Line and point defects in nonlinear anisotropic solids

  • Ashkan Golgoon
  • Arash YavariEmail author


In this paper, we present some analytical solutions for the stress fields of nonlinear anisotropic solids with distributed line and point defects. In particular, we determine the stress fields of (i) a parallel cylindrically symmetric distribution of screw dislocations in infinite orthotropic and monoclinic media, (ii) a cylindrically symmetric distribution of parallel wedge disclinations in an infinite orthotropic medium, (iii) a distribution of edge dislocations in an orthotropic medium, and (iv) a spherically symmetric distribution of point defects in a transversely isotropic spherical ball.


Transversely isotropic solids Orthotropic solids Monoclinic solids Defects Disclinations Dislocations Nonlinear elasticity 

Mathematics Subject Classification

74B20 70G45 74E10 15A72 74Fxx 



This work was partially supported by NSF—Grant No. CMMI 1561578, ARO Grant No. W911NF-18-1-0003, and AFOSR—Grant No. FA9550-12-1-0290.


  1. 1.
    Acharya, A.: A model of crystal plasticity based on the theory of continuously distributed dislocations. J. Mech. Phys. Solids 49(4), 761–784 (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Amar, M.B., Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids 53(10), 2284–2319 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bilby, B., Bullough, R., Smith, E.: Continuous distributions of dislocations: a new application of the methods of non-riemannian geometry. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 231, pp 263–273. The Royal Society (1955)Google Scholar
  4. 4.
    Brake, M.: An analytical elastic-perfectly plastic contact model. Int. J. Solids Struct. 49(22), 3129–3141 (2012)CrossRefGoogle Scholar
  5. 5.
    Brake, M.: An analytical elastic plastic contact model with strain hardening and frictional effects for normal and oblique impacts. Int. J. Solids Struct. 62, 104–123 (2015)CrossRefGoogle Scholar
  6. 6.
    Clayton, J.: Defects in nonlinear elastic crystals: differential geometry, finite kinematics, and second-order analytical solutions. ZAMM J. Appl. Math. Mech. Z. Angew. Math. Mech. 95(5), 476–510 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Demirkoparan, H., Merodio, J.: Bulging bifurcation of inflated circular cylinders of doubly fiber-reinforced hyperelastic material under axial loading and swelling. Math. Mech. Solids 22(4), 666–682 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Derezin, S.V., Zubov, L.M.: Disclinations in nonlinear elasticity. Z. Angew. Math. Mech. (ZAMM) 91(6), 433–442 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    do Carmo, M.: Riemannian Geometry. Mathematics: Theoryand Applications. Birkhäuser, Boston (1992). ISBN 1584883553CrossRefGoogle Scholar
  10. 10.
    Doyle, T., Ericksen, J.: Nonlinear elasticity. Adv. Appl. Mech. 4, 53–115 (1956)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ehret, A.E., Itskov, M.: Modeling of anisotropic softening phenomena: application to soft biological tissues. Int. J. Plast. 25(5), 901–919 (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Epstein, M., Elzanowski, M.: Material Inhomogeneities and Their Evolution: A Geometric Approach. Springer, Berlin (2007)zbMATHGoogle Scholar
  13. 13.
    Eshelby, J.: LXXXII. Edge dislocations in anisotropic materials. Lond. Edinb. Dublin Philos. Mag. J. Sci. 40(308), 903–912 (1949)CrossRefzbMATHGoogle Scholar
  14. 14.
    Eshelby, J., Read, W., Shockley, W.: Anisotropic elasticity with applications to dislocation theory. Acta metall. 1(3), 251–259 (1953)CrossRefGoogle Scholar
  15. 15.
    Gairola, B.: Nonlinear Elastic Problems. In: Nabarro, F.R.N. (ed.) Dislocations in Solids. North-Holland Publishing Co., Amsterdam (1979)Google Scholar
  16. 16.
