Symplectic Elasticity

Yao, Weian; Zhong, Wanxie; Lim, C.W.

doi:10.1142/6656

Cited by 161 publications

(150 citation statements)

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“…Equation (15) satisfies the four conditions of the symplectic inner product [9]. Hence, vector v forms a symplectic geometry space in accordance with the definition of the symplectic inner product (15).…”

Section: Derivation Of Symplectic Systemmentioning

confidence: 78%

“…It is hard to have explicit expression of the particular solution for the complex domain load q or nonhomogeneous boundary conditions at y = ±b. But an expanding form of that expression is possible by applying adjoint symplectic orthogonality property to eigenfunctions [9].…”

Section: Symplectic Eigenfunction Expansionmentioning

confidence: 99%

“…A new symplectic dual solution method can be used to solve elasticity in symplectic space via separation of variables and eigenfunction expansion [8,9]. Yao et al [10] studied an elastic wedge to reveal paradoxical characteristics.…”

Section: Introductionmentioning

confidence: 99%

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Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

Yao

Xiao

2011

Acta Mech Sin

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This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation. Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space. The operator matrix of the equation set is proven to be a Hamilton operator matrix. Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition. There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions, with opposite sides simply supported and opposite sides clamped. Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given. Analytical solutions using two examples are presented to show the use of the new methods described in this paper. new technique has good accuracy and better convergence speed than other methods, especially in relation to internal forces. A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods, with solutions compared to those produced by the Galerkin method.

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Section: Derivation Of Symplectic Systemmentioning

confidence: 78%

Section: Symplectic Eigenfunction Expansionmentioning

confidence: 99%

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Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

Yao

Xiao

2011

Acta Mech Sin

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“…Gorman and Ding proposed a semi-inverse superposition method based on the use of Lévy-type single series to obtain the free vibration solutions of rectangular thin plates [13]. Lim et al [14,15] developed the analytic solutions for free vibration of Lévy-type rectangular thin plates by the symplectic elasticity approach, which was originated by Zhong's group [16,17] and has been applied in many research fields [18], including elasticity [17,19], symplectic numerical methods [20,21], fracture mechanics [22,23], piezoelectricity [24], functionally graded effects [25], magneto-electro-elasticity [26]. Hu et al [27] further developed the method to obtain the vibration solutions of rectangular orthotropic plates with combinations of clamped and simply supported edges.…”

Section: Edgesmentioning

confidence: 99%

Analytic solutions for the free vibration of rectangular thin plates with two adjacent corners point-supported

Wang

2015

Arch Appl Mech

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Analytic free vibration solutions of rectangular thin plates with two adjacent corners point-supported are obtained for the first time by using an up-to-date symplectic superposition method, which demonstrates the ability to yield solutions via step-by-step derivations from the governing equations, without predetermining any forms of solutions. The essential difference between the current method and the other ones is that the solution procedure is conducted in the symplectic space rather than in the classical Euclidean space. This extends the scope of application of eigen expansion and paves the way for more analytic solutions of similar problems. Comprehensive highly accurate numerical results are presented to show the validity of the current solutions and to provide the benchmarks for future comparison and verification of other methods.

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“…For example, the SaintVenant torsion solution [1][2][3] of an elliptic bar is a typical case which was obtained by using the semi-inverse method. To derive the analytical solution in a rational manner, Zhong and co-workers [4] presented a new approach based on the Hamiltonian principle and the symplectic duality system. Here, we focus on the approach of boundary integral equation method (BIEM) to derive the semi-analytical solution in a logical manner.…”

Section: Introductionmentioning

confidence: 99%

Torsional rigidity of an elliptic bar with multiple elliptic inclusions using a null-field integral approach

Chen

Lee

2010

Comput Mech

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Following the success of using the null-field integral approach to determine the torsional rigidity of a circular bar with circular inhomogeneities (Chen and Lee in Comput Mech 44(2):221-232, 2009), an extension work to an elliptic bar containing elliptic inhomogeneities is done in this paper. For fully utilizing the elliptic geometry, the fundamental solutions are expanded into the degenerate form by using the elliptic coordinates. The boundary densities are also expanded by using the Fourier series. It is found that a Jacobian term may exist in the degenerate kernel, boundary density or boundary contour integral and cancel out to each other. Null-field points can be exactly collocated on the real boundary free of facing the principal values using the bump contour approach. After matching the boundary condition, a linear algebraic system is constructed to determine the unknown coefficients. An example of an elliptic bar with two inhomogeneities under the torsion is given to demonstrate the validity of the present approach after comparing with available results.

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Symplectic Elasticity

Cited by 161 publications

References 0 publications

Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

Analytic solutions for the free vibration of rectangular thin plates with two adjacent corners point-supported

Torsional rigidity of an elliptic bar with multiple elliptic inclusions using a null-field integral approach

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