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

810 YING LI AND CHANG-JUN CHENG
ui(x1, t), volume fraction ϕ(xi, t) and temperature θ(xi, t) make the following functional arrive at the
stationary value
�(ui, ϕ, θ) =
� t
0
(T + W + D − U) dt + �B, (19)
where H = T + W + D − U is a generalized Hamilton function.
Applying a variational calculation to (19) (whose detailed formulas are given in the Appendix) and
substituting the results obtained into the variational equation of (19), that is, δ� = 0, then integrating
(19) with regard to time from 0 to the final time t, and observing the beam has the appointed motions at
the initial and final time, as well as the arbitrariness of the variables δui, δMϕ, δMθ on the interval [0, l],
we obtain the differential equations of motion in terms of ui, Mϕ, Mθ, in which the balance of entropy
has been differentiated relative to t1:
�
�
D0hb
u1,1 + (u2,1) 2 +(u3,1) 2
�� D0hb u1,1 + (u2,1) 2 +(u3,1) 2
2
�� D0hb
2
u1,1 + (u2,1) 2 + (u3,1) 2
2
+ q1 = ρhbü1,
,1
�
u2,1
�
− D0Izu2,1111
,1
+ Iz
Iy + Iz
� �
u3,1 − D0Iyu3,1111
,1
(bv Mϕ − β Mθ),11 + q2 − N0u2,11 = ρhbü2 − ρ Izü2,11,
+ Iy
(bv Mϕ − β Mθ),11 + q3 − N0u3,11 = ρhbü3 − ρ Iyü3,11,
Iy + Iz
αv Mϕ,11 + bv(Izu2,11 + Iyu3,11) − ξv Mϕ + qϕ + mv Mθ = ρχ ¨Mϕ,
K Mθ,11 + βT0(Iz ˙u2,11 + Iy ˙u3,11) − mvT0 ˙Mϕ + qθ = ρce ˙Mθ.
This is a set of coupled nonlinear equations for ui, Mϕ and Mθ, in which the effects of the axial forces
N0, the neutral layer inertia ρhbü1, and the rotation inertias ρ Izü2,11 and ρ Iyü3,11 are included.
It can be also seen that the boundary conditions at the end designated forces may be derived from the
boundary virtual work equation in the variational equation δ� = 0. If we only consider a clamped-beam
without axial forces, the boundary conditions at the ends (x1 = 0, l) are
(20)
ui = 0, u2,1 = 0, u3,1 = 0, Mϕ,1 = 0, Mθ = 0. (21)
Observing that formula (18) is included in (19), the initial conditions at the initial time are given as
ui = u 0 i , Mϕ = M 0 ϕ , Mθ = M 0 θ , ˙ui = ˙u 0 i , ˙Mϕ = ˙M 0 ϕ , (22)
in which u 0 i , M0 ϕ , M0 θ , ˙u0 i , ˙M 0 ϕ are the known functions of x1. Especially, if the beam is at rest at the
initial time, these functions are equal to zeros.
4. Solution method
As application of the mathematical model above, the nonlinear mechanical characteristics of a two end
fixed beam without the axial force are investigated, and the influences of parameters are considered. For