含参数辛元与热弹性复合材料层合板分析
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摘要
为了提高计算热弹性复合材料层合板应力的精度,基于相关文献的非协调辛元理论,将弹性材料的广义H-R变分原理扩展为热弹性材料的广义H-R变分原理,并提出了相应的修正原理。建立了含参数非协调辛元。该单元的主要优点之一是避免了传统混合元中系数矩阵主对角线上存在零元素的问题,因而保证了有限元线性方程组数值结果的稳定性。同时,该单元关于广义的位移变量和应力变量对称,因此其对应的有限元线性方程组对称保辛。实例结果表明:与精确解对比,该单元的广义位移和广义应力的数值结果不仅精度一致,而且精度高;与位移有限元法的结果比较,收敛速度快。
In order to improve the stress accuracy of thermal elastic composite laminates,the generalized H-R variational principle of elastic material was extended to the generalized H-R variational principle of thermal elastic material based on the symplectic element theory of related references,and the corresponding modified principle was proposed.The parametered symplectic element was established.One of the main advantages of this element is that there is no zeros in the leading diagonal of coefficient matrix compared to the traditional mixed element.Consequently,the stability of the numerical results of finite element linear system of equations is guaranteed.At the same time,the element is symmetric for both the displacement variable and the stress variable,so its corresponding finite element linear system of equations is symmetric and symplectic conservation.Compared with the exact solution,the numerical results show that the accurate order of the generalized displacements and stresses are consistent,and hold high accuracy.
In order to improve the stress accuracy of thermal elastic composite laminates,the generalized H-R variational principle of elastic material was extended to the generalized H-R variational principle of thermal elastic material based on the symplectic element theory of related references,and the corresponding modified principle was proposed.The parametered symplectic element was established.One of the main advantages of this element is that there is no zeros in the leading diagonal of coefficient matrix compared to the traditional mixed element.Consequently,the stability of the numerical results of finite element linear system of equations is guaranteed.At the same time,the element is symmetric for both the displacement variable and the stress variable,so its corresponding finite element linear system of equations is symmetric and symplectic conservation.Compared with the exact solution,the numerical results show that the accurate order of the generalized displacements and stresses are consistent,and hold high accuracy.
引文
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[12]FELIPPA C A.Parametrized multifield variational principles in elasticity II:Hybrid functionals and the free formulation[J].Communications in Applied Numerical Methods,1989,5(2):89-98.
[13]陈万吉.一个高精度八结点六面体单元[J].力学学报,1982,18(3):262-271.CHEN W J.A high-precision eight-node hexahedron element[J].Acta Mechanica Sinica,1982,18(3):262-271(in Chinese).
[14]QING G H,MAO J H,LIU Y H.Generalized mixed finite element method for 3Delasticity problems[J].Acta Mechanica Sinica,2018,34(2):371-380.
[15]范家让.强厚叠层板壳的精确理论[M].北京:科学出版社,1996.FAN Ji R.Exact theory of laminated thick plates and shells[M].Beijing:Science Press,1996(in Chinese).
[2]徐翔民,张予东,李宾杰,等.纳米SiO2/尼龙66复合材料的力学性能和热性能[J].复合材料学报,2008,25(4):56-61.XU X M,ZHANG Y D,LI B J,et al.Mechanical and thermal properties of nano-SiO2/PA66composite[J].Acta Materiae Compositae Sinica.2008,25(4):56-61(in Chinese).
[3]OUKIT A,PIERRE R.Mixed finite element for the linear plate problem:The Hermann-Miyoshi model revisited[J].Numerische Mathematik,1996,74(4):453-477.
[4]ATLURI S N,GALLAGHER R H,ZIENKIEWICZ O C.Hybrid and mixed finite element methods[M].John Wiley&Sons,1983.
[5]钟万勰.应用力学的辛数学方法[M].北京:高等教育出版社,2006.ZHONG W X.Symplectic solution methodology in applied mechanics[M].Bejing:China Higher Education Press,2006(in Chinese).
[6]ZOU G P,TANG L M.Semi-analytical solution for laminated composite plates in Hamiltonian system[J].Computer Methods in Applied Mechanics and Engineering,1995,128(3-4):395-404.
[7]QING G H,QIU J J,LIU Y H.Free vibration analysis of stiffened laminated plates[J].International Journal of Solids and Structures,2006,43(6):1357-1371.
[8]QING G,TIAN J.Highly accurate symplectic element based on two variational principles[J].Acta Mechanica Sinica,2018,34(1):151-161.
[9]刘艳红.三类热弹性体的广义H-R变分原理和齐次向量方程[D].天津:天津大学,2009.LIU Y H.Generalized H-R variation principles of three thermoelastic solids and homogeneous vector equations[D].Tianjin:Tianjin University,2009(in Chinese).
[10]顾元宪,陈飚松,张洪武.结构动力方程的增维精细积分法[J].力学学报,2000,32(4):447-456.GU Y X,CHEN Y S,ZHANG H W.Precise time-intgration with dimension expanding method[J].Acta Mechanica Sinica,2000,32(4):447-456(in Chinese).
[11]CHIEN W Z.Method of high-order Lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals[J].Applied Mathematics and Mechanics,1983,4(2):143-157.
[12]FELIPPA C A.Parametrized multifield variational principles in elasticity II:Hybrid functionals and the free formulation[J].Communications in Applied Numerical Methods,1989,5(2):89-98.
[13]陈万吉.一个高精度八结点六面体单元[J].力学学报,1982,18(3):262-271.CHEN W J.A high-precision eight-node hexahedron element[J].Acta Mechanica Sinica,1982,18(3):262-271(in Chinese).
[14]QING G H,MAO J H,LIU Y H.Generalized mixed finite element method for 3Delasticity problems[J].Acta Mechanica Sinica,2018,34(2):371-380.
[15]范家让.强厚叠层板壳的精确理论[M].北京:科学出版社,1996.FAN Ji R.Exact theory of laminated thick plates and shells[M].Beijing:Science Press,1996(in Chinese).