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Continuum Mechanics and Thermodynamics

, Volume 30, Issue 4, pp 825–860 | Cite as

An anisotropic elastoplastic constitutive formulation generalised for orthotropic materials

  • M. K. Mohd Nor
  • N. Ma’at
  • C. S. Ho
Original Article
  • 70 Downloads

Abstract

This paper presents a finite strain constitutive model to predict a complex elastoplastic deformation behaviour that involves very high pressures and shockwaves in orthotropic materials using an anisotropic Hill’s yield criterion by means of the evolving structural tensors. The yield surface of this hyperelastic–plastic constitutive model is aligned uniquely within the principal stress space due to the combination of Mandel stress tensor and a new generalised orthotropic pressure. The formulation is developed in the isoclinic configuration and allows for a unique treatment for elastic and plastic orthotropy. An isotropic hardening is adopted to define the evolution of plastic orthotropy. The important feature of the proposed hyperelastic–plastic constitutive model is the introduction of anisotropic effect in the Mie–Gruneisen equation of state (EOS). The formulation is further combined with Grady spall failure model to predict spall failure in the materials. The proposed constitutive model is implemented as a new material model in the Lawrence Livermore National Laboratory (LLNL)-DYNA3D code of UTHM’s version, named Material Type 92 (Mat92). The combination of the proposed stress tensor decomposition and the Mie–Gruneisen EOS requires some modifications in the code to reflect the formulation of the generalised orthotropic pressure. The validation approach is also presented in this paper for guidance purpose. The \({\varvec{\psi }}\) tensor used to define the alignment of the adopted yield surface is first validated. This is continued with an internal validation related to elastic isotropic, elastic orthotropic and elastic–plastic orthotropic of the proposed formulation before a comparison against range of plate impact test data at 234, 450 and \({\mathrm {895\,ms}}^{\mathrm {-1}}\) impact velocities is performed. A good agreement is obtained in each test.

Keywords

Elastoplastic deformation Shockwave propagation Spall failure Orthotropic materials 

