Continuum Mechanics and Thermodynamics

, Volume 30, Issue 2, pp 421–455 | Cite as

Continuum modeling of twinning, amorphization, and fracture: theory and numerical simulations

Original Article

Abstract

A continuum mechanical theory is used to model physical mechanisms of twinning, solid-solid phase transformations, and failure by cavitation and shear fracture. Such a sequence of mechanisms has been observed in atomic simulations and/or experiments on the ceramic boron carbide. In the present modeling approach, geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors. After development of the general nonlinear theory, a first problem class considers simple shear deformation of a single crystal of this material. For homogeneous fields or stress-free states, algebraic systems or ordinary differential equations are obtained that can be solved by numerical iteration. Results are in general agreement with atomic simulation, without introduction of fitted parameters. The second class of problems addresses the more complex mechanics of heterogeneous deformation and stress states involved in deformation and failure of polycrystals. Finite element calculations, in which individual grains in a three-dimensional polycrystal are fully resolved, invoke a partially linearized version of the theory. Results provide new insight into effects of crystal morphology, activity or inactivity of different inelasticity mechanisms, and imposed deformation histories on strength and failure of the aggregate under compression and shear. The importance of incorporation of inelastic shear deformation in realistic models of amorphization of boron carbide is noted, as is a greater reduction in overall strength of polycrystals containing one or a few dominant flaws rather than many diffusely distributed microcracks.

