Generalized Fractional Derivative Anisotropic Viscoelastic Characterization.

Area of Science:

• Continuum Mechanics
• Materials Science
• Viscoelasticity

Background:

• Fractional calculus offers advanced modeling for complex material responses.
• Generalized Kelvin models are foundational for viscoelastic behavior analysis.
• Anisotropic and functionally graded materials present unique constitutive challenges.

Purpose of the Study:

• To formulate and examine isotropic and anisotropic fractional derivative constitutive relations for viscoelastic materials.
• To develop computationally efficient integral constitutive relations from fractional differential ones.
• To establish protocols for fitting fractional derivative models to experimental data.

Main Methods:

• Formulation of isotropic linear and nonlinear fractional derivative constitutive relations.
• Analytical extension to anisotropic, homogeneous, non-homogeneous, and functionally graded materials.
• Derivation of equivalent integral constitutive relations and development of approximate Fourier transform inversions.
• Development and evaluation of approximate protocols for curve fitting to experimental data.

Main Results:

• Fractional derivative constitutive relations were formulated and extended to diverse material behaviors.
• Equivalent integral constitutive relations and anisotropic shift functions were established.
• Approximate methods for Fourier transform inversion and experimental data fitting were developed and evaluated.
• Comparison of integer and fractional derivative constitutive relations for real material analysis.

Conclusions:

• Fractional derivative models provide a powerful framework for describing complex viscoelasticity.
• Integral forms offer computational advantages for viscoelastic analysis.
• Case-by-case examination is necessary to determine the optimal constitutive model for specific materials.