  Braginskii MHD provides a more accurate description of many plasma
environments than classical MHD since it actively treats the stress tensor
using a closure derived from physical principles. Stress tensor effects
nonetheless remain relatively unexplored for solar MHD phenomena, especially in
nonlinear regimes. This paper analytically examines nonlinear damping and
longitudinal flows of propagating shear Alfv\'en waves. Most previous studies
of MHD waves in Braginskii MHD considered the strict linear limit of vanishing
wave perturbations. We show that those former linear results only apply to
Alfv\'en wave amplitudes in the corona that are so small as to be of little
interest, typically a wave energy less than $10^{-11}$ times the energy of the
background magnetic field. For observed wave amplitudes, the Braginskii viscous
dissipation of coronal Alfv\'en waves is nonlinear and a factor around $10^9$
stronger than predicted by the linear theory. Furthermore, the dominant damping
occurs through the parallel viscosity coefficient $\eta_0$, rather than the
perpendicular viscosity coefficient $\eta_2$ in the linearized solution. This
paper develops the nonlinear theory, showing that the wave energy density
decays with an envelope $(1+z/L_d)^{-1}$. The damping length $L_d$ exhibits an
optimal damping solution, beyond which greater viscosity leads to lower
dissipation as the viscous forces self-organise the longitudinal flow to
suppress damping. Although the nonlinear damping greatly exceeds the linear
damping, it remains negligible for many coronal applications.
