Skip to main content
Log in

The transition of qualitative behaviour between rigid perfectly plastic and viscoplastic solutions

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Alexandrov S, Richmond O (2001) Singular plastic flow fields near surfaces of maximum friction stress. Int J Non-Linear Mech 36:1–11

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandrov S, Lyamina E (2002) Singular solutions for plane plastic flow of pressure-dependent materials. Dokl Phys 47:308–311

    Article  ADS  MathSciNet  Google Scholar 

  3. Alexandrov S, Mishuris G (2009) Qualitative behaviour of viscoplastic solutions in the vicinity of maximum-friction surfaces. J Eng Math 65:143–156

    Article  MathSciNet  MATH  Google Scholar 

  4. Alexandrov S, Jeng Y-R (2013) Singular rigid/plastic solutions in anisotropic plasticity under plane strain conditions. Cont Mech Therm 25:685–689

    Article  ADS  MathSciNet  Google Scholar 

  5. Alexandrov S, Richmond O (2001) Couette flows of rigid/plastic solids: analytical examples of the interaction of constitutive and frictional laws. Int J Mech Sci 43:653–665

    Article  MATH  Google Scholar 

  6. Alexandrov S, Mishuris G (2007) Viscoplasticity with a saturation stress: distinguished features of the model. Arch Appl Mech 77:35–47

    Article  MATH  Google Scholar 

  7. Alexandrov S, Harris D (2006) Comparison of solution behaviour for three models of pressure-dependent plasticity: a simple analytical example. Int J Mech Sci 48:750–762

    Article  MATH  Google Scholar 

  8. Alexandrov S, Pirumov A, Chesnikova O (2008) Plastic flow of porous materials in friction contact area. Powder Metall Met Ceram 47:512–517

    Article  Google Scholar 

  9. Alexandrov S (2011) Behavior of anisotropic plastic solutions in the vicinity of maximum-friction surfaces. J Appl Mech Techn Phys 52:483–490

    Article  ADS  MATH  Google Scholar 

  10. Alexandrov S, Lyamina E (2003) Compression of a mean-stress sensitive plastic material by rotating plates. Mech Solids 38:40–48

    Google Scholar 

  11. Alexandrov S, Jeng Y-R (2009) Compression of viscoplastic material between rotating plates. Trans ASME J Appl Mech 76: Paper 031017

  12. Alexandrov S (2009) Specific features of solving the problem of compression of an orthotropic plastic material between rotating plates. J Appl Mech Tech Phys 50:886–890

    Article  ADS  MATH  Google Scholar 

  13. Alexandrov S, Pirumov A, Chesnikova O (2009) Compression of a plastic porous material between rotating plates. J Appl Mech Techn Phys 50:651–657

    Article  ADS  MATH  Google Scholar 

  14. Alexandrov S, Harris D (2010) An exact solution for a model of pressure-dependent plasticity in an un-steady plane strain process. Eur J Mech A 29:966–975

    Article  MathSciNet  Google Scholar 

  15. Miszuris W (2009) Plane-strain compression of a three layer strip containing viscoplastic material with saturation stress. Mater Sci Forum 623:89–103

    Article  Google Scholar 

  16. Hill R (1950) The mathematical theory of plasticity. Oxford University Press, New York

    MATH  Google Scholar 

  17. Dealy JM, Wissbrun KF (1990) Melt rheology and its role in plastic processing: theory and applications. Van Nostrand Reinhold, New York

    Book  Google Scholar 

  18. Sinczak J, Kusiak J, Lapkowski W, Okon R (1992) The influence of deformation conditions on the flow of strain rate sensitive materials. J Mater Process Technol 34:219–224

    Article  Google Scholar 

  19. Mitsoulis E, Hatzikirakos SG (2013) Capillary extrusion flow of a fluoropolymer melt. Int J Mater Form 6:29–40

    Article  Google Scholar 

  20. Voce E (1948) The relationship between stress and strain for homogeneous deformation. J Inst Met 74:537–562

    Google Scholar 

  21. Jaspers SPFC, Dautzenberg JH (2002) Material behaviour in metal cutting: strains, strain rates and temperatures in chip formation. J Mater Process Technol 121:123–135

    Article  Google Scholar 

  22. Alexandrov S, Alexandrova N (2000) On the maximum friction law in viscoplasticity. Mech Time 4:99–104

    Article  MATH  Google Scholar 

  23. Atkins AG (1969) Consequences of high strain rates in cold working. J Inst Metals 97:289–298

    Google Scholar 

  24. Webster WDJ, Davis RL (1982) Development of a friction element for metal forming analysis. Trans ASME J Eng Ind 104:253–256

    Article  Google Scholar 

  25. Chen J-S, Pan C, Roque CMOL, Wang H-P (1998) A Lagrangian reproducing kernel particle method for metal forming analysis. Comput Mech 22:289–307

