基于状态向量法分析功能梯度一维六方压电准晶层合板的静态响应（英文）
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摘要
目的:功能梯度准晶材料有助于减缓层合板界面处的应力集中现象,提高层间粘接强度,从而提升层合板表面的耐磨性。本文旨在建立功能梯度压电准晶层合板的力学模型,并研究功能梯度变化和叠放顺序对层合板的影响。创新点:1.首次将状态向量法推广到功能梯度压电准晶板的分析中;2.假设功能梯度函数的变化形式为幂函数和指数函数;3.在准晶层中观察到声子场应力和电势的不敏感点。方法:1.通过联立三大基本方程,推导出准晶板的状态方程,并求解该微分方程,得到单层准晶板的解析解;2.通过引入功能梯度函数,使解析解中的描述各材料特性的值能够沿厚度方向呈现梯度变化;3.采用传递矩阵法,求出多层准晶板的解析解;4.通过仿真模拟,将所得结果与已有文献进行对比,验证所提方法的可行性和有效性。结论:1.准晶层合板中的功能梯度效应随着梯度参数的增加而增大,且材料参数的变化对声子场、相位子场以及电场的响应均产生影响。2.在功能梯度效应下,从准晶层中观察到了声子场应力和电势的不敏感点。3.与准晶作为中间层相比,准晶作为表层时机械载荷引起的位移响应更小。研究结果可以为压电准晶元器件的设计提供理论参考。
The effect of the non-homogeneity of material properties has been considered the important variation mechanism in the static responses of quasicrystal structures, but the existing theoretical model for it is unable to simulate the material change format beyond the exponential function. In this paper, we create a new model of functionally graded multilayered 1 D piezoelectric quasicrystal plates using the state vector approach, in which varying functionally graded electro-elastic properties can be extended from exponential to linear and higher order in the thickness direction. Based on the state equations, an analytical solution for a single plate has been derived, and the result for the corresponding multilayered case is obtained utilizing the propagator matrix method. The present study shows, in particular, that coefficient orders of two varying functions(the power function and the exponential function) of the material gradient provide the ability to tailor the mechanical behaviors in the system's phonon, phason, and electric fields. Moreover, the insensitive points of phonon stress and electric potential under functionally graded effects in the quasicrystal layer are observed. In addition, the influences of stacking sequences and discontinuity of horizontal stress are explored in the simulation by the new model. The results are very useful for the design and understanding of the characterization of functionally graded piezoelectric quasicrystal materials in their applications to multilayered systems.
The effect of the non-homogeneity of material properties has been considered the important variation mechanism in the static responses of quasicrystal structures, but the existing theoretical model for it is unable to simulate the material change format beyond the exponential function. In this paper, we create a new model of functionally graded multilayered 1 D piezoelectric quasicrystal plates using the state vector approach, in which varying functionally graded electro-elastic properties can be extended from exponential to linear and higher order in the thickness direction. Based on the state equations, an analytical solution for a single plate has been derived, and the result for the corresponding multilayered case is obtained utilizing the propagator matrix method. The present study shows, in particular, that coefficient orders of two varying functions(the power function and the exponential function) of the material gradient provide the ability to tailor the mechanical behaviors in the system's phonon, phason, and electric fields. Moreover, the insensitive points of phonon stress and electric potential under functionally graded effects in the quasicrystal layer are observed. In addition, the influences of stacking sequences and discontinuity of horizontal stress are explored in the simulation by the new model. The results are very useful for the design and understanding of the characterization of functionally graded piezoelectric quasicrystal materials in their applications to multilayered systems.
引文
Alibeigloo A,2018.Thermo elasticity solution of functionally graded,solid,circular,and annular plates integrated with piezoelectric layers using the differential quadrature method.Mechanics of Advanced Materials and Structures,25(9):766-784.https://doi.org/10.1080/15376494.2017.1308585
Altay G,D?kmeci MC,2012.On the fundamental equations of piezoelasticity of quasicrystal media.International Journal of Solids and Structures,49(23-24):3255-3262.https://doi.org/10.1016/j.ijsolstr.2012.06.016
Chan KC,Qu NS,Zhu D,2002.Fabrication of graded nickelquasicrystal composite by electrodeposition.Transactions of the IMF,80(6):210-213.https://doi.org/10.1080/00202967.2002.11871470
Chen WQ,Lee KY,2003.Alternative state space formulations for magnetoelectric thermoelasticity with transverse isotropy and the application to bending analysis of nonhomogeneous plates.International Journal of Solids and Structures,40(21):5689-5705.https://doi.org/10.1016/S0020-7683(03)00339-1
Ding DH,Yang WG,Hu CZ,et al.,1993.Generalized elasticity theory of quasicrystals.Physical Review B,48(10):7003-7010.https://doi.org/10.1103/PhysRevB.48.7003
Dubois JM,2005.Useful Quasicrystals.World Scientific,Singapore,Singapore,p.45-56.
