8 Interaction of Harmonic Waves of Different Types …
117
u ∗ =
p ∗ xd
ρc
2
1 N r 1
H
(2)
1 (k 1 r 1 )e
iωt
, w ∗ =
p ∗ d(z + d)
ρc
2
1 N r 1
H
(2)
1 (k 1 r 1 )e
iωt
,
σ 11∗ =
p ∗ d
N r
2
1
(1 + κ)r 1 H
(2)
1 (k 1 r 1 ) − k 1 r
2
11 H
(2)
2 (k 1 r 1 )
e
iωt
,
σ 13∗ = −(1 − κ)
p ∗ dk 1
N r
2
1
x(z + d)H
(2)
2 (k 1 r 1 )e
iωt
,
σ 33∗ =
p ∗ d
N r
2
1
(1 + κ)r 1 H
(2)
1 (k 1 r 1 ) − k 1 r
2
33 H
(2)
2 (k 1 r 1 )
e
iωt
.
A ϕ = −
p ∗ d
ρωc 1 N
, N = k 1 d H
(2)
0 (k 1 d) − (1 − κ)H
(2)
1 (k 1 d).
(8.22)
On plane surface in z = 0, these values are changing to:
u ∗ | z=0 =
p ∗ xd
ρc
2
1 N r 10
H
(2)
1 (k 1 r 10 ), w ∗ | z=0 =
d
2 p ∗
ρc
2
1 N r 10
H
(2)
1 (k 1 r 10 ),
σ 11∗ | z=0 =
p ∗ d
N r
2
10
(1 + κ)r 10 H
(2)
1 (k 1 r 10 ) − k 1 r
2
110 H
(2)
2 (k 1 r 10 )
,
σ 13∗ | z=0 = −(1 − κ)
p ∗ d
2 k 1 x
N r
2
10
H
(2)
2 (k 1 r 10 ),
σ 33∗ | z=0 =
p ∗ d
N r
2
10
(1 + κ)r 10 H
(2)
1 (k 1 r 10 ) − k 1 r
2
330 H
(2)
2 (k 1 r 10 )
.
(8.23)
8.5 The Plate Geometry
The plate consists of three layers, two bearing layers, and the filling one. Bearing
layers are isotropic and have thickness t. The filling layer of thickness h has
honeycomb structure. The plate is simply supported on the contour.
The plate motion is described by Paimushin V.N. equation system (Ivanov and
Paimushin 1995a, b), which takes into account structural features of the plate.
B
∂
2
∂ x 2
u
a
1
+ 2q
1
+ ω
2
ρ a u
a
1 = 0,
− D
∂
4
∂ x 4 w c + 2k 1 q
1
,x + p 1 − p 2 + ω
2
ρ c w c = 0,
− D
∂
4
∂ x 4 w a − 2c 3 w a + p 1 + p 2 + ω
2
ρ aw w a = 0,
u
a
1 − k 1 w c,x − k 2
q
1
,x
,x
+ k 3 q
1
= 0.
(8.24)