Keywords

Continuum mechanics Geometry Phase field Fracture Twinning Phase transformation Finsler space 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chen, M., McCauley, J.W., Hemker, K.J.: Shock-induced localized amorphization in boron carbide. Science 299, 1563–1566 (2003)ADSCrossRefGoogle Scholar
  2. 2.
    Yan, X.Q., Tang, Z., Zhang, L., Guo, J.J., Jin, C.Q., Zhang, Y., Goto, T., McCauley, J.W., Chen, M.W.: Depressurization amorphization of single-crystal boron carbide. Phys. Rev. Lett. 102, 075505 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Fanchini, G., McCauley, J.W., Chhowalla, M.: Behavior of disordered boron carbide under stress. Phys. Rev. Lett. 97, 035502 (2006)ADSCrossRefGoogle Scholar
  4. 4.
    Taylor, D.E., McCauley, J.W., Wright, T.W.: The effects of stoichiometry on the mechanical properties of icosahedral boron carbide under loading. J. Phys. Condens. Matter. 24, 505402 (2012)CrossRefGoogle Scholar
  5. 5.
    Taylor, D.E.: Shock compression of boron carbide: a quantum mechanical analysis. J. Am. Ceram. Soc. 98, 3308–3318 (2015)CrossRefGoogle Scholar
  6. 6.
    An, Q., Goddard, W.A., Cheng, T.: Atomistic explanation of shear-induced amorphous band formation in boron carbide. Phys. Rev. Lett. 113(9), 095501 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    An, Q., Goddard, W.A.: Atomistic origin of brittle failure of boron carbide from large-scale reactive dynamics simulations: suggestions toward improved ductility. Phys. Rev. Lett. 115, 105051 (2015a)Google Scholar
  8. 8.
    Grady, D.E.: Adiabatic shear failure in brittle solids. Int. J. Impact Eng. 38, 661–667 (2011)CrossRefGoogle Scholar
  9. 9.
    Clayton, J.D.: Towards a nonlinear elastic representation of finite compression and instability of boron carbide ceramic. Philos. Mag. 92, 2860–2893 (2012a)ADSCrossRefGoogle Scholar
  10. 10.
    Clayton, J.D.: Mesoscale modeling of dynamic compression of boron carbide polycrystals. Mech. Res. Commun. 49, 57–64 (2013)CrossRefGoogle Scholar
  11. 11.
    Clayton, J.D., Tonge, A.: A nonlinear anisotropic elastic-inelastic constitutive model for polycrystalline ceramics and minerals with application to boron carbide. Int. J. Solids Struct. 64–65, 191–207 (2015)CrossRefGoogle Scholar
  12. 12.
    Li, Y., Zhao, Y.H., Liu, W., Zhang, Z.H., Vogt, R.G., Lavernia, E.J., Schoenung, J.M.: Deformation twinning in boron carbide particles within nanostructured Al 5083/B\(_4\)C metal matrix composites. Philos. Mag. 90, 783–792 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    Sano, T., Randow, C.L.: The effect of twins on the mechanical behavior of boron carbide. Metall. Mater. Trans. A 42, 570–574 (2011)CrossRefGoogle Scholar
  14. 14.
    Clayton, J.D.: Nonlinear Mechanics of Crystals. Springer, Dordrecht (2011)CrossRefMATHGoogle Scholar
  15. 15.
    Clayton, J.D., McDowell, D.L.: A multiscale multiplicative decomposition for elastoplasticity of polycrystals. Int. J. Plast. 19, 1401–1444 (2003)CrossRefMATHGoogle Scholar
  16. 16.
    Kalidindi, S.R.: Incorporation of deformation twinning in crystal plasticity models. J. Mech. Phys. Solids 46, 267–290 (1998)ADSCrossRefMATHGoogle Scholar
  17. 17.
    Barton, N.R., Winter, N.W., Reaugh, J.E.: Defect evolution and pore collapse in crystalline energetic materials. Model. Simul. Mater. Sci. Eng. 17, 035003 (2009)ADSCrossRefGoogle Scholar
  18. 18.
    Clayton, J.D.: A continuum description of nonlinear elasticity, slip and twinning, with application to sapphire. Proc. R. Soc. Lond. A 465, 307–334 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Clayton, J.D.: Deformation, fracture, and fragmentation in brittle geologic solids. Int. J. Fract. 163, 151–172 (2010a)CrossRefMATHGoogle Scholar
  20. 20.
    Clayton, J.D.: Modeling nonlinear electromechanical behavior of shocked silicon carbide. J. Appl. Phys. 107, 013520 (2010b)ADSCrossRefGoogle Scholar
  21. 21.
    Aslan, O., Cordero, N.M., Gaubert, A., Forest, S.: Micromorphic approach to single crystal plasticity and damage. Int. J. Eng. Sci. 49, 1311–1325 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Voyiadjis, G.Z., Kattan, P.I.: Damage Mechanics. CRC Press, Boca Raton (2005)CrossRefMATHGoogle Scholar
  23. 23.
    Bammann, D.J., Solanki, K.N.: On kinematic, thermodynamic, and kinetic coupling of a damage theory for polycrystalline material. Int. J. Plast. 26, 775–793 (2010)CrossRefMATHGoogle Scholar
  24. 24.
    Xu, X.-P., Needleman, A.: Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1434 (1994)ADSCrossRefMATHGoogle Scholar
  25. 25.
    Clayton, J.D.: Dynamic plasticity and fracture in high density polycrystals: constitutive modeling and numerical simulation. J. Mech. Phys. Solids 53, 261–301 (2005)ADSCrossRefMATHGoogle Scholar
  26. 26.