Fan TY,2010.Mathematical Theory of Elasticity of Quasicrystals and Its Applications.Science Press,Beijing,China,p.118-120(in Chinese).
Fan TY,2013.Mathematical theory and methods of mechanics of quasicrystalline materials.Engineering,5(4):407-448.https://doi.org/10.4236/eng.2013.54053
Fujiwara T,de Laissardière GT,Yamamoto S,1994.Electronic structure and electron transport in quasicrystals.Materials Science Forum,150-151:387-394.https://doi.org/10.4028/www.scientific.net/msf.150-151.387
Gao Y,Zhao BS,2009.General solutions of three-dimensional problems for two-dimensional quasicrystals.Applied Mathematical Modelling,33(8):3382-3391.https://doi.org/10.1016/j.apm.2008.11.001
Guo JH,Chen JY,Pan EN,2016.Size-dependent behavior of functionally graded anisotropic composite plates.International Journal of Engineering Science,106:110-124.https://doi.org/10.1016/j.ijengsci.2016.05.008
Hu CZ,Wang RH,Ding DH,et al.,1997.Piezoelectric effects in quasicrystals.Physical Review B,56(5):2463-2468.https://doi.org/10.1103/PhysRevB.56.2463
Hu WF,Liu YH,2015.A new state space solution for rectangular thick laminates with clamped edges.Chinese Journal of Theoretical and Applied Mechanics,47(5):762-771(in Chinese).https://doi.org/10.6052/0459-1879-15-033
Levinson M,Cooke DW,1983.Thick rectangular plates-I:the generalized Navier solution.International Journal of Mechanical Sciences,25(3):199-205.https://doi.org/10.1016/0020-7403(83)90093-0
Li LH,Liu GT,2012.Stroh formalism for icosahedral quasicrystal and its application.Physics Letters A,376(8-9):987-990.https://doi.org/10.1016/j.physleta.2012.01.027
Li XF,Xie LY,Fan TY,2013.Elasticity and dislocations in quasicrystals with 18-fold symmetry.Physics Letters A,377(39):2810-2814.https://doi.org/10.1016/j.physleta.2013.08.033
Li XY,Ding HJ,Chen WQ,2006.Pure bending of simply supported circular plate of transversely isotropic functionally graded material.Journal of Zhejiang University SCIENCE A,7(8):1324-1328.https://doi.org/10.1631/jzus.2006.A1324
Li XY,Li PD,Wu TH,et al.,2014.Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect.Physics Letters A,378(10):826-834.https://doi.org/10.1016/j.physleta.2014.01.016
Li Y,Yang LZ,Gao Y,2017.An exact solution for a functionally graded multilayered one-dimensional orthorhombic quasicrystal plate.Acta Mechanica,in press.https://doi.org/10.1007/s00707-017-2028-8
Louzguine-Luzgin DV,Inoue A,2008.Formation and properties of quasicrystals.Annual Review of Materials Research,38:403-423.https://doi.org/10.1146/annurev.matsci.38.060407.130318
Mikaeeli S,Behjat B,2016.Three-dimensional analysis of thick functionally graded piezoelectric plate using EFGmethod.Composite Structures,154:591-599.https://doi.org/10.1016/j.compstruct.2016.07.067
Móricz F,1989.OnΛ2-strong convergence of numerical sequences and Fourier series.Acta Mathematica Hungarica,54(3-4):319-327.https://doi.org/10.1007/BF01952063
Pan E,Han F,2005.Exact solution for functionally graded and layered magneto-electro-elastic plates.International Journal of Engineering Science,43(3-4):321-339.https://doi.org/10.1016/j.ijengsci.2004.09.006
Qing GH,Wang L,Zhang XH,2017.Analytical solution of composite laminates with two opposite sides clamped and other sides free boundary.Machinery Design&Manufacture,(2):161-164(in Chinese).https://doi.org/10.3969/j.issn.1001-3997.2017.02.045
Shechtman D,Blech I,Gratias D,et al.,1984.Metallic phase with long-range orientational order and no translational symmetry.Physical Review Letters,53(20):1951-1953.https://doi.org/10.1103/PhysRevLett.53.1951
Sheng HY,Wang H,Ye JQ,2007.State space solution for thick laminated piezoelectric plates with clamped and electric open-circuited boundary conditions.International Journal of Mechanical Sciences,49(7):806-818.https://doi.org/10.1016/j.ijmecsci.2006.11.012
