A novel high-precision non-classical method to solve fractional rheology and viscoelastic vibration: Linear computational complexity and experimental verification

The theory of fractional calculus can precisely depict the non-integer order characteristics of materials, especially when it comes to predicting viscoelastic behaviors such as rheology and viscoelastic damping. Despite the marked superiority of fractional calculus in the realm of viscoelastic material modeling, its numerical processing encounters numerous challenges. This is because traditional computational methods for handling such problems must deal with a large amount of historical data, thus leading to low efficiency. Moreover, existing non-classical methods typically find it arduous to simultaneously take into account both computational accuracy and efficiency. In light of the aforementioned issues, this study presents an innovative non-classical computational approach. Through the implementation of the piecewise processing strategy, this study effectively addresses the inherent limitation of weak algebraic decay in the infinite state representation associated with non-classical methods. This innovative approach not only achieves a substantial improvement in computational accuracy but also maintains an efficient linear computational complexity, thereby striking an optimal balance between precision and computational efficiency. This method has been successfully applied to the solution of multi- component fractional viscoelastic constitutive equations and verified through experiments. Furthermore, based on the nonlocal strain gradient theory and the fractional-order constitutive relation, the nonlinear motion equation of the fractional viscoelastic nanobeam is derived. Comparative analyses of the vibration responses of the linear and nonlinear models are conducted, revealing the nonlinear viscoelastic damping characteristics of the system. The research outcomes indicate that this method is applicable to addressing fractional viscoelastic mechanics problems and holds the potential to extend to a broader category of fractional differential equations, being capable of providing computational support for multiple disciplines.