Geodesic
The method of separation of variables for linear viscoelastic anisotropic body problems
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 84 (2023), pp. 123-138

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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 operators. 
@article{VTGU_2023_84_a9,
     author = {A. A. Svetashkov and N. A. Kupriyanov and M. S. Pavlov},
     title = {The method of separation of variables for linear viscoelastic anisotropic body problems},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {123--138},
     publisher = {mathdoc},
     number = {84},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2023_84_a9/}
}
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A. A. Svetashkov; N. A. Kupriyanov; M. S. Pavlov. The method of separation of variables for linear viscoelastic anisotropic body problems. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 84 (2023), pp. 123-138. http://geodesic.mathdoc.fr/item/VTGU_2023_84_a9/