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.

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.

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.

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).

Mixed State Hamiltonian Element for Plane Transversely Isotropic Magnetoelectroelastic Solids

2013 ◽  
Vol 275-277 ◽  
pp. 1978-1983
Author(s):  
Xiao Chuan Li ◽  
Jin Shuang Zhang
Keyword(s):  
Mixed State ◽  
Hamiltonian Formulation ◽  
Transversely Isotropic ◽  
Laminated Plates ◽  
Governing Equations ◽  
Time Coordinate ◽  
First Order ◽  
Linear Elements ◽  
Propagation Matrix ◽  
Magnetoelectroelastic Solids

Hamiltonian dual equation of plane transversely isotropic magnetoelectroelastic solids is derived from variational principle and mixed state Hamiltonian elementary equations are established. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate. Then the x-direction is discreted with the linear elements to obtain the state-vector governing equations, which are a set of first order differential equations in z and are solved by the analytical approach. Because present approach is analytic in z direction, there is no restriction on the thickness of plate through the use of the present element. Using the propagation matrix method, the approach can be extended to analyze the problems of magnetoelectroelastic laminated plates. Present semi-analytical method of mixed Hamiltonian element has wide application area.

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.

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.

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.

The Free Vibration of the Magneto-Electro-Elastic Materials Laminated Circular Plate

2011 ◽  
Vol 374-377 ◽  
pp. 2193-2199 ◽  
Author(s):  
Qun Guan
Keyword(s):  
Free Vibration ◽  
Circular Plate ◽  
Single Layer ◽  
Transversely Isotropic ◽  
State Variables ◽  
Vibration Properties ◽  
Elastic Circular Plate ◽  
Symmetric Deformation ◽  
Laminated Circular Plate ◽  
The Given

In this paper, on the basis of the dynamic equation of material which contains transversely isotropic piezoelectric、piezomagnetic and elastic media, the state variables equation of piezoelectric、piezomagnetic and elastic circular plate in axial symmetric deformation is deduced. Under the given boundary conditions, according to the theorem of Caylay-Hamilton and applying the transfer matrix method, the solutions of state variables equation which are about free vibration of piezoelectric、piezomagnetic and elastic circular plate of single layer and multilayered are obtained. The calculation example indicates that from the varying law of the first natural frequency of the plate depending on the thickness-to-radius radio of the plate under varied conditions, the vibration properties of piezoelectric、piezomagnetic and laminated plate is related to the laminatation order of the material is gotten to know.

scholarly journals Direct integration method in three-dimensional elasticity and thermoelasticity problems for inhomogeneous transversely isotropic solids: governing equations in terms of stresses

2019 ◽  
pp. 102-105
Author(s):  
R. M. Kushnir ◽  
Y. V. Tokovyy ◽  
D. S. Boiko
Keyword(s):  
Stress Tensor ◽  
Integral Equations ◽  
Integration Method ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Solution Technique ◽  
Direct Integration ◽  
Governing Equations ◽  
Direct Integration Method

An efficient technique for thermoelastic analysis of inhomogeneous anisotropic solids is suggested within the framework of three-dimensional formulation. By making use of the direct integration method, a system of governing equations is derived in order to solve three-dimensional problems of elasticity and thermoelasticity for transversely isotropic inhomogeneous solids with elastic and thermo-physical properties represented by differentiable functions of the variable in the direction that is transversal to the plane of isotropy. By implementing the relevant separation of variables, the obtained equations can be uncoupled and reduced to second-kind integral equations for individual stress-tensor components and the total stress, which represents the trace of the stress tensor. The latter equations can be attempted by any of the numerical, analyticalnumerical, or analytical means available for the solution of the second-kind integral equations. In order to construct the solutions in an explicit form, an advanced solution technique can be developed on the basis of the resolvent-kernel method implying the series representation by the recurring kernels, computed iteratively by the original kernel of an integral equation.

Three-Dimensional Stress Singularities at Conical Notches and Inclusions in Transversely Isotropic Materials

1986 ◽  
Vol 53 (1) ◽  
pp. 89-96 ◽  
Author(s):  
Nihal Somaratna ◽  
T. C. T. Ting
Keyword(s):  
General Solution ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Solution Procedure ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Stress Singularities ◽  
Isotropic Materials ◽  
Transversely Isotropic Materials

This study examines analytically the possible existence of stress singularities of the form σ = ρδf(θ,φ) at the apex of axisymmetric conical boundaries in transversely isotropic materials. (ρ, θ, φ) refer to spherical coordinates with the origin at the apex. The problems of one conical boundary and of two conical boundaries with a common apex are considered. The boundaries are either rigidly clamped or traction free. Separation of variables enables the general solution to be expressed in terms of Legendre functions of the first and second kind. Imposition of boundary conditions leads to an eigenequation that would determine possible values of δ. The degenerate case that arises when the eigenvalues of the elasiticity constants are identical is also discussed. Isotropic materials constitute only a particular case in this class of degenerate materials and previously reported eigenequations corresponding to isotropic materials are shown to be recoverable from the present results. Numerical results corresponding to a few selected cases are also presented to illustrate the solution procedure.

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