Analytical Solutions for Laminated Transversely Isotropic Magnetoelectroelastic Plates

2013 ◽  
Vol 634-638 ◽  
pp. 2425-2431
Author(s):  
Xiao Chuan Li ◽  
Qing Li
Keyword(s):  
Symplectic Geometry ◽  
Separation Of Variables ◽  
Electric Displacement ◽  
Transversely Isotropic ◽  
Hamiltonian Operator ◽  
Hamiltonian Matrix ◽  
Operator Matrix ◽  
Zero Eigenvalue ◽  
Dual Variables ◽  
Eigen Solutions

The theory of Hamiltonian system is introduced for the problems of laminated transversely isotropic magnetoelectroelastic plates. The partial differential equations of the magnetoelectroelastic solids are derived corresponding to the Lagrange density function and Legendre’s transformation. These equations are a set of the first-order Hamiltonian equations and expressed with displacements, electric potential and magnetic potential, as well as their dual variables--lengthways stress, electric displacement and magnetic induction in the symplectic geometry space. To obtain the solutions of the equations, the schemes of separation of variables and expansion of eigenvector of Hamiltonian operator matrix in the polar direction are implemented. The homogenous solutions of the equations consist of zero eigen-solutions and nonzero eigen-solutions. All the eigen-solutions of zero eigenvalue are obtained in the symmetric deformation. These solutions give the classical Saint-Venant’s solutions because the Hamiltonian matrix is symplectic. The method is rational, analytical method and does not require any trial functions.

Saint Venant Solutions in Symmetric Deformation for Rectangular Transversely Isotropic Magnetoelectroelastic Solids

2011 ◽  
Vol 284-286 ◽  
pp. 2243-2250 ◽  
Author(s):  
Xiao Chuan Li
Keyword(s):  
Separation Of Variables ◽  
Electric Displacement ◽  
Transversely Isotropic ◽  
Rectangular Domain ◽  
Generalized Variational Principle ◽  
Hamiltonian Matrix ◽  
Operator Matrix ◽  
Zero Eigenvalue ◽  
Symmetric Deformation ◽  
Magnetoelectroelastic Solids

Hamiltonian system used in dynamics is introduced to formulate the transversely isotropic magnetoelectroelastic solids plane problem in rectangular domain and symplectic dual equation is derived corresponding to the generalized variational principle of the magnetoelectroelastic solids. The equation is expressed with displacements, electric potential and magnetic potential, as well as their duality variables--lengthways stress, electric displacement and magnetic induction in the symplectic geometry space. Since the x-coordinate is treated as time variable so that z becomes the independent coordinate in the Hamiltonian matrix operator. The symplectic dual approach enables the separation of variables to work and all the Saint Venant solutions in the symmetric deformation are obtained directly via the zero eigenvalue solutions and all their Jordan normal form of the corresponding Hamiltonian operator matrix and the boundary condition. An example is presented to illustrate the proposed approach.

scholarly journals Symplectic Analytical Solutions for the Magnetoelectroelastic Solids Plane Problem in Rectangular Domain

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Xiao-Chuan Li ◽  
Wei-An Yao
Keyword(s):  
Plane Problem ◽  
Analytical Solutions ◽  
Symplectic Geometry ◽  
Complete Solution ◽  
Electric Displacement ◽  
Transversely Isotropic ◽  
Local Effect ◽  
Rectangular Domain ◽  
Zero Eigenvalue ◽  
Magnetoelectroelastic Solids

The transversely isotropic magnetoelectroelastic solids plane problem in rectangular domain is derived to Hamiltonian system. In symplectic geometry space with the origin variables—displacements, electric potential, and magnetic potential, as well as their duality variables—lengthways stress, electric displacement, and magnetic induction, on the basis of the obtained eigensolutions of zero-eigenvalue, the eigensolutions of nonzero-eigenvalues are also obtained. The former are the basic solutions of Saint-Venant problem, and the latter are the solutions which have the local effect, decay drastically with respect to distance, and are covered in the Saint-Venant principle. So the complete solution of the problem is given out by the symplectic eigensolutions expansion. Finally, a few examples are selected and their analytical solutions are presented.

