The value of mixed elements splitting factor and analysis of static problems in laminated plates

doi:10.19936/j.cnki.2096-8000.20250428.006

Abstract

Abstract: Compared to the traditional mixed variational principles, the superiority of generalized mixed variational principles with splitting factors is very obvious. At present, the selection of splitting factor values in generalized mixed models is an open problem. Based on the theory of strain energy equal to complementary energy and strain energy and complementary energy equal to half of the mixed energy in the linear elastic problem, an error analysis method for selecting splitting factors is proposed. This method provides a new approach for obtaining appropriate splitting factors for the compatible generalized mixed elements. Numerical example analyses show that the splitting factor value of the compatible mixed element obtained by the method in this paper can adapt to the 3D composite laminated plate problem, and more accurate numerical results can be obtained under the same mesh model.

Key words: generalized mixed elements, splitting factor, strain energy, complementary energy, composites

CLC Number:

TB332

HUANG Xuanhong, WANG Yonggang. The value of mixed elements splitting factor and analysis of static problems in laminated plates[J]. COMPOSITES SCIENCE AND ENGINEERING, 2025, 0(4): 47-56.

References

[1] 荣廷玉. 弹性力学分裂模量变分原理. 西南交通大学学报, 1981(1): 48-56.
[2] RONG T Y. Generalized mixed variational principles and new fem models in solid mechanics. International Journal of Solids & Structures, 1988, 24(11): 1131-1140.
[3] RONG T Y, LU A Q. Parametrized lagrange multiplier method and construction of generalized mixed variational principles for computational mechanics. Computer Methods in Applied Mechanics & Engineering, 1998, 164(3-4): 287-296.
[4] CAI L, RONG T, CHEN D. Generalized mixed variational methods for reissner plate and its applications. Computational Mechanics, 2002, 30(1): 29-37.
[5] DANG F, RONG T, SUN X. Splitting elastic modulus variational principle for incompressible or nearly incompressible materials and its application. Communications in Numerical Methods in Engineering, 2000, 16(9): 649-656.
[6] RONG T-Y, LU A-Q. Generalized mixed variational principles and solutions of ill-conditioned problems in computational mechanics. Part Ⅱ: shear locking. Computer Methods in Applied Mechanics & Engineering, 2003, 192(44-46): 4981-5000.
[7] QING G H, MAO J H, LIU Y H. Generalized mixed finite element method for 3D elasticity problems. Acta Mechanica Sinica, 2018, 34: 371-380.
[8] QING G H, TIAN J. Highly accurate symplectic element based on two variational principles. Acta Mechanica Sinica, 2018, 34: 151-161.
[9] QING G H, MAO J H, LIU Y H. Highly accurate noncompatible generalized mixed finite element method for 3D elasticity problems. Journal of Mechanics of Materials and Structures, 2017, 12(4): 505-519.
[10] LIU W S, YU F, HE Z P, et al. A high-precision progressive damage model based on generalized mixed finite element method. Archive of Applied Mechanics, 2020, 90(6): 559-571.
[11] 王聿航, 卿光辉. 基于广义混合有限元的压电复合材料层合板的数值分析. 复合材料学报, 2022, 39(6): 2987-2996.
[12] 王燮, 卿光辉. 基于非协调广义部分混合元的层合板边界效应分析. 复合材料学报, 2022, 39(8): 4172-4178.
[13] LAGE R G, SOARES C M M, SOARES C A M, et al. Layerwise partial mixed finite element analysis of magneto-electro-elastic plates. Computers & Structures, 2004, 82(17-19): 1293-1301.
[14] LEZGY-NAZARGAH M, MESHKANI Z. An efficient partial mixed finite element model for static and free vibration analyses of FGM plates rested on two-parameter elastic foundations. Structural Engineering & Mechanics, 2018, 66(5): 665-676.
[15] FELIPPA C A. Parametrized multifield variational principles in elasticity:Ⅰ. mixed functionals. Communications in Applied Numerical Methods, 1989, 5(2): 79-88.
[16] DANG F N, RONG T Y, SUN X F. Splitting elastic modulus finite element method and its application. Communications in Numerical Methods in Engineering, 2001, 17(10): 675-690.
[17] 党发宁, 赖新芳, 郑娅娜. 变刚度混合型有限元. 岩土力学, 2004(增刊2): 81-84, 92.
[18] LEZGY-NAZARGAH M. A high-performance parametrized mixed finite element model for bending and vibration analyses of thick plates. Acta Mechanica, 2016, 227(12): 3429-3450.
[19] ARNOLD D N, WINTHER R. Mixed finite elements for elasticity. Numerische Mathematik, 2002, 92(3): 401-419.
[20] ADAMS S, COCKBURN B. A mixed finite element method for elasticity in three dimensions. Journal of Scientific Computing, 2005, 25(3): 515-521.
[21] TIMOSHENKO S P, GOODIER J N. Theory of elasticity: 3rd edition. New York: McGraw-Hiu, 1970: 30-32.
[22] 吴家龙. 弹性力学. 北京: 高等教育出版社, 2016.