Anna Stankiewicz

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Abstract. Fractional calculus is a mathematical approach dealing with derivatives and integrals of arbitrary and also complex orders. Therefore, it adds a new means to understand and describe the nature and behavior of complex dynamical systems. Here we use the fractional calculus for modeling mechanical viscoelastic properties of materials. In the present work, after reviewing some of the main viscoelastic fractional models, a new parallel model is employed, connecting in parallel two Scott-Blair models with additional multiplicative weight functions. The model is presented in terms of two power functions weighted by Debye-type functions extend representation, understanding and description of complex systems viscoelastic properties. Monotonicity of the model relaxation modulus is studied and some upper bounds for the minimal time value, above which the model relaxation modulus is monotonically decreasing are given and compared both analytically and numerically. The comparison with the results of relaxation tests executed on some real phenomena has shown that the parallel Scott-Blair model involving fractional derivatives has been in a good agreement.

Key words: fractional calculus, viscoelasticity, relaxation modulus, Scott-Blair fractional model

Parallel Scott-Blair fractional model of viscoelastic biological materials

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