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Homogenized Gurson-type behavior equations for strain rate sensitive materials

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Abstract

In this paper, the classical Gurson model for ductile porous media is extended for strain-rate-dependent materials. Based on micromechanical considerations, approximate closed-form macroscopic behavior equations are derived to describe the viscous response of a ductile metallic material. To this end, the analysis of the expansion of a long cylindrical void in an ideally plastic solid introduced by McClintock (J Appl Mech 35:363, 1968) is revisited. The classical Gurson yield locus has been modified to explicitly take into account the strain rate sensitivity parameter for strain rate power-law solids. Two macroscopic approaches are proposed in this work. Both models use the first term of a Taylor series expansion to approximate integrals to polynomial functions. The first proposed closed-form approach is analytically more tractable than the second one. The second approach is more accurate. In order to compare the proposed approximate Gurson-type macroscopic functions with the behavior of the original Gurson yield locus, numerical finite element analyses for cylindrical cells have been conducted for a wide range of porosities, triaxialities, and strain rate sensitivity parameters. The results presented evidence that, for large values of the rate sensitivity parameters, the proposed extended Gurson-type models have the important quality to better predict the behavior of rate sensitive materials than the classical one. They also provide simpler and accurate alternatives to more traditional viscoplastic models.

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References

  1. McClintock, F.A.: A criterion for ductile fracture by the growth of holes. J. Appl. Mech. 35, 363 (1968)

    Article  Google Scholar 

  2. Rice, J.R., Tracey, D.M.: On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17, 201 (1969)

    Article  Google Scholar 

  3. Gurson, A.: Continuum theory of ductile rupture by void nucleation and growth: part I—yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99, 2 (1977)

    Article  Google Scholar 

  4. Chu, C., Needleman, A.: Void nucleation effects in biaxially stretched sheets. J. Eng. Mater. Technol. 102, 249 (1980)

    Article  Google Scholar 

  5. Tvergaard, V., Needleman, A.: Analysis of cup-cone fracture in a round tensile bar. Acta Metall. 32, 157 (1984)

    Article  Google Scholar 

  6. Gologanu, M., Leblond, J., Devaux, J.: Approximate models for ductile metals containing nonspherical voids-case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723 (1993)

    Article  MATH  Google Scholar 

  7. Gologanu, M., Leblond, J., Devaux, J.: Approximate models for ductile metals containing nonspherical voids-case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Technol. 116, 290 (1994)

    Article  MATH  Google Scholar 

  8. Pardoen, T., Hutchinson, J.: An extended model for void growth and coalescence. J. Mech. Phys. Solids 48, 2467 (2000)

    Article  MATH  Google Scholar 

  9. Madou, K., Leblond, J.: A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids I: limit-analysis of some representative cell. J. Mech. Phys. Solids 60, 1020 (2012)

    Article  MathSciNet  Google Scholar 

  10. Leblond, J.B., Perrin, G., Devaux, J.: An improved Gurson-type model for hardenable ductile metals. Eur. J. Mech. A Solids 14, 499 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Benzerga, A., Besson, J.J., Pineau, A.: Anisotropic ductile fracture. Part II: theory. Acta Mater. 52, 4639 (2004)

    Article  Google Scholar 

  12. Cazacu, O., Stewart, J.: Analytic plastic potential for porous aggregates with matrix exhibiting tension–compression asymmetry. J. Mech. Phys. Solids 57, 325 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Stewart, J., Cazacu, O.: Analytical yield criterion for an anisotropic material containing spherical voids and exhibiting tension-compression asymmetry. Int. J. Solids Struct. 48, 357 (2011)

    Article  MATH  Google Scholar 

  14. Benallal, A., Desmorat, R., Fournage, M.: An assessment of the role of the third stress invariant in the Gurson approach for ductile fracture. Eur. J. Mech. A Solids 47, 400 (2014)

    Article  MathSciNet  Google Scholar 

  15. Vadillo, G., Reboul, J., Fernández-Sáez, J.: A modified Gurson model to account for the influence of the Lode parameter at high triaxialities. Eur. J. Mech. A Solids 56, 31 (2016)

    Article  MathSciNet  Google Scholar 

  16. Perrin, G.: Homogenized behavior equations for porous Bingham viscoplastic material. J. Eng. Mech. 128, 885 (2002)

    Article  Google Scholar 

  17. Ponte-Castañeda, P.: The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39, 45 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Willis, J.: On methods for bounding the overall properties of nonlinear composites. J. Mech. Phys. Solids 39, 73 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ponte-Castañeda, P., Zaidman, M.: Constitutive models for porous materials with evolving microstructure. J. Mech. Phys. Solids 42, 1459 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Danas, K., Idiart, M., Ponte-Castañeda, P.: A homogenization-based constitutive model for isotropic viscoplastic porous media. Int. J. Solids Struct. 45, 3392 (2008)

    Article  MATH  Google Scholar 

  21. Idiart, M.: The macroscopic behavior of power-law and ideally plastic materials with elliptical distribution of porosity. Mech. Res. Commun. 35(8), 583 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Leblond, J., Perrin, G., Suquet, P.: Exact results and approximate models for porous viscoplastic solids. Int. J. Plast. 10(3), 213 (1994)

