The Hamiltonian system method for the stress analysis in axisymmetric problems of viscoelastic solids. (English) Zbl 1251.74038

Summary: With the use of the Laplace integral transformation and state space formalism, the classical axial symmetric quasistatic problem of viscoelastic solids is discussed. By employing the method of separation of variables, the governing equations under Hamiltonian system are established, and hence, general solutions including the zero eigensolutions and nonzero eigensolutions are obtained analytically. Due to the completeness property of the general solutions, their linear combinations can describe various boundary conditions. Simply by applying the adjoint relationships of the symplectic orthogonality, the eigensolution expansion method for boundary condition problems is given. In the numerical examples, stress distributions of a circular cylinder under the end and lateral boundary conditions are obtained. The results exhibit that stress concentrations appear due to the displacement constraints, and that the effects are seriously confined near the constraints, decreasing rapidly with the distance from the boundary.


74S30 Other numerical methods in solid mechanics (MSC2010)
74D10 Nonlinear constitutive equations for materials with memory
Full Text: DOI


[1] B. Miled, I. Doghri, and L. Delannay, “Coupled viscoelastic-viscoplastic modeling of homogeneous and isotropic polymers: numerical algorithm and analytical solutions,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 47-48, pp. 3381-3394, 2011. · Zbl 1230.74049
[2] V. V. Pokazeev, “The flutter of an elastic or a viscoelastic cantilevered fixed strip,” Journal of Applied Mathematics and Mechanics, vol. 72, no. 4, pp. 446-451, 2008.
[3] A. D. Drozdov, E. A. Jensen, and J. D. C. Christiansen, “Thermo-viscoelastic response of nanocomposite melts,” International Journal of Engineering Science, vol. 46, no. 2, pp. 87-104, 2008. · Zbl 1213.82120
[4] D. Dureisseix and H. Bavestrello, “Information transfer between incompatible finite element meshes: application to coupled thermo-viscoelasticity,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 44-47, pp. 6523-6541, 2006. · Zbl 1124.74046
[5] M. Bottoni, C. Mazzotti, and M. Savoia, “A finite element model for linear viscoelastic behaviour of pultruded thin-walled beams under general loadings,” International Journal of Solids and Structures, vol. 45, no. 3-4, pp. 770-793, 2008. · Zbl 1167.74582
[6] H. H. Zhang, G. Rong, and L. X. Li, “Numerical study on deformations in a cracked viscoelastic body with the extended finite element method,” Engineering Analysis with Boundary Elements, vol. 34, no. 6, pp. 619-624, 2010. · Zbl 1267.74118
[7] A. D. Mesquita and H. B. Coda, “A boundary element methodology for viscoelastic analysis: part II without cells,” Applied Mathematical Modelling, vol. 31, no. 6, pp. 1171-1185, 2007. · Zbl 1141.74044
[8] W. X. Zhong, “The reciprocal theorem and the adjoint symplectic orthogonality relation,” Acta Mechanica Sinica, vol. 24, no. 4, pp. 432-437, 1992 (Chinese).
[9] W. X. Zhong and F. W. Williams, “Physical interpretation of the symplectic orthogonality of the eigensolutions of a Hamiltonian or symplectic matrix,” Computers & Structures, vol. 49, no. 4, pp. 749-750, 1993. · Zbl 0797.70011
[10] X. S. Xu, W. X. Zhong, and H. W. Zhang, “The Saint-Venant problem and principle in elasticity,” International Journal of Solids and Structures, vol. 34, no. 22, pp. 2815-2827, 1997. · Zbl 0942.74560
[11] W. X. Zhang and X. S. Xu, “The symplectic approach for two-dimensional thermo-viscoelastic analysis,” International Journal of Engineering Science, vol. 50, pp. 56-69, 2012. · Zbl 1423.74182
[12] S. Y. Zhang and Z. C. Deng, “Lie group integration for constrained generalized Hamiltonian system with dissipation by projection method,” Applied Mathematics and Mechanics, vol. 25, no. 4, pp. 424-429, 2004. · Zbl 1082.65134
[13] W. A. Yao and X. C. Li, “Symplectic duality system on plane magnetoelectroelastic solids,” Applied Mathematics and Mechanics, vol. 27, no. 2, pp. 195-205, 2006. · Zbl 1145.74011
[14] C. W. Lim and X. S. Xu, “Symplectic elasticity: theory and applications,” Applied Mechanics Reviews, vol. 63, no. 5, Article ID 050802, 10 pages, 2010.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.