Classical and thermodynamically consistent fractional Burgers models are examined in creep and stress relaxation tests. Using the Laplace transform method, the creep compliance and relaxation modulus are obtained in integral form, that yielded, when compared to the thermodynamical requirements, the narrower range of model parameters in which the creep compliance is a Bernstein function while the relaxation modulus is completely monotonic. Moreover, the relaxation modulus may even be oscillatory function with decreasing amplitude. The asymptotic analysis of the creep compliance and relaxation modulus is performed near the initial time-instant and for large time as well.

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Thermodynamically consistent fractional Burgers constitutive models for viscoelastic media, divided into two classes according to model behavior in stress relaxation and creep tests near the initial time instant, are coupled with the equation of motion and strain forming the fractional Burgers wave equations. Cauchy problem is solved for both classes of Burgers models using integral transform method and analytical solution is obtained as a convolution of the solution kernels and initial data. The form of solution kernel is found to be dependent on model parameters, while its support properties implied infinite wave propagation speed for the first class and finite for the second class. Spatial profiles corresponding to the initial Dirac delta displacement with zero initial velocity display features which are not expected in wave propagation behavior.

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A primary mass attached through a solid horizontal rod to the wall, able to slide without friction along the horizontal line under a periodic force modeled by a sine function, and an added mass attached to the main one through another horizontal solid rod, also able to slide along the same line without friction, represent a problem encountered in almost all textbooks on mechanical vibrations. However, many of the books consider conditions ensuring zero steady-state amplitude of the primary mass and just several of them consider conditions ensuring either reduction of the primary mass amplitude or cutting one down as much as possible. Once again, in the whole class of books, one can find the rods of either Hookean or the Kelvin-Voigt type, i.e. linear springs or linear springs connected in parallel to dashpots. In this work, the vibration absorbing conditions ensuring the reduction of the primary mass steady-state amplitude will be stated for the Kelvin-Zener model of viscoelastic rod and its fractional generalization. The obtained conditions will be related to the restrictions on coefficients in these models that follow from the Clausius-Duhem inequality. The proposed model could be used for the study of energy dissipation in mechanical systems incorporating polymers, elastomers, living tissues and other real materials.