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**On the fractional calculus model of viscoelastic behavior.**
*(English)*
Zbl 0613.73034

It is considered a mathematical model of the viscoelastic phenomenon employing derivatives of fractional order, which is examined with respect to its consistency with thermodynamic principles. The constraints on the parameters of the model are revealed to ensure that the model predicts a nonnegative rate of energy dissipation and nonnegative internal work. Such analysis is fulfilled for an initially five parameter constitutive relation describing the one-dimensional state of stress. Examination of the sinusoidal response shows in particular that the steady-state response predicts a positive material loss factor at all frequencies.

Analyzing the relaxation and creep responses the authors make the conclusion that satisfaction of thermodynamic constraints leads to a well behaved mathematical description of the viscoelastic phenomenon and is necessary for proper reflection of these responses. A good correlation with the experimental data is also noted. The authors mention in particular that the considered one-dimensional model, parameters of which are calculated on the basis of experiments by the least squares method, yields specific models which are accurate for several decades of material properties.

Analyzing the relaxation and creep responses the authors make the conclusion that satisfaction of thermodynamic constraints leads to a well behaved mathematical description of the viscoelastic phenomenon and is necessary for proper reflection of these responses. A good correlation with the experimental data is also noted. The authors mention in particular that the considered one-dimensional model, parameters of which are calculated on the basis of experiments by the least squares method, yields specific models which are accurate for several decades of material properties.

Reviewer: V.Korneev

### MSC:

74D99 | Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) |

74D05 | Linear constitutive equations for materials with memory |

74D10 | Nonlinear constitutive equations for materials with memory |

74A20 | Theory of constitutive functions in solid mechanics |