  A new mathematical representation for nonlinear viscoelasticity is presented
based on application of the Volterra series expansion to the general nonlinear
relationship between shear stress and shear strain history. This theoretical
and experimental framework, which we call Medium Amplitude Parallel
Superposition (MAPS) Rheology, reveals a new material property, the third order
complex modulus, which describes completely the weakly nonlinear response of a
viscoelastic material in an arbitrary simple shear flow. In this first part, we
discuss several theoretical aspects of this mathematical formulation and new
material property. For example, we show how MAPS measurements can be performed
in strain- or stress-controlled contexts and provide relationships between the
weakly nonlinear response functions measured in each case. We show that the
MAPS response function is a super-set of the response functions that have been
previously reported in medium amplitude oscillatory shear and parallel
superposition rheology experiments. We also show how to exploit inherent
symmetries of the MAPS response function to reduce it to a minimal domain for
straightforward measurement and visualization. We compute this material
property for a few constitutive models to illustrate the potential richness of
the data sets generated by MAPS experiments. Finally, we discuss the MAPS
framework in the context of some other nonlinear, time-dependent rheological
probes and explain how the MAPS methodology has a distinct advantage over these
others because it generates data embedded in a very high dimensional space
without driving fluid mechanical instabilities, and is agnostic to the flow
protocol.