    Article  MATH  Google Scholar 

  26. Kwon K-C, Youn SK (2006) The least-squares meshfree method for rigid-plasticity with frictional contact. Int J Solids Struct 43:7450–7481

    Article  MATH  Google Scholar 

  27. Hager C, Wohlmuth BI (2009) Nonlinear complementarity functions for plasticity problems with frictional contact. Comput Meth Appl Mech Eng 198:3411–3427

    Article  MathSciNet  MATH  Google Scholar 

  28. Temizer I (2013) A mixed formulation of mortar-based contact with friction. Comput Meth Appl Mech Eng 255:183–195

    Article  MathSciNet  MATH  Google Scholar 

  29. Thaitirarot A, Flicek RC, Hills DA, Barber JR (2014) The use of static reduction in the finite element solution of two-dimensional frictional contact problems. Proc IMechE Part C 228:1474–1487

    Article  Google Scholar 

  30. Kobayashi S, Oh SI, Atlan T (1989) Metal forming and the finite-element method. Oxford University Press, New York

    Google Scholar 

  31. Druyanov B (1993) Technological mechanics of porous bodies. Clarendon Press, New-York

    MATH  Google Scholar 

  32. Xu W, Di S, Thomson P (2003) Rigid-plastic/rigid-viscoplastic FE simulation using linear programming for metal forming. Int J Numer Meth Eng 56:487–506

    Article  MATH  Google Scholar 

  33. Huang J, Xu W, Thomson P, Di S (2003) A general rigid-plastic/rigid-viscoplastic FEM for metal-forming processes based on the potential reduction interior point method. Int J Mach Tools Manufact 43:379–389

    Article  Google Scholar 

  34. Lu P, Zhao G, Guan Y, Wu X (2008) Bulk metal forming process simulation based on rigid-plastic/viscoplastic element free Galerkin method. Mater Sci Eng A479:197–212

    Article  Google Scholar 

  35. Fries T-P, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Meth Eng 84:253–304

    MathSciNet  MATH  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sergei Alexandrov.

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

$$\begin{aligned} \frac{\mathrm{d}G}{\mathrm{d}\varphi }= & {} -\frac{2G\sin 2\varphi }{\cos 2\varphi +A_0 },\end{aligned}$$
(68)
$$\begin{aligned} \frac{\mathrm{d}\theta }{\mathrm{d}\varphi }= & {} \frac{\cos 2\varphi }{\cos 2\varphi +A_0 },\end{aligned}$$
(69)
$$\begin{aligned} \frac{\mathrm{d}f}{\mathrm{d}\varphi }= & {} -\frac{G\cos 2\varphi }{\cos 2\varphi +A_0 }. \end{aligned}$$
(70)

Integrating (68) gives

$$\begin{aligned} G=G_0 \left( {\cos 2\varphi +A_0 } \right) , \end{aligned}$$
(71)

where \(G_0 \) is constant. Substituting (71) into (70) and integrating leads to

$$\begin{aligned} f=-\frac{G_0 }{2}\sin 2\varphi . \end{aligned}$$
(72)

This solution satisfies boundary condition (22) owing to (18). Integrating (69) and using boundary condition (18) results in

$$\begin{aligned} \theta =\left\{ {{\begin{array}{ll} \varphi -\frac{A_0 }{\sqrt{A_0^2 -1}}\hbox {arctan}\left( {\sqrt{\frac{A_0 -1}{A_0 +1}}\tan \varphi } \right) &{} \quad \hbox {if}\quad A_0 >1, \\ \varphi -\frac{1}{2}\tan \varphi &{} \quad \hbox {if}\quad A_0 =1, \\ \varphi -\frac{A_0 }{\sqrt{1-A_0^2 }}\hbox {arctanh}\left( {\sqrt{\frac{1-A_0 }{1+A_0 }}\tan \varphi } \right) &{} \quad \hbox {if}\quad \left| {A_0 } \right| <1. \\ \end{array} }} \right. \end{aligned}$$
(73)

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

$$\begin{aligned} G=2\left( {\cos 2\varphi +A_0 } \right) . \end{aligned}$$
(74)

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

$$\begin{aligned} \alpha =\left\{ {{\begin{array}{ll} \frac{\pi }{4}-\frac{A_0 }{\sqrt{A_0^2 -1}}\hbox {arctan}\left( {\sqrt{\frac{A_0 -1}{A_0 +1}}} \right) &{} \quad \hbox {if}\quad A_0 >1, \\ \frac{\pi }{4}-\frac{1}{2} &{} \quad \hbox {if}\quad A_0 =1, \\ \frac{\pi }{4}-\frac{A_0 }{\sqrt{1-A_0^2 }}\hbox {arctanh}\left( {\sqrt{\frac{1-A_0 }{1+A_0 }}} \right) &{} \quad \hbox {if}\quad 0\le A_0 <1. \\ \end{array} }} \right. \end{aligned}$$
(75)

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-015-9797-7

Keywords

Mathematics Subject Classification

Navigation