Skip to main content
SpringerLink
Log in

Approximate analytical equations for the deformation of an inclusion in a viscoplastic matrix

  • Original Papers
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

The strain rate in an inclusion embedded in a matrix undergoing large strains is investigated theoretically. The inclusion is either a spheroid (axisymmetric loading), or an elliptical cylinder (plane strain loading), and both are assumed to be made of incompressible power law viscous materials, including the limit case of perfect plasticity. Localization factors (i.e., normalized strain rates) derived from various approaches (viz, linearization models, variational principle, finite element method) are first compared for a sphere and a circular cylinder. The influence of the inclusion aspect ratio is then investigated by means of the variational approach. Finally, an original and practical purpose of the paper is to propose approximate analytical equations to fit the dependence of the localization factor on the strain rate sensitivity parameter, as well as on the inclusion hardness and aspect ratios.

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

Similar content being viewed by others

References

  1. Shen, Y. L., Finot, M., Needlemann, A., Suresh, S.: Effective plastic response of two-phase composites. Acta Metall. Mater.43, 1701–1722 (1995).

    Google Scholar 

  2. Lebensohn, R. A., Tomé, C. N.: A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. Acta Metall. Mater.41, 2611–2624 (1993).

    Google Scholar 

  3. Montheillet, F., Briottet, L.: Modelling the effects of morphology and topology on strain inhomogeneity in two-phase materials. In: Int. Conf. on the Quantitative Description of Materials Microstructure (Q-MAT'97) (Wojnar, L. et al., eds.), pp. 37–48. Warsaw 1997.

  4. Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London Ser.A241, 376–396 (1957).

    Google Scholar 

  5. Eshelby, J. D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. London Ser.A252, 561–569 (1959).

    Google Scholar 

  6. Kinoshita, N., Mura, T.: Elastic fields of inclusions in anisotropic media. Phys. Stat. Solids5, 759–768 (1971).

    Google Scholar 

  7. Laws, N.: The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material. J. Elasticity7, 91–97 (1977).

    Google Scholar 

  8. Bilby, B. A., Eshelby, J. D., Kundu, A. K.: The change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity. Tectonophysics28, 265–274 (1975).

    Google Scholar 

  9. Howard, I. C., Brierley, P.: On the finite deformation of an inhomogeneity in a viscous liquid. Int. J. Eng. Sci.14, 1151–1159 (1976).

    Google Scholar 

  10. Gilormini, P., Montheillet, F.: Champs de vitesses et de contraintes autour d'une inclusion cylindrique en traction ou compression plane (calcul analytique). J. Méc. Théor. Appl.3, 563–576 (1984).

    Google Scholar 

  11. Gilormini, P., Montheillet, F.: Deformation of an inclusion in a viscous matrix and induced stress concentrations. J. Mech. Phys. Solids34, 97–123 (1986).

    Google Scholar 

  12. Mura, T.: Micromechanics of defects in solids. Dordrecht: Martinus Nijhoff 1987.

    Google Scholar 

  13. Duva, J. M.: A self-consistent analysis of the stiffening effect of rigid inclusions on a power-law material. J. Eng. Mater. Technol.106, 317–321 (1984).

    Google Scholar 

  14. Weng, G. J.: Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions. Int. J. Engl. Sci.22, 845–856 (1984).

    Google Scholar 

  15. Lee, B. J., Mear, M. E.: Effective properties of power-law solids containing elliptical inhomogeneities. Part I: Rigid inclusions. Mech. Mater.13, 313–335 (1992).

    Google Scholar 

  16. Zbib, H. M., Zhu, H. T.: On the mechanics of plastic deformation in metal matrix composites. In: Inelasticity and micromechanics of metal matrix composites (Voyiadjis, G. Z., Ju, J. W., eds.) pp. 97–118. The Netherlands: Elsevier 1994.

    Google Scholar 

  17. Zhu, H. T., Zbib, H. M.: A macroscopic model for plastic flow in metalmatrix composites. Int. J. Plasticity11, 471–499 (1995).

    Google Scholar 

  18. Zaidman, M., Ponte Castañeda, P.: The finite deformation of nonlinear composite materials—II. Evolution of the microstructure. Int. J. Solids Struct.33, 1287–1303 (1996).

    Google Scholar 

  19. Bao, G., Hutchinson, J. W., McMeeking, R. M.: Particle reinforcement of ductile matrices against plastic flow and creep. Acta Metall. Mater.39, 1871–1882 (1991).

    Google Scholar 

  20. Bao, G., Hutchinson, J. W., McMeeking, R. M.: The flow stress of dual phase, no hardening solids. Mech. Mater.12, 85–94 (1991).

    Google Scholar 

  21. Dragone, T. L., Nix, W. D.: Geometric factors affecting the international stress distribution and high temperature creep rate of discontinuous fiber reinforced metals. Acta Metall. Mater.38, 1941–1953 (1990).

    Google Scholar 

  22. Van Houtte, P.: Heterogeneity of plastic strain around an ellipsoidal inclusion in an ideal plastic matrix. Acta Metall. Mater.43, 2859–2879 (1995).

    Google Scholar 

  23. Duva, J. M., Hutchinson, J. W.: Constitutive potentials for dilutely voided nonlinear materials. Mech. Mater.3, 41–54 (1984).