Sladek J,Sladek V,Pan E,2013.Bending analyses of 1Dorthorhombic quasicrystal plates.International Journal of Solids and Structures,50(24):3975-3983.https://doi.org/10.1016/j.ijsolstr.2013.08.006
Sun TY,Guo JH,Zhang XY,2018.Static deformation of a multilayered one-dimensional hexagonal quasicrystal plate with piezoelectric effect.Applied Mathematics and Mechanics(English Edition),39(3):335-352.https://doi.org/10.1007/s10483-018-2309-9
Suresh S,Mortensen A,1998.Fundamentals of Functionally Graded Materials:Processing and Thermomechanical Behavior of Graded Metals and Metal-ceramic Composites.IOM Communications,London,UK,p.156-163.
Timoshenko SP,Goodier JN,1970.Theory of Elasticity.McGraw-Hill,New York,USA,p.78-82.
Wang JG,Chen LF,Fang SS,2003.State vector approach to analysis of multilayered magneto-electro-elastic plates.International Journal of Solids and Structures,40(7):1669-1680.https://doi.org/10.1016/S0020-7683(03)00027-1
Wang X,Zhang JQ,Guo XM,2005.Two kinds of contact problems in decagonal quasicrystalline materials of point group 10 mm.Acta Mechanica Sinica,37(2):169-174(in Chinese).https://doi.org/10.3321/j.issn:0459-1879.2005.02.007
Xu WS,Wu D,Gao Y,2017.Fundamental elastic field in an infinite plane of two-dimensional piezoelectric quasicrystal subjected to multi-physics loads.Applied Mathematical Modelling,52:186-196.https://doi.org/10.1016/j.apm.2017.07.014
Yang B,Ding HJ,Chen WQ,2012.Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported.Applied Mathematical Modelling,36(1):488-503.https://doi.org/10.1016/j.apm.2011.07.020
Yang LZ,Gao Y,Pan EN,et al.,2015.An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate.Acta Mechanica,226(11):3611-3621.https://doi.org/10.1007/s00707-015-1395-2
Yaslan H?,2013.Equations of anisotropic elastodynamics in3D quasicrystals as a symmetric hyperbolic system:deriving the time-dependent fundamental solutions.Applied Mathematical Modelling,37(18-19):8409-8418.https://doi.org/10.1016/j.apm.2013.03.039
Ying J,LüCF,Lim CW,2009.3D thermoelasticity solutions for functionally graded thick plates.Journal of Zhejiang University SCIENCE A,10(3):327-336.https://doi.org/10.1631/jzus.A0820406
Zhao MH,Dang HY,Fan CY,et al.,2017.Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material.Part 1:theoretical solution.Engineering Fracture Mechanics,179:59-78.https://doi.org/10.1016/j.engfracmech.2017.04.019
Zhao MH,Li Y,Fan CY,et al.,2018.Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects.Part I:theoretical solutions.Applied Mathematical Modelling,57:583-602.https://doi.org/10.1016/j.apm.2017.07.023
Zhou YB,Li XF,2018.Two collinear mode-III cracks in one-dimensional hexagonal piezoelectric quasicrystal strip.Engineering Fracture Mechanics,189:133-147.https://doi.org/10.1016/j.engfracmech.2017.10.030
Altay G,D?kmeci MC,2012.On the fundamental equations of piezoelasticity of quasicrystal media.International Journal of Solids and Structures,49(23-24):3255-3262.https://doi.org/10.1016/j.ijsolstr.2012.06.016
Chan KC,Qu NS,Zhu D,2002.Fabrication of graded nickelquasicrystal composite by electrodeposition.Transactions of the IMF,80(6):210-213.https://doi.org/10.1080/00202967.2002.11871470
Chen WQ,Lee KY,2003.Alternative state space formulations for magnetoelectric thermoelasticity with transverse isotropy and the application to bending analysis of nonhomogeneous plates.International Journal of Solids and Structures,40(21):5689-5705.https://doi.org/10.1016/S0020-7683(03)00339-1
Ding DH,Yang WG,Hu CZ,et al.,1993.Generalized elasticity theory of quasicrystals.Physical Review B,48(10):7003-7010.https://doi.org/10.1103/PhysRevB.48.7003
Dubois JM,2005.Useful Quasicrystals.World Scientific,Singapore,Singapore,p.45-56.
