Skip to main content
SpringerLink
Log in

Deformation Twinning in Single Crystals

  • Chapter
  • First Online:
Nonlinear Elastic and Inelastic Models for Shock Compression of Crystalline Solids

Part of the book series: Shock Wave and High Pressure Phenomena ((SHOCKWAVE))

  • 1068 Accesses

Abstract

A continuum theory is formulated to simultaneously address thermoelasticity, plasticity, and twinning in anisotropic single crystals subjected to arbitrarily large deformations. Dislocation glide and deformation twinning are dissipative mechanisms, while energy storage mechanisms associated with dislocation lines and twin boundaries are described via scalar internal state variables. In the inelastic regime, for highly symmetric orientations and rate independent shear strength, the Rankine–Hugoniot conditions and constitutive relations can be reduced to a set of algebraic equations to be solved for the material response. In a case study, the model describes the thermomechanical behavior of single crystals of alumina, i.e., sapphire. Resolved shear stresses necessary for glide or twin nucleation are estimated from nonlinear elastic calculations, theoretical considerations of Peierls barriers and stacking fault energies, and observations from both quasi-static and shock compression experiments. Residual elastic volume changes, predicted from nonlinear elastic considerations and approximated dislocation line energies, are positive and proportional to the dislocation line density and twin boundary area density. Analytical solutions to the planar shock problem are presented for c-axis compression of sapphire wherein rhombohedral twinning modes are activated.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Asaro, R.: Crystal plasticity. J. Appl. Mech. 50, 921–934 (1983)

    Article  ADS  MATH  Google Scholar 

  2. Azhdari, A., Nemat-Nasser, S.: Experimental and computational study of fracturing in an anisotropic brittle solid. Mech. Mater. 28, 247–262 (1998)

    Article  Google Scholar 

  3. Barton, N., Winter, N., Reaugh, J.: Defect evolution and pore collapse in crystalline energetic materials. Model. Simul. Mater. Sci. Eng. 17, 035003 (2009)

    Article  ADS  Google Scholar 

  4. Bell, R., Cahn, R.: The dynamics of twinning and the interrelation of slip and twinning in zinc crystals. Proc. R. Soc. Lond. A 239, 494–521 (1957)

    Article  ADS  Google Scholar 

  5. Beltz, G., Rice, J., Shih, C., Xia, L.: A self-consistent model for cleavage in the presence of plastic flow. Acta Mater. 44, 3943–3954 (1996)

    Article  Google Scholar 

  6. Bilby, B., Crocker, A.: The theory of the crystallography of deformation twinning. Proc. R. Soc. Lond. A 288, 240–255 (1965)

    Article  ADS  Google Scholar 

  7. Boiko, V., Garber, R., Kosevich, A.: Reversible Crystal Plasticity. AIP Press, New York (1994)

    Google Scholar 

  8. Bond, W.e.a.: Standards on piezoelectric crystals. Proc. Inst. Radio Eng. 37, 1378–1395 (1949)

    Google Scholar 

  9. Bourne, N., Millett, J., Chen, M., McCauley, J., Dandekar, D.: On the Hugoniot elastic limit in polycrystalline alumina. J. Appl. Phys. 102, 073514 (2007)

    Article  ADS  Google Scholar 

  10. Castaing, J., Cadoz, J., Kirby, S.: Prismatic slip of Al2O3 single crystals below 1000∘C in compression under hydrostatic pressure. J. Am. Ceram. Soc. 64, 504–511 (1981)

    Article  Google Scholar 

  11. Castaing, J., Munoz, A., Rodriguez, A.: Hardening of rhombohedral twinning in sapphire (α-Al2O3) by basal slip dislocations. Philos. Mag. A 82, 1419–1431 (2002)

    ADS  Google Scholar 

  12. Castaing, J., He, A., Lagerlöf, K., Heuer, A.: Deformation of sapphire (α-Al2O3) by basal slip and basal twinning below 700∘c. Philos. Mag. 84, 1113–1125 (2004)

    Article  ADS  Google Scholar 

  13. Castanet, R.: Selected data on the thermodynamic properties of α-alumina. High Temp. High Pressures 16, 449–457 (1984)

    Google Scholar 

  14. Chang, Y., Lloyd, J., Becker, R., Kochmann, D.: Modeling microstructure evolution in magnesium: Comparison of detailed and reduced-order kinematic models. Mech. Mater. 108, 40–57 (2017)