    Vogler, T.J., Clayton, J.D.: Heterogeneous deformation and spall of an extruded tungsten alloy: plate impact experiments and crystal plasticity modeling. J. Mech. Phys. Solids 56, 297–335 (2008)ADSCrossRefGoogle Scholar
  27. 27.
    Foulk, J.W., Vogler, T.J.: A grain-scale study of spall in brittle materials. Int. J. Fract. 163, 225–242 (2010)CrossRefMATHGoogle Scholar
  28. 28.
    Hou, T.Y., Rosakis, P., LeFloch, P.: A level-set approach to the computation of twinning and phase-transition dynamics. J. Comput. Phys. 150, 302–331 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Del Piero, G., Lancioni, G., March, R.: A variational model for fracture mechanics: numerical experiments. J. Mech. Phys. Solids 55, 2513–2537 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kuhn, C., Müller, R.: A continuum phase field model for fracture. Eng. Fract. Mech. 77, 3625–3634 (2010)CrossRefGoogle Scholar
  31. 31.
    Clayton, J.D., Knap, J.: A geometrically nonlinear phase field theory of brittle fracture. Int. J. Fract. 189, 139–148 (2014)CrossRefGoogle Scholar
  32. 32.
    Weinberg, K., Hesch, C.: A high-order finite deformation phase-field approach to fracture. Contin. Mech. Thermodyn. 29, 935–945 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Levitas, V.A., Levin, V.A., Zingerman, K.M., Freiman, E.I.: Displacive phase transitions at large strains: phase-field theory and simulations. Phys. Rev. Lett. 103, 025702 (2009)ADSCrossRefGoogle Scholar
  34. 34.
    Clayton, J.D.: Phase field theory and analysis of pressure-shear induced amorphization and failure in boron carbide ceramic. AIMS Mater. Sci. 1, 143–158 (2014a)CrossRefGoogle Scholar
  35. 35.
    Schmitt, R., Kuhn, C., Müller, R.: On a phase field approach for martensitic transformations in a crystal plastic material at a loaded surface. Contin. Mech. Thermodyn. 29, 957–968 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Clayton, J.D., Knap, J.: A phase field model of deformation twinning: nonlinear theory and numerical simulations. Phys. D 240, 841–858 (2011a)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Clayton, J.D., Knap, J.: Phase field modeling of twinning in indentation of transparent single crystals. Model. Simul. Mater. Sci. Eng. 19, 085005 (2011b)ADSCrossRefGoogle Scholar
  38. 38.
    Clayton, J.D., Knap, J.: Phase field analysis of fracture induced twinning in single crystals. Acta Mater. 61, 5341–5353 (2013)CrossRefGoogle Scholar
  39. 39.
    Hildebrand, F.E., Miehe, C.: A phase field model for the formation and evolution of martensitic laminate microstructure at finite strains. Philos. Mag. 92, 4250–4290 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    Padilla, C.A.H., Markert, B.: A coupled ductile fracture phase-field model for crystal plasticity. Contin. Mech. Thermodyn. 29, 1017–1026 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Clayton, J.D.: Finsler geometry of nonlinear elastic solids with internal structure. J. Geom. Phys. 112, 118–146 (2017a)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Saczuk, J.: Finslerian Foundations of Solid Mechanics. Zeszyty naukowe Instytutu Maszyn Przeplywowych Polskiej Akademii Nauk w Gdansku, Wydawnictwo IMP PAN, Gdansk (1996)MATHGoogle Scholar
  43. 43.
    Stumpf, H., Saczuk, J.: A generalized model of oriented continuum with defects. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 80, 147–169 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Clayton, J.D.: Finsler-geometric continuum mechanics. Technical Report ARL-TR-7694, US Army Research Laboratory, Aberdeen Proving Ground MD (2016a)Google Scholar
  45. 45.
    Clayton, J.D.: Generalized finsler geometric continuum physics with applications in fracture and phase transformations. Zeitschrift fur Angewandte Mathematik und Physik (ZAMP) 68, 9 (2017b)Google Scholar
  46. 46.
    Clayton, J.D.: Finsler-geometric continuum mechanics and the micromechanics of fracture in crystals. J. Micromech. Mol. Phys. 1, 164003 (2016b)CrossRefGoogle Scholar
  47. 47.
    Clayton, J.D.: Finsler-geometric continuum dynamics and shock compression. Int. J. Fract. 208, 53–78 (2017c)Google Scholar
  48. 48.
    Clayton, J.D., Kraft, R.H., Leavy, R.B.: Mesoscale modeling of nonlinear elasticity and fracture in ceramic polycrystals under dynamic shear and compression. Int. J. Solids Struct. 49, 2686–2702 (2012)CrossRefGoogle Scholar
  49. 49.
    Clayton, J.D., Knap, J.: Phase field modeling of directional fracture in anisotropic polycrystals. Comput. Mater. Sci. 98, 158–169 (2015a)CrossRefGoogle Scholar
  50. 50.
    Clayton, J.D., Knap, J.: Phase field modeling of coupled fracture and twinning in single crystals and polycrystals. Comput. Methods Appl. Mech. Eng. 312, 447–467 (2016)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    An, Q., Goddard, W.A.: Boron suboxide and boron subphosphide crystals: hard ceramics that shear without brittle failure. Chem. Mater. 27, 2855–2860 (2015b)CrossRefGoogle Scholar
  52. 52.
