多层复合壳体三维振动分析的谱-微分求积混合法
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摘要
对于较厚的多层复合壳体,其振动位移沿厚度方向呈锯齿形变化且层间剪切和拉、压应力呈三维耦合状态,采用传统的等效单层理论分析已不能满足精度要求.建立不受结构厚度、铺层材料性质和铺层方式限制的三维分析方法具有重要的研究价值.本文以独立铺层为建模对象,结合广义谱方法与微分求积技术建立了一种适用一般边界条件和铺层方式的多层复合壳体三维分析新方法——谱-微分求积混合法.该方法应用三维弹性理论对独立铺层进行精确建模,有效克服了二维简化理论对横向变形以及层间应力估计不确切的缺点;引入微分求积技术对铺层进行数值离散,将三维偏微分问题转化为二维偏微分问题,降低了求解维度和难度;应用广义谱方法近似地表述离散计算面上的场变量,将获取的二维偏微分方程转化为以场变量谱展开系数为未知量的线性代数方程组,避免了对超越方程的求解.数值验证结果表明该方法收敛性好,计算精度高.
The equivalent single layer(ESL) theories can be grossly in error for predicting vibration characteristics of thick multilayered shells because the vibration displacement and stress field of such shells under vibration are in full 3-D coupling condition. It is necessary to develop more accurate and efficient methods which are capable of dealing with multilayered structures with different boundary conditions, general laminations as well as arbitrary thickness universally.In order to overcome the drawback of the existing three-dimensional methods that are only confined for very limited cases such as cross-ply laminated rectangular plates under simply-supported boundary conditions, a general spectral-differential quadrature method is proposed. This method is undertaken by the exact 3-D elasticity theory so that it's able to study very well the dynamic behavior of thick multilayered structures which cannot be provided by the 2-D ESL theories. In each individual layer, the transverse domain is discretized by the differential quadrature technique. The displacement fields of the discretized surfaces are selected as fundamental unknowns. Then, each fundamental unknown is invariantly expanded by the general spectral method as a series of complete, orthogonal polynomials. The problems are stated in a variational form by the aid of penalty parameters which provides complete flexibilities to describe any prescribed boundary conditions. The current method can successfully avoid solving a highly nonlinear transcendental equation that is rely on roots-locating numerical method and all the modal information can be obtained just by solving linear algebraic equation systems. Numerical verification shows that the proposed method has high calculation precision. The method can be directly extend to the static and dynamic analysis of multilayered shells as well.
The equivalent single layer(ESL) theories can be grossly in error for predicting vibration characteristics of thick multilayered shells because the vibration displacement and stress field of such shells under vibration are in full 3-D coupling condition. It is necessary to develop more accurate and efficient methods which are capable of dealing with multilayered structures with different boundary conditions, general laminations as well as arbitrary thickness universally.In order to overcome the drawback of the existing three-dimensional methods that are only confined for very limited cases such as cross-ply laminated rectangular plates under simply-supported boundary conditions, a general spectral-differential quadrature method is proposed. This method is undertaken by the exact 3-D elasticity theory so that it's able to study very well the dynamic behavior of thick multilayered structures which cannot be provided by the 2-D ESL theories. In each individual layer, the transverse domain is discretized by the differential quadrature technique. The displacement fields of the discretized surfaces are selected as fundamental unknowns. Then, each fundamental unknown is invariantly expanded by the general spectral method as a series of complete, orthogonal polynomials. The problems are stated in a variational form by the aid of penalty parameters which provides complete flexibilities to describe any prescribed boundary conditions. The current method can successfully avoid solving a highly nonlinear transcendental equation that is rely on roots-locating numerical method and all the modal information can be obtained just by solving linear algebraic equation systems. Numerical verification shows that the proposed method has high calculation precision. The method can be directly extend to the static and dynamic analysis of multilayered shells as well.
