A novel approach to nonlinear variable-order fractional viscoelasticity

This paper addresses nonlinear viscoelastic behaviour of fractional systems with variable time-dependent fractional order. In this case, the main challenge is that the Boltzmann linear superposition principle, i.e. the theoretical basis on which linear viscoelastic fractional operators are formulated, does not apply in standard form because the fractional order is not constant with time. Moving from this consideration, the paper proposes a novel approach where the system response is derived by a consistent application of the Boltzmann principle to an equivalent system, built at every time instant based on the fractional order at that instant and the response at all the previous ones. The approach is readily implementable in numerical form, to calculate either stress or strain responses of any fractional system where fractional order may change with time. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.