Microscopic formulation of fractional calculus theory of viscoelasticity based on lattice dynamics

A microscopic fractional calculus theory of viscoelasticity is developed on the basis of lattice dynamics by generalizing the standard model of a chain of coupled simple harmonic oscillators to a chain of coupled fractional oscillators by generalizing the integral equations of motion of a chain of simple harmonic oscillators into ones involving fractional integrals. This set of integral equations of motion pertaining to the chain of coupled fractional oscillators is solved by using Laplace transforms. In the continuum limit the time-fractional diffusion-wave equation in one dimension is obtained. The response of the system to the sinusoidal forcing in this limit consists of a transient part and an attenuated steady part. Expressions for the absorption coefficient and the specific dissipation function are derived and numerical applications are discussed.