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Title: Symplectic methods for intelligent devices and singularity problems
Other Titles: Xin fang fa zai zhi neng zhuang zhi he qi yi xing wen ti zhong de ying yong
Authors: Zheng, Jianjun (鄭建軍)
Department: Department of Building and Construction
Degree: Doctor of Philosophy
Issue Date: 2007
Publisher: City University of Hong Kong
Subjects: Symplectic geometry.
Piezoelectric materials.
Notes: xv, 160 leaves : ill. 30 cm.
Thesis (Ph.D.)--City University of Hong Kong, 2007.
Includes bibliographical references (leaves 142-149)
CityU Call Number: QA665 .Z44 2007
Type: thesis
Abstract: The main work of the present research includes two parts: extending the symplectic approach to piezoelectric material and doing investigations of intelligent devices; introducing integral boundary condition and studying discontinuous problem and singularity problem.. The symplectic approach was introduced to study the elasticity problems in conservative Hamiltonian systems by Zhong (1991). Displacements and stresses are introduces as dual variables so that the second order differential equations in Hamiltonian systems can be transferred to the first order differential equations by Legendre transformation. The method of separation of variables is subsequently applied to solve the equations. The eigensolutions of zero and nonzero eigenvalues are presented with the former representing basic mechanical properties and the latter describing the localized solutions. Their linear combinations can cover all kinds of solutions with any boundary conditions along the edges. The method shows an analytical and rigorous process which is different from the traditional semi-inverse approaches. In the thesis, the symplectic method is developed to studying piezoelectric material. Electrical potential and electrical displacement are introduced as another couple of dual variables added into the symplectic phase space. Three chains of eigensolutions of zero eigenvalue exist. Each of them presents the elastic and piezoelectric behaviors of the piezoelectric materials, respectively. The coupling effects can also be clearly observed for the eigensolutions. Numerical examples of typical piezoelectric structures, such as single layer piezoelectric beam, bimorph structure and sandwich structure of piezoelectric materials, are studied. The analytical solutions which are seldom maintained in the past research papers are obtained from the symplectic approach. Compared with the results obtaining by other methods, the present results appear to be quite encouraging. The symplectic method is also used to investigate singularity problems and problems with discontinuous boundary conditions. Eigensolutions solved from Hamiltonian equations constitute a complete symplectic phase space. The analytical solutions of singularity problems can be solved from the symplectic phase space by the mixed energy variational principle. Numerical examples are presented with the results compared with existing one. Good agreement is achieved. The integral path along boundaries in the mixed energy variational principle is modified when studying the problems with discontinuous boundary conditions. Because of the properties of the eigensolutions in the symplectic phase space, the solutions are stable and automatically convergent. Crack problems are used as examples to test efficiency of the method. Stress intensity factor is solved and compared with those by finite element methods. They agree well with each other.
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