  In view of the increasing attention to the time responses of complex fluids
described by power-laws in association with the need to capture inertia effects
that manifest in high-frequency microrheology, we compute the five basic
time-response functions of in-series or in-parallel connections of two
elementary fractional derivative elements known as the Scott-Blair (springpot)
element. The order of fractional differentiation in each Scott-Blair element is
allowed to exceed unity reaching values up to 2 and at this limit-case the
Scott-Blair element becomes an inerter--a mechanical analogue of the electric
capacitor that its output force is proportional only to the relative
acceleration of its end-nodes. With this generalization, inertia effects may be
captured beyond the traditional viscoelastic behavior. In addition to the
relaxation moduli and the creep compliances, we compute closed form expressions
of the memory functions, impulse fluidities (impulse response functions) and
impulse strain-rate response functions of the generalized fractional derivative
Maxwell fluid, the generalized fractional derivative Kelvin-Voigt element and
their special cases that have been implemented in the literature. Central to
these calculations is the fractional derivative of the Dirac delta function
which makes possible the extraction of singularities embedded in the fractional
derivatives of the two-parameter Mittag-Leffler function that emerges
invariably in the time-response functions of fractional derivative rheological
modes.