    Article  Google Scholar 

  15. Chin, G., Hosford, W., Mendorf, D.: Accommodation of constrained deformation in FCC metals by slip and twinning. Proc. R. Soc. Lond. A 309, 433–456 (1969)

    Article  ADS  Google Scholar 

  16. Christian, J., Mahajan, S.: Deformation twinning. Prog. Mater. Sci. 39, 1–157 (1995)

    Article  Google Scholar 

  17. Chung, P., Clayton, J.: Multiscale modeling of point and line defects in cubic crystals. Int. J. Multiscale Comput. Eng. 5, 203–226 (2007)

    Article  Google Scholar 

  18. Clarebrough, L., Hargreaves, M., West, G.: The density of dislocations in compressed copper. Acta Metall. 5, 738–740 (1957)

    Article  Google Scholar 

  19. Clayton, J.: Dynamic plasticity and fracture in high density polycrystals: constitutive modeling and numerical simulation. J. Mech. Phys. Solids 53, 261–301 (2005)

    Article  ADS  MATH  Google Scholar 

  20. Clayton, J.: A continuum description of nonlinear elasticity, slip and twinning, with application to sapphire. Proc. R. Soc. Lond. A 465, 307–334 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Clayton, J.: Modeling finite deformations in trigonal ceramic crystals with lattice defects. Int. J. Plast. 26, 1357–1386 (2010)

    Article  MATH  Google Scholar 

  22. Clayton, J.: A nonlinear thermomechanical model of spinel ceramics applied to aluminum oxynitride (AlON). J. Appl. Mech. 78, 011013 (2011)

    Article  ADS  Google Scholar 

  23. Clayton, J.: Nonlinear Mechanics of Crystals. Springer, Dordrecht (2011)

    Book  MATH  Google Scholar 

  24. Clayton, J.: Analysis of shock compression of strong single crystals with logarithmic thermoelastic-plastic theory. Int. J. Eng. Sci. 79, 1–20 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Clayton, J.: Finite strain analysis of shock compression of brittle solids applied to titanium diboride. Int. J. Impact Eng. 73, 56–65 (2014)

    Article  Google Scholar 

  26. Clayton, J.: Mesoscale models of interface mechanics in crystalline solids. J. Mater. Sci. 53, 5515–5545 (2018)

    Article  ADS  Google Scholar 

  27. Clayton, J.: Nonlinear thermomechanics for analysis of weak shock profile data in ductile polycrystals. J. Mech. Phys. Solids 124, 714–757 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  28. Clayton, J., Bammann, D.: Finite deformations and internal forces in elastic-plastic crystals: interpretations from nonlinear elasticity and anharmonic lattice statics. J. Eng. Mater. Technol. 131, 041201 (2009)

    Article  Google Scholar 

  29. Clayton, J., Becker, R.: Elastic-plastic behavior of cyclotrimethylene trinitramine single crystals under spherical indentation: modeling and simulation. J. Appl. Phys. 111, 063512 (2012)

    Article  ADS  Google Scholar 

  30. Clayton, J., Chung, P.: An atomistic-to-continuum framework for nonlinear crystal mechanics based on asymptotic homogenization. J. Mech. Phys. Solids 54, 1604–1639 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Clayton, J., Knap, J.: A phase field model of deformation twinning: nonlinear theory and numerical simulations. Phys. D 240, 841–858 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Clayton, J., Knap, J.: Phase field modeling of twinning in indentation of transparent single crystals. Model. Simul. Mater. Sci. Eng. 19, 085005 (2011)

    Article  ADS  Google Scholar 

  33. Clayton, J., Knap, J.: Phase field analysis of fracture induced twinning in single crystals. Acta Mater. 61, 5341–5353 (2013)

    Article  Google Scholar 

  34. Clayton, J., Bammann, D., McDowell, D.: Anholonomic configuration spaces and metric tensors in finite strain elastoplasticity. Int. J. Non Linear Mech. 39, 1039–1049 (2004)

    Article  MATH  Google Scholar 

  35. Clayton, J., Bammann, D., McDowell, D.: A geometric framework for the kinematics of crystals with defects. Philos. Mag. 85, 3983–4010 (2005)