    Article  MATH  Google Scholar 

  23. Gărăjeu, M., Michel, M.J., Suquet, P.: A micromechanical approach of damage in viscoplastic materials by evolution in size shape and distribution of voids. Comput. Methods Appl. Mech. Eng. 183, 223 (2000)

    Article  MATH  Google Scholar 

  24. Koplik, J., Needleman, A.: Void growth and coalescence in porous plastic solids. Int. J. Solids Struct. 24, 835 (1988)

    Article  Google Scholar 

  25. Brocks, W., Sun, D.Z., Hönig, A.: Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials. Int. J. Plast. 11, 971 (1995)

    Article  MATH  Google Scholar 

  26. Steglich, D., Brocks, W.: Micromechanical modelling of the behaviour of ductile materials including particles. Comput. Mater. Sci. 9, 7 (1997)

    Article  Google Scholar 

  27. Faleskog, J., Shih, C.: Micromechanics of coalescence I. Synergistic effects of elasticity, plastic yielding and multi-size-scale voids. J. Mech. Phys. Solids 45, 21 (1997)

    Article  Google Scholar 

  28. Fabrégue, D., Pardoen, T.: A constitutive model for elastoplastic solids containing primary and secondary voids. J. Mech. Phys. Solids 56, 719 (2008)

    Article  MATH  Google Scholar 

  29. Fritzen, F., Forest, S., Böhlke, T., Kondo, D., Kanit, T.: Computational homogenization of elasto-plastic porous metals. Int. J. Plast. 29, 102 (2012)

    Article  Google Scholar 

  30. Keralavarma, S., Hoelscher, S., Benzerga, A.: Void growth and coalescence in anisotropic plastic solids. Int. J. Solids Struct. 48, 1696 (2011)

    Article  MATH  Google Scholar 

  31. Gao, X., Kim, J.: Modeling of ductile fracture: significance of void coalescence. Int. J. Solids Struct. 43, 6277 (2006)

    Article  MATH  Google Scholar 

  32. Tvergaard, V.: Behaviour of voids in a shear field. Int. J. Fract. 158, 41 (2009)

    Article  MATH  Google Scholar 

  33. Scheyvaerts, F., Onck, P., Tekoglu, C., Pardoen, T.: The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension. J. Mech. Phys. Solids 59, 373 (2011)

    Article  MATH  Google Scholar 

  34. Srivastava, A., Needleman, A.: Porosity evolution in a creeping single crystal. Model. Simul. Mater. Sci. Eng. 20, 035010 (2012)

    Article  Google Scholar 

  35. Faleskog, J., Gao, X., Shih, C.: Cell model for nonlinear fracture analysis I. Micromech. Calibration Int. J. Fract. 89, 355 (1998)

    Article  Google Scholar 

  36. Gao, X., Faleskog, J., Shih, C.F.: Ductile tearing in part-through cracks: experiments and cell-model predictions. Eng. Fract. Mech. 59, 761 (1998)

    Article  Google Scholar 

  37. Lecarme, L., Tekog̃lu, C., Pardoen, T.: Void growth and coalescence in ductile solids with stage III and stage IV strain hardening. Int. J. Plast. 27, 1203 (2011)

    Article  MATH  Google Scholar 

  38. ABAQUS/Standard, Simulia, User’s Manual, version 6.15 edn. Dassault Systèmes, Providence (2015)

  39. Kim, J., Gao, X., Srivatsan, T.S.: Modeling of void growth in ductile solids: effects of stress triaxiality and initial porosity. Eng. Fract. Mech. 71, 379 (2004)

    Article  Google Scholar 

  40. Vadillo, G., Fernández-Sáez, J.: An analysis of Gurson model with parameters dependent on triaxiality based on unitary cells. Eur. J. Mech. A Solids 28, 417 (2009)

    Article  MATH  Google Scholar 

  41. Gosh, A.: Tensile instability and necking in materials with strain hardening and strain-rate hardening. Acta Metall. 25, 1413 (1977)

    Article  Google Scholar 

  42. Klöcker, H., Tvergaard, V.: Growth and coalescence of non-spherical voids in metals deformed at elevated temperature. Int. J. Mech. Sci. 40, 1995 (2003)

    MATH  Google Scholar 

  43. Vadillo, G., Zaera, R., Fernández-Sáez, J.: Consistent integration of the constitutive equations of Gurson materials under adiabatic conditions. Comput. Methods Appl. Mech. Eng. 197, 1280 (2008)

    Article  MATH  Google Scholar 

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  1. Department of Continuum Mechanics and Structural Analysis, University Carlos III of Madrid, Avda. de la Universidad, 30, 28911, Leganés, Madrid, Spain

    J. Reboul & G. Vadillo

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  1. J. Reboul
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  2. G. Vadillo
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Reboul, J., Vadillo, G. Homogenized Gurson-type behavior equations for strain rate sensitive materials. Acta Mech 229, 3517–3536 (2018). https://doi.org/10.1007/s00707-018-2189-0

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  • DOI: https://doi.org/10.1007/s00707-018-2189-0

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