7.7 Mathematical Model of Three-Layer Micro- and Nano-Beams
277
d
2 V
dx 2 = 0,
(7.155)
Dγ
2
1 +
3
2(1 + ν 3 )
(1 − ϑ) γ 3 t
2
3
γ 2
l 3
h 3
2
γ
d
2
α
dx 2 −
− (1 − ϑ)
1 +
3γ 3 t
2
3
γ γ
l 3
h 3
2
d
3 w
dx 3
− (1 − ϑ) G 3 bht 3 γ α = 0,
(7.156)
D
1 +
3
2 (1 + ν 3 )
γ 3 t 2
3
γ γ
l 3
h 3
2
γ
d 3 α
dx 3 −
−
1 +
6
γ 1 t 2
1
1 + ν 1
l 1
h 1
2
+
γ 3 t 2
3
1 + ν 3
l 3
h 3
2
+
γ 2 t 2
2
1 + ν 2
l 2
h 2
2
d 4 w
dx 4
+ qb = 0.
(7.157)
After employment of the following simplifications
a
2
h =
3
2 (1 + ν 3 )
(1 − ϑ) γ 3 l
2
3
γ 2
, b
2
h =
3γ 3 l
2
3
γ γ
,
c
2
h =
3
2 (1 + ν 3 )
γ 3 l
2
3
γ γ
, d
2
h =
6
γ 1 l
2
1
1 + ν 1
+
γ 3 l
2
3
1 + ν 3
+
γ 2 l
2
2
1 + ν 2
,
(7.158)
Equations (7.156), (7.157) take the following form
Dγ
2
1 +
a h
h
2
γ
d
2
α
dx 2 − (1 − ϑ)
1 +
b h
h
2
d
3 w
dx 3
−
− (1 − ϑ) G 3 bht 3 γ α = 0,
(7.159)
D
1 +
c h
h
2
γ
d
3
α
dx 3 −
1 +
d h
h
2
d
4 w
dx 4
+ qb = 0.
(7.160)
Therefore, the system of equilibrium equations (7.155)–(7.157) split into
Eq. (7.155) and the system of Eqs. (7.159)–(7.160) regarding the functions γ α and w.
It is suitable for a further analysis to reduce the whole problem to only one equation
by introducing w and γ α as a function χ being differentiable the required times:
w =
1 −
h
2
β
1 +
a h
h
2
d
2
dx 2
χ,
γ α = − (1 − ϑ)
h
2
β
1 +
b h
h
2
d
3
χ
dx 3 .
(7.161)
Substituting the expression (7.161) into Eq. (7.159), we find