 ). The corresponding inhomogeneity parameters are calculated by Eqs. (2), (3), (5) and are equal to ε h = –0.5 - –0.2, ω h = –1.5 - –2.2, γ h = –2.3. For this material combination, all of the listed material parameters, namely, thermal expansion coefficient, Young's modulus and fracture toughness, decrease towards the upper part of the FGC/H structure. The thermal conductivity of this material is also decreased in the direction to the ceramic top, from 16 to 2 (Wm -1 K -1 ).