  A general framework for Maxwell-Oldroyd type differential constitutive models
is examined, in which an unspecified nonlinear function of the stress and
rate-of-deformation tensors is incorporated into the well-known corotational
version of the Jeffreys model discussed by Oldroyd. For medium amplitude simple
shear deformations, the recently developed mathematical framework of medium
amplitude parallel superposition (MAPS) rheology reveals that this generalized
nonlinear Maxwell model can produce only a limited number of distinct
signatures, which combine linearly in a well-posed basis expansion for the
third order complex viscosity. This basis expansion represents a library of
MAPS signatures for distinct constitutive models that are contained within the
generalized nonlinear Maxwell model. We describe a framework for quantitative
model identification using this basis expansion, and discuss its limitations in
distinguishing distinct nonlinear features of the underlying constitutive
models from medium amplitude shear stress data. The leading order contributions
to the normal stress differences are also considered, revealing that only the
second normal stress difference provides distinct information about the weakly
nonlinear response space of the model. After briefly considering the conditions
for time-strain separability within the generalized nonlinear Maxwell model, we
apply the basis expansion of the third order complex viscosity to derive the
medium amplitude signatures of the model in specific shear deformation
protocols. Finally, we use these signatures for estimation of model parameters
from rheological data obtained by these different deformation protocols,
revealing that three-tone oscillatory shear deformations produce data that is
readily able to distinguish all features of the medium amplitude, simple shear
response space of this generalized class of constitutive models.
