均匀热载荷和机械载荷共同作用下具有温度依赖材料特性的层合梁的弹性力学解（英文）
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摘要
本文基于二维热弹性力学理论,考虑材料物性参数的温度依赖性,研究了均匀热荷载和机械荷载联合作用下简支叠层梁的精确解。首先将叠层梁沿各组分材料的界面划分为一系列单层,以保证每一单层材料的物理性能均匀。利用等效模型,将作用在单层内的均匀热荷载转化为作用于层两端的法向面力和横向热荷载。其次,利用状态空间法求得任一层内位移和应力的一般解。第三,根据相邻层界面的连续性条件,使用传递矩阵法递推得到顶层和底层位移和应力的关系,由作用于梁上下表面的机械荷载确定未知系数。研究了本文解的收敛性,并给出了与Timoshenko的解和有限元解的比较。最后通过数值算例,详细分析了随温度变化的材料物理参数对叠层梁力学性能的影响。
An exact solution for simply-supported laminated beams with material properties variable with temperature under a combination of uniform thermo-load and mechanical loads was investigated, based on the two-dimensional(2-D) thermo-elasticity theory. Firstly, the beam was divided into a series of layers with uniform material properties along the interfaces of the beam. The uniform thermo-load acted on each layer was transformed into a combination of the normal surface forces acted at the two ends and the transverse thermo-load. Secondly, the state space method was employed to obtain the general solutions of displacements and stresses in an arbitrary layer. Thirdly, based on the interfacial continuity conditions between adjacent layers, the relations of displacement and stress components between the top and bottom layers of the beam were recursively derived by use of the transfer-matrix method. The unknowns in the solutions can be solved by the mechanical loads acted on the top and bottom surfaces. The convergence of the present solutions was checked. The comparative study of the present solutions with the Timoshenko's solutions and the finite element(FE) solutions was carried out. The effects of material properties variable with temperature on the thermo-elastic behavior of laminated beams were discussed in detail.
An exact solution for simply-supported laminated beams with material properties variable with temperature under a combination of uniform thermo-load and mechanical loads was investigated, based on the two-dimensional(2-D) thermo-elasticity theory. Firstly, the beam was divided into a series of layers with uniform material properties along the interfaces of the beam. The uniform thermo-load acted on each layer was transformed into a combination of the normal surface forces acted at the two ends and the transverse thermo-load. Secondly, the state space method was employed to obtain the general solutions of displacements and stresses in an arbitrary layer. Thirdly, based on the interfacial continuity conditions between adjacent layers, the relations of displacement and stress components between the top and bottom layers of the beam were recursively derived by use of the transfer-matrix method. The unknowns in the solutions can be solved by the mechanical loads acted on the top and bottom surfaces. The convergence of the present solutions was checked. The comparative study of the present solutions with the Timoshenko's solutions and the finite element(FE) solutions was carried out. The effects of material properties variable with temperature on the thermo-elastic behavior of laminated beams were discussed in detail.
引文
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[10]LI Guo-qiang,JIANG Shou-chao,YIN Ying-zhi,CHEN Kai,LI Ming-fei.Experimental studies on the properties of constructional steel at elevated temperatures[J].Journal of Structural Engineering,2003,129(12):1717-1721.DOI:10.1061/(ASCE)0733-9445(2003)129:12(1717).
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[14]REDDY J N,ARCINIEGA R A.Shear deformation plate and shell theories:From Stavsky to present[J].Mechanics of Advanced Materials and Structures,2004,11(6):535-582.DOI:10.1080/15376490490452777.
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[16]SAYYAD A S,GHUGAL Y M.Bending,buckling and free vibration of laminated composite and sandwich beams:Acritical review of literature[J].Composite Structures,2017,171:486-504.DOI:10.1016/j.compstruct.2017.03.053.
[17]TIMOSHENKO S.Analysis of bi-metal thermostats[J].Journal of the Optical Society of America,1925,11(3):233-255.DOI:10.1364/JOSA.11.000233.
[18]MOHANDES M,GHASEMI A R.Modified couple stress theory and finite strain assumption for nonlinear free vibration and bending of micro/nanolaminated composite Euler-Bernoulli beam under thermal loading[J].Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,2016,231(21):4044-4056.DOI:10.1177/0954406216656884.
[19]FU Y,WANG J,HU S.Analytical solutions of thermal buckling and postbuckling of symmetric laminated composite beams with various boundary conditions[J].Acta Mechanica,2014,225(1):13-29.DOI:10.1007/s00707-013-0941-z.
