An application of the symplectic system in two-dimensional viscoelasticity

Xü, Xinsheng; Zhang, Weixiang; Li, Xue; Wang, Gaping

doi:10.1016/j.ijengsci.2006.06.006

Cited by 22 publications

(6 citation statements)

References 40 publications

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“…Letσ i j denotes the stress andε i j denotes the strain components, then their deviatoric components are as followsS i j =σ i j −σ δ i j ,ẽ i j =ε i j −ẽδ i j , whereσ =σ kk /3 andẽ =ε kk /3 stand for mean stress and mean strain, respectively. The constitutive relations of three-dimensional viscoelasticity can be described uniformly as [17] ⎧ ⎨…”

Section: The Fundamental Problem and The Hamiltonian Systemmentioning

confidence: 99%

“…However, this method can not be applied directly into viscoelasticity for the energy non-conservation. Xu et al [17] discussed plane problems of viscoelasticity in the Hamiltonian system on the basis of the correspondence principle and analytical inverse Laplace transform, and studied the decay property of non-zero eigenvectors, the creep and relaxation characters of two-dimensional viscoelastic materials and local effect near the boundary. In this paper, based on the investigation of the character of viscoelastic material, the Hamiltonian system is applied into three-dimensional viscoelasticity, and the dual equations of the system are constructed.…”

Section: Introductionmentioning

confidence: 99%

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Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

Zhang

2008

Arch Appl Mech

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The traditional Saint-Venant problem of three-dimensional viscoelasticity is discussed under the Hamiltonia system with the use of the Laplace integral transformation, and the original problem is transformed into finding eigenvalues and eigenvectors of the Hamiltonia operator matrix. Since local effect near the boundary is usually neglected, all solutions of Saint-Venant problems can be obtained directly by the combinations of zero eigenvectors. Moreover, the adjoint relationships of the symplectic orthogonality of zero eigenvectors in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigenvector space of the time domain, and the iterative application of Laplace transformation is not needed. Simply by applying the adjoint relationships of the symplectic orthogonality, an effective method for boundary condition is given. Based on this method, some typical examples are discussed, in which the whole character of total creep and relaxation of viscoelasticity is clearly revealed.

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Section: The Fundamental Problem and The Hamiltonian Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

Zhang

2008

Arch Appl Mech

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“…The semi-inverse method method has at least three limitations: (1) the solution is not complete, and can only approximately satisfy the boundary conditions. For example, when the fixed end of the cantilever beam is subjected to displacement constraints, the stress concentration will occur when external force are imposed [3]. The semi-inverse method cannot be explained this kind of concentration.…”

Section: Introductionmentioning

confidence: 99%

Elastic bending solutions in the Hamiltonian system

Zhang¹,

Chen²

2017

Proceedings of the 2017 6th International Conference on Energy and Environmental Protection (ICEEP 2017)

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Abstract. In this paper, we introduce the Hamiltonian system to study bending problems of elastic cantilever-beam. The variational method and the variable seperation method are employed to establish the dual governing equations. Due to the completeness property of the eigenvectors, the solution can be expressed by the linear combination of these eigenvectors, and the coefficients of the combination are determined by the boundary conditions. In the numerical results, stress distributions of bending deformation are discussed.

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“…China; Tel: 18623717351; E-mail: weixiang_zh@163.com elasticity are the same as those of elasticity in mathematics form. Xu et al [8] analyzed two-dimensional problems of viscoelastic solids with the use of Laplace transform and the Hamiltonian formulation, and discussed a numerical method to deal with problems of displacement or stress boundary conditions and non-homogeneous governing equations.…”

Section: Introductionmentioning

confidence: 99%

A Hamiltonian System Method for Three-Dimensional Viscoelastic Solids

Zhang¹,

Bai²

2015

TOCSJ

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Abstract:The Hamiltonian system in quasi-static problems of 3D viscoelastic solids has been introduced in this paper. Based on the principle of elastic-viscoelastic correspondence, the problem of solving partial differential equations is reduced to finding general eigensolutions of the dual equations and all the analytical fundamental eigensolutions and their corresponding Jordan forms are derived. After the establishment of symplectic adjoint relation, the final solution is expressed by linear combinations of the general eigensolutions, and the combinations are determined by the given boundary conditions. For its applications, problems of various boundary conditions and the inhomogeneous governing equations are discussed.

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An application of the symplectic system in two-dimensional viscoelasticity

Cited by 22 publications

References 40 publications

Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

Elastic bending solutions in the Hamiltonian system

A Hamiltonian System Method for Three-Dimensional Viscoelastic Solids

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