  The FENE-P and sPTT viscoelastic models are widely used for modelling of
complex fluids. Although they are derived from distinct micro-structural
theories, these models can become mathematically identical in steady and
homogeneous flows with a particular choice of the value of the model
parameters. However, even with this choice of parameter values, the model
responses are known to differ from each other in transient flows. In this work,
we investigate the responses of the FENE-P and sPTT constitutive models in
Large Amplitude Oscillatory Shear (LAOS). In steady-shear, the shear stress
scales with the non-dimensional group $Wi/(aL) \ (Wi\sqrt{\epsilon})$ for the
FENE-P (sPTT) model, where $Wi$ is the Weissenberg number, $L^2$ is the limit
of extensibility in the FENE-P model ($a$ being $L^2/(L^2-3)$) and $\epsilon$
is the extensibility parameter in the sPTT model. Our numerical and analytical
results show that, in LAOS, the FENE-P model only shows this universality for
large values of $L^2$ whereas the sPTT model shows this universality for all
values of$\epsilon$. In the strongly non-linear region, there is a drastic
difference between the responses of the two models, with the FENE-P model
exhibiting strong shear stress overshoots which manifest as self-intersecting
secondary loops in the viscous Lissajous curves. We quantify the non-linearity
exhibited by each constitutive model using the Sequence of Physical Processes
framework. Despite the high degree of non-linearity exhibited by the FENE-P
model, we also show using fully non-linear 1D simulations that it does not
shear band in LAOS within the range of conditions studied.