引文
1陈建霖,励争,储鹏程.大开口复合材料层合板强度破坏研究.力学学报,2016,48(7):1326-1333(Chen Jianlin,Li Zheng,Chu Pengcheng.Strength analysis of fiber reinforced composite laminates with big cutouts.Chinese Journal of Theoretical and Applied Mechanics,2016,48(7):1326-1333(in Chinese))
2 马强,周凤玺,刘杰.梯度波阻板的地基振动控制研究.力学学报,2017,49(7):1360-1369(Ma Qiang,Zhou Fengxi,Liu Jie.Analysis of ground vibration control by graded wave impeding block.Chinese Journal of Theoretical and Applied Mechanics,2017,49(7):1360-1369(in Chinese))
3 郭洪宝,王波,贾普荣等.平纹编织陶瓷基复合材料面内剪切细观损伤行为研究.力学学报,2016,48(3):361-368(Guo Hongbao,Wang Bo,Jia Purong,et al.Mesoscopic damage behaviors of plain woven ceramic composite under in-plane shear loading.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):361- 368 (in Chinese))
4 Reddy JN.Mechanics of Laminated Composite Plates and Shells:Theory and Analysis.Florida:CRC Press,2003
5 Jin G,Ye T,Wang X,et al.A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions.Composites Part B Engineering,2016,89:230-252
6 Ye T,Jin G.Elasticity solution for vibration of generally laminated beams by a modified Fourier expansion-based sampling surface method.Computers&Structures,2016,167:115-130
7 瞿叶高,华宏星,谌勇等.复合材料旋转壳自由振动分析的新方法力学学报,2013,45(1):139-143(Qu Yegao,Hua Hongxing,Chen Yong,et al.Vibroacoustic performance of simply supported honeycomb sandwich panels.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):139-143(in Chinese))
8 Carrera E.Historical review of zig-zag theories for multilayered plates and shells.Applied Mechanics Reviews,2003,56(3):287-308
9任树伟,辛锋先,卢天健.蜂窝层芯夹层板结构振动与传声特性研究.力学学报,2013,45(3):349-358(Ren Shuwei,Xin Fengxian,Lu Tianjian.Vibroacoustic performance of simply supported honeycomb sandwich panels.Chinese Journal of Theoretical and Applied Mechanics,2013,45(3):349-358(in Chinese))
10 刘人怀,薛江红.复合材料层合板壳非线性力学的研究进展.力学学报,2017,49(3):487-506(Liu Renhuai,Xue Jianghong.Development of non-linear theories for laminated composite plates and shells.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):139-143(in Chinese))
11 Liu B,Xing Y.Exact solutions for free vibrations of orthotropic rectangular Mindlin plates.Composite Structures,2011,93:1664-1672
12 吴振.高性能整体-局部高阶理论及高阶层合板单元.[博士论文].大连:大连理工大学,2007(Wu Zhen.Higher-performance global-local higher-order theory and higher-order laminated plate elements.[Ph D Thesis].Dalian:Dalian University of Technology,2001(in Chinese))
13 Lo SH,Zhen W,Cheung YK,et al.Hygrothermal effects on multilayered composite plates using a refined higher order theory.Composite Structures,2010,92(3):633-646
14 吕朝锋.基于状态空间架构的微分求积法及其应用.[博士论文].杭州:浙江大学,2006(L¨u Chaofeng.State-space-based differential quadrature method and its applications.[Ph D Thesis].Hangzhou:Zhejiang University,2006(in Chinese))
15 Saada AS.Elasticity:Theory and Applications,second ed.Florida:Ross Publishing Inc,2009
16 Ye T,Jin G,Su Z.Three-dimensional vibration analysis of sandwich and multilayered plates with general ply stacking sequences by a spectral-sampling surface method.Composite Structures,2017,176:1124-1142
17 Reddy JN.Energy Principles and Variational Methods in Applied Mechanic.New York:John Wiley&Sons,1984
18 Chen WQ,L¨u CF.3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported.Composite Structures,2005,69(1):77-87
19 Qu Y,Wu S,Li H,et al.Three-dimensional free and transient vibration analysis of composite laminated and sandwich rectangular parallelepipeds:Beams,plates and solids.Composites Part B:Engineering,2015,73:96-110
20 Rao MK,Desai YM.Analytical solutions for vibrations of laminated and sandwich plates using mixed theory.Composite Structures,2004,63(3):361-373