    Article  ADS  Google Scholar 

  36. Coleman, B., Gurtin, M.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)

    Article  ADS  Google Scholar 

  37. Eshelby, J.: Elastic inclusions and inhomogeneities. In: Sneddon, I., Hill, R. (eds.) Progress in Solid Mechanics, vol. 2, pp. 89–140. North-Holland, Amsterdam (1961)

    Google Scholar 

  38. Farber, Y., Yoon, S., Lagerlöf, K., Heuer, A.: Microplasticity during high temperature indentation and the Peierls potential in sapphire (α-Al2O3) single crystals. Phys. Status Solidi A 137, 485–498 (1993)

    Article  ADS  Google Scholar 

  39. Frenkel, J.: Zur theorie der elastizitätsgrenze und der festigkeit kristallinischer körper. Zeitschrift für Physik 37, 572–609 (1926)

    Article  ADS  MATH  Google Scholar 

  40. Friedel, J.: Dislocations. Pergamon, Oxford (1964)

    MATH  Google Scholar 

  41. Fuller, H., Winey, J., Gupta, Y.: Inelastic deformation in shocked sapphire single crystals. J. Appl. Phys. 113, 226102 (2013)

    Article  ADS  Google Scholar 

  42. Germain, P., Lee, E.: On shock waves in elastic-plastic solids. J. Mech. Phys. Solids 21, 359–382 (1973)

    Article  ADS  MATH  Google Scholar 

  43. Gieske, J., Barsch, G.: Pressure dependence of the elastic constants of single crystalline aluminum oxide. Phys. Status Solidi B 29, 121–131 (1968)

    Article  ADS  Google Scholar 

  44. Gilman, J.: Resistance to shock-front propagation in solids. J. Appl. Phys. 50, 4059–4064 (1979)

    Article  ADS  Google Scholar 

  45. Grady, D.: Shock-wave compression of brittle solids. Mech. Mater. 29, 181–203 (1998)

    Article  ADS  Google Scholar 

  46. Graham, R.: Determination of third- and fourth-order longitudinal elastic constants by shock compression techniques–application to sapphire and fused quartz. J. Acoust. Soc. Am. 51, 1576–1581 (1972)

    Article  ADS  Google Scholar 

  47. Graham, R., Brooks, W.: Shock-wave compression of sapphire from 15 to 420 kbar. The effects of large anisotropic compressions. J. Phys. Chem. Solids 32, 2311–2330 (1971)

    Article  Google Scholar 

  48. Gurrutxaga-Lerma, B., Balint, D., Dini, D., Eakins, D., Sutton, A.: Attenuation of the dynamic yield point of shocked aluminum using elastodynamic simulations of dislocation dynamics. Phys. Rev. Lett. 114, 174301 (2015)

    Article  ADS  Google Scholar 

  49. Hankey, R., Schuele, D.: Third-order elastic constants of Al2O3. J. Acoust. Soc. Am. 48, 190–202 (1970)

    Article  ADS  Google Scholar 

  50. Heuer, A.: Deformation twinning in corundum. Philos. Mag. 13, 379–393 (1966)

    Article  ADS  Google Scholar 

  51. Heuer, A., Lagerlöf, K., Castaing, J.: Slip and twinning dislocations in sapphire (α-Al2O3). Philos. Mag. A 78, 747–763 (1998)

    Article  ADS  Google Scholar 

  52. Hirth, J., Lothe, J.: Theory of Dislocations. Wiley, New York (1982)

    MATH  Google Scholar 

  53. Holder, J., Granato, A.: Thermodynamic properties of solids containing defects. Phys. Rev. 182, 729–741 (1969)

    Article  ADS  Google Scholar 

  54. Horie, Y.: Thermodynamics of dislocations and shock compression of solids. Phys. Rev. B 21, 5549–5557 (1980)

    Article  ADS  Google Scholar 

  55. Hou, T., Rosakis, P., LeFloch, P.: A level-set approach to the computation of twinning and phase-transition dynamics. J. Comput. Phys. 150, 302–331 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Hull, D., Bacon, D.: Introduction to Dislocations. Butterworth-Heinemann, Oxford (1984)

    Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  58. James, R.: Finite deformation by mechanical twinning. Arch. Ration. Mech. Anal. 77, 143–176 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  59. Jog, C.: The explicit determination of the logarithm of a tensor and its derivative. J. Elast. 93, 141–148 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  60. Johnson, J.: Calculation of plane-wave propagation in anisotropic elastic-plastic solids. J. Appl. Phys. 43, 2074–2082 (1972)