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References

  1. 1.
    Ahmadi, S.F., Eskandari, M.: Vibration analysis of a rigid circular disk embedded in a transversely isotropic solid. J. Eng. Mech. 140(7), 04014048 (2013)CrossRefGoogle Scholar
  2. 2.
    Ahmadi, S.F., Eskandari, M.: Rocking rotation of a rigid disk embedded in a transversely isotropic half-space. Civil Eng. Infrastruct. J. 47(1), 125–128 (2014)Google Scholar
  3. 3.
    Anderson, C.E., Cox, P.A., Johnson, G.R., Maudlin, P.J.: A constitutive model for anisotropic materials suitable for wave propagation computer program-II. Comput. Mech. 15, 201–223 (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Aravas, N.: Finite-strain anisotropic plasticity and the plastic spin. Model. Simul. Mater. Sci. 2, 483–504 (1994)ADSCrossRefGoogle Scholar
  5. 5.
    Asay, J.R., Shahinpoor, M.: High-Pressure shock compression of solids. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  6. 6.
    Banabic, D.: Sheet Metal Forming Processes. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Barlat, F.: Crystallographic texture, anisotropic yield surface and forming limits of sheet metals. Mater. Sci. Eng. 91, 55 (1987)CrossRefGoogle Scholar
  8. 8.
    Barlat, F., Lian, J.: Plastic behaviour and stretchability of sheet metals. Part I: a yield function for orthotropic sheets under plane stress conditions. Int. J. Plast. 5, 51–66 (1989)CrossRefGoogle Scholar
  9. 9.
    Belytschko, T., Liu, W.K., Moran, B.: Nonlinear Finite Elements for Continua and Structures. Wiley, Chichester (2000)zbMATHGoogle Scholar
  10. 10.
    Boehler, J.P.: On irreducible representations for isotropic scalar functions. ZAMM 57, 323–327 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Butcher, B.M.: Behaviour of Dense Media Under High Dynamic Pressure, p. 245. Gordon and Breach, New York (1968)Google Scholar
  12. 12.
    Campbell, J.: Lagrangian hydrocode modeling of hypervelocity impact on spacecraft. Ph.D. thesis, Cranfield University, Cranfield, UK (1998)Google Scholar
  13. 13.
    Davison, L., Graham, R.A.: Shock compression of solids. Phys. Rep. 55, 255–379 (1979)ADSCrossRefGoogle Scholar
  14. 14.
    De Vuyst, T.A.: Hydrocode Modelling of Water Impact. Ph.D. thesis, Cranfield University, Cranfield, UK (2003)Google Scholar
  15. 15.
    Drumheller, D.S.: Introduction to wave propagation in nonlinear fluids and solids. Cambridge University Press, Cambridge, UK (1998)CrossRefGoogle Scholar
  16. 16.
    Eidel, B., Gruttmann, F.: Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation. Comput. Mater. Sci. 28, 732–742 (2003)CrossRefGoogle Scholar
  17. 17.
    Eliezer, S., Ghatak, A., Hora, H., Teller, E.: An introduction to equations of state, theory and applications. Cambridge University Press, Cambridge (1986)Google Scholar
  18. 18.
    Eskandari, M., Shodja, H.M., Ahmadi, S.F.: Lateral translation of an inextensible circular membrane embedded in a transversely isotropic half-space. Eur. J. Mech. A Solids 39, 134–143 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fountzoulas, C.G., Gazonas, G.A., Cheeseman, B.A.: Computational modeling of tungsten carbide sphere impact and penetration into high-strength-low-alloy (HSLA)-100 steel targets. J. Mech. Mater. Struct. 2(10), 1965 (2007)CrossRefGoogle Scholar
  20. 20.
    Grady, D.E.: The spall strength of condensed matter. J. Mech. Phys. Solid 36, 353–384 (1988)ADSCrossRefGoogle Scholar
  21. 21.
    Grady, D.E., Kipp, M.E.: Fragmentation properties of metals. Int. J. Impact Eng. 20(1–5), 293–308 (1997)CrossRefGoogle Scholar
  22. 22.
    Gray, G.T., Bourne, N.K., Millett, J.C.F.: Shock response of tantalum: lateral stress and shear strength through the front. J. Appl. Phys. 94(10), 6430–6436 (2003)ADSCrossRefGoogle Scholar
  23. 23.
    Gruneisen, E.: The State of Solid Body, NASA R19542 (1959)Google Scholar
  24. 24.
    Hill, R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Ser. A 193, 281–297 (1948)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hallquist, J.: Theoretical manual for DYNA3D. Technical report, Lawrence Livermore National Laboratory (1983)Google Scholar
  26. 26.
    Holzapfel, G.A.: Nonlinear Solid Mechanics, A Continuum Approach for Engineering. Wiley, Chichester (2007)Google Scholar
  27. 27.
    Itskov, M.: On the application of the additive decomposition of generalized strain measures in large strain plasticity. Mech. Res. Commun. 31, 507–517 (2004)CrossRefzbMATHGoogle Scholar
  28. 28.
    Itskov, M., Aksel, N.: A constitutive model for orthotropic elasto-plasticity at large strains. Arch. Appl. Mech. 74, 75–91 (2004)ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Khan, A.S., Kazmi, R., Farrokh, B.: Multiaxial and non-proportional loading responses, anisotropy and modeling of Ti–6Al–4V titanium alloy over wide ranges of strain rates and temperatures. Int. J. Plast. 23(6), 931–950 (2007a)CrossRefzbMATHGoogle Scholar
  30. 30.
    Khan, A.S., Kazmi, R., Farrokh, B., Zupan, M.: Effect of oxygen content and microstructure on the thermo-mechanical response of three Ti–6Al–4V alloys: experiments and modeling over a wide range of strain-rates and temperatures. Int. J. Plast. 23(7), 1105–1125 (2007b)CrossRefzbMATHGoogle Scholar
  31. 31.
    Khan, A.S., Kazmi, R., Pandey, A., Stoughton, T.: Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: a very low work hardening aluminum alloy (Al6061-T6511). Int. J. Plast. 25(9), 1611–1625 (2009)CrossRefzbMATHGoogle Scholar
  32. 32.
    Lin, J.I.: DYNA3D: A Nonlinear, Explicit, Three-Dimensional Finite Element Code for Solid and Structural Mechanics User Manual. Lawrence Livermore National Laboratory, Livermore (2004)Google Scholar
