形状记忆合金纤维复合材料梁非线性变形、热屈曲和振动
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摘要
研究具有形状记忆合金(SMA)纤维的复合材料梁非线性静变形、热屈曲和振动。采用Euler-Bernoulli梁理论、Timoshenko梁理论和Reddy高阶理论进行结构建模;根据Von-Kármán应变场理论描述梁的几何非线性;采用Brinson热力学本构方程计算SMA纤维的受限回复特性;基于Hamilton原理导出梁的非线性偏微分控制方程;采用Galerkin法导出两端简支对称铺层SMA纤维复合材料梁的非线性静变形、热屈曲和振动近似解。通过数值计算揭示SMA纤维含量、激励温度和初始应变对非线性静变形、热屈曲和振动的影响规律。研究表明,当长厚比较大时,剪切变形的影响很小,上述理论均可适用;但长厚比较小时,Euler-Bernoulli和Timoshenko梁理论的结果与Reddy高阶理论的结果相差较大,剪切变形的影响是显著的。
This paper presents an investigation into the nonlinear deformation,thermal buckling and vibration of the composite beams embedded with shape memory alloy(SMA)fibers.The beams were modeled by using Euler-Bernoulli,Timoshenko and Reddy higher-order beam theories.The Von Kármán type nonlinear strain-displacement equations were employed to describe the geometric nonlinearity of the beam.One-dimensional thermo-mechanical constitutive equation proposed by Brinson was utilized to calculate the recovery stress of the constrained SMA fibers.The nonlinear partial differential equations of the beams were derived based on the Hamilton's principle.The approximate solutions to the nonlinear deformation,thermo-buckling and vibration were obtained by using the Galerkin approach.Numerical calculation was carried out to show the effect of the volume fraction,actuation temperature,and the initial strain of SMA fibers on the nonlinear deformation,thermal buckling and vibration of the SMA fiber composite beams.Results of this study show that for a larger ratio of length to thickness,the shear-deformation effect is very insignificant and the three theories mentioned above are all applicable.However,for a low ratio oflength to thickness,the results of Euler-Bernoulli,Timoshenko and Reddy higher-order theories differ greatly,and the shear deformation is shown to have an important effect on the deformation,thermal buckling and vibration behaviors.
This paper presents an investigation into the nonlinear deformation,thermal buckling and vibration of the composite beams embedded with shape memory alloy(SMA)fibers.The beams were modeled by using Euler-Bernoulli,Timoshenko and Reddy higher-order beam theories.The Von Kármán type nonlinear strain-displacement equations were employed to describe the geometric nonlinearity of the beam.One-dimensional thermo-mechanical constitutive equation proposed by Brinson was utilized to calculate the recovery stress of the constrained SMA fibers.The nonlinear partial differential equations of the beams were derived based on the Hamilton's principle.The approximate solutions to the nonlinear deformation,thermo-buckling and vibration were obtained by using the Galerkin approach.Numerical calculation was carried out to show the effect of the volume fraction,actuation temperature,and the initial strain of SMA fibers on the nonlinear deformation,thermal buckling and vibration of the SMA fiber composite beams.Results of this study show that for a larger ratio of length to thickness,the shear-deformation effect is very insignificant and the three theories mentioned above are all applicable.However,for a low ratio oflength to thickness,the results of Euler-Bernoulli,Timoshenko and Reddy higher-order theories differ greatly,and the shear deformation is shown to have an important effect on the deformation,thermal buckling and vibration behaviors.
引文
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[6]MAJEWSKA K,ZAK A,OSTACHOWICZ W M.Active vibration control of a cracked composite beam by MSM actuators[J].Key Engineering Materials,2007,347(S9):697-702.
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[12]KANG K W,KIM H J,KIM J K.The role of shape memory alloy on impact response of glass/epoxy laminates under low temperature[J].Journal of Mechanical Science and Technology,2007,21(10):1682-1688.
[13]BRINSON L C,HUANG M S,BOLLER C,et al.Analysis of controlled beam deflections using SMA wires[J].Journal of Intelligent Material Systems and Structures,1997,8(1):12-25.