    Article  ADS  Google Scholar 

  61. Johnson, J.: Wave velocities in shock-compressed cubic and hexagonal single crystals above the elastic limit. J. Phys. Chem. Solids 43, 609–616 (1974)

    Article  ADS  Google Scholar 

  62. Johnson, J., Jones, O., Michaels, T.: Dislocation dynamics and single-crystal constitutive relations: shock-wave propagation and precursor decay. J. Appl. Phys. 41, 2330–2339 (1970)

    Article  ADS  Google Scholar 

  63. Kalidindi, S.: Incorporation of deformation twinning in crystal plasticity models. J. Mech. Phys. Solids 46, 267–290 (1998)

    Article  ADS  MATH  Google Scholar 

  64. Kenway, P.: Calculated stacking-fault energies in α-Al2O3. Philos. Mag. B 68, 171–183 (1993)

    Article  ADS  Google Scholar 

  65. Kocks, U., Argon, A., Ashby, M.: Thermodynamics and kinetics of slip. Prog. Mater. Sci. 19, 1–291 (1975)

    Article  Google Scholar 

  66. Kronberg, M.: Plastic deformation of single crystals of sapphire: basal slip and twinning. Acta Metall. 5, 507–524 (1957)

    Article  Google Scholar 

  67. Kuhlmann-Wilsdorf, D.: Frictional stress acting on a moving dislocation in an otherwise perfect crystal. Phys. Rev. 120, 773–781 (1960)

    Article  ADS  MathSciNet  Google Scholar 

  68. Lagerlöf, K., Mitchell, T., Heuer, A., Riviere, J., Cadoz, J., Castaing, J., Phillips, D.: Stacking fault energy in sapphire (α-Al2O3). Acta Metall. 32, 97–105 (1984)

    Article  Google Scholar 

  69. Lagerlöf, K., Heuer, A., Castaing, J., Riviere, J., Mitchell, T.: Slip and twinning in sapphire (α-Al2O3). J. Am. Ceram. Soc. 77, 385–397 (1994)

    Article  Google Scholar 

  70. Lankford, J., Predebon, W., Staehler, J., Subhash, G., Pletka, B., Anderson, C.: The role of plasticity as a limiting factor in the compressive failure of high strength ceramics. Mech. Mater. 29, 205–218 (1998)

    Article  Google Scholar 

  71. Lebensohn, R., Tomé, C.: A study of the stress state associated with twin nucleation and propagation in anisotropic materials. Philos. Mag. A 67, 187–206 (1993)

    Article  ADS  Google Scholar 

  72. Lee, J., Yoo, M.: Elastic strain energy of deformation twinning in tetragonal crystals. Metall. Trans. A 21, 2521–2530 (1990)

    Article  Google Scholar 

  73. Lloyd, J., Priddy, M.: Simulating strain localization in rolled magnesium. Acta Mater. 129, 149–158 (2017)

    Article  Google Scholar 

  74. Lloyd, J., Clayton, J., Austin, R., McDowell, D.: Shock compression modeling of metallic single crystals: comparison of finite difference, steady wave, and analytical solutions. Adv. Model. Simul. Eng. Sci. 2, 14 (2015)

    Article  Google Scholar 

  75. McCauley, J., Strassburger, E., Patel, P., Paliwal, B., Ramesh, K.: Experimental observations on dynamic response of selected transparent armor materials. Exp. Mech. 53, 3–29 (2013)

    Article  Google Scholar 

  76. Meyers, M., Vöhringer, O., Lubarda, V.: The onset of twinning in metals: a constitutive description. Acta Mater. 49, 4025–4039 (2001)

    Article  Google Scholar 

  77. Meyers, M., Gregori, F., Kad, B., Schneider, M., Kalantar, D., Remington, B., Ravichandran, G., Boehly, T., Wark, J.: Laser-induced shock compression of monocrystalline copper: characterization and analysis. Acta Mater. 51, 1211–1228 (2003)

    Article  Google Scholar 

  78. Molinari, A., Ravichandran, G.: Fundamental structure of steady plastic shock waves in metals. J. Appl. Phys. 95, 1718–1732 (2004)