    Google Scholar 

  24. Huang, Y.: Accurate dilatation rates for spherical voids in triaxial stress fields. J. Appl. Mech.58, 1084–1086 (1991).

    Google Scholar 

  25. Lee, B. J., Mear, M. E.: Axisymmetric deformation of power law solids containing a dilute concentration of aligned spheroidal voids. J. Mech. Phys. Solids40, 1805–1836 (1992).

    Google Scholar 

  26. Klöcker, H., Montheillet, F.: Velocity, strain rate and stress fields around a spheroidal cavity in a linearly viscous material. Eur. J. Mech. A/Solids15, 397–422 (1996).

    Google Scholar 

  27. Briottet, L.: Etude théorique de l'évolution du comportement de matériaux viscoplastiques endommagés. Thesis, Ecole des Mines de Saint-Etienne (1994).

  28. Vernusse, Ph.: Comportement mécanique d'un matériau microhétérogéne. Applications à la déformation à chaud d'aciers à deux phases ductiles. Thesis, Ecole des Mines de Saint-Etienne (1993).

  29. Dajno, D.: Rhéologie globale et structurale des alliages de titane TA6 V etBetacez dans les domaines α+β et β. Thesis, Ecole des Mines de Saint-Etienne (1991).

  30. Berveiller, M., Zaoui, A.: A simplified self-consistent scheme for two-phase metals plasticity. Res. Mech. Lett.1, 119–124 (1981).

    Google Scholar 

  31. Hill, R.: Continuum micromechanics of elastoplastic polycrystals. J. Mech. Phys. Solids13, 89–101 (1965).

    Google Scholar 

  32. Hutchinson, J. W.: Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. R. Soc. London Ser.A 348, 101–127 (1976).

    Google Scholar 

  33. Molinari, A., Canova, G. R., Ahzi, S.: A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metall.35, 2983–2994 (1987).

    Google Scholar 

  34. Molinari, A., Tóth, L. S.: Tuning a self consistent viscoplastic model by finite element results —I. Modeling. Acta Metall. Mater.42, 2453–2458 (1994).

    Google Scholar 

  35. Gilormini, P., Germain, Y.: A finite element analysis of the inclusion problem for power law viscous materials. Int. J. Solids Struct.23, 413–437 (1987).

    Google Scholar 

  36. Hill, R.: New horizons in the mechanics of solids. J. Mech. Phys. Solids5, 66–74 (1956).

    Google Scholar 

  37. Budiansky, B., Hutchinson, J. W., Slutsky, S.: Void growth and collapse in viscous solids. In: Mechanics of solids (Hopkins, H. G., Sewell, M. J., eds.), pp. 13–44. Oxford: Pergamon Press 1982.

    Google Scholar 

  38. Lee, B. J., Mear, M. E.: Effect of inclusion shape on the stiffness of nonlinear two phase composites. J. Mech. Phys. Solids39, 627–649 (1991).

    Google Scholar 

  39. Montheillet, F., Briottet, L.: Prévision des hétérogénéités de déformation dans un agrégat de deux phases viscoplastiques. 6éme Colloque Franco-Polonais «Hétérogénéités de déformation plastique», Ecole des Mines, Saint-Etienne (1995), Rev. Métall.-Science et Génie des Matériaux (in press).

  40. Juganaru, M., Maurice, Cl., Sakho, I., Montheillet, F.: A topology based model for large strain deformation of polycrystals. In: Int. Conf. on the Quantitative Description of Materials Microstructure (Q-MAT'97) (Wojnar, L., et al., eds.), pp. 331–336. Warsaw 1997.

Download references

Author information

Author notes

  1. L. Briottet

    Present address: Ingénieur de Recherche, Centre d'Etudes Nucléaires de Grenoble, CEREM/DEM/SGM, 17 rue des Martyrs, 38054, Grenoble Cedex 9

  2. P. Gilormini (Directeur de Recherche au CNRS)

    Present address: Laboratoire de Mécanique et Technologie, Ecole Normale Supérieure de Cachan, Université Paris VI, URA CNRS 860, 61 avenue du Président Wilson, 94235, Cachan Cedex

  3. F. Montheillet (Directeur de Recherche au CNRS)

    Present address: Département Microstructures et Mise en Forme, Ecole des Mines de Saint-Etienne (Centre SMS), URA CNRS 1884, 158 cours Fauriel, 42023, Saint-Etienne Cedex 2, France

Authors and Affiliations

    Authors
    1. L. Briottet
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. P. Gilormini
      View author publications

      You can also search for this author in PubMed Google Scholar

    3. F. Montheillet
      View author publications

      You can also search for this author in PubMed Google Scholar

    Rights and permissions

    Reprints and permissions

    About this article

    Cite this article

    Briottet, L., Gilormini, P. & Montheillet, F. Approximate analytical equations for the deformation of an inclusion in a viscoplastic matrix. Acta Mechanica 134, 217–234 (1999). https://doi.org/10.1007/BF01312656

    Download citation

    • Received:

    • Revised:

    • Issue Date:

    • DOI: https://doi.org/10.1007/BF01312656

    Keywords

    Search

    Navigation