  The large amplitude oscillatory shear (LAOS) has been extensively studied for
understanding the rheological responses of yield stress fluids. However, the
employed methodology for determining the yield stress remains uncertain albeit
the fact that many classical or plausible methods exist in the literature.
Along these lines, herein, based on Fourier transform (FT) rheology, stress
decomposition, and stress bifurcation, a new straightforward method termed as
algebraic stress bifurcation was developed. More specifically, the main goal
was to determine the yield stress and investigate the solid-liquid transition
of fluids in LAOS. A simple and efficient mathematical framework was
established and verified by the KVHB, Saramito, Giesekus models, and FT
rheology. The main strength of this approach is that only the data from the
stress/strain sweep are required instead of Lissajous curves. Alternative
curves based on the first harmonic were constructed to demonstrate the
non-critical role of both higher harmonics and phenomenological Lissajous
curves in determining yield stress. The determined start and end yield points
in the solid-liquid transition were compared with the already existing methods.
Furthermore, the resulting solid-liquid transition region was analyzed by FT
rheology, stress decomposition, and sequence of the physical process to obtain
information on nonlinearity and intracycle/intercycle yielding. Our work
provides fruitful insights for explaining and reducing the complexities of the
stress bifurcation technique by using an easy-to-understand and implement
format. Therefore, a concise theoretical framework was introduced for
understanding the concept of yield stress, the intercycle yielding process, and
the rational choice of yield stress measurement techniques.
