Hierarchical analogues to fractional relaxation equations

Recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena, such as stress-strain relationships in polymeric materials. In the rheological constitutive equations this implies the replacement of the first-order derivatives by fractional derivatives. Here we show that such procedures have hierarchical mechanical analogues. We focus on the generalized dashpot and the generalized Maxwell model and display the corresponding arrangements. Our models allow a transparent interpretation of the parameters which enter the fractional equations, and reveal that the internal dynamics are hierarchically constrained.