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On the monotonicity of the relaxation spectrum of fractional Maxwell model of viscoelastic materials

Stankiewicz A.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

2018

Vol. 7, No 1

117--125

This article focuses on the relaxation spectrum of fractional Maxwell model, which is a generalization of classic viscoelastic Maxwell model to non-integer order derivatives. The analytical formula for the spectrum of relaxation frequencies is derived. Theoretical analysis of the relaxation spectrum monotonicity is conducted by using simple analytical methods and illustrated by means of numerical examples. The necessary and sufficient conditions for the existence and uniqueness of the maximum of relaxation spectrum are stated and proved. The analytical formulas for minimum and maximum of the relaxation spectrum are derived. Also, a few useful properties concerning the relaxation spectrum monotonicity and concavity are given in the mathematical form of simple inequalities expressed directly in terms of the fractional Maxwell model parameters, which can be used to simplify the calculations and analysis.