## 4 Asymptotic Analysis  
A necessary condition to satisfy Equation (28) or (32) as \(\mu\to 0\) is \(\Lambda\to\infty\) at some point of the interval \(\theta_{0}\leq\theta\leq\pi/2\) or \(\theta_{0}\leq\theta\leq\theta_{c}\), respectively. By assumption, \(\Phi\) is a monotonically increasing function of its argument. It is seen from (23) that the argument is a monotonically decreasing function of \(\theta\). Therefore, \(\Lambda\) may approach infinity only in the vicinity of \(\theta=\theta_{0}\).  
Equation (28) can be rewritten as  
\[1=\mu I_{1}+\mu I_{2} \tag{33}\]  
where  
\[I_{1}=\int\limits_{\theta_{0}}^{(1+\delta)\theta_{0}}\frac{1}{\sin\theta} \Lambda\bigg{(}\frac{C_{1}}{\sin^{2}\theta}\bigg{)}d\theta,\ I_{2}=\int \limits_{(1+\delta)\theta_{0}}^{\pi/2}\frac{1}{\sin\theta}\Lambda\bigg{(} \frac{C_{1}}{\sin^{2}\theta}\bigg{)}d\theta \tag{34}\]  
and \(0<\delta<<1\). If \(\alpha>0\) then it follows from (20) and (23) that  
\[\Lambda\bigg{(}\frac{C_{1}}{\sin^{2}\theta}\bigg{)}=A^{-1/\alpha}\bigg{(} \frac{C_{1}}{\sin^{2}\theta}\bigg{)}^{1/\alpha} \tag{35}\]  
in the vicinity of the surface \(\theta=\theta_{0}\). It is then seen from (34) that \(I_{1}\to\infty\) as \(C_{1}\to\infty\). Therefore, Equation (34) has a solution at any small value of \(\mu\) and the regime of sticking at the surface \(\theta=\theta_{0}\) is always possible.  
If \(\alpha<0\), then it follows from (20) and (23) that  
\[\Lambda\bigg{(}\frac{C_{1}}{\sin^{2}\theta}\bigg{)}=\bigg{[}-\frac{1}{A} \bigg{(}\frac{k_{\rm s}}{k_{0}}-\frac{C_{1}}{\sin^{2}\theta}\bigg{)}\bigg{]} ^{1/\alpha} \tag{36}\]  
in the vicinity of the surface \(\theta=\theta_{0}\). It is seen from this equation that the maximum possible value of \(C_{1}\) is  
\[C_{m}=\frac{k_{\rm s}\sin^{2}\theta_{0}}{k_{0}}. \tag{37}\]  
Therefore, the integral \(I_{2}\) is bounded at any value of \(C_{1}\). Using (36) and (37) the integral \(I_{1}\) at \(C_{1}=C_{m}\) can be written as  
\[I_{1}=\bigg{(}-\frac{1}{A}\frac{k_{\rm s}}{k_{0}}\bigg{)}^{1/\alpha}\int \limits_{\theta_{0}}^{(1+\delta)\theta_{0}}\frac{1}{\sin\theta}\bigg{[}\bigg{(} 1-\frac{\sin^{2}\theta_{0}}{\sin^{2}\theta}\bigg{)}\bigg{]}^{1/\alpha}d\theta. \tag{38}\]The integrand is represented as  
\[\frac{1}{\sin\theta}\left[\left(1-\frac{\sin^{2}\theta_{0}}{\sin^{2}\theta}\right) \right]^{1/\alpha}=B(\theta-\theta_{0})^{1/\alpha}+\varrho\left[\left(\theta- \theta_{0}\right)^{1/\alpha}\right] \tag{39}\]  
as \(\theta\rightarrow\theta_{0}\). Here \(B\) is a function of \(\theta_{0}\). It is evident from (38) and (39) that the integral \(I_{1}\) is divergent if  
\[\frac{1}{\alpha}+1\leq 0 \tag{40}\]  
and convergent if  
\[\frac{1}{\alpha}+1>0 \tag{41}\]  
Therefore, the regime of sticking at the surface \(\theta=\theta_{0}\) is always possible if inequality (40) is satisfied. On the other hand, the regime of sliding occurs at \(C_{1}=C_{m}\) if inequality (41) is satisfied. It will be seen later that this qualitative difference in solution behavior predicts the qualitative difference in the dependence between two measurable quantities, \(\omega\) and the torque.  
It is seen from (25) and (26) that \(\xi_{\theta\varphi}\) is proportional to \(\Lambda\). Then, it follows from (39) that  
\[\xi_{\theta\varphi}=O\left[\left(\theta-\theta_{0}\right)^{1/\alpha}\right] \tag{42}\]  
as \(\theta\rightarrow\theta_{0}\) if \(C_{1}=C_{m}\). It is seen from this equation that the velocity gradient approaches infinity near the friction surface if \(\alpha<0\). On the other hand, the shear stress is bounded everywhere, as follows from (18). Then, it is evident from (42) that the plastic work rate is given by  
\[E=\xi_{\theta\varphi}\sigma_{\theta\varphi}=O\left[\left(\theta-\theta_{0} \right)^{1/\alpha}\right] \tag{43}\]  
as \(\theta\rightarrow\theta_{0}\). Then, the inequality (41) and (43) ensure that the triple integral \(\underset{V}{\iint}\,EdV\) converges. Here, \(V\) is the volume of the sample.  
The case corresponding to Equation (32) can be treated in a similar manner.