21 Qu Y,Meng G.Dynamic analysis of composite laminated and sandwich hollow bodies of revolution based on three-dimensional elasticity theory.Composite Structures,2014,112:378-396
2 马强,周凤玺,刘杰.梯度波阻板的地基振动控制研究.力学学报,2017,49(7):1360-1369(Ma Qiang,Zhou Fengxi,Liu Jie.Analysis of ground vibration control by graded wave impeding block.Chinese Journal of Theoretical and Applied Mechanics,2017,49(7):1360-1369(in Chinese))
3 郭洪宝,王波,贾普荣等.平纹编织陶瓷基复合材料面内剪切细观损伤行为研究.力学学报,2016,48(3):361-368(Guo Hongbao,Wang Bo,Jia Purong,et al.Mesoscopic damage behaviors of plain woven ceramic composite under in-plane shear loading.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):361- 368 (in Chinese))
4 Reddy JN.Mechanics of Laminated Composite Plates and Shells:Theory and Analysis.Florida:CRC Press,2003
5 Jin G,Ye T,Wang X,et al.A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions.Composites Part B Engineering,2016,89:230-252
6 Ye T,Jin G.Elasticity solution for vibration of generally laminated beams by a modified Fourier expansion-based sampling surface method.Computers&Structures,2016,167:115-130
7 瞿叶高,华宏星,谌勇等.复合材料旋转壳自由振动分析的新方法力学学报,2013,45(1):139-143(Qu Yegao,Hua Hongxing,Chen Yong,et al.Vibroacoustic performance of simply supported honeycomb sandwich panels.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):139-143(in Chinese))
8 Carrera E.Historical review of zig-zag theories for multilayered plates and shells.Applied Mechanics Reviews,2003,56(3):287-308
9任树伟,辛锋先,卢天健.蜂窝层芯夹层板结构振动与传声特性研究.力学学报,2013,45(3):349-358(Ren Shuwei,Xin Fengxian,Lu Tianjian.Vibroacoustic performance of simply supported honeycomb sandwich panels.Chinese Journal of Theoretical and Applied Mechanics,2013,45(3):349-358(in Chinese))
10 刘人怀,薛江红.复合材料层合板壳非线性力学的研究进展.力学学报,2017,49(3):487-506(Liu Renhuai,Xue Jianghong.Development of non-linear theories for laminated composite plates and shells.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):139-143(in Chinese))
11 Liu B,Xing Y.Exact solutions for free vibrations of orthotropic rectangular Mindlin plates.Composite Structures,2011,93:1664-1672
12 吴振.高性能整体-局部高阶理论及高阶层合板单元.[博士论文].大连:大连理工大学,2007(Wu Zhen.Higher-performance global-local higher-order theory and higher-order laminated plate elements.[Ph D Thesis].Dalian:Dalian University of Technology,2001(in Chinese))
13 Lo SH,Zhen W,Cheung YK,et al.Hygrothermal effects on multilayered composite plates using a refined higher order theory.Composite Structures,2010,92(3):633-646
14 吕朝锋.基于状态空间架构的微分求积法及其应用.[博士论文].杭州:浙江大学,2006(L¨u Chaofeng.State-space-based differential quadrature method and its applications.[Ph D Thesis].Hangzhou:Zhejiang University,2006(in Chinese))
15 Saada AS.Elasticity:Theory and Applications,second ed.Florida:Ross Publishing Inc,2009
16 Ye T,Jin G,Su Z.Three-dimensional vibration analysis of sandwich and multilayered plates with general ply stacking sequences by a spectral-sampling surface method.Composite Structures,2017,176:1124-1142
17 Reddy JN.Energy Principles and Variational Methods in Applied Mechanic.New York:John Wiley&Sons,1984
18 Chen WQ,L¨u CF.3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported.Composite Structures,2005,69(1):77-87
19 Qu Y,Wu S,Li H,et al.Three-dimensional free and transient vibration analysis of composite laminated and sandwich rectangular parallelepipeds:Beams,plates and solids.Composites Part B:Engineering,2015,73:96-110
20 Rao MK,Desai YM.Analytical solutions for vibrations of laminated and sandwich plates using mixed theory.Composite Structures,2004,63(3):361-373
21 Qu Y,Meng G.Dynamic analysis of composite laminated and sandwich hollow bodies of revolution based on three-dimensional elasticity theory.Composite Structures,2014,112:378-396