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

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818 YING LI AND CHANG-JUN CHENG For TEVB and TEB, the amplitudes of deflection all decrease slightly with time as the parameter a11 increases. Besides, the variations of the void moment and thermal moment with a11 are similar to a10. 6. Conclusion In this paper, the equation for the balance of entropy is firstly converted to an equivalent form without the first-order time-derivative by integral, and introducing the moments caused by the change of the void volume fraction and temperature, the Hamilton variational principle is extended to the three-dimensional thermoelastic beams with voids under finite deformations. A nonlinear theory of thermoelastic beams with voids is established based on the Kirchhoff–Love hypothesis, in which the influences of the axial forces, neutral layer inertia and rotation inertia are considered. The theory presents a set of coupling nonlinear differential equations of three displacements, one void moment and one thermal moment, in which not only the accelerations of the deflection and void moment are included, but also the effects of the neutral layer inertia and rotation inertia. So the theory is a complete nonlinear mathematical model of thermoelastic beams with voids under the Kirchhoff–Love hypothesis. To illustrate the correctness of the theory and method of this paper, the plane bending problem of a simply supported elastic beam with small deformations is first studied, and the results are compared with the analytic solution. One can see that the numerical solution obtained from the DQM is accordant with the analytic solution, and the DQ solution has good convergence also. As application, the static and dynamic responses of the plane bending of a fixed-fixed beam are studied. It is difficult to obtain the analytic solution of the nonlinear problem, so one kind of numerical methods is adopted to solve the problem. Firstly, the DQM is used to discrete the nonlinear system on the spatial domain, a set of nonlinear algebraic equations or ordinary differential equations are obtained for the static or dynamic problem, respectively. Then, the Newton–Raphson method and Runge–Kutta method are adopted to calculate the static and dynamical system, respectively. Mechanical characteristics of the system are investigated for four beams, that is, TEVB, TEB, EVB and EB, and the influences of parameters are all investigated. The general conclusion is that the presence of voids enlarges beam deflection. The deformation-void coupling parameters mainly bring changes in the deflection and void moment. The effect of the deformation-heat coupling parameters is significant. The void-heat coupling parameters change the void moment and thermal moment greatly, but the effect on the deflection is little. Beam deflection increases with an increase in a1, a3, a5 but decreases with an increase in a2, a4, a8. The effect of other parameters on the deflection is not obvious. For the dynamical problem, the effects of the temperature on the deflection and void moment are less obvious than the static case, which maybe the external work is transformed to the thermal energy partially and to be dissipated. Appendix: Derivation of Hamilton variational principle of thermoelastic beam with voids Letting u2(x1, t) ≡ 0 in the equations above corresponds to considering only plane bending of the beam. Applying a variational calculation to (19), we then obtain the variational formulas �� δT1 = − {ρh(ü1δu1 + ü3δu3) + ρχ ¨MϕδMϕ} ∂ d� + � � ∂t {ρh( ˙u1δu1 + ˙u3δu3) + ρχ ˙MϕδMϕ} d�,
� δT2 = Iρ �� δW = A NONLINEAR MODEL OF THERMOELASTIC BEAMS WITH VOIDS, WITH APPLICATIONS 819 ∂� � ∂ ˙u3 ∂x1 � � � � ∂ � ∂2 ˙u3 l δ ˙u3 ds − Iρ � ∂t ∂x 2 � δu3 − 1 ∂2ü3 ∂x 2 � δu3 d�, 1 (q1δu1 + q3δu3) d� + Nx0δu1(0) − Nxaδu1(a), � �� t1 δD(Mθ) = − � 0 K � h ¯h Mθ,11δMθ + + T0 2T0 K � � � � t1 12 K MθδMθ dt1 d�+ (l Mθ,1δMθ) dt1 ds, T0 h2 ∂� 0 T0 � � � 0 ∂ε11 δU = −(λ + 2µ)h δu1 + � ∂x1 ∂ � ε ∂x1 0 � � �� ∂u3 (λ + 2µ)h3 ∂ 11 δu3 d� + ∂x1 12 � 4u3 ∂x 4 δu3 d� 1 �� + −αv Mϕ,11δMϕ + ξv MϕδMϕ + � 12αv � �� MϕδMϕ d� − mv(MϕδMθ + MθδMϕ) d� h2 � − h3 �� (bvδMϕ − βδMθ) 12 � ∂2u3 ∂x 2 � ∂ + bv 1 2Mϕ ∂x 2 −β 1 ∂2 Mθ ∂x 2 � � �� ρce δu3 d� − MθδMθ d� + δUB, 1 � T0 in which � � δUB = (λ + 2µ)h lε ∂� 0 11δu1 + lε 0 � � � ∂u3 (λ + 2µ)h3 11 δu3 ds + l ∂x1 12 ∂� ∂2u3 ∂x 2 δ 1 ∂u3 − l ∂x1 ∂3u3 ∂x 3 � δu3 ds 1 � � � ∂ � Mϕ + αv l δMϕ − ∂� ∂x1 h3 � l(bv Mϕ − β Mθ)δ 12 ∂u3 � + ∂x1 h3 � � � l(bv Mϕ − β Mθ),1 δu3 ds, 12 � t�� δ�B = T αδuα ds + (uα − ūα)δTα ds + MhδMϕ ds + (Mϕ − Mϕ)δMh ds � + ∂VQ 0 ∂Vσ � 1 M QδMθ ds + T0 �� + � ∂Vθ ∂Vu T0 ∂Vh � �� 1 (Mθ − Mθ)δMQ ds dt + ρχh � (Mϕ|t=0 − M 0 ϕ )δ ˙Mϕ − ˙M 0 ϕδMϕ �� d� + � h � �� t1 ∂Vϕ ρh � (uα|t=0 − u 0 α )δ ˙uα − ˙u 0 αδuα � d� ρce (Mθ|t=0 − M 0 θ )δMθ � dt d�. Substituting these variational formulas into δ� = 0, we can obtain the differential equations of motion in terms of u1, u3, Mϕ, Mθ, that is, Equations (20). Boundary conditions can be derived from the following boundary virtual work equation in variational equation: � � (λ + 2µ)h lε ∂� 0 11δu1 + lε 0 � � � ∂u3 (λ + 2µ)h3 11 δu3 ds + l ∂x1 12 ∂� ∂2u3 ∂x 2 δ 1 ∂u3 − l ∂x1 ∂3u3 ∂x 3 � δu3 ds 1 � � � ∂ � Mϕ + αv l δMϕ − ∂� ∂x1 h3l 12 (bv Mϕ − β Mθ)δ ∂u3 + ∂x1 h3l 12 (bv � Mϕ − β Mθ),1δu3 ds � � t1 K + (l Mθ,1δMθ) dt1 ds − Nxaδu1(a) + Nx0δu1(0) = 0. ∂� 0 T0 References [Bellman and Casti 1971] R. E. Bellman and J. Casti, “Differential quadrature and long-term integration”, J. Math. Anal. Appl. 34:2 (1971), 235–238. 0 T0
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