Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

808 YING LI AND CHANG-JUN CHENG
3. The Hamilton principle and mathematical model of thermoelastic beams with voids
In order to present the nonlinear mathematical model of the problem, we first derive a generalized Hamilton
principle for thermoelastic beams with voids under geometric nonlinearity.
Let
� �
��
Mϕ = ϕ(xi, t)x2x3 dA and Mθ = θ(xi, t)x2x3 dA (6)
A
A
be the moments caused by the change of the volume fraction ϕ(xi, t) and temperature θ(xi, t), respectively.
Set
� �
qϕ = αv x2x3(ϕ,22 + ϕ,33) dA
A
�� b/2
= αv
−b/2
qθ = K
� �
�
(x3ϕ,3)| h/2
�
� h/2 �
−h/2 − ϕ|h/2
−h/2
x2 dx2 + (x2ϕ,2)|
−h/2
b/2
�
�
−b/2 − ϕ|b/2
−b/2
x3 dx3 ,
x2x3(θ,22 + θ,33) dA
T0 A
�� b/2
= K
T0
−b/2
�
(x3θ,3)| h/2
�
� h/2 �
−h/2 − θ|h/2
−h/2
x2 dx2 + (x2θ,2)|
−h/2
b/2
�
�
−b/2 − θ|b/2
−b/2
x3 dx3 .
According to [Cowin and Nunziato 1983], on the surfaces of the beam, ϕ(xi, t) needs to satisfy the
conditions ϕ,2|x2=±b/2 = 0 and ϕ,3|x3=±h/2 = 0, so the expression for qϕ simplifies to
� � b/2
� h/2
qϕ = −αv
(ϕ| b/2
−b/2 )x3
�
dx3
(8)
(ϕ|
−b/2
h/2
−h/2 )x2 dx2 +
−h/2
The thermal boundary conditions of the beam may be given as
K θ,2|x2=b/2 = ¯h(T∞ − θb/2), K θ,2|x2=−b/2 = −¯h(T∞ − θ−b/2),
K θ,3|x3=h/2 = ¯h(T∞ − θh/2), K θ,3|x3=−h/2 = −¯h(T∞ − θ−h/2),
where ¯h is the heat transfer coefficient, θb/2, θ−b/2, θh/2, and θ−h/2 denote the temperatures on the
surfaces x2 = ±b/2 and x3 = ±h/2, and T∞ is the temperature of surrounding medium. If we assume
that T∞ = 0, the absolute temperature is equal to the reference temperature. The second expression of
(7) then simplifies to
� � b/2
h ¯h/2 + K
qθ = −
(θ| h/2
−h/2 )x2 dx2 + b¯h/2
� h/2 + K
(θ| b/2
−b/2 )x3
�
dx3 . (10)
T0
−b/2
To evaluate (8) and (10), we express the change of the volume fraction and temperature as the series
ϕ(xi, t) =
∞�
Substituting (11) into (6) yields
m=0 n=0
T0
∞�
ϕmn(x1, t)x m 2 xn 3 ,θ(xi, t) =
∞�
−h/2
m=0 n=0
∞�
ϑmn(x1, t)x m 2 xn 3
(7)
(9)
(11)
Mϕ = Aϕ11(x1, t) + o(b 3 h 3 ) , Mθ = Aϑ11(x1, t) + o(b 3 h 3 ), (12)