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

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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)
A NONLINEAR MODEL OF THERMOELASTIC BEAMS WITH VOIDS, WITH APPLICATIONS 809 where ϕ11(x1, t) and ϑ11(x1, t) are the terms of (11) when m = n = 1. The term o(b3h3 ), indicating higher-order quantities than b3h3 , can be omitted in the calculation. From (8), (10), (11), (12), we have qϕ ≈ − (Iy � + Iz)αv (h ¯h/2) + K Iz Mϕ, qθ ≈ − A T0 A + (b¯h/2) � + K Iy Mθ, (13) T0 A where A = �� A (x2x3) 2 dA = (b3 /12)(h3 /12), Iy = �� A x2 3 dA = h3b/12, and Iz = �� A x2 2 dA = b3h/12. According to the theory of beams, the strain energy of a beam in terms of displacements, void moment and thermal moment may be expressed as U = 1 � l �� l 1 D0ε11ε11 dA dx1 + (αv Mϕ,1Mϕ,1+ξv M 2 2(Iy + Iz) 2 ϕ − qϕ Mϕ − ρce M 2 θ ) dx1 0 A − 1 Iy + Iz � l 0 0 (bv Mϕ − β Mθ)(Izu2,11 + Iyu3,11) dx1 − 1 Iy + Iz � l 0 T0 mv Mϕ Mθ dx1. (14) Assume that the total kinetic energy of the beam is T = T1 +T2, in which T1 is caused by displacements and void moment, T2 is caused by rotation; these terms are given as T1 = ρhb 2 T2 = ρ 2 Define the symbol � l 0 � l � l ( ˙u 0 2 1 + ˙u2 2 + ˙u2 3 ) dx1 1 + ρχ ˙Mϕ 2(Iy + Iz) 0 ˙Mϕ dx1, � � � (x2 ˙u2,1) 2 + (x3 ˙u3,1) 2� dA dx1 = ρ � l� Iz( ˙u2,1) 2 2 + Iy( ˙u3,1) 2� dx1. A 1 D(Mθ) = 2(Iy + Iz) � l� t1 0 0 � K Mθ,1Mθ,1 − qθ Mθ T0 0 (15) � dt1 dx1. (16) Assume that the beam is subjected to an arbitrary transverse distributed load qi in the xi-direction, that the two ends are subjected to an axial force N0, and that all the forces are conservative. In the absence of external equilibrated force and heat source, the external work may be given as Let � l �B = 0 � l + 0 ρχ Iy + Iz W = � l 0 qiui dx1 + N0 2 ρhb � (u1|t=0 − u 0 1 ) ˙u1 − ˙u 0 1u1 + (u3|t=0 − u 0 3 ) ˙u3 − ˙u 0 3u3 � dx1 � l � (Mϕ|t=0−M 0 ϕ ) ˙Mϕ− ˙M 0 ϕ Mϕ � � l dx1+ 0 0 (u 2 2,1 + u2 3,1 ) dx1 + N0u1(0) − N0u1(a) (17) 1 Iy + Iz �� t1 0 ρce T0 (Mθ|t=0−M 0 θ )Mθ � dt dx1. (18) Generalized Hamilton principle for thermoelastic beams with voids. In all possible displacement fields ui(x1, t), volume fraction field ϕ(xi, t)and temperature field θ(xi, t) satisfying the geometric constraint conditions and having the appointed motions at the initial and final time, the actual displacements
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