[20]MORADI S,MANSOURI M H.Thermal buckling analysis of shear deformable laminated orthotropic plates by differential quadrature[J].Steel and Composite Structures,2012,12(2):129-147.DOI:10.12989/scs.2012.12.2.129.
[21]SHATERZADEH A R,ABOLGHASEMI S,REZAEI R.Finite element analysis of thermal buckling of rectangular laminated composite plates with circular cut-out[J].Journal of Thermal Stresses,2014,37(5):604-623.DOI:10.1080/01495739.2014.885322.
[22]HAN J W,KIM J S,CHO M.New enhanced first-order shear deformation theory for thermo-mechanical analysis of laminated composite and sandwich plates[J].Composites Part B,2017,116:422-450.DOI:10.1016/j.compositesb.2016.10.087.
[23]KHDEIR A A,REDDY J N.Thermal stresses and deflections of cross-ply laminated plates using refined plate theories[J].Journal of Thermal Stresses,1991,14(4):419-438.DOI:10.1080/01495739108927077.
[24]NAJAFIZADEH M M,HEYDARI H R.Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory[J].European Journal of Mechanics-A/Solids,2004,23(6):1085-1100.DOI:10.1016/j.euromechsol.2004.08.004.
[25]SINGH S,SINGH J,SHUKLA K K.Buckling of laminated composite plates subjected to mechanical and thermal loads using meshless collocations[J].Journal of Mechanical Science and Technology,2013,27(2):327-336.DOI:10.1007/s12206-012-1249-y.
[26]KAPURIA S,DUMIR P C,AHMED A.An efficient higher order zigzag theory for composite and sandwich beams subjected to thermal loading[J].International Journal of Solids and Structures,2003,40(24):6613-6631.DOI:10.1016/j.ijsolstr.2003.08.014.
[27]GHUGAL Y M,KULKARNI S K.Thermal response of symmetric cross-ply laminated plates subjected to linear and non-linear thermo-mechanical loads[J].Journal of Thermal Stresses,2013,36(5):466-479.DOI:10.1080/01495739.2013.770664.
[28]GHUGAL Y M,KULKARNI S K.Flexural analysis of cross-ply laminated plates subjected to nonlinear thermal and mechanical loadings[J].Acta Mechanica,2013,224(3):675-690.DOI:10.1007/s00707-012-0774-1.
[29]SAYYAD A S,SHINDE B M,GHUGAL Y M.Thermoelastic bending analysis of orthotropic plates using hyperbolic shear deformation theory[J].Composites:Mechanics,Computations,Applications:An International Journal,2013,4(3):257-278.DOI:10.1615/Comp Mech Comput Appl Int J.v4.i3.50.
[30]SAYYAD A S,GHUGAL Y M.,MHASKE B A.Afour-variable plate theory for thermoelastic bending analysis of laminated composite plates[J].Journal of Thermal Stresses,2015,38(8):904-925.DOI:10.1080/01495739.2015.1040310.
[31]XU Ye-peng,ZHOU Ding.Two-dimensional thermoelastic analysis of beams with variable thickness subjected to thermo-mechanical loads[J].Applied Mathematical Modelling,2012,36(12):5818-5829.DOI:10.1016/j.apm.2012.01.048.
[32]QIAN Hai,ZHOU Ding,LIU Wei-qing,FANG Hai.Elasticity solution of laminated beams subjected to thermo-loads[J].Journal of Central South University,2015,22:2297-2305.DOI:10.1007/s11771-015-2754-9.
[33]QIAN Hai,ZHOU Ding,LIU Wei-qing,FANG Hai,LUWei-dong.3-D elasticity solutions of layered rectangular plates subjected to thermo-loads[J].Journal of Thermal Stresses,2015,38(4):377-398.DOI:10.1080/01495739.2014.985570.
[34]LV Chao-feng,CHEN Wei-qiu,ZHONG Zheng.Twodimensional thermoelasticity solution for functionally graded thick beams[J].Science in China Series G:Physics,Mechanics and Astronomy,2006,49(4):451-460.DOI:10.1007/s11433-006-0451-2.
[35]PILIPCHUK V N,BERDICHEVSKY V L,IBRAHIM R A.Thermo-mechanical coupling in cylindrical bending of sandwich plates[J].Composite Structures,2010,92(11):2632-2640.DOI:10.1016/j.compstruct.2010.03.007.
[36]ALIBEIGLOO A.Thermo elasticity solution of sandwich circular plate with functionally graded core using generalized differential quadrature method[J].Composite Structures,2016,136:229-240.DOI:10.1016/j.compstruct.2015.10.012.
