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Calculation of discrete spectrum of relaxation of viscoelastic media using experimentally determined relaxation modulus. Statement of the problem and algorithms of its solution

Bassistov Yu.A., Teplukhin A.V., Yanovsky Yu.G.

#### Abstract:

The problem of calculation of spectra of relaxation time periods of viscoelastic media is one of the most topical problems of mechanics. This problem belongs to the class of ill-posed inverse problems. Currently, several algorithms for solution of such problems is known: the method of regularization by Tikhonov for the Fredholm equations of the first kind, the method of approximation based on application of series of exponential functions, and the approaches under development by the authors, which are based on the methods of mathematical statistics. In order to make the comparison of the known algorithms and to choose the better one, we propose an algorithm based on the Proni method (the Proni method is widely used for spectrum analysis of time sequences). It is revealed that solution of inverse problems based on an application of exponential series is unstable even in a root-mean-square metric and requires regularization. At that, small number of exponents can be used for the regularization of solution and the number of exponents itself is a parameter of regularization. It is necessary to point out that the algorithm based on the linear exponential model requires additional modernization of the Proni method in order to obtain stable solution for wide range of relaxation time intervals. On the other hand, such modernization has no perspectives, since it is difficult to apply it to solution of nonlinear problems.

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