dЄ_{H}/dt = 2 ΩR/L_{0}
where R is the radius of the equal dimension
windup drums, and L_{0} is the fixed, unsupported length
(referred to as the "stretch zone") of the specimen sample being stretched,
which is equal to the centerline
distance between the master and slave drums.
The material's resistance to stretch imparts
a tangential force, F, on the surface of the drums which is
then translated as a resultant torque, T,
that can either be resolved as a driving torque on the rotating drive
shaft [F] (as in the case of when the SER is accommodated on
a Controlled Stress Rheometer) or as a twisting moment transmitted
through the chassis to the stationary torque shaft [G] (as in the case
of when the SER is mounted on a Controlled Rate Rheometer).
This resultant torque, T, can easily be determined from a summation
of moments about the primary axis of rotation, which yields:
T = 2 (F + F_{F})
R
where T is the resultant torque measured by the
torque sensor and F_{F} is the frictional contribution
from the bearings and intermeshing gears. With the precision
bearings and gears outfitted on the SER3 unit, the frictional term is
typically less than 1% of the measured torque signal and can be
neglected such that the expression for the measured torque can be
simplified to:
T = 2 F R
If there is no deviation between the nominal and
actual strain rates, the instantaneous crosssectional area, A(t),
of the stretched specimen changes exponentially with time, t, for a
constant Hencky strain rate experiment and can be expressed
as:
A(t) = A_{0} exp[
(dЄ_{H}/dt) t]
where A_{0} is the initial
crosssectional area of the unstretched specimen. For a constant
Hencky strain rate, the transient extensional viscosity (also referred
to as the tensile stress growth function),
η_{E}^{+}(t), of the stretched sample can then be expressed
as:
η_{E}^{+}(t) = F(t) / [A(t) (dЄ_{H}/dt)]
where F(t) is the instantaneous extensional
force at time t exerted by the sample as it resists stretch as
determined from the measured torque signal, T. Hence, for a given
rate of
extensional deformation,
the measured torque signal is
directly related to
the extensional
viscosity of the specimen
being stretched in the isolated
"stretch zone" of length L_{0} defined
by the tangent plane
between
the drums.
