关键词:多层复合壳体;三维振动;谱--微分求积混合法;一般边界条件 Abstract 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.
多层复合壳体的边界条件可以分为3大类:几何边界条件、自然边界条件和混合边界条件. 在实际求解中,只有几何形状规则、至少一对边简支、铺层对称的极少数多层复合结构才能找到既满足控制微分方程又满足边界约束条件的解析解. 对于其他情况,可将结构的边界条件方程和控制微分方程转化成能量泛函,然后通过求解泛函的极值问题进行求解. 本文考虑几何边界条件下层合壳体的振动问题,为了适用一般边界条件,通过引入罚函数,将边界约束条件改写为如下泛函 式中 为罚参数. 上述处理方法克服了传统解析方法难以找到满足边界条件展开式的 困难,极大拓宽了展开式的选择范围. 表1给出了不同边界条件下罚参数的取值情况. Table 1 表1 表1不同边界条件下罚参数的取值情况(边) Table 1Penalty parameters for various classical boundaries (at the end of )
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多层复合壳体三维分析新方法——谱--微分求积混合法. 该方法的特点为:(1)应用三维弹性理论对独立铺层进行精确建模;(2)引入微分求积技术对铺层进行数值离散,从而将三维偏微分问题转化为二维偏微分问题;(3)应用广义谱方法近似地表述离散计算面上的场变量,从而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. 与传统三维求解方法相比,谱--微分求积混合法在处理不同边界条件、铺层方式时具有很好的普适性和灵活性. 针对给定的不同边界条件和铺层方式,仅需要通过改变相关参数设置来完成边界约束的施加和铺层方式的配置而不需要逐一重新编程处理. 数值验证结果表明该方法收敛性好,计算精度高. The authors have declared that no competing interests exist.
(ChenJianlin, LiZheng, ChuPengcheng.Strength analysis of fiber reinforced composite laminates with big cutouts . Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(7): 1326-1333 (in Chinese)) [本文引用: 1]
(MaQiang, ZhouFengxi, LiuJie.Analysis of ground vibration control by graded wave impeding block . Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(7): 1360-1369 (in Chinese)) [本文引用: 1]
(GuoHongbao, WangBo, JiaPurong, 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)) [本文引用: 1]
[4]
ReddyJN.Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. Florida: CRC Press, 2003 [本文引用: 1]
[5]
JinG, YeT, WangX, 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 [本文引用: 1]
[6]
YeT, JinG.Elasticity solution for vibration of generally laminated beams by a modified Fourier expansion-based sampling surface method . Computers & Structures, 2016, 167: 115-130 [本文引用: 1]
(QuYegao, HuaHongxing, ChenYong, et al.Vibroacoustic performance of simply supported honeycomb sandwich panels . Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(1): 139-143 (in Chinese)) [本文引用: 1]
[8]
CarreraE.Historical review of zig-zag theories for multilayered plates and shells . Applied Mechanics Reviews, 2003, 56(3): 287-308 [本文引用: 1]
(RenShuwei, XinFengxian, LuTianjian.Vibroacoustic performance of simply supported honeycomb sandwich panels . Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(3): 349-358 (in Chinese)) [本文引用: 1]
(LiuRenhuai, XueJianghong.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)) [本文引用: 1]
[11]
LiuB, XingY.Exact solutions for free vibrations of orthotropic rectangular Mindlin plates . Composite Structures, 2011, 93: 1664-1672 [本文引用: 1]
(WuZhen.Higher-performance global-local higher-order theory and higher-order laminated plate elements. [PhD Thesis]. Dalian: Dalian University of Technology, 2001 (in Chinese)) [本文引用: 2]
[13]
LoSH, ZhenW, CheungYK, et al.Hygrothermal effects on multilayered composite plates using a refined higher order theory . Composite Structures, 2010, 92(3): 633-646 [本文引用: 1]
(LüChaofeng.State-space-based differential quadrature method and its applications. [PhD Thesis]. Hangzhou: Zhejiang University, 2006 (in Chinese)) [本文引用: 1]
[15]
SaadaAS.Elasticity: Theory and Applications, second ed. Florida: Ross Publishing Inc, 2009 [本文引用: 1]
[16]
YeT, JinG, SuZ.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]
ReddyJN.Energy Principles and Variational Methods in Applied Mechanic. New York: John Wiley & Sons, 1984 [本文引用: 2]
[18]
ChenWQ, LüCF.3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported . Composite Structures, 2005, 69(1): 77-87 [本文引用: 3]
[19]
QuY, WuS, LiH, 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 [本文引用: 3]
[20]
RaoMK, DesaiYM.Analytical solutions for vibrations of laminated and sandwich plates using mixed theory . Composite Structures, 2004, 63(3): 361-373 [本文引用: 2]
[21]
QuY, MengG.Dynamic analysis of composite laminated and sandwich hollow bodies of revolution based on three-dimensional elasticity theory . Composite Structures, 2014, 112: 378-396 [本文引用: 4]