Constitutive behavior modeling and fractional derivatives

Publisher Summary This chapter reveals that fractional calculus allows a physically sound generalization of classical models from the linear theory of viscoelasticity. From a mathematical point of view, the operation of fractional integrodifferentiation is well defined and can be easily handled in Fourier or Laplace space. Integrodifferentiation allows, for instance, to interpolate smoothly between Hooke's and Newton's laws. The simplest form of a fractional rheological model is introduced, that is, fractional element (FE), and several mechanical analogs are presented, namely arrangements made out of springs and dashpots. In these (infinite) networks the order of fractional integration or differentiation can be adjusted in several ways; such as, by varying the material constants of the springs and dashpots involved, or by changing the structure of the arrangement. It is shown that parallel or serial combinations of FEs lead to more complex models; in particular, the extensions of the Maxwell, the Kelvin-Voigt, the Zener and the Poynting-Thomson models are studied. The representation of generalized viscoelastic models by fractional analogues also allows a deeper insight into the physics behind fractional stress-strain relations.