    Article  ADS  Google Scholar 

  79. Munson, D., Lawrence, R.: Dynamic deformation of polycrystalline alumina. J. Appl. Phys. 50, 6272–6282 (1979)

    Article  ADS  Google Scholar 

  80. Nabarro, F.: Dislocations in a simple cubic lattice. Proc. Phys. Soc. 59, 256–272 (1947)

    Article  ADS  Google Scholar 

  81. Paxton, A., Gumbsch, P., Methfessel, M.: A quantum mechanical calculation of the theoretical strength of metals. Philos. Mag. Lett. 63, 267–274 (1991)

    Article  ADS  Google Scholar 

  82. Peierls, R.: The size of a dislocation. Proc. Phys. Soc. 52, 34–37 (1940)

    Article  ADS  Google Scholar 

  83. Perrin, G., Delannoy-Coutris, M.: Analysis of plane elastic-plastic shock-waves from the fourth-order anharmonic theory. Mech. Mater. 2, 139–153 (1983)

    Article  Google Scholar 

  84. Pletka, B., Heuer, A., Mitchell, T.: Work-hardening in sapphire (α-Al2O3). Acta Metall. 25, 25–33 (1977)

    Article  Google Scholar 

  85. Price, P.: Nucleation and growth of twins in dislocation-free zinc crystals. Proc. R. Soc. Lond. A 260, 251–262 (1961)

    Article  ADS  Google Scholar 

  86. Proust, G., Tomé, C., Jain, A., Agnew, S.: Modeling the effect of twinning and detwinning during strain-path changes of magnesium alloy AZ31. Int. J. Plast. 25, 861–880 (2009)

    Article  MATH  Google Scholar 

  87. Regueiro, R., Bammann, D., Marin, E., Garikipati, K.: A nonlocal phenomenological anisotropic finite deformation plasticity model accounting for dislocation defects. J. Eng. Mater. Technol. 124, 380–387 (2002)

    Article  Google Scholar 

  88. Reinhart, W., Chhabildas, L., Vogler, T.: Investigating phase transitions and strength in single-crystal sapphire using shock-reshock loading techniques. Int. J. Impact Eng. 33, 655–669 (2006)

    Article  Google Scholar 

  89. Rodríguez, M., Castaing, J., Munoz, A., Veyssiere, P., Rodríguez, A.D.: Analysis of a kink pair model applied to a Peierls mechanism in basal and prism plane slips in sapphire (α-Al2O3) deformed between 200 and 1800∘C. J. Am. Ceram. Soc. 91, 1612–1617 (2008)

    Article  Google Scholar 

  90. Rohatgi, A., Vecchio, K.: The variation of dislocation density as a function of the stacking fault energy in shock-deformed fcc materials. Mater. Sci. Eng. A 328, 256–266 (2002)

    Article  Google Scholar 

  91. Rosakis, P., Tsai, H.: Dynamic twinning processes in crystals. Int. J. Solids Struct. 32, 2711–2723 (1995)

    Article  MATH  Google Scholar 

  92. Sarkar, S., Ballabh, T., Middya, T., Basu, A.: T-matrix approach to effective nonlinear elastic constants of heterogeneous materials. Phys. Rev. B 54, 3926–3931 (1996)

    Article  ADS  Google Scholar 

  93. Schultz, R., Jensen, M., Bradt, R.: Single crystal cleavage of brittle materials. Int. J. Fract. 65, 291–312 (1994)

    Article  Google Scholar 

  94. Scott, W., Orr, K.: Rhombohedral twinning in alumina. J. Am. Ceram. Soc. 66, 27–32 (1983)

    Article  Google Scholar 

  95. Seeger, A., Haasen, P.: Density changes of crystals containing dislocations. Philos. Mag. 3, 470–475 (1958)

    Article  ADS  Google Scholar 

  96. Snow, J., Heuer, A.: Slip systems in Al2O3. J. Am. Ceram. Soc. 56, 153–157 (1973)

    Article  Google Scholar 

  97. Staroselsky, A., Anand, L.: Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning. J. Mech. Phys. Solids 46, 671–696 (1998)

    Article  ADS  MATH  Google Scholar 

  98. Staroselsky, A., Anand, L.: A constitutive model for hcp materials deforming by slip and twinning: application to magnesium alloy AZ31B. Int. J. Plast. 19, 1843–1864 (2003)