[2]SPERANZINI E,AGNETTI S.Strengthening of glass beams with steel reinforced polymer(SRP)[J].Composites Part B:Engineering,2014,67:280-289.DOI:10.1016/j.compositesb.2014.06.035.
[3]JHA D K,KANT T,SINGH R K.A critical review of recent research on functionally graded plates[J].Composite Structures,2013,96:833-849.DOI:10.1016/j.compstruct.2012.09.001.
[4]LIU Wu-xiang.Analysis of steady heat conduction for 3Daxisymmetric functionally graded circular plate[J].Journal of Central South University,2013,20(6):1616-1622.DOI:10.1007/s11771-013-1654-0.
[5]SU H,BANERJEE J R,CHEUNG C W.Dynamic stiffness formulation and free vibration analysis of functionally graded beams[J].Composite Structures,2013,106:854-862.DOI:10.1016/j.compstruct.2013.06.029.
[6]TRINH L C,VO T P,THAI H T,NGUYEN T K.An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads[J].Composites Part B:Engineering,2016,100:152-163.DOI:10.1016/j.compositesb.2016.06.067.
[7]ABBAS I A.Generalized thermoelastic interaction in functional graded material with fractional order three-phase lag heat transfer[J].Journal of Central South University,2015,22(5):1606-1613.DOI:10.1007/s11771-015-2677-5.
[8]CHANG Y F,CHEN Y H,SHEU M S,YAO G C.Residual stress-strain relationship for concrete after exposure to high temperatures[J].Cement and Concrete Research,2006,36(10):1999-2005.DOI:10.1016/j.cemconres.2006.05.029.
[9]YANG Xiu-ying,ZHAO Jin-cheng.Experimental study on stress-strain-temperature models for structural steel[J].Journal of Harbin Institute of Technology(New Series),2011,18(1):6-10.DOI:10.11916/j.issn.1005-9113.2011.01.002.
[10]LI Guo-qiang,JIANG Shou-chao,YIN Ying-zhi,CHEN Kai,LI Ming-fei.Experimental studies on the properties of constructional steel at elevated temperatures[J].Journal of Structural Engineering,2003,129(12):1717-1721.DOI:10.1061/(ASCE)0733-9445(2003)129:12(1717).
[11]TAUCHERT T R.Thermally induced flexure,buckling,and vibration of plates[J].Applied Mechanics Reviews,1991,44(8):347-360.DOI:10.1115/1.3119508.
[12]GHUGAL Y M,SHIMPI R P.A review of refined shear deformation theories of isotropic and anisotropic laminated beams[J].Journal of Reinforced Plastics and Composites,2001,20(3):255-272.DOI:10.1106/N95G-ERA1-A1CM-RD7E.
[13]GHUGAL Y M,SHIMPI R P.A review of refined shear deformation theories of isotropic and anisotropic laminated plates[J].Journal of Reinforced Plastics and Composites,2002,21(9):775-813.DOI:10.1177/073168402128988481.
[14]REDDY J N,ARCINIEGA R A.Shear deformation plate and shell theories:From Stavsky to present[J].Mechanics of Advanced Materials and Structures,2004,11(6):535-582.DOI:10.1080/15376490490452777.
[15]CARRERA E,BRISCHETTO S.A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates[J].Applied Mechanics Reviews,2009,62(1):010803.DOI:10.1115/1.3013824.
[16]SAYYAD A S,GHUGAL Y M.Bending,buckling and free vibration of laminated composite and sandwich beams:Acritical review of literature[J].Composite Structures,2017,171:486-504.DOI:10.1016/j.compstruct.2017.03.053.
[17]TIMOSHENKO S.Analysis of bi-metal thermostats[J].Journal of the Optical Society of America,1925,11(3):233-255.DOI:10.1364/JOSA.11.000233.
[18]MOHANDES M,GHASEMI A R.Modified couple stress theory and finite strain assumption for nonlinear free vibration and bending of micro/nanolaminated composite Euler-Bernoulli beam under thermal loading[J].Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,2016,231(21):4044-4056.DOI:10.1177/0954406216656884.
[19]FU Y,WANG J,HU S.Analytical solutions of thermal buckling and postbuckling of symmetric laminated composite beams with various boundary conditions[J].Acta Mechanica,2014,225(1):13-29.DOI:10.1007/s00707-013-0941-z.
