Abstract
An incompressible rigid viscoplastic material is confined between two planar plates which are inclined at an angle \(2\alpha \). In a certain sense, the viscoplastic model adopted approaches rigid perfectly plastic models as the equivalent strain rate approaches zero and infinity. The plates intersect in a hinged line, and the angle \(\alpha \) slowly decreases from an initial value. The maximum-friction law is assumed at the plates. A boundary value problem for the flow of the material is formulated, and its asymptotic analysis is carried out near the friction surface. The solution may exhibit sliding or sticking at the plates. Solutions which exhibit sticking may have a rigidly rotating zone in the region adjacent to the plates. Solutions which exhibit sliding are singular. Solutions which exhibit sticking may also be singular under certain conditions. In general, there are several critical values of \(\alpha \) at which changes in the qualitative behaviour of the solution occur. Qualitative features of the solution found are compared with those of the solution for the classical rigid plastic model.
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Acknowledgments
WM acknowledges funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement PIAPP-GA-284544-PARM-2. The research described was also supported by Grants RFBR-14-01-00087 (Russia) and NSH-1275.2014.1 (Russia).
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Appendix
Appendix
The rigid perfectly plastic solution is obtained from Eqs. (31) to (33) assuming that \(\Phi =k=\hbox {constant}, A_0 =A/k\) and \(\mu =0\). Then these equations become represented as
Integrating (68) gives
where \(G_0 \) is constant. Substituting (71) into (70) and integrating leads to
This solution satisfies boundary condition (22) owing to (18). Integrating (69) and using boundary condition (18) results in
It is evident from (69) that the derivative \({\mathrm{d}\varphi }/{\mathrm{d}\theta }\) is negative in the range \(0<\varphi <\pi /4\) if \(A_0 \le -1\). This contradicts the friction law since \(\sigma _{r\theta } \) given by (15) must be negative at the friction surface. Assume that the regime of sticking occurs at \(\theta =\alpha \) and \(A_0 \ne 0\). Then it follows from (24) and (71) that \(A_0 =-\cos 2\varphi _\mathrm{f}\), where \(\varphi _\mathrm{f} \) is the value of \(\varphi \) at the friction surface. It is evident that \(\varphi _\mathrm{f} <\pi /4\) since \(A_0 \ne 0\). Substituting this value of \(A_0 \) into (73) shows that \(\alpha \rightarrow \infty \), which makes no sense. Therefore, the regime of sticking is impossible, unless \(A_0 =0\). In the case of sliding, boundary condition (17) is valid. Therefore, it follows from (72) and boundary condition (23) that \(G_0 =2\) and then, from (71), that
The radial velocity must be positive at \(\theta =\alpha \) at sliding. Therefore, it is seen from (74) that \(A_0 \ge 0\) and \(A_0 =0\) corresponds to a regime of sticking. The equation for \(A_0 \) follows from (17) and (73) in the form
The first equation of this system covers the range \(0<\alpha <\pi /4-1/2\), and the third equation covers the range \(\pi /4-1/2<\alpha \le \pi /4\). Therefore, the only way to obtain a solution at \(\alpha >\pi /4\) is to assume that there is a rigid zone in the domain \(\pi /4\le \theta \le \alpha \). In the plastic zone, \(0\le \theta \le \pi /4\), the solution found at \(A_0 =0\) is valid.
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Alexandrov, S., Miszuris, W. The transition of qualitative behaviour between rigid perfectly plastic and viscoplastic solutions. J Eng Math 97, 67–81 (2016). https://doi.org/10.1007/s10665-015-9797-7
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DOI: https://doi.org/10.1007/s10665-015-9797-7