#### 539.3 Modeling the viscoelastic characteristics of materials based on the numerical inversion of the Laplace transform

##### Valishin A. A. (Bauman Moscow State Technical University), Tinyaev M. A. (Bauman Moscow State Technical University)

###### VISCOELASTICITY, RELAXATION, CREEP, RELAXATION AND CREEP NUCLEI, LAPLACE TRANSFORM, FOURIER METHOD, QUADRATURE FORMULA METHOD

doi: 10.18698/2309-3684-2020-3-321

When designing products made of composite materials intended for operation in difficult conditions of inhomogeneous deformations and temperatures, it is important to take into account the viscoelastic properties of the binder and fillers. The article analyzes the relationship between relaxation and creep characteristics. All known creep and re-laxation kernels in the literature are considered. The problem of transformation of creep characteristics into relaxation characteristics and vice versa is discussed in de-tail. There is a simple relationship between the creep and relaxation functions in the Laplace image space. However, when returning to the space of the originals, in many cases there are great difficulties in reversing the Laplace transform. Two numerical methods for inverting the Laplace transform are considered: the use of the Fourier series in sine and the method of quadrature formulas. Algorithms and computer programs for realization of these methods are made. It is shown that the operating time of a computer program implementing the Fourier method by sine is almost 2 times less than the operating time of a computer program implementing the quadrature formula method. However, the first method is inferior to the latter method in accuracy of calculations: the relaxation functions and relaxation rates, it is advisable to find the first method, since the computational error is almost indiscernible, and the functions of creep and creep speed, the second way, because for most functions, the result obtained by the second method is much more accurate than the result obtained by the first method. The Gavrillac-Negami creep function and the Gavrillac-Negami creep rate function could not be constructed due to a complex recursive formula for the series coefficients, but using both methods, these functions can still be obtained and compared with each other.

[1] Dimitrienko Yu.I. Nelineynaya mekhanika sploshnoy sredy [Nonlinear continuum mechanics]. Moscow, Fizmatlit Publ., 2009, 624 p.
[2] Kristensen R. Vvedenie v teoriyu vyazkouprugosti sredy [sredy [Nonlinear continuum mechanics]. Moscow, Mir Publ., 1974, 339 p.
[3] Blend D. Teoriya linejnoj vyazkouprugosti [Theory of linear viscoelasticity]. Moscow, Mir Publ., 1965, 199 p.
[4] Birger I.A., Panovko Ya.G. Prochnost', ustojchivost', kolebaniya. Spravochnik v tryoh tomah, tom 1 [Strength, stability, oscillations. Handbook in three volumes, volume 1]. Moscow, Mashinostroenie Publ., 1968, 831 p.
[5] Dimitrienko Yu. I. Mekhanika sploshnoy sredy. Tom 4. Osnovy mekhaniki tver­dogo tela [Continuum Mechanics. Vol. 4. Fundamentals of solid mechanics]. Moscow, BMSTU Publ., 2013, 624 p.
[6] Il'yushin A.A., Pobedrya B.E. Osnovy matematicheskoj teorii termovyazkouprugosti [Fundamentals of the mathematical theory of thermovyazcoelasticity]. Moscow, Nauka Publ., 1970, 356 p.
[7] Rabotnov Yu.N. Polzuchest' elementov konstrukcij [Creep of structural elements]. Moscow, Nauka Publ., 1966, 752 p.
[8] Koltunov M.A. Polzuchest' i relaksaciya [Creep and relaxation]. Moscow, Vysshaya shkola Publ., 1976, 276 p.
[9] Moskvitin B.V. Soprotivlenie vyazkouprugih materialov [Resistance of viscoelastic materials]. Moscow, Nauka Publ., 1972, 328 p.
[10] Rzhanicyn A.R. Teoriya polzuchesti [Theory of creep]. Moscow, Stroyizdat Publ., 1968, 416 p.
[11] Bartenev G.M., Frenkel S.Ya. Fizika polimerov [Physics of polymers]. Leningrad, Chemistry Publ., 1990, 432 p.
[12] Prohorov A.M. Fizicheskaya enciklopediya. T.1. Aaronovo — Dlinnye [Physical Encyclopedia. Vol. 1. Aaronovo — Dlinnye]. Moscow, Soviet Encyclopedia Publ., 1988, 703 p.
[13] Dimitrienko Y.I., Gubareva E.A., Sborschikov S.V. Finite element modulation of effective viscoelastic properties of unilateral composite materials. Маthematical Modeling and Computational Methods, 2014, no. 2, pp. 28–48.
[14] Dimitrienko Yu.I., Minin V.V., Syzdykov E.K. Modeling of the thermomechanical processes in composite shells in local radiation heating. Composites: Mechanics, Computations, Applications, 2011, vol. 2, iss. 2, pp. 147–169.
[15] Dimitrienko Yu.I., Gubareva E.A., Pichugina A.E. Theory of composite cylindrical shells under quasistatic vibrations, based on an asymptotic analysis of the general viscoelasticity theory equations. IOP Conference Series: Material Science and Engineering, 2019, vol. 683, no. 012013. DOI: 10.1088/1757-899X/683/1/012013
[16] Dimitrienko Y.I., Koryakov M.N., Zakharov A.A., Stroganov A.S. Computational modeling of conjugated gasdynamic and thermomechanical processes in composite structures of high speed aircraft. Маthematical Modeling and Coтputational Methods, 2014, no. 3, pp. 3–24.
[17] Janke E., Emde F., Lesh F. Special'nye funkcii [Special functions]. Moscow, Fizmatgiz Publ., 1968, 344 p.
[18] Pagurova V.I. Tablicy nepolnoj gamma-funkcii [Tables of incomplete gamma function]. Moscow, Fizmatgiz Publ., 1963, 235 p.
[19] Krylov V.I., Skoblya N.S. Metody priblizhennogo preobrazovaniya Fur'e i obrashcheniya preobrazovaniya Laplasa [Methods of the approximate Fourier transform and inversion of the Laplace transform]. Moscow, Nauka Publ., 1974, 223 p.

Валишин А.А., Тиняев М.А. Моделирование вязкоупругих характеристик материалов на основе численного обращения преобразования Лапласа. Математическое моделирование и численные методы, 2020, № 3, с. 3–21.