Applied Mathematics and Mechanics

, Volume 13, Issue 11, pp 1017–1022

# Hamiltonian system and simpletic geometry in mechanics of composite materials (I) — Fundamental theory

• Zhong Wan-xie
• Ouyang Hua-jiang
Article

## Abstract

For the first time, Hamiltonian system used in dynamics is introduced to formulate statics and Hamiltonian equation is derived corresponding to the original governing equation, which enables separation of variables to work and eigen function to be obtained for the boundary problem. Consequently, analytical and semi-analytical solutions can be got. The method is especially suitable to solve rectangular plane problem and spatial prism in elastic mechanics.

The paper presents a new idea to solve partially differential equation in solid mechanics. The flexural problem and plane stress problem of laminated plate are studied in detail.

### Key words

Hamiltonian system simpletic geometry analytical solution semi-analytical solution

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### References

1. [1]
Sokolnikoff, I. S.,Mathematical Theory of Elasticity, (2nd edition), Mac Graw-Hill (1956).Google Scholar
2. [2]
Courant, R. and D. Hilbert,Mathemato-Physical Methods, translated by M. Qian and D. Guo, Science Press, Beijing (1957). (Chinese version)Google Scholar
3. [3]
Zhong, W. and X. Zhong, Computational structural mechanics, optimal control and semi—analytical method for PDE,Computer and Structures,37, 6 (1990).
4. [4]
Zhong, W. and J. Lin, et al., The eigenvalue problem of the chain of identical substructures and the expansion method solution based on the eigenvectors,Acta Mechanica Sinica,23, 1 (1991).Google Scholar
5. [5]
Zhong, W., Eigen solution of discrete LQ control problems,Comput. Struct. Mech. Appl.,7, 2 (1990). (in Chinese)Google Scholar
6. [6]
Zhong, W. and Z. Yang, Algorithm of main eigen pairs of continuous LQ control problems,Applied Math. and Mech.,12, 1 (1991).
7. [7]
Xie, X.,Optimal Control Theory and Applications, Qinghua Univ. Press, Beijing (1986). (in Chinese)Google Scholar
8. [8]
Zhong, W., Plane elasticity problem in strip domain and Hamiltonian system,J. Dalian Univ. of Tech.,31, 4 (1991). (in Chinese)
9. [9]
Arnold, V. I.,Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, Inc. (1978).Google Scholar
10. [10]
Jones, R. M.,Mechanics of Composite Materials, translated by Y. Zhu, et al., Shanghai Science and Technology Press (1981). (Chinese version)Google Scholar
11. [11]
Qin, M., Simpletic geometry and computational Hamiltonian mechanics,Mechanics and Practice,12, 6 (1990). (in Chinese)Google Scholar