  33. 33.
    Lubarda, V.A., Krajcinovic, D.: Some fundamental issues in the rate theory of damage-elastoplasticity. Int. J. Plast. 11, 763–797 (1995)CrossRefzbMATHGoogle Scholar
  34. 34.
    Man, C.: On the correlation of elastic and plastic anisotropy in sheet metals. J Elast 39(2), 165–173 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mandel, J.: Plasticité Classiqueet Viscoplastié, CISM Lecture Notes. Springer, Wien (1972)Google Scholar
  36. 36.
    Meyers, M.A.: Dynamic Behaviour of Materials. Wiley, Inc., New York (1994)Google Scholar
  37. 37.
    Mohd Nor M.K.: Modelling Rate Dependent Behaviour of Orthotropic Metals. Ph.D. thesis, Cranfield University, Cranfield, UK (2012)Google Scholar
  38. 38.
    Mohd Nor, M.K., Vignjevic, R., Campbell, J.: Modelling of shockwave propagation in orthotropic materials. Appl. Mech. Mater. 315, 557–561 (2013a)CrossRefGoogle Scholar
  39. 39.
    Mohd Nor, M.K., Vignjevic, R., Campbell, J.: Plane-stress analysis of the new stress tensor decomposition. Appl. Mech. Mater. 315, 635–639 (2013b)CrossRefGoogle Scholar
  40. 40.
    Mohd Nor, M.K., Mohamad Suhaimi, I.: Effects of temperature and strain rate on commercial aluminum alloy AA5083. Appl. Mech. Mater. 660, 332–336 (2014)CrossRefGoogle Scholar
  41. 41.
    Mont’ans, F.J., Bathe, K.J.: Towards a model for large strain anisotropic elasto-plasticity. In: Onate, E., Owen, R. (eds.) Computational Plasticity, pp. 13–36. Springer, Berlin (2007)CrossRefGoogle Scholar
  42. 42.
    Nakamachi, E., Tam, N.N., Morimoto, H.: Multi-scale finite element analyses of sheet metals by using SEM-EBSD measured crystallographic RVE models. Int. J. Plast. 23(3), 450–489 (2007)CrossRefzbMATHGoogle Scholar
  43. 43.
    Panov, V.: Modelling of behaviour of metals at high strain rates. Ph.D. dissertation, Cranfield University, Cranfield (2006)Google Scholar
  44. 44.
    Reese, S., Vladimirov, I.N.: Anisotropic modelling of metals in forming processes. IUTAM Symposium on Theoretical Computational and Modelling Aspects of Inelastic Media, vol. 11, pp. 175–184 (2008)Google Scholar
  45. 45.
    Schmidt, R.M., Davies, F.W., Lempriere, B.M.: Temperature dependent spall threshold of four metal alloys. J. Phys. Chem. Solids 39(4), 375–385 (1978)ADSCrossRefGoogle Scholar
  46. 46.
    Schröder, J., Gruttmann, F., Löblein, J.: A simple orthotropic finite elasto-plasticity model based on generalized stress-strain measures. Comput. Mech. 30, 48–64 (2002)CrossRefzbMATHGoogle Scholar
  47. 47.
    Schröder, J., Hackl, K.: Plasticity and Beyond: Microstructures, Crystal-Plasticity and Phase Transitions. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  48. 48.
    Sinha, S., Ghosh, S.: Modeling cyclic ratcheting based fatigue life of HSLA steels using crystal plasticity FEM simulations and experiments. Int. J. Fatigue 28(12), 1690–1704 (2006)CrossRefGoogle Scholar
  49. 49.
    Sitko, M., Skoczeń, B., Wróblewski, A.: FCC-BCC phase transformation in rectangular beams subjected to plastic straining at cryogenic temperatures. Int. J. Mech. Sci. 52(7), 993–1007 (2010)CrossRefGoogle Scholar
  50. 50.
    Smallman, R.E.: Modern Physical Metallurgy, 4th edn. Butterworths, London (1985)Google Scholar
  51. 51.
    Stevens, A.L., Tuler, F.R.: Effect of shock precompression on the dynamic fracture strength of 1020 steel and 6061-T6 aluminum. J. Appl. Phys. 42(13), 5665 (1971)ADSCrossRefGoogle Scholar
  52. 52.
    Steinberg D.J.: Equation of State and Strength Properties of Selected Materials, Report No. UCRL-MA-106439, Lawrence Livermore National Laboratory, Livermore, CA (1991)Google Scholar
  53. 53.
    Vignjevic, R., Bourne, N.K., Millett, J.C.F., De Vuyst, T.: Effects of orientation on the strength of the aluminum alloy 7010-T6 during shock loading: experiment and simulation. J. Appl. Phys. 92(8), 4342–4348 (2002)ADSCrossRefGoogle Scholar
  54. 54.
    Vignjevic, R., Campbell, J., Bourne, N.K., Djordjevic, N.: Modelling shock waves in orthotropic elastic materials. Conference on Shock Compression of Condensed Matter, Hawaii, June (2007)Google Scholar
  55. 55.
    Vignjevic, R., Djordjevic, N., Panov, V.: Modelling of dynamic behaviour of orthotropic metals including damage and failure. Int. J. Plast. 38, 47–85 (2012)CrossRefGoogle Scholar
  56. 56.
    Vladimirov, I.N., Pietryga, M.P., Reese, S.: On the modelling of non-linear kinematic hardening at finite strains with application to springback—comparison of time integration algorithms. Int. J. Numer. Methods Eng. 75(1), 1–28 (2008)CrossRefzbMATHGoogle Scholar
  57. 57.
    Wilson, L.T., Reedal, D.R., Kuhns, L.D., Grady, D.E., Kipp, M.E.: Using a numerical fragmentation model to understand the fracture and fragmentation of naturally fragmenting munitions of differing materials and geometries. In: 19th International Symposium of Ballistics, pp. 7–11, Interlaken, Switzerland (2001)Google Scholar
  58. 58.
    Zel’dovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and High temperature Hydrodynamic Phenomena, vols. 1 and 2. Academic Press, New York (1966)Google Scholar
  59. 59.
    Zheng, Q.S.: Theory of representations for tensor functions-A unified invariant approach to constitutive equations. Appl. Mech. Rev. 47(11), 545 (1994)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Crashworthiness and Collisions Research Group, Mechanical Failure Prevention and Reliability Research Center, Faculty of Mechanical and Manufacturing EngineeringUniversiti Tun Hussein Onn MalaysiaParit RajaMalaysia

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