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An application of the symplectic system in two-dimensional viscoelasticity. (English) Zbl 1213.74077
Summary: This paper redescribes fundamental problem of the two-dimensional viscoelasticity in symplectic system. With the aid of the symplectic character and integral transformation, solutions of duality equations are obtained, or Saint-Venant solutions of extension and bend and local solutions of boundary effects. Thus the original problem is reduced to finding zero eigenvalue eigensolutions and non-zero eigenvalue eigensolutions. Meanwhile, adjoint relationships of the symplectic orthogonality in the Laplace domain are generalized to in the time domain. After obtaining fundamental eigensolutions, the problem can be discussed in the eigensolution space of the time domain without the need of the Laplace transformation and inverse one. As its application, a direct method is shown and some examples are discussed, which reveal relations between the creep or relaxation and eigensolutions. The symplectic method and numerical method provide an idea for other researching as well.

MSC:
74D05 Linear constitutive equations for materials with memory
74G50 Saint-Venant’s principle
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[1] Saint-Venant, B., Memoire sur la torsion des prismes, Paris memoir. savants etrangers, 14, 233-560, (1856)
[2] Saint-Venant, B., Memoire sur la flexion des prismes, J. math. pures. appl., 1, 89-189, (1856)
[3] Knowles, J.K.; Horgan, C.O., On the exponential decay of stresses in circular elastic cylinders subject to axisymmetric self-equilibrated end loads, Int. J. solid struct., 5, 33-50, (1969) · Zbl 0164.55302
[4] Horgan, C.O.; Knowles, J.K., Recent developments concerning saint-venant’s principle, (), 179-269 · Zbl 0569.73010
[5] Stephen, N.G.; Wang, M.Z., Decay rates for the hollow circular cylinder, J. appl. mech., 59, 747-753, (1992) · Zbl 0770.73019
[6] Ladeveze, P., Saint-Venant principle in elasticity, J. mec. theor. appl., 2, 161-184, (1983) · Zbl 0529.73006
[7] Chirita, S.; Romania, Iasi, Saint-Venant problem and semi-inverse solutions in linear viscoelasticity, Acta mech., 94, 221-232, (1992) · Zbl 0753.73034
[8] Chirita, S.; Ciarletta, M.; Fabrizio, M., Saint-venant’s principle in linear viscoelasticity, Int. J. eng. sci., 35, 1221-1236, (1997) · Zbl 0917.73028
[9] Lin, R.M.; Lim, M.K., Complex eigensensitivity-based characterization of structures with viscoelastic damping, J. acoust. soc. am., 100, 3182-3191, (1996)
[10] Muravyov, A.; Hutton, S.G., Closed-form solutions and the eigenvalue problem for vibration of discrete viscoelastic systems, J. appl. mech. ASME, 64, 684-691, (1997) · Zbl 0899.73138
[11] Mokeyev, V.V., A generalized complex eigenvector method for dynamic analysis of heterogeneous viscoelastic structures, Int. J. numer. meth. eng., 50, 2271-2282, (2001) · Zbl 1017.74026
[12] Barrett, K.E.; Gotts, A.C., FEM for one- and two-dimensional viscoelastic materials with spherical and rotating domains using FFT, Comput. struct., 82, 181-192, (2004)
[13] Lotfi, A.; Molnarka, G., The method of asymptotic expansion for plate problem in the linear theory of viscoelasticity, Z. angew. math. mech., 80, S391-S392, (2000) · Zbl 0979.74043
[14] Nayfeh, S.A., Damping of flexural vibration in the plane of lamination of elastic-viscoelastic sandwich beams, J. sound vibr., 276, 689-711, (2004)
[15] Hryniewicz, Z., Dynamic analysis of system with deterministic and stochastic viscoelastic dampers, J. sound vibr., 278, 1013-1023, (2004)
[16] Lee, S.S., Free-edge stress singularity in a two-dimensional unidirectional viscoelastic laminate model, J. appl. mech. ASME, 64, 408-414, (1997) · Zbl 0892.73032
[17] Rossikhin, Y.A.; Shitikova, M.V., New method for solving dynamic problems of fractional derivative viscoelasticity, Int. J. eng. sci., 39, 149-176, (2001)
[18] Schmidt, A.; Gaul, L., Finite element formulation of viscoelastic constitutive equations using fractional time derivatives, Nonlinear dyn., 29, 37-55, (2002) · Zbl 1028.74013
[19] Haneczok, G.; Weller, M., A fractional model of viscoelastic relaxation, Mater. sci. eng., 370, 209-212, (2004)
[20] Batra, R.C.; Yu, J.H., Torsion of a viscoelastic cylinder, J. appl. mech. ASME, 67, 424-426, (2000) · Zbl 1110.74334
[21] Tzeng, J.T., Viscoelastic analysis of composite cylinders subjected to rotation, J. compos. mater., 36, 229-239, (2002)