    Article  MATH  Google Scholar 

  99. Swegle, J., Grady, D.: Shock viscosity and the prediction of shock wave rise times. J. Appl. Phys. 58, 692–701 (1985)

    Article  ADS  Google Scholar 

  100. Taylor, G.: The mechanism of plastic deformation of crystals. part i. theoretical. Proc. R. Soc. Lond. A 145, 362–387 (1934)

    Google Scholar 

  101. Teodosiu, C.: Elastic Models of Crystal Defects. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  102. Teodosiu, C., Sidoroff, F.: A theory of finite elastoviscoplasticity of single crystals. Int. J. Eng. Sci. 14, 165–176 (1976)

    Article  MATH  Google Scholar 

  103. Thurston, R.: Waves in solids. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VI, pp. 109–308. Springer, Berlin (1974)

    Google Scholar 

  104. Tomé, C., Lebensohn, R., Kocks, U.: A model for texture development dominated by deformation twinning: application to zirconium alloys. Acta Metall. et Materialia 39, 2667–2680 (1991)

    Article  Google Scholar 

  105. Toupin, R., Rivlin, R.: Dimensional changes in crystals caused by dislocations. J. Math. Phys. 1, 8–15 (1960)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  106. Tressler, R., Barber, D.: Yielding and flow of c-axis sapphire filaments. J. Am. Ceram. Soc. 57, 13–19 (1974)

    Article  Google Scholar 

  107. Tymiak, N., Gerberich, W.: Initial stages of contact-induced plasticity in sapphire. I. Surface traces of slip and twinning. Philos. Mag. 87, 5143–5168 (2007)

    Google Scholar 

  108. Van Houtte, P.: Simulation of the rolling and shear texture of brass by the Taylor theory adapted for mechanical twinning. Acta Metall. 26, 591–604 (1978)

    Article  Google Scholar 

  109. Wallace, D.: Thermodynamics of Crystals. Wiley, New York (1972)

    Book  Google Scholar 

  110. Wiederhorn, S.: Fracture of sapphire. J. Am. Ceram. Soc. 52, 485–491 (1969)

    Article  Google Scholar 

  111. Winey, J., Gupta, Y., Hare, D.: R-axis sound speed and elastic properties of sapphire single crystals. J. Appl. Phys. 90, 3109–3111 (2001)

    Article  ADS  Google Scholar 

  112. Wu, X., Kalidindi, S., Necker, C., Salem, A.: Prediction of crystallographic texture evolution and anisotropic stress–strain curves during large plastic strains in high purity α-titanium using a Taylor-type crystal plasticity model. Acta Mater. 55, 423–432 (2007)

    Article  Google Scholar 

  113. Xu, D.S., Chang, J.P., Li, J., Yang, R., Li, D., Yip, S.: Dislocation slip or deformation twinning: confining pressure makes a difference. Mater. Sci. Eng. A 387, 840–844 (2004)

    Article  Google Scholar 

  114. Yoo, M., Lee, J.: Deformation twinning in HCP metals and alloys. Philos. Mag. A 63, 987–1000 (1991)

    Article  ADS  Google Scholar 

  115. Zanzotto, G.: The Cauchy–Born hypothesis, nonlinear elasticity and mechanical twinning in crystals. Acta Crystallogr. A 52, 839–849 (1996)

    Article  Google Scholar 

  116. Zhang, C., Kalia, R., Nakano, A., Vashishta, P.: Hypervelocity impact induced deformation modes in α-alumina. Appl. Phys. Lett. 91, 071906 (2007)

    Article  ADS  Google Scholar 

  117. Zouboulis, E., Grimsditch, M.: Refractive index and elastic properties of single-crystal corundum (α-Al2O3) up to 2100 K. J. Appl. Phys. 70, 772–776 (1991)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Impact Physics CCRL-WMP-C, United States Army Research Laboratory, Aberdeen, USA

    John D. Clayton

  2. University of Maryland, College Park, MD, USA

    John D. Clayton

Authors
  1. John D. Clayton
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Clayton, J.D. (2019). Deformation Twinning in Single Crystals. In: Nonlinear Elastic and Inelastic Models for Shock Compression of Crystalline Solids. Shock Wave and High Pressure Phenomena. Springer, Cham. https://doi.org/10.1007/978-3-030-15330-4_9

Download citation

Publish with us

Policies and ethics

Search

Navigation