Mikhail Aleksandrovich Tinyaev (Bauman Moscow State Technical University) :


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)

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.

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.