Energies of Mechanical Fractional-Order Elements: Causal Concept and Kernel Effects

Abstract

The energies of the classical Maxwell mechanical model of viscoelastic behavior have been studied as a template with a variety of relaxation kernels in light of a causal formulation of the force–displacement relationship. The starting point uses the Lorenzo–Hartley model with the time-fractional Riemann–Liouville derivative. This approach has been reformulated based on critical analysis, allowing for the application of a variety of relaxation (memory) functions mainly based on the Mittag-Leffler family, in order to meet the need for broader modeling of viscoelastic behavior. The examples provided include cases of the types of forces used by Lorenzo and Hartley as well as a new family of force approximations such as a general power-law ramp, polynomials, and the Prony series.

Keywords

• Maxwell mechanical model
• causality
• force–displacement relationship
• energies of fractional models
• Mittag-Leffler function