**On secondary loops in
LAOS via self-intersection of Lissajous-Bowditch
curves**

by Randy H. Ewoldt and Gareth H. McKinley

When the shear stress measured in Large
Amplitude Oscillatory Shear (LAOS) deformation is represented as a two-dimensional
Lissajous-Bowditch curve, the corresponding trajectory
can appear to self- intersect and form secondary loops. This self-intersection
is a general consequence of a strongly nonlinear material response to the
imposed oscillatory forcing and can be observed for various material systems and
constitutive models. We derive the mathematical criteria for the formation of
secondary loops, quantify the location of the apparent intersection, and
furthermore suggest a qualitative physical understanding for the associated nonlinear
material behavior. We show that when secondary loops appear in the viscous projection
of the stress response (the 2-D plot of stress vs. strain-rate) they are best interpreted by understanding
the corresponding elastic response (the 2-D projection of stress vs. strain)
The analysis shows clearly that sufficiently strong *elastic *nonlinearity is required to observe secondary
loops on the conjugate *viscous
*projection. Such a strong
elastic nonlinearity physically corresponds to a nonlinear viscoelastic
shear stress overshoot in which existing stress is unloaded more quickly than
new deformation is accumulated. This general understanding of secondary loops
in LAOS flows can be applied to various molecular
configurations and microstructures such as polymer solutions, polymer melts, soft
glassy materials and other structured fluids.