10 The Effect of Preheating on the Thermoelastic …
147
K
(0)
3 j (α, x 3 ) =
1
0
3
k=1
f
(0)
3k jk+6 e
σ
(0)
k x 3 ,
(10.26)
K
(0)
4 j (α, x 3 ) =
1
0
3
k=1
s f
(0)
4k jk+6 e
σ
(0)
k x 3 .
In Eqs. (10.25)–(10.26) the notations: 0 is the matrix L determinant, j,k is the
algebraic complement of the element with the index jk of the matrix L. The indexes
j = 1, 3, 4 are surface load, respectively.
Thus, relations (10.23)–(10.26) determine the displacement (i = 1, 3) or temperature (i = 4) of a thermoelastic structurally inhomogeneous half-space arbitrary
point under the action of a given in the region oscillating load q j0 (x 1 ) taking into
account the presence of initial deformations and preheating.
10.5 Results Discussion
To study the influence of the initial deformed states, we consider the problem of
vibrations in cadmium sulfide (CdS) layer rigidly coupled with magnesium oxide
(MgO) half-space. The initial prestress state created by using initial deformation and
preheating conditions. There is a mechanical and thermal load q 0 (x 1 )e
−iωt on the top
of layer surface distributed in the area x 1 ∈ [−1, 1]. The following are their physical
characteristics:
c
(1)
1111 = 9.07 × 10
10 N/m
2 , c
(1)
1133 = 5.1 × 10
10 N/m
2 , c
(1)
3333 = 9.38 × 10
10 N/m
2 ,
c
(1)
1331 = c
(1)
3113 = 1.50 × 10
10 N/m
2 , β
(1)
11 = 7.09 × 10
5 N/K/m
2 ,
β
(1)
33 = 7.24 × 10
5 N/K/m
2 , c
(1)
ε = 329 J/kg/K, λ
(1)
11 = λ
(1)
33 = 20.0 W/m/K,
ρ
(1)
= 4820 kg/m
3 , τ 0 = 280 K,
c
(0)
1111 = 30.0 × 10
10 N/m
2 , c
(0)
1133 = 10.1 × 1–0
10 N/m
2 , c
(0)
3333 = 30.0 × 10
10 N/m
2 ,
c
(0)
1331 = c
(0)
3113 = 15.75 × 10
10 N/m
2 , β
(0)
11 = β
(0)
33 = 4.4 × 10
6 N/K/m
2 ,
c
(0)
ε = 875 J/kg/K, λ
(0)
11 = λ
(0)
33 = 58.0 W/m/K, ρ
(0)
= 3576 kg/m
3 .
Figure 10.1 shows the Green’s function K poles distribution of a layered thermoelastic half-space, in the absence of initial stresses. A special feature of problems for
Fig. 10.1 Real poles of
Green’s function K
147
K
(0)
3 j (α, x 3 ) =
1
0
3
k=1
f
(0)
3k jk+6 e
σ
(0)
k x 3 ,
(10.26)
K
(0)
4 j (α, x 3 ) =
1
0
3
k=1
s f
(0)
4k jk+6 e
σ
(0)
k x 3 .
In Eqs. (10.25)–(10.26) the notations: 0 is the matrix L determinant, j,k is the
algebraic complement of the element with the index jk of the matrix L. The indexes
j = 1, 3, 4 are surface load, respectively.
Thus, relations (10.23)–(10.26) determine the displacement (i = 1, 3) or temperature (i = 4) of a thermoelastic structurally inhomogeneous half-space arbitrary
point under the action of a given in the region oscillating load q j0 (x 1 ) taking into
account the presence of initial deformations and preheating.
10.5 Results Discussion
To study the influence of the initial deformed states, we consider the problem of
vibrations in cadmium sulfide (CdS) layer rigidly coupled with magnesium oxide
(MgO) half-space. The initial prestress state created by using initial deformation and
preheating conditions. There is a mechanical and thermal load q 0 (x 1 )e
−iωt on the top
of layer surface distributed in the area x 1 ∈ [−1, 1]. The following are their physical
characteristics:
c
(1)
1111 = 9.07 × 10
10 N/m
2 , c
(1)
1133 = 5.1 × 10
10 N/m
2 , c
(1)
3333 = 9.38 × 10
10 N/m
2 ,
c
(1)
1331 = c
(1)
3113 = 1.50 × 10
10 N/m
2 , β
(1)
11 = 7.09 × 10
5 N/K/m
2 ,
β
(1)
33 = 7.24 × 10
5 N/K/m
2 , c
(1)
ε = 329 J/kg/K, λ
(1)
11 = λ
(1)
33 = 20.0 W/m/K,
ρ
(1)
= 4820 kg/m
3 , τ 0 = 280 K,
c
(0)
1111 = 30.0 × 10
10 N/m
2 , c
(0)
1133 = 10.1 × 1–0
10 N/m
2 , c
(0)
3333 = 30.0 × 10
10 N/m
2 ,
c
(0)
1331 = c
(0)
3113 = 15.75 × 10
10 N/m
2 , β
(0)
11 = β
(0)
33 = 4.4 × 10
6 N/K/m
2 ,
c
(0)
ε = 875 J/kg/K, λ
(0)
11 = λ
(0)
33 = 58.0 W/m/K, ρ
(0)
= 3576 kg/m
3 .
Figure 10.1 shows the Green’s function K poles distribution of a layered thermoelastic half-space, in the absence of initial stresses. A special feature of problems for
Fig. 10.1 Real poles of
Green’s function K