Fan TY,2010.Mathematical Theory of Elasticity of Quasicrystals and Its Applications.Science Press,Beijing,China,p.118-120(in Chinese).
Fan TY,2013.Mathematical theory and methods of mechanics of quasicrystalline materials.Engineering,5(4):407-448.https://doi.org/10.4236/eng.2013.54053
Fujiwara T,de Laissardière GT,Yamamoto S,1994.Electronic structure and electron transport in quasicrystals.Materials Science Forum,150-151:387-394.https://doi.org/10.4028/www.scientific.net/msf.150-151.387
Gao Y,Zhao BS,2009.General solutions of three-dimensional problems for two-dimensional quasicrystals.Applied Mathematical Modelling,33(8):3382-3391.https://doi.org/10.1016/j.apm.2008.11.001
Guo JH,Chen JY,Pan EN,2016.Size-dependent behavior of functionally graded anisotropic composite plates.International Journal of Engineering Science,106:110-124.https://doi.org/10.1016/j.ijengsci.2016.05.008
Hu CZ,Wang RH,Ding DH,et al.,1997.Piezoelectric effects in quasicrystals.Physical Review B,56(5):2463-2468.https://doi.org/10.1103/PhysRevB.56.2463
Hu WF,Liu YH,2015.A new state space solution for rectangular thick laminates with clamped edges.Chinese Journal of Theoretical and Applied Mechanics,47(5):762-771(in Chinese).https://doi.org/10.6052/0459-1879-15-033
Levinson M,Cooke DW,1983.Thick rectangular plates-I:the generalized Navier solution.International Journal of Mechanical Sciences,25(3):199-205.https://doi.org/10.1016/0020-7403(83)90093-0
Li LH,Liu GT,2012.Stroh formalism for icosahedral quasicrystal and its application.Physics Letters A,376(8-9):987-990.https://doi.org/10.1016/j.physleta.2012.01.027
Li XF,Xie LY,Fan TY,2013.Elasticity and dislocations in quasicrystals with 18-fold symmetry.Physics Letters A,377(39):2810-2814.https://doi.org/10.1016/j.physleta.2013.08.033
Li XY,Ding HJ,Chen WQ,2006.Pure bending of simply supported circular plate of transversely isotropic functionally graded material.Journal of Zhejiang University SCIENCE A,7(8):1324-1328.https://doi.org/10.1631/jzus.2006.A1324
Li XY,Li PD,Wu TH,et al.,2014.Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect.Physics Letters A,378(10):826-834.https://doi.org/10.1016/j.physleta.2014.01.016
Li Y,Yang LZ,Gao Y,2017.An exact solution for a functionally graded multilayered one-dimensional orthorhombic quasicrystal plate.Acta Mechanica,in press.https://doi.org/10.1007/s00707-017-2028-8
Louzguine-Luzgin DV,Inoue A,2008.Formation and properties of quasicrystals.Annual Review of Materials Research,38:403-423.https://doi.org/10.1146/annurev.matsci.38.060407.130318
Mikaeeli S,Behjat B,2016.Three-dimensional analysis of thick functionally graded piezoelectric plate using EFGmethod.Composite Structures,154:591-599.https://doi.org/10.1016/j.compstruct.2016.07.067
Móricz F,1989.OnΛ2-strong convergence of numerical sequences and Fourier series.Acta Mathematica Hungarica,54(3-4):319-327.https://doi.org/10.1007/BF01952063
Pan E,Han F,2005.Exact solution for functionally graded and layered magneto-electro-elastic plates.International Journal of Engineering Science,43(3-4):321-339.https://doi.org/10.1016/j.ijengsci.2004.09.006
Qing GH,Wang L,Zhang XH,2017.Analytical solution of composite laminates with two opposite sides clamped and other sides free boundary.Machinery Design&Manufacture,(2):161-164(in Chinese).https://doi.org/10.3969/j.issn.1001-3997.2017.02.045
