The method of separation of variables for linear viscoelastic anisotropic body problems
Nowadays, polymers are widely used in various fields. Such materials often exhibit viscoelastic properties. Engineering analysis considering viscoelasticity is laborious and requires certain expertize. This paper proposes a method for solving linear viscoelastic problems in a simpler way and presents a variant of the solution extension to an anisotropic case. The Volterra correspondence principle allows one to analyze viscoelastic bodies on the basis of the analytical solution like an elastic problem. The developed method is described in a similar way. It allows determining of some functions of time and material constants whose values at a certain point in time can be used as elastic constants. The solutions to these two problems are identical. To substantiate this statement, the authors consider the conditions for maximum equivalence of specific potential energy functionals of strain and stress (for the cases of kinematic and force boundary conditions, respectively) of viscoelastic and reference elastic media. The functions satisfying these conditions have been found, and a new method for solving the problems of linear viscoelasticity of an anisotropic body has been shown using several examples.
Keywords
effective modules of Lagrange and Castilian types,
variational problem,
anisotropy,
orthotropy,
integral operatorsAuthors
| Svetashkov Aleksandr A. | Tomsk Polytechnic University | svetashkov@tpu.ru |
| Kupriyanov Nikolay A. | Tomsk Polytechnic University | kupriyanov@tpu.ru |
| Pavlov Mikhail S. | Tomsk Polytechnic University; Tomsk State University | mspavlov@tpu.ru |
Всего: 3
References
Maxwell J.C. On the dynamical theory of gases // Philosophical Transactions. 1867. V. 157. P. 49-88.
Boltzman L. Zur theorie der elastischen nachwirkung // Wiener Berichte. 1874. Is. 70. P. 275 10.1002/andp. 18782411107.
Volterra V. Lecons sur Les Fonctions de Lignes. Paris: Gautierr Villars, 1912. 230 p.
Volterra V. Theory of Functionals and of Integral and integrodifferential Equations. London; Glasgow: Blackie & Son Limited, 1930. 226 p.
Cristensen R.M. Theory of Viscoelasticity: An Introduction. New York: Academic, 1980. 364 p.
Работнов Ю.Н. Ползучесть элементов конструкций. М.: Наука, 1966. 752 с.
Ильюшин А.А., Победря Б.Е. Основы математической теории термо-вязкоупругости. М.: Наука, 1970. 280 с.
Schapery R.A. Stress analysis of viscoelastic composite materials // Journal of Composite Materials. 1967. V. 1, is. 3. P. 228-267.
Wu W., Jiang G., Huang S., Leo C.J. Vertical dynamic response of pile embedded in layered transversely isotropic soil // Mathematical Problems in Engineering. 2014. V. 2014. Art. 126916. 12 p.
Kaloerov S.A., Koshkin A.A. Solving the problem of linear viscoelasticity for piecewise-homogeneous anisotropic plates // International Applied Mechanics. 2017. V. 53, is. 6. P. 11231129.
Kaminskii A.A., Selivanov M.F. A Method for solving boundary-value problems of linear viscoelasticity for anisotropic composites // International Applied Mechanics. 2003. V. 39, is. 11. P. 1294-1304. doi: 10.1023/B:INAM.0000015599.90700.86.
Holzapfel G.A., Gasser T.C. A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications // Computer Methods in Applied Mechanics and Engineering. 2003. V. 190, is. 34. P. 4379-4403.
Simo J.C. On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects // Computer Methods in Applied Mechanics and Engineering. 1987. V. 60. P. 153-173.
Галин Л.А. Контактные задачи теории упругости и вязкоупругости. М.: Наука, 1980. 304 с.
Атам М.Н.М., Победря Б.Е. К решению квазистатических задач анизотропной вязкоупругости // Известия Академии наук Армянской ССР. Механика. 1978. № 2. С. 19-27.
Svetashkov A.A., Kupriyanov N.A., Pavlov M.S., Vakurov A.A. Variable separation method for solving boundary value problems of isotropic linearly viscoelastic bodies // Acta Mechanica. 2020. V. 231, is. 9. P. 3583-3606.
Победря Б.Е. Механика композиционных материалов. М.: Изд-во МГУ, 1984. 336 с.
Svetashkov A., Kupriyanov N., Manabaev K. Modification of the iterative method for solving linear viscoelasticity boundary value problems and its implementation by finite element method // Acta Mechanica. 2018. V. 229, is. 6. P. 2539-2559.
Лехницкий С.Г. Теория упругости анизотропного тела. М.: Наука, 1977. 416 с.