[14]RAGHAVAN J,BARTKIEWICZ T,BOYKO S,et al.Damping,tensile,and impact properties of superelastic shapememory alloy(SMA)fiber-reinforced polymer composites[J].Composites:Part B,2010,41(3):214-222.
[15]DU X W,SUN G,SUN S S.A study on the deflection of shape memory alloy(SMA)reinforced thermoviscoelastic beam[J].Composites Science and Technology,2004,64(9):1375-1381.
[16]MOHAMMAD M B,MORTEZA D,ALIREZA F.Vibration analysis of a beam with embedded shape memory alloy wires[J].Acta Mechanica Solida Sinica,2013,26(5):536-550.
[17]KHDEIR A A,REDDY J N.Buckling of cross-ply laminated beams with arbitrary boundary conditions[J].Composite Structures,1997,37(1):1-3.
[18]SONG S J,WAS A M.Effect of shear deformation on buckling and free vibration of laminated composite beams[J].Composite Structures,1997,37(1):33-43.
[19]GOYAL V K,KAPANIA R K.A shear-deformable beam element for the analysis of laminated composites[J].Finite Elements in Analysis&Design,2007,43:463-477.
[20]AYDOGDU M.Buckling analysis of cross-ply laminated beams with general boundary conditions by Ritz method[J].Composites Science and Technology,2006,66(10):1248-1255.
[21]AYDOGDU M.Thermal buckling analysis of cross-ply laminated composite beams with general boundary conditions[J].Composites Science and Technology,2007,67(6):1096-1104.
[22]EMAM S A.Analysis of shear-deformable composite beams in postbuckling[J].Composite Structures,2011,94(1):24-30.
[23]ASADI H,BODAGHI M,SHAKERI M,et al.An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams[J].European Journal of Mechanics A/Solids,2013,42(6):454-468.
[24]ASADI H,BODAGHI M,SHAKERI M,et al.Nonlinear dynamics of SMA-fiber-reinforced composite beams subjected to a primary/secondary-resonance excitation[J].Acta Mechanica,2015,266(2):437-455.
[25]REN Y S,DU C G,SHI Y Y.Nonlinear free and forced Vibration behavior of shear-deformable composite beams with shape memory alloy fibers[J/OL].Shock and Vibration,2016:ID 1056087.
[26]ZHONG Z W,CHEN R R,MEI C,et al.Buckling and postbuckling of shape memory alloy fiber-reinforced composite plates[C]/Proceedings of the ASME Aerospace Division,Mechanical Engineering Congress and Exposition,Chicago,1994:115-32.
[27]BRINSON L C.One-dimensional constitutive behavior of shape memory alloys:Thermomechanical derivation with non-constant material functions and redefined variable.[J].Intelligent Material Systems and Structures,1993,4(2):229-242.
[28]KE L L,YANG J,KITIPORNCHAI S.An analytical study on the nonlinear vibration of functionally graded beams[J].Meccanica,2010,45(6):743-752.
[2]LAU K,ZHOU L,TAO X.Control of natural frequencies of a clamped-clamped composite beam with embedded shape memory alloy wires[J].Composite Structures,2002,58(1):39-47.
[3]TSAI X Y,CHEN L W.Dynamic stability of a shape memory alloy wire reinforced composite beam[J].Composite Structures,2002,56(3):235-241.
[4]AOKI T,SHIMAMOTO A.Active vibration control using cantilever beam of smart matrix composite with embedded shape memory alloy[J].Key Engineering Materials,2004,270-273:2187-2192.
[5]ZHANG Y,ZHAO Y P.A study of composite beam with shape memory alloy arbitrarily embedded under thermal and mechanical loadings[J].Materials&Design,2007,28(4):1096-1115.
[6]MAJEWSKA K,ZAK A,OSTACHOWICZ W M.Active vibration control of a cracked composite beam by MSM actuators[J].Key Engineering Materials,2007,347(S9):697-702.
[7]LEE C,ZHUO H,HSU C.Lateral vibration of a composite stepped beam consisted of SMA helical spring based on equivalent Euler-Bernoulli beam theory[J].Journal of Sound&Vibration,2009,324(1):179-193.
[8]DOS REIS R,ROCHA SOUTO C,DE ARAUJO C,et al.Vibration attenuation in an epoxy smart composite beam with embedded NiTi shape memory wires[J].Material Science Forum,2010,643:7-13.