[22] Bonet, J., Large strain viscoelastic constitutive models, Int. J. solids struct., 38, 2953-2968, (2001) · Zbl 1058.74027
[23] Adolfsson, K.; Enelund, M., Fractional derivative viscoelasticity at large deformations, Nonlinear dyn., 33, 301-321, (2003) · Zbl 1065.74015
[24] Sogabe, Y.; Nakano, M., Finite element analysis of dynamic behavior of viscoelastic materials using FFT, JSME int. J. ser. A, 39, 71-77, (1996)
[25] Sim, W.J.; Lee, S.H., Finite element analysis of transient dynamic viscoelastic problems in time domain, J. mech. sci. tech., 19, 61-71, (2005)
[26] Akoz, Y.; Kadioglu, F., The mixed finite element method for the quasi-static and dynamic analysis of viscoelastic Timoshenko beams, Int. J. numer. meth. eng., 44, 1909-1932, (1999) · Zbl 0932.74064
[27] Leung, A.; Zhu, B., Two-dimensional viscoelastic vibration by analytic Fourier p-elements, Thin-walled struct., 41, 1159-1170, (2003)
[28] Schanz, M.; Antes, H., A new visco- and elastodynamic time domain boundary element formulation, Comput. mech., 20, 452-459, (1997) · Zbl 0898.73071
[29] Mesquita, A.D.; Coda, H.B., New methodology for the treatment of two dimensional viscoelastic coupling problems, Comput. meth. appl. mech. eng., 192, 1911-1927, (2003) · Zbl 1140.74564
[30] Mesquita, A.D.; Coda, H.B., A two-dimensional BEM/FEM coupling applied to viscoelastic analysis of composite domains I, Int. J. numer. meth. eng., 57, 251-270, (2003) · Zbl 1062.74638
[31] Brilla, J., Laplace transform and new mathematical theory of viscoelasticity, Meccanica, 32, 187-195, (1997) · Zbl 0892.73014
[32] Folch, A.; Fernandez, J., Ground deformation in a viscoelastic medium composed of a layer overlying a half-space: a comparison between point and extended sources, Geophysical J. int., 140, 37-50, (2000)
[33] Wang, J.Z.; Zhou, Y.H., Computation of the Laplace inverse transform by application of the wavelet theory, Commun. numer. meth. eng., 19, 959-975, (2003) · Zbl 1035.65159
[34] De Chant, L.J., Impulsive displacement of a quasi-linear viscoelastic material through accurate numerical inversion of the Laplace transform, Comput. math. appl., 43, 1161-1170, (2002) · Zbl 1156.74392
[35] Ezzat, M.A.; El-Karamany, A.S., The relaxation effects of the volume properties of viscoelastic material in generalized thermoelasticity with thermal relaxation, J. thermal stresses, 26, 671-690, (2003) · Zbl 1211.74047
[36] Temel, B.; Calim, F.F., Quasi-static and dynamic response of viscoelastic helical rods, J. sound vibr., 271, 921-935, (2004)
[37] Papargyri-Beskou, S.; Beskos, D.E., Response of gradient-viscoelastic bar to static and dynamic axial load, ACTA mech., 170, 199-212, (2004) · Zbl 1079.74037
[38] Zhang, W.F., A numerical method for wave propagation in viscoelastic stratified porous media, Transp. porous media, 61, 15-24, (2005)
[39] Huang, Y.; Crouch, S.L., A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions, Eng. anal. boundary elements, 29, 725-737, (2005) · Zbl 1182.74220
[40] Schanz, M.; Antes, H., Convolution quadrature boundary element method for quasi-static visco- and poroelastic continua, Comput. struct., 83, 673-684, (2005)
[41] Syngellakis, S., Boundary element methods for polymer analysis, Eng. anal. boundary elements, 27, 125-135, (2003) · Zbl 1080.74564
[42] Schanz, M., A boundary element formulation in time domain for viscoelastic solids, Commun. numer. methods eng., 15, 799-809, (1999) · Zbl 0952.74080
[43] Ding, R.; Zhu, Z.Y.; Cheng, C.J., Boundary element method for solving dynamical response of viscoelastic thin plate (I), Appl. math. mech., 18, 229-235, (1997) · Zbl 0889.73075
[44] Gaul, L.; Schanz, M., Boundary element calculation of transient response of viscoelastic solids based on inverse transformation, Meccanica, 32, 171-178, (1997) · Zbl 0908.73085
[45] Zhong, W.X., Duality system in applied mechanics and optimal control, (2004), Kluwer Academic Publishers Dordrecht
[46] Xu, X.S.; Zhong, W.X.; Zhang, H.W., The Saint-Venant problem and principle in elasticity, Int. J. solids struct., 34, 2815-2827, (1997) · Zbl 0942.74560
[47] Christensen, R.M., Theory of viscoelasticity, (1982), Academic Press New York · Zbl 0456.73021
[48] Lakes, R.S., Viscoelastic solids, (1999), CRC Press Boca Raton, FL · Zbl 1098.74013
[49] Drozdov, A.D., Mechanics of viscoelastic solids, (1998), Wiley New York · Zbl 0910.73021
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