A novel approach to nonlinear variable-order fractional viscoelasticity

Area of Science:

• Materials Science
• Applied Mathematics
• Nonlinear Dynamics

Background:

• Fractional calculus is used to model viscoelastic materials.
• The Boltzmann superposition principle is fundamental for linear viscoelasticity.
• Time-dependent fractional orders pose challenges to standard modeling due to the principle's limitations.

Purpose of the Study:

• To develop a novel approach for analyzing nonlinear viscoelastic behavior in systems with variable time-dependent fractional orders.
• To address the limitations of the standard Boltzmann superposition principle when the fractional order changes over time.
• To provide a numerically implementable method for calculating stress and strain responses in such systems.

Main Methods:

• A novel approach is proposed by applying the Boltzmann superposition principle to an equivalent system.
• The equivalent system is constructed at each time instant, considering the instantaneous fractional order and prior responses.
• The method is designed for straightforward numerical implementation.

Main Results:

• The proposed method successfully derives system responses for fractional systems with time-varying fractional orders.
• It allows for the calculation of stress or strain responses.
• The approach is validated for its applicability in complex material modeling.

Conclusions:

• The developed method offers a robust solution for modeling nonlinear viscoelasticity with dynamic fractional orders.
• This work extends the applicability of fractional calculus in advanced materials modeling.
• The numerical implementation facilitates practical applications in engineering and material science.