    Subhash, G., Awasthi, A.P., Kunka, C., Jannotti, P., DeVries, M.: In search of amorphization-resistant boron carbide. Scr. Mater. 123, 158–162 (2016)CrossRefGoogle Scholar
  53. 53.
    Bejancu, A.: Finsler Geometry and Applications. Ellis Horwood, New York (1990)MATHGoogle Scholar
  54. 54.
    Rund, H.: A divergence theorem for Finsler metrics. Monatshefte fur Mathematik 79, 233–252 (1975)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Clayton, J.D.: Differential Geometry and Kinematics of Continua. World Scientific, Singapore (2014b)CrossRefMATHGoogle Scholar
  56. 56.
    Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)CrossRefGoogle Scholar
  57. 57.
    Levitas, V.I.: Phase field approach to martensitic phase transformations with large strains and interface stresses. J. Mech. Phys. Solids 70, 154–189 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Clayton, J.D.: On anholonomic deformation, geometry, and differentiation. Math. Mech. Solids 17, 702–735 (2012b)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Clayton, J.D., Bammann, D.J., McDowell, D.L.: Anholonomic configuration spaces and metric tensors in finite strain elastoplasticity. In. J. Non-Linear Mech. 39, 1039–1049 (2004)CrossRefMATHGoogle Scholar
  60. 60.
    Weyl, H.: Space-Time-Matter, fourth edn. Dover, New York (1952)Google Scholar
  61. 61.
    Clayton, J.D., Bammann, D.J., McDowell, D.L.: A geometric framework for the kinematics of crystals with defects. Philos. Mag. 85, 3983–4010 (2005)ADSCrossRefGoogle Scholar
  62. 62.
    Clayton, J.D., McDowell, D.L., Bammann, D.J.: Modeling dislocations and disclinations with finite micropolar elastoplasticity. Int. J. Plast. 22, 210–256 (2006)CrossRefMATHGoogle Scholar
  63. 63.
    Tjahjanto, D.D., Turteltaub, S., Suiker, A.S.J.: Crystallographically based model for transformation-induced plasticity in multiphase carbon steels. Contin. Mech. Thermodyn. 19, 399–422 (2008)ADSCrossRefMATHGoogle Scholar
  64. 64.
    Clayton, J.D., Knap, J.: Nonlinear phase field theory for fracture and twinning with analysis of simple shear. Philos. Mag. 95, 2661–2696 (2015b)ADSCrossRefGoogle Scholar
  65. 65.
    Takaki, T., Hasebe, T., Tomita, Y.: Two-dimensional phase-field simulation of self-assembled quantum dot formation. J. Crystal Growth 287, 495–499 (2006)ADSCrossRefGoogle Scholar
  66. 66.
    Boiko, V.S., Garber, R.I., Kosevich, A.M.: Reversible Crystal Plasticity. AIP Press, New York (1994)Google Scholar
  67. 67.
    Hirth, J.P., Lothe, J.: Theory of Dislocations. Wiley, New York (1982)MATHGoogle Scholar
  68. 68.
    Rice, J.R.: Mathematical analysis in the mechanics of fracture. In: Liebowitz, H. (ed.) Fracture: An Advanced Treatise, pp. 191–311. Academic Press, New York (1968)Google Scholar
  69. 69.
    Beaudet, T.D., Smith, J.R., Adams, J.W.: Surface energy and relaxation in boron carbide \((10\bar{1}1)\) from first principles. Solid State Communications 219, 43–47 (2015)ADSCrossRefGoogle Scholar
  70. 70.
    Dandekar, D. P.: Shock response of boron carbide. Technical Report ARL-TR-2456, US Army Research Laboratory, Aberdeen Proving Ground MD (2001)Google Scholar
  71. 71.
    Ferdjani, H., Abdelmoula, R., Marigo, J.-J.: Insensitivity to small defects of the rupture of materials governed by the Dugdale model. Contin. Mech. Thermodyn. 19, 191–210 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Bourne, N.K.: The relation of failure under 1D shock to the ballistic performance of brittle materials. Int. J. Impact Eng. 35, 674–683 (2008)CrossRefGoogle Scholar
  73. 73.
    Clayton, J.D.: Penetration resistance of armor ceramics: dimensional analysis and property correlations. Int. J. Impact Eng. 85, 124–131 (2015)CrossRefGoogle Scholar
  74. 74.
    Clayton, J.D.: Dimensional analysis and extended hydrodynamic theory applied to long-rod penetration of ceramics. Def. Technol. 12, 334–342 (2016c)CrossRefGoogle Scholar
  75. 75.
    Vogler, T.J., Reinhart, W.D., Chhabildas, L.C.: Dynamic behavior of boron carbide. J. Appl. Phys. 95, 4173–4183 (2004)ADSCrossRefGoogle Scholar
  76. 76.
    Paliwal, B., Ramesh, K.T.: Effect of crack growth dynamics on the rate-sensitive behavior of hot-pressed boron carbide. Scr. Mater. 57, 481–484 (2007)CrossRefGoogle Scholar
  77. 77.
    Tang, B., An, Q., Goddard, W.A.: Improved ductility of boron carbide by microalloying with boron suboxide. J. Phys. Chem. C 119, 24649–24656 (2015)CrossRefGoogle Scholar

Copyright information

© This is a U.S. government work and its text is not subject to copyright protection in the United States; however, its text may be subject to foreign copyright protection 2017

Authors and Affiliations

  1. 1.Impact PhysicsUS ARLAberdeenUSA
  2. 2.A. James Clark School of EngineeringUniversity of MarylandCollege ParkUSA
  3. 3.Computational and Engineering SciencesUS ARLAberdeenUSA

Personalised recommendations