PSI - Issue 40

I.Yu. Smolin et al. / Procedia Structural Integrity 40 (2022) 385–391 I. Yu. Smolin et al./ Structural Integrity Procedia 00 (2022) 000 – 000

387

3

ν ρ

Poisson's ratio mass density

σ ij

stress tensor components

2. Problem description Let us consider the problem of residual internal stress arising in a layered ceramic composite composed of layers of ultrahigh-temperature ceramics upon cooling from the sintering temperature to room temperature. Take, for example, the composite composition described in the paper by Burlachenko et al. (2019). The composite sample is a disc 5 mm thick and 30 mm in diameter. The disc consists of five layers of the same thickness of 1 mm but of different compositions in accordance with Fig. 1. The lower layer consists of pure zirconia and the upper layer is a composite of ZrB 2 – 20% SiC. The three intermediate layers consist of mixtures providing a gradual transfer from the lower layer to the upper layer. Ceramic layers are connected by sintering at a temperature of 1900 °C and, after which they gradually cool down to room temperature 20 °C. ZrB 2 – 20% SiC z

90% (ZrB 2 – 20% SiC) – 10% ZrO 2 70% (ZrB 2 – 20% SiC) – 30% ZrO 2 30% (ZrB 2 – 20% SiC) – 70% ZrO 2 ZrO 2

r

Fig. 1. Structure of the layered ceramic composite ZrB 2 – SiC – ZrO 2 under study.

We assume that all layers are homogeneous. Due to the cylindrical symmetry of the research object, the problem can be considered in an axisymmetric two-dimensional formulation. The axis of symmetry z is directed perpendicular to the plane of the disk, while the axis r runs along a radius of the disk (Fig. 1). Along the third axis θ, all parameters of the state of stress and strain are considered unchanged due to the homogeneity of the layers. 3. Mathematical formulation 3.1. Equations During cooling of the sintered composite, the deformation and stress fields depend on the temperature field, but, in turn, the changes in stress and strain do not affect the temperature field (heat is not produced in this case). So, to solve the problem, we can perform uncoupled thermal-stress analysis. In this case, a sequential analysis is carried out — first the formation of the temperature field and then the analysis of the stress and strain. Since all material parameters (elastic moduli, heat capacity, coefficient of thermal conductivity, linear coefficient of thermal expansion) depend on temperature, the heat transfer problem should be treated as a transient one. In this case, modeling can be performed within the framework of quasi-static uncoupled problems of thermoelasticity (Nowacki, 1986). According to this approach, the system of equations includes the heat conduction equation, Eq. (1); the equilibrium equations, Eq. (2); the strain-displacement relations, Eq. (3); and the Duhamel – Neumann relations as the constitutive equations, Eq. (4).

= c T 1  









  

 +  

  

  

 r rk T

 z k T

,

(1)

t

 r r

z





 

 



 − 

 

 

rz

zz

rz

0 =

rr

zr

rr

+

+



,

,

(2)

0 =

+

+

r

z

r

r

z

r