[9]REN Y S,YANG S L,WANG X H.Structural modeling of SMA fiber hybrid active thin-walled composite beams[J].Composite Structures,2009,91(1):120-130.
[10]任勇生,杜向红,姚文莉,等.旋转SMA纤维混杂复合材料薄壁梁的自由振动[J].固体力学学报,2011,32(3):258-575.REN Yongsheng,DU Xianghong,YAO Wenli,et al.Free vibration analysis of rotating composite thin-walled closed-section beams with SMA fibers[J].Acta Mechanica Solida Sinica,2011,32(3):258-575.
[11]SOHN J W,HAN Y M,CHOI S B,et al.Vibration and position tracking control of a flexible beam using SMA wire actuators[J].Journal of Vibration&Control,2009,15(2):263-281.
[12]KANG K W,KIM H J,KIM J K.The role of shape memory alloy on impact response of glass/epoxy laminates under low temperature[J].Journal of Mechanical Science and Technology,2007,21(10):1682-1688.
[13]BRINSON L C,HUANG M S,BOLLER C,et al.Analysis of controlled beam deflections using SMA wires[J].Journal of Intelligent Material Systems and Structures,1997,8(1):12-25.
[14]RAGHAVAN J,BARTKIEWICZ T,BOYKO S,et al.Damping,tensile,and impact properties of superelastic shapememory alloy(SMA)fiber-reinforced polymer composites[J].Composites:Part B,2010,41(3):214-222.
[15]DU X W,SUN G,SUN S S.A study on the deflection of shape memory alloy(SMA)reinforced thermoviscoelastic beam[J].Composites Science and Technology,2004,64(9):1375-1381.
[16]MOHAMMAD M B,MORTEZA D,ALIREZA F.Vibration analysis of a beam with embedded shape memory alloy wires[J].Acta Mechanica Solida Sinica,2013,26(5):536-550.
[17]KHDEIR A A,REDDY J N.Buckling of cross-ply laminated beams with arbitrary boundary conditions[J].Composite Structures,1997,37(1):1-3.
[18]SONG S J,WAS A M.Effect of shear deformation on buckling and free vibration of laminated composite beams[J].Composite Structures,1997,37(1):33-43.
[19]GOYAL V K,KAPANIA R K.A shear-deformable beam element for the analysis of laminated composites[J].Finite Elements in Analysis&Design,2007,43:463-477.
[20]AYDOGDU M.Buckling analysis of cross-ply laminated beams with general boundary conditions by Ritz method[J].Composites Science and Technology,2006,66(10):1248-1255.
[21]AYDOGDU M.Thermal buckling analysis of cross-ply laminated composite beams with general boundary conditions[J].Composites Science and Technology,2007,67(6):1096-1104.
[22]EMAM S A.Analysis of shear-deformable composite beams in postbuckling[J].Composite Structures,2011,94(1):24-30.
[23]ASADI H,BODAGHI M,SHAKERI M,et al.An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams[J].European Journal of Mechanics A/Solids,2013,42(6):454-468.
[24]ASADI H,BODAGHI M,SHAKERI M,et al.Nonlinear dynamics of SMA-fiber-reinforced composite beams subjected to a primary/secondary-resonance excitation[J].Acta Mechanica,2015,266(2):437-455.
[25]REN Y S,DU C G,SHI Y Y.Nonlinear free and forced Vibration behavior of shear-deformable composite beams with shape memory alloy fibers[J/OL].Shock and Vibration,2016:ID 1056087.
[26]ZHONG Z W,CHEN R R,MEI C,et al.Buckling and postbuckling of shape memory alloy fiber-reinforced composite plates[C]/Proceedings of the ASME Aerospace Division,Mechanical Engineering Congress and Exposition,Chicago,1994:115-32.
[27]BRINSON L C.One-dimensional constitutive behavior of shape memory alloys:Thermomechanical derivation with non-constant material functions and redefined variable.[J].Intelligent Material Systems and Structures,1993,4(2):229-242.
[28]KE L L,YANG J,KITIPORNCHAI S.An analytical study on the nonlinear vibration of functionally graded beams[J].Meccanica,2010,45(6):743-752.