PSI - Issue 40

I.Yu. Smolin et al. / Procedia Structural Integrity 40 (2022) 385–391

388 4

I. Yu. Smolin et al./ Structural Integrity Procedia 00 (2022) 000 – 000

   +  = dr du z rz 2 1

du

r u r  = 

dr du r

dz du z

  

r

,

,

,

,

(3)

rr  =

zz  =

dz



)  0

(

.

(4)

3 ij ij  =  +   −  − 2 K T T kk ij

Here density, temperature, stress, strain, and displacement are functions of spatial coordinates and time. Material parameters (heat capacity, coefficient of thermal conductivity, elastic moduli, linear coefficient of thermal expansion), as well as density, are functions of coordinates and temperature and depend on time implicitly (via the temperature).

3.2. Initial and boundary conditions

The initial conditions correspond to the absence of stress and strain at a uniform sintering temperature of 1900 °C all over the disk. The boundary conditions for the solid mechanics equations correspond to the conditions of axial symmetry at r = 0, the free boundary conditions at the outer edge of the disk at r = 15 mm and on the planes of the disk at z = 0 and z = 5 mm. For the heat conduction equation, Newton's law of cooling for the heat flux was specified on all faces except r = 0 where the symmetry condition was adopted: ( ) r q T T = − − . (5) The surface film coefficient, β , was set as 100 W/(m 2 ⋅ K) during cooling assuming forced convection in air. 3.3. Temperature-dependent material parameters Data on the temperature dependence of physical and mechanical parameters of the composite constituents were determined from literature data. For zirconium dioxide, the data were taken from the paper by Yang (2015). They are as follows:

( ) 2.072 3.656 10 , ( ) 0.3 3.2 10 [GPa], ( ) 274.1 0.027 4 5  + = −   = + = − − − T k T T T T E T

7 2 T

−

4.347 10 

K)], [W/(m 

(6)

4 2 T

7 3 T

−

−

( ) 274 0.795 = + T c T

6.19 10 

1.71 10 

K)], [J/(kg 

−

+

6

9

12 2

−

−

−

( ) 7.091 10 2.532 10   −  = T

2.262 10 

[1/K],

T

T

+

(

( T T +  −

) ) ]. 293 [kg/m 3

( ) 5600 1 3 ( )  = T

For the ZrB 2 – 20% SiC composite, the temperature dependences were determined using the rule of mixtures. To do this, the data for ZrB 2 and SiC separately were taken from different sources (Zimmermann et al., 2008; Lugovy et al., 2016, Skripnyak et al., 2017; Gash et al, 2005; Zhang et al, 2009; Chakraborty et al, 2014, Iikubo et al., 2010). As a result, we get the following formulas: