**6. Singularity**

It is seen from (18) that the derivative *dϕ*/*dθ* approaches infinity as *ϕ* → *π*/4 if *K*0 = 0. If *m* < 1 then *ϕw* < *π*/4 and the solution is not singular. If the regime of sticking occurs then *ϕw* = *π*/4 but *K*0 = 0. Therefore, the solution may be singular only if *m* = 1 and the regime of sliding occurs. It follows from (18) that:

$$\frac{d\varrho}{d\theta} = \frac{K\_0 \sqrt{1-c}}{2(\pi/4 - \varrho)} + O(1) \tag{46}$$

as *ϕ* → *π*/4. Integrating and using the boundary condition *ϕ* = *π*/4 at *θ* = *α* yields:

$$\frac{\pi}{4} - \varphi = \sqrt{K\_0 \sqrt{1 - \alpha}} \sqrt{\theta - \alpha} + o\left(\sqrt{\theta - \alpha}\right) \tag{47}$$

as *θ* → *α* .

> Consider the stress field. Differentiating (11) with respect to *θ* yields:

$$\frac{\partial \sigma\_{\Pi}}{\partial \theta} = \frac{\partial \sigma}{\partial \theta} - 2T\sqrt{1 - c}\sin 2\rho \frac{d\rho}{d\theta}, \; \frac{\partial \sigma\_{\theta\theta}}{\partial \theta} = \frac{\partial \sigma}{\partial \theta} + 2T\sqrt{1 - c}\sin 2\rho \frac{d\rho}{d\theta}, \; \frac{\partial \sigma\_{r\theta}}{\partial \theta} = -2T\cos 2\rho \frac{d\rho}{d\theta}.\tag{48}$$

Eliminating the derivative *dϕ*/*dθ* in these equations by means of (18) gives:

$$\begin{split} \frac{\partial \sigma\_{\rm tr}}{\partial \theta} &= \frac{\partial \sigma}{\partial \theta} - \frac{2T(1 - c)(K\_0 + \cos 2\rho)\sin 2\rho}{\cos 2\rho}, \frac{\partial \sigma\_{\rm \partial \theta}}{\partial \theta} = \frac{\partial \sigma}{\partial \theta} + \frac{2T(1 - c)(K\_0 + \cos 2\rho)\sin 2\rho}{\cos 2\rho}, \\ &\frac{\partial \sigma\_{\rm r\theta}}{\partial \theta} = -2T\sqrt{1 - c}(K\_0 + \cos 2\rho). \end{split} \tag{49}$$

It is evident that the derivative *∂σr<sup>θ</sup>*/*∂θ* is of a finite magnitude at *ϕ* = *π*/4 (or *θ* = *α*). The derivative *∂σ*/*∂θ* involved in (49) is determined from (18), (19) and (20) as:

$$\frac{\partial \sigma}{\partial \theta} = 2T \left[ 1 - \frac{(1 - c)(\mathcal{K}\_0 + \cos 2\varphi)}{\cos 2\varphi} \right] \sin 2\varphi. \tag{50}$$

Equations (49) and (50) combine to give:

$$\frac{\partial \sigma\_{rr}}{\partial \theta} = 2T \left[ 1 - \frac{2(1 - c)(K\_0 + \cos 2\varphi)}{\cos 2\varphi} \right] \sin 2\varphi, \quad \frac{\partial \sigma\_{\theta\theta}}{\partial \theta} = 2T \sin 2\varphi. \tag{51}$$

It is evident that the derivative *∂σθθ*/*∂θ* is of a finite magnitude at *ϕ* = *π*/4 (or *θ* = *α*). Expanding the right-hand side of the first equation in (51) in a series in the vicinity of *ϕ* = *π*/4 results in

$$\frac{\partial \sigma\_{rr}}{\partial \theta} = -\frac{2TK\_0(1-c)}{\left(\pi/4-\varrho\right)} + o\left[\left(\pi/4-\varrho\right)^{-1}\right] \tag{52}$$

as *ϕ* → *π*/4. Equations (47) and (52) combine to give:

$$\frac{\partial \sigma\_{rr}}{\partial \theta} = -\frac{2TK\_0(1-c)}{\sqrt{K\_0\sqrt{1-c}\sqrt{\theta-a}}} + o\left[\left(\theta - a\right)^{-1/2}\right] \tag{53}$$

as *θ* → *α* . It is seen from this equation that the derivative *∂σrr*/*∂θ* approaches infinity (or negative infinity) in the vicinity of the friction surface and follows an inverse square root rule.

Consider the strain rate field. It follows from the definition for *G*, (27) and (31), that *ξrr* = −*ξθθ* = *<sup>ω</sup>G*0(*<sup>K</sup>*0 + cos <sup>2</sup>*ϕ*)/2. It is evident from this equation that the normal strain rates in the polar coordinate system are bounded at the friction surface. The shear strain rate is determined from (18), (27) and (31) as:

$$\zeta\_{r\theta}^{x} = -\frac{\omega \,\mathrm{G}\_{0} \sqrt{1 - c} (K\_{0} + \cos 2\varphi) \tan 2\varphi}{2}. \tag{54}$$

It is seen from this equation that |*ξr<sup>θ</sup>* | → ∞ as *ϕ* → *π*/4. Expanding the right-hand side of (54) in a series in the vicinity of *ϕ* = *π*/4 results in:

$$\zeta\_{r\theta}^{\mathbb{Z}} = -\frac{\omega G\_0 K\_0 \sqrt{1-\varepsilon}}{4} \left(\frac{\pi}{4} - \varphi\right)^{-1} + o\left[\left(\frac{\pi}{4} - \varphi\right)^{-1}\right] \tag{55}$$

as *ϕ* → *π*/4. Equations (47) and (55) combine to give:

$$\zeta\_{r\theta}^{\pi} = -\frac{\omega \mathcal{G}\_0 \sqrt{K\_0 \sqrt{1-c}}}{4\sqrt{\theta - a}} + o\left[\left(\theta - a\right)^{-1/2}\right] \tag{56}$$

as *θ* → *α* . It is seen from this equation that the shear strain rate in the polar coordinate system follows an inverse square root rule in the vicinity of the friction surface. This result is in agreemen<sup>t</sup> with the general theory developed in [9].

Some models of anisotropic plasticity (for example, [18]) involve the material spin. Therefore, it is of interest to understand the asymptotic behavior of the only non-zero spin component, *ωrθ*, near the friction surface. By definition,

$$
\omega\_{r\theta} = \frac{1}{2} \left( \frac{1}{r} \frac{\partial u\_r}{\partial \theta} - \frac{\partial u\_\theta}{\partial r} - \frac{u\_\theta}{r} \right). \tag{57}
$$

Equations (26) and (57) combine to give:

$$
\omega\_{r\theta} = \frac{\omega}{4} \left( \frac{d^2 \mathcal{g}}{d\theta^2} + 4\mathfrak{g} \right). \tag{58}
$$

Using the definition for *G*, (18), (31) and (34) Equation (58) can be rewritten as:

$$
\omega\_{r\theta} = \frac{\omega \,\mathrm{G}\_0 \sin 2\varrho}{4} \left[ \frac{1}{\sqrt{1-c}} - \frac{2\sqrt{1-c}}{\cos 2\varrho} (\mathrm{K}\_0 + \cos 2\varrho) \right]. \tag{59}
$$

It is seen from this equation that |*<sup>ω</sup>r<sup>θ</sup>* | → ∞ as *ϕ* → *π*/4. Expanding the right-hand side of (59) in a series in the vicinity of *ϕ* = *π*/4 results in:

$$
\omega\_{r\theta} = -\frac{\omega\sqrt{1-\epsilon}G\_0K\_0}{4} \left(\frac{\pi}{4} - \phi\right)^{-1} + o\left[\left(\frac{\pi}{4} - \phi\right)^{-1}\right] \tag{60}
$$

as *ϕ* → *π*/4. Equations (47) and (60) combine to give:

$$
\omega\_{r\theta} = -\frac{\omega\sqrt{\sqrt{1-\varepsilon}K\_0}G\_0}{4\sqrt{\theta-a}} + o\left[\left(\theta-a\right)^{-1/2}\right] \tag{61}
$$

as *θ* → *α* . The qualitative behavior of the material spin near the friction surface that its magnitude approaches infinity should be taken into account in material models that involve this quantity. A similar approach has been used in visco-plasticity [19], where the qualitative behavior of the quadratic invariant of the strain tensor near the friction surface, that its magnitude approaches infinity has been taken into account.