Symplectic Solutions in Singularity Problems of Anisotropic Beams

2011 ◽  
Vol 378-379 ◽  
pp. 90-93
Author(s):  
You Zhen Yang
Keyword(s):  
Separation Of Variables ◽  
Mathematical Physics ◽  
Symplectic Space ◽  
Variation Principle ◽  
Symplectic Method ◽  
Stress Distributions ◽  
Zero Eigenvalue ◽  
Numerical Examples ◽  
Localized Solutions ◽  
Dual Variables

Based on the two-dimensional elasticity,the symplectic method is applied to study analytically the stress distributions of anisotropic beam.Using variation principle and introducing separation of variables, dual equations were presented.Then in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of sepatation of variables and eigenfunction vector expansion.So the original problems come down to solve the eigensolutions of zero eigenvalue and non-zeroes eigenvalue that describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle. Completed numerical examples are newly given to compare with established results.

Hamiltonian Duality Equation on Three-Dimensional Problems of Magnetoelectroelastic Solids

2012 ◽  
Vol 268-270 ◽  
pp. 1099-1104
Author(s):  
Xiao Chuan Li ◽  
Jin Shuang Zhang
Keyword(s):  
Variational Principle ◽  
Differential Equations ◽  
Separation Of Variables ◽  
Electric Displacement ◽  
Hamiltonian Formulation ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Coefficient Matrix ◽  
Symplectic Space ◽  
Magnetoelectroelastic Solids

Hamiltonian system used in dynamics is introduced to formulate the three-dimensional problems of the transversely isotropic magnetoelectroelastic solids. The Hamiltonian dual equations in magnetoelectroelastic solids are developed directly from the modified Hellinger-Reissner variational principle derived from generalized Hellinger-Ressner variational principle with two classes of variables. These variables not only include such origin variables as displaces, electric potential and magnetic potential, but also include such their dual variables as lengthways stress, electric displacement and magnetic induction in the symplectic space. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate so that the method of separation of variables can be used. The governing equations are a set of first order differential equations in z, and the coefficient matrix of the differential equations is Hamiltonian in (x, y).

Symplectic Solutions in Singularity Problems of Anisotropic Beams

2011 ◽  
Vol 147 ◽  
pp. 136-139
Author(s):  
You Zhen Yang
Keyword(s):  
Separation Of Variables ◽  
Mathematical Physics ◽  
Symplectic Space ◽  
Variation Principle ◽  
Symplectic Method ◽  
Stress Distributions ◽  
Zero Eigenvalue ◽  
Numerical Examples ◽  
Localized Solutions ◽  
Dual Variables

Based on the two-dimensional elasticity,the symplectic method is applied to study analytically the stress distributions of anisotropic beam.Using variation principle and introducing separation of variables, dual equations were presented.Then in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of sepatation of variables and eigenfunction vector expansion.So the original problems come down to solve the eigensolutions of zero eigenvalue and non-zeroes eigenvalue that describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle. Completed numerical examples are newly given to compare with established results.

Fundamental solutions of two 3D rectangular semi-permeable cracks in transversely isotropic piezoelectric media based on the non-local theory

2020 ◽  
Vol 16 (6) ◽  
pp. 1497-1520
Author(s):  
Haitao Liu ◽  
Liang Wang
Keyword(s):  
Electric Displacement ◽  
Transversely Isotropic ◽  
Elastic Behavior ◽  
Classical Solutions ◽  
Local Theory ◽  
Dual Integral Equations ◽  
Piezoelectric Media ◽  
Content Type ◽  
Present Solution ◽  
Non Local

PurposeThe paper aims to present the non-local theory solution of two three-dimensional (3D) rectangular semi-permeable cracks in transversely isotropic piezoelectric media under a normal stress loading.Design/methodology/approachThe fracture problem is solved by using the non-local theory, the generalized Almansi's theorem and the Schmidt method. By Fourier transform, this problem is formulated as three pairs of dual integral equations, in which the elastic and electric displacements jump across the crack surfaces. Finally, the non-local stress and the non-local electric displacement fields near the crack edges in piezoelectric media are derived.FindingsDifferent from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack edges in piezoelectric media.Originality/valueAccording to the literature survey, the electro-elastic behavior of two 3D rectangular cracks in piezoelectric media under the semi-permeable boundary conditions has not been reported by means of the non-local theory so far.

