TITLE:
The Equilibrium of Fractional Derivative and Second Derivative: The Mechanics of a Power-Law Visco-Elastic Solid
AUTHORS:
Franz Konstantin Fuss
KEYWORDS:
Fractional Derivative, Non-Linear Visco-Elasticity, Power Law, Mittag-Leffler Function, Underdamping, Overdamping, Innovative Design of Material Damping
JOURNAL NAME:
Applied Mathematics,
Vol.7 No.16,
October
14,
2016
ABSTRACT: This paper investigates the equilibrium of
fractional derivative and 2nd derivative, which occurs if the original function
is damped (damping of a power-law viscoelastic solid with viscosities η of 0 ≤
η ≤ 1), where the fractional derivative corresponds to a force applied to the
solid (e.g. an impact force), and the second derivative corresponds to
acceleration of the solid’s centre of mass, and therefore to the inertial
force. Consequently, the equilibrium satisfies the principle of the force
equilibrium. Further-more, the paper provides a new definition of under- and
overdamping that is not exclusively disjunctive, i.e. not either under- or
over-damped as in a linear Voigt model, but rather exhibits damping phases
co-existing consecutively as time progresses, separated not by critical
damping, but rather by a transition phase. The three damping phases of a
power-law viscoelastic solid—underdamping, transition and overdamping—are
characterized by: underdamping—centre of mass oscillation about zero line; transition—centre
of mass reciprocation without crossing the zero line; overdamping—power decay.
The innovation of this new definition is critical for designing non-linear
visco-elastic power-law dampers and fine-tuning the ratio of under- and
overdamping, considering that three phases—underdamping, transition, and
overdamping—co-exist consecutively if 0