Skip to main content
SpringerLink
Log in

An accurate numerical method of solving singular boundary value problems for the stationary flow of granular materials and its application

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

The rigid/plastic solutions are singular near certain surfaces. A special numerical method is required to solve such boundary value problems. The present paper develops such a method for two models of pressure-dependent plasticity. Both are based on the Mohr–Coulomb yield criterion. Stationary planar flows are considered. The numerical method is characteristics-based. Its distinguishing feature is employing the extended R–S method. The output of numerical solutions, in addition to stress and velocity fields, is the strain rate intensity factor, which controls the magnitude of the shear strain rate near the singular surface. The method applies to finding a solution for the flow of granular material through a wedge-shaped die. The accuracy of the solution is verified by comparison with an analytical solution for the flow through an infinite channel and an available numerical solution for pressure-independent material. An applied aspect of this study is that the strain rate intensity factor can be used in non-traditional constitutive equations for predicting the evolution of material properties near surfaces with high friction.

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

Access this article

Log in via an institution

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Hill, J.M., Selvadurai, A.P.S.: Mathematics and mechanics of granular materials. J. Eng. Math. 52, 1–9 (2005)

    Article  MathSciNet  Google Scholar 

  2. Cox, G.M., Thamwattana, N., McCue, S.W., Hill, J.M.: Coulomb-Mohr granular materials: quasi-static flows and the highly frictional limit. Appl. Mech. Rev. 61, 060802 (2008)

    Article  ADS  Google Scholar 

  3. Spencer, A.J.M.: A theory of the kinematics of ideal soils under plane strain conditions. J. Mech. Phys. Solids 12, 337–351 (1964)

    Article  MathSciNet  ADS  Google Scholar 

  4. Harris, D., Grekova, E.F.: A hyperbolic well-posed model for the flow of granular materials. J. Eng. Math. 52, 107–135 (2005)

    Article  MathSciNet  Google Scholar 

  5. Ahadi, A., Krenk, S.: Implicit integration of plasticity models for granular materials. Comp. Meth. Appl. Mech. Eng. 192, 3471–3488 (2003)

    Article  MathSciNet  Google Scholar 

  6. Williams, M.L.: Stress singularities from various boundary conditions in angular corner of plates in extension. Trans. ASME J. Appl. Mech. 19, 526–528 (1952)

    Article  ADS  Google Scholar 

  7. Rice, J.R., Rosengren, G.F.: Plane strain deformation near a crack tip in a power-law hardening material. J. Mech. Phys. Solids 16, 1–12 (1968)

    Article  ADS  Google Scholar 

  8. Hutchinson, J.M.: Singular behaviour at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids 16, 13–31 (1968)

    Article  ADS  Google Scholar 

  9. Rahman, M., Hancock, J.W.: Elastic perfectly-plastic asymptotic mixed mode crack tip fields in plane stress. Int. J. Solids Struct. 43, 3692–3704 (2006)

    Article  Google Scholar 

  10. Yuan, H.: Singular stress fields at V-notch tips in elastoplastic pressure-sensitive materials. Acta Mech. 118, 151–170 (1996)

    Article  Google Scholar 

  11. Papanastasiou, P., Durban, D.: Singular crack-tip plastic fields in Tresca and Mohr-Coulomb solids. Int. J. Solids Struct. 136–137, 250–258 (2018)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  16. Alexandrov, S., Mustafa, Y.: Singular solutions in viscoplasticity under plane strain conditions. Meccanica 48, 2203–2208 (2013)

    Article  MathSciNet  Google Scholar 

  17. Chen, J.-C., Pan, C., Rogue, C.M.O.L., Wang, H.-P.: A Lagrangian reproducing kernel particle method for metal forming analysis. Comput. Mech. 22, 289–307 (1998)

    Article  Google Scholar 

  18. Facchinetti, M., Miszuris, W.: Analysis of the maximum friction condition for green body forming in an ANSYS environment. J. Eur. Ceram. Soc. 36, 2295–2302 (2016)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Hill, R.: Basic stress analysis of hyperbolic regimes in plastic media. Math Proc Camb Phil Soc. 88, 359–369 (1980)

    Article  MathSciNet  Google Scholar 

  21. Alexandrov, S., Kuo, C.-Y., Jeng, Y.-R.: A numerical method for determining the strain rate intensity factor under plane strain conditions. Cont Mech Therm. 28, 977–992 (2016)

    Article  MathSciNet  Google Scholar 

  22. Hill, R., Lee, E.H., Tupper, S.J.: A method of numerical analysis of plastic flow in plane strain and its application to the compression of a ductile material between rough plates. Trans ASME J Appl Mech. 18, 46–52 (1951)

    Article  MathSciNet  Google Scholar 

  23. Alexandrov, S.: Geometry of plane strain characteristic fields in pressure-dependent plasticity. ZAMM 95, 1296–1301 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  24. Harris, D.: On the numerical integration of the stress equilibrium equations governing the ideal plastic plane deformation of a granular material. Acta Mech. 55, 219–238 (1985)

    Article  MathSciNet  Google Scholar 

  25. Harris, D.: A mathematical model for the waste region of a long-wall mineworking. J. Mech. Phys. Solids 33, 489–524 (1985)

    Article  ADS  Google Scholar 

  26. Mehrabadi, M.M., Cowin, S.C.: Initial planar deformation of dilatant granular materials. J. Mech. Phys. Solids 26, 269–284 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  27. Alexandrov, S., Jeng, Y.-R., Kuo, C.-Y., Chen, C.-Y.: Towards the theoretical/experimental description of the evolution of material properties near frictional interfaces in metal forming processes. Trib Int. 171 (2022) Article 107518

