Johnson-Segalman negative-slope shear threshold (${\mu }_0/({\alpha }^2{G}_0\tau )<1/8$)

For nonzero slip parameter with $|\alpha |<1$, scale shear rate as $x=\tau \sqrt{1-{\alpha }^2}\dot{\gamma }$ and stress as $S=x[b+1/(1+x^2)]$, where $b={\mu }_0/({\alpha }^2{G}_0\tau )$. The derivative is $b+(1-x^2)/(1+x^2{)}^2$ and its least value is $b-1/8$, attained at $x^2=3$. A decreasing homogeneous constitutive segment therefore exists exactly under the displayed condition. It permits different shear rates at the same stress and motivates shear banding; the local curve alone does not select a band-interface stress.