10 The Effect of Preheating on the Thermoelastic …
141
10.3 The 3D Linear Thermoelasticity Equations
We assumed that the body is in the orthogonal Lagrange coordinate system x 1 , x 2 , x 3
associated with the natural state of the medium. The axis x 3 is directed vertically
upward from the body. Thermoelastic layer |x 1 |, |x 2 | < ∞, −h ≤ x 3 ≤ 0 is rigidly
coupled with a thermoelastic substrate |x 1 |, |x 2 | < ∞, −∞ ≤ x 3 ≤ −h. The
governing equations for transversely isotropic thermoelastic materials taking into
account the presence of an initially deformed state and preheating are (Levi et al.
2015; Levi et al. 2019; Belyankova et al. 2012; Belyankova et al. 2016):
σ
∗(n)
1
= c
∗(n)
11
∂u
(n)
1
∂ x 1
+ c
∗(n)
12
∂u
(n)
2
∂ x 2
+ c
∗(n)
13
∂u
(n)
3
∂ x 3
− β
∗(n)
1 u
(n)
4 ,
σ
∗(n)
2
= c
∗(n)
12
∂u
(n)
1
∂ x 1
+ c
∗(n)
22
∂u
(n)
2
∂ x 2
+ c
∗(n)
23
∂u
(n)
3
∂ x 3
− β
∗(n)
2 u
(n)
4 ,
σ
∗(n)
3
= c
∗(n)
13
∂u
(n)
1
∂ x 1
+ c
∗(n)
32
∂u
(n)
2
∂ x 2
+ c
∗(n)
33
∂u
(n)
3
∂ x 3
− β
∗(n)
3 u
(n)
4 ,
(10.1)
σ
∗(n)
12 = c
∗(n)
1212
∂u
(n)
1
∂ x 2
+ c
∗(n)
1221
∂u
(n)
2
∂ x 1
, σ
∗(n)
21 = c
∗(n)
2112
∂u
(n)
1
∂ x 2
+ c
∗(n)
1212
∂u
(n)
2
∂ x 1
,
σ
∗(n)
13 = c
∗(n)
1313
∂u
(n)
1
∂ x 3
+ c
∗(n)
1331
∂u
(n)
3
∂ x 1
, σ
∗(n)
31 = c
∗(n)
3113
∂u
(n)
1
∂ x 3
+ c
∗(n)
1313
∂u
(n)
3
∂ x 1
,
σ
∗(n)
23 = c
∗(n)
2323
∂u
(n)
2
∂ x 3
+ c
∗(n)
2332
∂u
(n)
3
∂ x 2
, σ
∗(n)
32 = c
∗(n)
3223
∂u
(n)
2
∂ x 3
+ c
∗(n)
2323
∂u
(n)
3
∂ x 2
,
where u
(n)
=
u
(n)
1 , u
(n)
2 , u
(n)
3 , u
(n)
4
is the extended displacement vector
u
(n)
1 , u
(n)
2 , u
(n)
3 in the direction of the corresponding coordinates and temperature
u
(n)
4 in the body, n = 0, 1 are the parameters of the substrate and the coating, respectively, c
∗(n)
i jkl , β
∗(n)
kl
= α
(n)
i j c
∗(n)
i jkl are the components of the elastic and thermoelastic
constant tensors of materials subjected to prestressing conditions.
The oscillations of a prestressed layered thermoelastic medium are described by
the equations of motion and thermal conductivity (Belyankova et al. 2012):
∂σ
∗(n)
1
∂ x 1
+
∂σ
∗(n)
12
∂ x 2
+
∂σ
∗(n)
13
∂ x 3
= ρ
(n)
0
∂
2 u
(n)
1
∂t 2
∂σ
∗(n)
21
∂ x 1
+
∂σ
∗(n)
2
∂ x 2
+
∂σ
∗(n)
23
∂ x 3
= ρ
(n)
0
∂
2 u
(n)
2
∂t 2
(10.2)