  28. Lyamina, E.: Prediction of a material property gradient near the friction surface in axisymmetric extrusion and drawing. Metals 12 (2022) Article 1310

  29. Uesugi, M., Kishida, H.: Frictional resistance at yield between dry sand and mild steel. Soils Found. 26, 139–149 (1986)

    Article  Google Scholar 

  30. Boulon, M.: Basic features of soil structure interface behaviour. Comp Geotechn. 7, 115–131 (1989)

    Article  Google Scholar 

  31. Boulon, M.: Modelling of soil-structure interface behaviour: a comparison between elastoplastic and rate type laws. Comp Geotechn. 9, 21–46 (1990)

    Article  Google Scholar 

  32. De Gennaro, V., Frank, R.: Elasto-plastic analysis of the interface behaviour between granular media and structure. Comp Geotechn. 29, 547–572 (2002)

    Article  Google Scholar 

  33. Ravera, E., Laloui, L.: Failure mechanism of fine-grained soil-structure interface for energy piles. Soil Found. 62 (2022) Article 101152

  34. Liu, H., Song, E., Ling, H.I.: Constitutive modeling of soil-structure interface through the concept of critical state soil mechanics. Mech Res Comm. 33, 515–531 (2006)

    Article  Google Scholar 

  35. Zheng, X.M., Hill, J.M.: Boundary effects for Couette flow of granular materials: dynamic modelling. Appl. Math. Model. 20, 82–92 (1996)

    Article  Google Scholar 

  36. Pemberton, C.S.: Flow of imponderable granular materials in wedge-shaped channels. J. Mech. Phys. Solids 13, 351–360 (1965)

    Article  ADS  Google Scholar 

  37. Jiang, M.J., Harris, D., Yu, H.-S.: A novel approach to examining double-shearing type of models for granular materials. Granular Matter 7, 157–168 (2005)

    Article  Google Scholar 

  38. Harris, D.: Some properties of a new model for slow flow of granular materials. Meccanica 41, 351–362 (2006)

    Article  MathSciNet  Google Scholar 

  39. Harris, D.: Double shearing and double rotation: a generalisation of the plastic potential model in the mechanics of granular materials. Int. J. Eng. Sci. 47, 1208–1215 (2009)

    Article  MathSciNet  Google Scholar 

  40. Marshall, E.A.: The compression of a slab of ideal soil between rough plates. Acta Mech. 3, 82–92 (1967)

    Article  Google Scholar 

  41. Spencer, A.J.M.: Plastic flow past a smooth cone. Acta Mech. 54, 63–74 (1984)

    Article  Google Scholar 

  42. Spencer, A.J.M.: Compression and shear of a layer of granular material. J. Eng. Math. 52, 251–264 (2005)

    Article  MathSciNet  Google Scholar 

  43. Alexandrov, S., Kuo, C.-Y., Jeng, Y.-R.: An accurate numerical solution for the singular velocity field near the maximum friction surface in plane strain extrusion. Int. J. Solids Struct. 150, 107–116 (2018)

  44. Dehdari, V., Ajdari, M.: Experimental study on shear strength parameters of municipal solid waste employing a large direct shear apparatus. Geomech Geoeng. 17, 1184–1199 (2022)

    Article  Google Scholar 

  45. Stutz, H., Masin, D., Sattari, A.S., Wuttke, F.: A general approach to model interfaces using existing soil constitutive models application to hypoplasticity. Comp Geotech. 87, 115–127 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

This publication has been supported by the RUDN University Scientific Projects Grant System, project No. 202247-2-000.

Funding

This work was made possible by the NCKU 100 program. It was financially supported by the Ministry of Science and Technology of Taiwan (MOST 106-2923-E-194-002-MY3, 108-2221-E-006-228-MY3 and 108-2119-M-006-010) and Air Force Office of Science Research (AFOSR) under contract no. FA4869- 06-1-0056 AOARD 064053. Professor Yeau-Ren Jeng would like to acknowledge Medical Device Innovation Center (MDIC) from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan and AC2T research GmbH (AC2T) in Austria (COMET InTribology, FFG-No.872176).

Author information

Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics RAS, 101-1 Vernadskogo Avenue, 119526, Moscow, Russia

    Sergei Alexandrov

  2. RUDN University, 6 Miklukho-Maklaya St, Moscow, 117198, Russia

    Sergei Alexandrov

  3. Research Center for Applied Sciences, Academia Sinica, Taipei, 115, Taiwan

    Chih-Yu Kuo

  4. Department of Biomedical Engineering, National Cheng Kung University (NCKU), Tainan, 70101, Taiwan

    Yeau-Ren Jeng

  5. Academy of Innovative Semiconductor and Sustainable Manufacturing, National Cheng Kung University (NCKU), Tainan, 70101, Taiwan

    Yeau-Ren Jeng

  6. Medical Device Innovation Center, National Cheng Kung University (NCKU), Tainan, 70101, Taiwan

    Yeau-Ren Jeng

Authors
  1. Sergei Alexandrov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chih-Yu Kuo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yeau-Ren Jeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yeau-Ren Jeng.

Ethics declarations

Authors’ contributions

All authors contributed equally to this paper. All authors reviewed the manuscript.

Availability of data and materials

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Conflict of interest

The authors declare no competing interests

Ethical Approval

Not applicable

Additional information

Communicated by Andreas Öchsner.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alexandrov, S., Kuo, CY. & Jeng, YR. An accurate numerical method of solving singular boundary value problems for the stationary flow of granular materials and its application. Continuum Mech. Thermodyn. 36, 171–195 (2024). https://doi.org/10.1007/s00161-023-01269-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-023-01269-x

Keywords

Search

Navigation