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A hyperbolic augmented elasto-plastic model for pressure-dependent yield

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Abstract

A three-dimensional elasto-plastic model for the deformation and flow of granular materials which generalises the plastic potential model and contains an additional term analogous to that appearing in the double-shearing model is presented. It is shown that for planar flows the resulting system of first-order partial differential equations is hyperbolic. This is in distinct contrast to both the non-associated plastic potential and double-shearing models, which fail to be hyperbolic. The ill-posedness of the Cauchy problem for the planar double-shearing model is due to the presence of the rotation rate of the principal axes of stress while that of the non-associated plastic potential model is due to distinct quasi-static spatial stress and velocity characteristics. The present model attains well-posedness by replacing the planar rotation rate of the principal stress axes by the vector intrinsic spin of a Cosserat continuum and using it to ensure identical spatial stress and velocity characteristics. Flows in which the intrinsic spin vector is constant in both space and time correspond to flows in an ordinary continuum. The model governing such flows is embedded into a Cosserat model in such a way that the characteristic structure is preserved.

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References

  1. Anand L.: Plane deformations of ideal granular materials. J. Mech. Phys. Solids 31, 105–122 (1983)

    Article  MATH  Google Scholar 

  2. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. A. Hermann, Paris (1909)

  3. de Borst R., Sluys L.J.: Localisation in a Cosserat continuum under static and dynamic loading conditions. Comp. Methods Appl. Mech. Eng. 90, 805–827 (1991)

    Article  Google Scholar 

  4. de Josselin de Jong, G.: Statics and Kinematics of the Failable Zone of a Granular Material. Uitgeverij Waltmann, Delft (1959)

  5. de Josselin de Jong G.: Mathematical elaboration of the double-sliding, free rotating model. Arch. Mech. 29, 561–591 (1977)

    Google Scholar 

  6. Drucker, D.C.: A more fundamental approach to plastic stress-strain relations. In: Sternberg, E. (ed.) Proceedings of the 1st. U.S. National Congress of Applied Mechanics, pp. 487–491. ASME, New York (1951)

  7. Drucker D.C., Prager W.: Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10, 157–165 (1952)

    MATH  MathSciNet  Google Scholar 

  8. Geniev, G.A.: Problems of the Dynamics of a Granular Medium (in Russian). Akad Stroit Archit SSSR, 3-121 Moscow (1958)

  9. Harris D.: Constitutive equations for planar deformations of rigid-plastic materials. J. Mech. Phys. Solids 41, 1515–1531 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Harris D.: A unified formulation for plasticity models of granular and other materials. Proc. R. Soc. Lond. A 450, 37–49 (1995)

    Article  MATH  Google Scholar 

  11. Harris D.: Ill- and well-posed models of granular flow. Acta Mech. 146, 199–225 (2001)

    Article  MATH  Google Scholar 

  12. Harris D.: Characteristic relations for a model for the flow of granular materials. Proc. R. Soc. Lond. A 457, 349–370 (2001)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Harris D.: Double-slip and spin: dilatant shear in a reduced Cosserat model. In: Wu, W., Yu, H.S. (eds.) Modern Trends in Geomechanics, pp. 329–346. Springer, Berlin (2006)

    Chapter  Google Scholar 

  15. Hill R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950)

    MATH  Google Scholar 

  16. Ilyushin A.A.: On the postulate of plasticity (in Russian). Prikl. Math. Mekh. 25, 503–507 (1961)

    Google Scholar 

  17. Jackson R.: Some mathematical and physical aspects of continuum models for the motion of the granular materials. In: Meyer, R.E. (ed.) Theory of Dispersed Multiphase Flow, pp. 291–337. Academic Press, New York (1983)

    Chapter  Google Scholar 

  18. Jiang M.J., Harris D., Yu H.S.: Kinematic models for non-coaxial granular materials. Part I: theory. Int. J. Num. Anal. Methods Geomech. 29, 643–661 (2005)

    Article  MATH  Google Scholar 

  19. Jiang M.J., Harris D., Yu H.S.: Kinematic models for non-coaxial granular materials. Part II: evaluation. Int. J. Num. Anal. Methods Geomech. 29, 663–689 (2005)

    Article  MATH  Google Scholar 

  20. Jiang M.J., Yu H.S., Harris D.: A novel discrete model for granular material incorporating rolling resistance. Comput. Geotech. 32, 340–357 (2005)

    Article  Google Scholar 

  21. Kruyt N.P.: An analysis of the generalized double-sliding models for cohesionless granular materials. J. Mech. Phys. Solids 38, 27–35 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mandel J.: Sur les lignes de glissement et le calcul des dé placements dans la déformation plastique. C. R. Acad. Sci. Paris 225, 1272–1273 (1947)

    MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Morrison H.L., Richmond O.: Application of Spencer’s ideal soil model to granular materials flow. J. Appl. Mech. 43, 49–53 (1976)

    Article  MATH  Google Scholar 

  26. Mroz, Z., Szymansk, C.: Non-associated flow rules in description of plastic flow of granular materials, CISM Course 217 (1974). In: Olszak, W., Sukjle, L. Limit Analysis and Rheological Approach in Soil Mechanics, Springer, Vienna (1979)

  27. Mühlhaus H.B., Vardoulakis I.: The thickness of shear bands in granular materials. Géotechnique 37, 271–283 (1987)

    Article  Google Scholar 

  28. Pemberton C.S.: Flow of imponderable granular material in wedge-shaped channels. J. Mech. Phys. Solids 13, 352–360 (1965)

    Article  Google Scholar 

  29. Pitman E.B., Schaeffer D.G.: Stability of time dependent compressible granular flow in two dimensions. Comm. Pure Appl. Math. 40, 421–447 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  30. Renardy M., Rogers R.C.: An Introduction to Partial Differential Equations. Springer, New York (2004)

    MATH  Google Scholar 

  31. Schaeffer D.G., Pitman E.B.: Ill-posedness in three dimensional plastic flow. Comm. Pure Appl. Math. 41, 879–890 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  32. Schaeffer, D.G.: Mathematical issues in the continuum formulation of slow granular flow. In: Joseph, D.D., Schaeffer, D.G. (eds.) Two Phase Waves in Fluidised Beds, Sedimentation and Granular Flows, pp. 118–129. Institute of Mathematics and its Applications, University of Minnesota, Minneapolis (1990)

  33. 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  MATH  MathSciNet  Google Scholar 

  34. Spencer, A.J.M.: Deformation of ideal granular materials. In: Hopkins, H.G., Sewell, M.J. (eds.) The Rodney Hill 60th Anniversary volume, pp. 607–652. Pergamon, Oxford (1982)

  35. Vardoulakis, I., Sulem, J.: Bifurcation Analysis in Geomechanics. Blackie, London (1995)

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  1. School of Mathematics, University of Manchester, Manchester, UK

    David Harris

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  1. David Harris
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Harris, D. A hyperbolic augmented elasto-plastic model for pressure-dependent yield. Acta Mech 225, 2277–2299 (2014). https://doi.org/10.1007/s00707-014-1129-x

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