Shechtman D,Blech I,Gratias D,et al.,1984.Metallic phase with long-range orientational order and no translational symmetry.Physical Review Letters,53(20):1951-1953.https://doi.org/10.1103/PhysRevLett.53.1951
Sheng HY,Wang H,Ye JQ,2007.State space solution for thick laminated piezoelectric plates with clamped and electric open-circuited boundary conditions.International Journal of Mechanical Sciences,49(7):806-818.https://doi.org/10.1016/j.ijmecsci.2006.11.012
Sladek J,Sladek V,Pan E,2013.Bending analyses of 1Dorthorhombic quasicrystal plates.International Journal of Solids and Structures,50(24):3975-3983.https://doi.org/10.1016/j.ijsolstr.2013.08.006
Sun TY,Guo JH,Zhang XY,2018.Static deformation of a multilayered one-dimensional hexagonal quasicrystal plate with piezoelectric effect.Applied Mathematics and Mechanics(English Edition),39(3):335-352.https://doi.org/10.1007/s10483-018-2309-9
Suresh S,Mortensen A,1998.Fundamentals of Functionally Graded Materials:Processing and Thermomechanical Behavior of Graded Metals and Metal-ceramic Composites.IOM Communications,London,UK,p.156-163.
Timoshenko SP,Goodier JN,1970.Theory of Elasticity.McGraw-Hill,New York,USA,p.78-82.
Wang JG,Chen LF,Fang SS,2003.State vector approach to analysis of multilayered magneto-electro-elastic plates.International Journal of Solids and Structures,40(7):1669-1680.https://doi.org/10.1016/S0020-7683(03)00027-1
Wang X,Zhang JQ,Guo XM,2005.Two kinds of contact problems in decagonal quasicrystalline materials of point group 10 mm.Acta Mechanica Sinica,37(2):169-174(in Chinese).https://doi.org/10.3321/j.issn:0459-1879.2005.02.007
Xu WS,Wu D,Gao Y,2017.Fundamental elastic field in an infinite plane of two-dimensional piezoelectric quasicrystal subjected to multi-physics loads.Applied Mathematical Modelling,52:186-196.https://doi.org/10.1016/j.apm.2017.07.014
Yang B,Ding HJ,Chen WQ,2012.Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported.Applied Mathematical Modelling,36(1):488-503.https://doi.org/10.1016/j.apm.2011.07.020
Yang LZ,Gao Y,Pan EN,et al.,2015.An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate.Acta Mechanica,226(11):3611-3621.https://doi.org/10.1007/s00707-015-1395-2
Yaslan H?,2013.Equations of anisotropic elastodynamics in3D quasicrystals as a symmetric hyperbolic system:deriving the time-dependent fundamental solutions.Applied Mathematical Modelling,37(18-19):8409-8418.https://doi.org/10.1016/j.apm.2013.03.039
Ying J,LüCF,Lim CW,2009.3D thermoelasticity solutions for functionally graded thick plates.Journal of Zhejiang University SCIENCE A,10(3):327-336.https://doi.org/10.1631/jzus.A0820406
Zhao MH,Dang HY,Fan CY,et al.,2017.Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material.Part 1:theoretical solution.Engineering Fracture Mechanics,179:59-78.https://doi.org/10.1016/j.engfracmech.2017.04.019
Zhao MH,Li Y,Fan CY,et al.,2018.Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects.Part I:theoretical solutions.Applied Mathematical Modelling,57:583-602.https://doi.org/10.1016/j.apm.2017.07.023
Zhou YB,Li XF,2018.Two collinear mode-III cracks in one-dimensional hexagonal piezoelectric quasicrystal strip.Engineering Fracture Mechanics,189:133-147.https://doi.org/10.1016/j.engfracmech.2017.10.030