The nonlinear yielding responses of three theoretical models, including the Bingham, a modified Bingham, and Giesekus models, to large-amplitude oscillatory shear are investigated under the framework proposed recently by Rogers et al. (2011). Under this framework, basis states are allowed to wax and wane throughout an oscillation, an approach that conflicts directly with the assumptions of all Fourier-like linear algebraic approaches. More physical yielding descriptions of the nonlinear waveforms are attained by viewing the responses as representing purely elastic to purely viscous sequences of physical processes. These interpretations are compared with, and contrasted with, results obtained from linear algebraic analysis methods: Fourier-transform rheology; and the Chebyshev description of the so-called elastic and viscous stress components sigma' and sigma ''. Further, we show that the discrepancies between the built-in model responses and parameters, and the interpretations of the Chebyshev and Fourier coefficients are directly related to misinterpretations of sigma' and sigma '' as being the elastic and viscous stress contributions. We extend these ideas and discuss how every linear algebraic analysis is likely to conflate information from predominantly elastic and viscous processes when a material yields. (C) 2012 The Society of Rheology. [DOI: 10.1122/1.3662962]