The Symplectic Solution for the Bi-Directional Functionally Graded Piezoelectric Materials

2012 ◽  
Vol 174-177 ◽  
pp. 131-134 ◽  
Author(s):  
Y Z Yang
Keyword(s):  
Material Properties ◽  
Piezoelectric Materials ◽  
Separation Of Variables ◽  
Functionally Graded ◽  
Symplectic Space ◽  
Variation Principle ◽  
Symplectic Method ◽  
Functionally Graded Piezoelectric Materials ◽  
Dual Variables ◽  
Functionally Graded Piezoelectric

The symplectic method is applied to study analytically the deflection field of bi-directional functionally graded piezoelectric materials in this paper. And the material properties varies exponentially both along the axial and transverse coordinates.The dual equations were presented by variation principle and introducing separation of variables used. Then in the symplectic space which consists of the original variables and their dual variables, the problem can be solved via symplectic expansion. This comparisons with experimental data were carried out to verify the validity of the symplectic method.

The Bending of Symmetrical Laminated Composite Plate Based on Symplectic Method

2011 ◽  
Vol 328-330 ◽  
pp. 1646-1649
Author(s):  
Y.Z. Yang
Keyword(s):  
Composite Plate ◽  
Separation Of Variables ◽  
Laminated Composite ◽  
Transcendental Equation ◽  
Laminated Plates ◽  
Symplectic Space ◽  
Variation Principle ◽  
Symplectic Method ◽  
Stress Distributions ◽  
Dual Variables

This paper applies a symplectic method to study analytically the stress distributions of Composite laminated plates. 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. The transcendental equation and eigen-vector are deduced. The results of cross-ply and angle-ply laminated graphite–epoxy composite plate are shown, which is compared with established results. The parameters’ influences on mechanical property are also discussed.

The Symplectic Method for the Bending of Symmetrical Laminated Composite Plates with Clamped Boundary

2012 ◽  
Vol 479-481 ◽  
pp. 282-285
Author(s):  
Jin Guo Song ◽  
You Zhen Yang
Keyword(s):  
Separation Of Variables ◽  
Laminated Composite ◽  
Composite Plates ◽  
Laminated Plates ◽  
Symplectic Space ◽  
Variation Principle ◽  
Symplectic Method ◽  
Stress Distributions ◽  
Dual Variables ◽  
Function Vector

The symplectic method is applied to study analytically the stress distributions of composite laminated plates in this paper applies. The dual equations were presented by variation principle and introducing separation of variables used. 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 separation of variables and Eigen function vector expansion. The equation and Eigen-vector are deduced. The results of cross-ply laminated graphite–epoxy composite plate with clamped boundary are shown, which is compared with established results. The parameters’ influences on mechanical property are also discussed.

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.

The Higher Order Crack Tip Fields for Anti-Plane Crack in Functionally Graded Piezoelectric Materials

2014 ◽  
Vol 472 ◽  
pp. 617-620 ◽  
Author(s):  
Yao Dai ◽  
Xiao Chong ◽  
Shi Min Li
Keyword(s):  
Material Properties ◽  
Piezoelectric Materials ◽  
Crack Problem ◽  
Functionally Graded ◽  
Plane Crack ◽  
Homogeneous Material ◽  
Plane Shear ◽  
Functionally Graded Piezoelectric Materials ◽  
Fundamental Significance ◽  
Functionally Graded Piezoelectric

The anti-plane crack problem is studied in functionally graded piezoelectric materials (FGPMs). The material properties of the FGPMs are assumed to be the exponential function of y. The crack is electrically impermeable and loaded by anti-plane shear tractions and in-plane electric displacements. Similar to the Williams solution of homogeneous material, the high order asymptotic fields are obtained by the method of asymptotic expansion. This investigation possesses fundamental significance as Williams solution.

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.

The Higher Order Crack Tip Fields for Anti-Plane Crack in Linear Functionally Graded Piezoelectric Materials

2014 ◽  
Vol 989-994 ◽  
pp. 715-718
Author(s):  
Yao Dai ◽  
Xiao Chong ◽  
Shi Min Li
Keyword(s):  
Material Properties ◽  
Piezoelectric Materials ◽  
Crack Tip ◽  
Expansion Method ◽  
Higher Order ◽  
Functionally Graded ◽  
Crack Tip Field ◽  
Crack Tip Fields ◽  
Functionally Graded Piezoelectric Materials ◽  
Functionally Graded Piezoelectric

The crack tip field in functionally graded piezoelectric materials (FGPMs) under mechanical and electrical loadings is studied. Different from previous analyses, all material properties of the functionally graded piezoelectric materials are assumed to be linear function of y perpendicular to the crack. The crack surfaces are supposed to be insulated electrically. Similar to the Williams’ solution of homogeneous elastic materials, the higher order crack tip fields of FGPMs are obtained by the eigen-expansion method.

Analysis of Elasticity Solution of Bi-Direction Functionally Graded Piezoelectric Beam Subject to Voltage

2012 ◽  
Vol 166-169 ◽  
pp. 824-827 ◽  
Author(s):  
Y Z Yang
Keyword(s):  
Hamiltonian System ◽  
Exact Solutions ◽  
Material Properties ◽  
Functionally Graded ◽  
Symplectic Method ◽  
Elasticity Solution ◽  
Numerical Examples ◽  
Piezoelectric Beam ◽  
Functionally Graded Piezoelectric

This paper presents symplectic method for the derivation of exact solutions of functionally graded piezoelectric beam with the material properties varying exponentially both along the axial and transverse coordinates. In the approach, the related equations and formulas are developed in terms of dual equations, which can be solved by variables separation and symplectic expansion in Hamiltonian system. To verify advantages of the method, numerical examples of bi-directional functionally piezoelectric beam are discussed.

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