Generalized Inverse of Upper Triangular Infinite Dimensional Hamiltonian Operators

2013 ◽  
Vol 20 (03) ◽  
pp. 395-402
Author(s):  
Junjie Huang ◽  
Xiang Guo ◽  
Yonggang Huang ◽  
Alatancang
Keyword(s):  
Explicit Expression ◽  
Generalized Inverse ◽  
Hamiltonian Operator ◽  
Operator Matrix ◽  
Infinite Dimensional ◽  
Symplectic Spaces ◽  
Hamiltonian Operators

In this paper, we deal with the generalized inverse of upper triangular infinite dimensional Hamiltonian operators. Based on the structure operator matrix J in infinite dimensional symplectic spaces, it is shown that the generalized inverse of an infinite dimensional Hamiltonian operator is also Hamiltonian. Further, using the decomposition of spaces, an upper triangular Hamiltonian operator can be written as a new operator matrix of order 3, and then an explicit expression of the generalized inverse is given.

A State Space Solution Approach for Problems of Cylindrical Tubes and Circular Plates

2015 ◽  
Vol 31 (5) ◽  
pp. 557-572 ◽  
Author(s):  
W.-D. Tseng ◽  
J.-Q. Tarn
Keyword(s):  
General Solution ◽  
State Space ◽  
Eigenfunction Expansion ◽  
Separation Of Variables ◽  
Transversely Isotropic ◽  
Circular Plates ◽  
Solution Approach ◽  
Space Solution ◽  
Cylindrical Tubes ◽  
Space Formalism

AbstractWe present a general solution approach for analysis of transversely isotropic cylindrical tubes and circular plates. On the basis of Hamiltonian state space formalism in a systematic way, rigorous solutions of the twisting problems are determined by means of separation of variables and symplectic eigenfunction expansion.

scholarly journals On Symplectic Analysis for the Plane Elasticity Problem of Quasicrystals with Point Group 12 mm

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Hua Wang ◽  
Jianrui Chen ◽  
Xiaoyu Zhang
Keyword(s):  
Mixed Boundary ◽  
Separation Of Variables ◽  
Point Group ◽  
Mixed Boundary Conditions ◽  
Hamiltonian Operator ◽  
Plane Elasticity ◽  
Elasticity Problem ◽  
Operator Matrices ◽  
Group 12 ◽  
Plane Elasticity Problem

The symplectic approach, the separation of variables based on Hamiltonian systems, for the plane elasticity problem of quasicrystals with point group 12 mm is developed. By introducing appropriate transformations, the basic equations of the problem are converted to two independent Hamiltonian dual equations, and the associated Hamiltonian operator matrices are obtained. The study of the operator matrices shows the feasibility of the method. Without any assumptions, the general solution is presented for the problem with mixed boundary conditions.

Plane Electromechanical Fieldes in a Piezoelectric Material with a Crack

2006 ◽  
Vol 324-325 ◽  
pp. 247-250
Author(s):  
Shu Hong Liu ◽  
Meng Wu ◽  
Shu Min Duan ◽  
Hong Jun Wang
Keyword(s):  
Boundary Condition ◽  
General Solution ◽  
Piezoelectric Material ◽  
Mechanical Load ◽  
Electric Displacement ◽  
Transversely Isotropic ◽  
Two Dimensional ◽  
Electric Boundary Condition ◽  
Boundary Collocation Method ◽  
Transversely Isotropic Piezoelectric Material

A two-dimensional electromechanical analysis is performed on a transversely isotropic piezoelectric material containing a crack based on the impermeable electric boundary condition. By introducing stress function, a general solution is provided in terms of triangle series. It is shown that the stress and electric displacement are all of 1/2 order singularity in front of the crack tip. In addition, the electromechanical fields in the vicinity of the crack when subjected to uniform tensile mechanical load are obtained using boundary collocation method.

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