[20]MORADI S,MANSOURI M H.Thermal buckling analysis of shear deformable laminated orthotropic plates by differential quadrature[J].Steel and Composite Structures,2012,12(2):129-147.DOI:10.12989/scs.2012.12.2.129.
[21]SHATERZADEH A R,ABOLGHASEMI S,REZAEI R.Finite element analysis of thermal buckling of rectangular laminated composite plates with circular cut-out[J].Journal of Thermal Stresses,2014,37(5):604-623.DOI:10.1080/01495739.2014.885322.
[22]HAN J W,KIM J S,CHO M.New enhanced first-order shear deformation theory for thermo-mechanical analysis of laminated composite and sandwich plates[J].Composites Part B,2017,116:422-450.DOI:10.1016/j.compositesb.2016.10.087.
[23]KHDEIR A A,REDDY J N.Thermal stresses and deflections of cross-ply laminated plates using refined plate theories[J].Journal of Thermal Stresses,1991,14(4):419-438.DOI:10.1080/01495739108927077.
[24]NAJAFIZADEH M M,HEYDARI H R.Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory[J].European Journal of Mechanics-A/Solids,2004,23(6):1085-1100.DOI:10.1016/j.euromechsol.2004.08.004.
[25]SINGH S,SINGH J,SHUKLA K K.Buckling of laminated composite plates subjected to mechanical and thermal loads using meshless collocations[J].Journal of Mechanical Science and Technology,2013,27(2):327-336.DOI:10.1007/s12206-012-1249-y.
[26]KAPURIA S,DUMIR P C,AHMED A.An efficient higher order zigzag theory for composite and sandwich beams subjected to thermal loading[J].International Journal of Solids and Structures,2003,40(24):6613-6631.DOI:10.1016/j.ijsolstr.2003.08.014.
[27]GHUGAL Y M,KULKARNI S K.Thermal response of symmetric cross-ply laminated plates subjected to linear and non-linear thermo-mechanical loads[J].Journal of Thermal Stresses,2013,36(5):466-479.DOI:10.1080/01495739.2013.770664.
[28]GHUGAL Y M,KULKARNI S K.Flexural analysis of cross-ply laminated plates subjected to nonlinear thermal and mechanical loadings[J].Acta Mechanica,2013,224(3):675-690.DOI:10.1007/s00707-012-0774-1.
[29]SAYYAD A S,SHINDE B M,GHUGAL Y M.Thermoelastic bending analysis of orthotropic plates using hyperbolic shear deformation theory[J].Composites:Mechanics,Computations,Applications:An International Journal,2013,4(3):257-278.DOI:10.1615/Comp Mech Comput Appl Int J.v4.i3.50.
[30]SAYYAD A S,GHUGAL Y M.,MHASKE B A.Afour-variable plate theory for thermoelastic bending analysis of laminated composite plates[J].Journal of Thermal Stresses,2015,38(8):904-925.DOI:10.1080/01495739.2015.1040310.
[31]XU Ye-peng,ZHOU Ding.Two-dimensional thermoelastic analysis of beams with variable thickness subjected to thermo-mechanical loads[J].Applied Mathematical Modelling,2012,36(12):5818-5829.DOI:10.1016/j.apm.2012.01.048.
[32]QIAN Hai,ZHOU Ding,LIU Wei-qing,FANG Hai.Elasticity solution of laminated beams subjected to thermo-loads[J].Journal of Central South University,2015,22:2297-2305.DOI:10.1007/s11771-015-2754-9.
[33]QIAN Hai,ZHOU Ding,LIU Wei-qing,FANG Hai,LUWei-dong.3-D elasticity solutions of layered rectangular plates subjected to thermo-loads[J].Journal of Thermal Stresses,2015,38(4):377-398.DOI:10.1080/01495739.2014.985570.
[34]LV Chao-feng,CHEN Wei-qiu,ZHONG Zheng.Twodimensional thermoelasticity solution for functionally graded thick beams[J].Science in China Series G:Physics,Mechanics and Astronomy,2006,49(4):451-460.DOI:10.1007/s11433-006-0451-2.
[35]PILIPCHUK V N,BERDICHEVSKY V L,IBRAHIM R A.Thermo-mechanical coupling in cylindrical bending of sandwich plates[J].Composite Structures,2010,92(11):2632-2640.DOI:10.1016/j.compstruct.2010.03.007.
[36]ALIBEIGLOO A.Thermo elasticity solution of sandwich circular plate with functionally graded core using generalized differential quadrature method[J].Composite Structures,2016,136:229-240.DOI:10.1016/j.compstruct.2015.10.012.
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