    Ghaednia, H., Marghitu, D.B.: Permanent deformation during the oblique impact with friction. Arch. Appl. Mech. 86(1–2), 121–134 (2016)CrossRefGoogle Scholar
  17. 17.
    Giordano, S., Palla, P., Colombo, L.: Nonlinear elasticity of composite materials. Eur. Phys. J. B 68(1), 89–101 (2009)CrossRefGoogle Scholar
  18. 18.
    Golgoon, A., Yavari, A.: On the stress field of a nonlinear elastic solid torus with a toroidal inclusion. J. Elast. 128(1), 115–145 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Golgoon, A., Yavari, A.: Nonlinear elastic inclusions in anisotropic solids. J. Elast. 130(2), 239–269 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Golgoon, A., Sadik, S., Yavari, A.: Circumferentially-symmetric finite eigenstrains in incompressible isotropic nonlinear elastic wedges. Int. J. Non-Linear Mech. 84, 116–129 (2016)CrossRefGoogle Scholar
  21. 21.
    Goriely, A., Moulton, D.E., Vandiver, R.: Elastic cavitation, tube hollowing, and differential growth in plants and biological tissues. Europhys. Lett. 91(1), 18001 (2010)CrossRefGoogle Scholar
  22. 22.
    Head, A.: Unstable dislocations in anisotropic crystals. Physica Status Solidi (b) 19(1), 185–192 (1967)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hooshmand, M., Mills, M., Ghazisaeidi, M.: Atomistic modeling of dislocation interactions with twin boundaries in ti. Model. Simul. Mater. Sci. Eng. 25(4), 045003 (2017)CrossRefGoogle Scholar
  24. 24.
    Indenbom, V.: Dislocations and internal stresses. In: Modern Problems in Condensed Matter Sciences, Vol. 31, pp. 1–174. Elsevier (1992)Google Scholar
  25. 25.
    Jackson, R.L., Green, I.: A finite element study of elasto-plastic hemispherical contact against a rigid flat. Trans. ASME-F-J. Tribol. 127(2), 343–354 (2005)CrossRefGoogle Scholar
  26. 26.
    Jackson, R.L., Ghaednia, H., Pope, S.: A solution of rigid-perfectly plastic deep spherical indentation based on slip-line theory. Tribol. Lett. 58(3), 47 (2015)CrossRefGoogle Scholar
  27. 27.
    Katanaev, M.: Introduction to the geometric theory of defects. arXiv preprint cond-mat/0502123 (2005)Google Scholar
  28. 28.
    Kinoshita, N., Mura, T.: Elastic fields of inclusions in anisotropic media. Physica Status Solidi (a) 5(3), 759–768 (1971)CrossRefGoogle Scholar
  29. 29.
    Knowles, J.K.: The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids. Int. J. Fract. 13(5), 611–639 (1977)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kondo, K.: Geometry of elastic deformation and incompatibility. Mem. Unifying Study Basic Probl. Eng. Sci. Means Geom. 1, 5–17 (1955a)Google Scholar
  31. 31.
    Kondo, K.: Non-Riemannian geometry of imperfect crystals from a macroscopic viewpoint. Mem. Unifying Study Basic Probl. Eng. Sci. Means Geom. 1, 6–17 (1955b)Google Scholar
  32. 32.
    Li, J.Y., Dunn, M.L.: Anisotropic coupled-field inclusion and inhomogeneity problems. Philos. Mag. A 77(5), 1341–1350 (1998)CrossRefGoogle Scholar
  33. 33.
    Liu, I., et al.: On representations of anisotropic invariants. Int. J. Eng. Sci. 20(10), 1099–1109 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lothe, J.: Dislocations in anisotropic media. In: Indenbom, V.L., Lothe, J. (eds.) Elastic Strain Fields and Dislocation Mobility, pp. 269–328. North-Holland, Amsterdam (1992)CrossRefGoogle Scholar
  35. 35.
    Lu, J., Papadopoulos, P.: A covariant constitutive description of anisotropic non-linear elasticity. Z. Angew. Math. Phys. (ZAMP) 51(2), 204–217 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice Hall, New York (1983)zbMATHGoogle Scholar
  37. 37.
    Mazzucato, A.L., Rachele, L.V.: Partial uniqueness and obstruction to uniqueness in inverse problems for anisotropic elastic media. J. Elast. 83(3), 205–245 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Merodio, J., Ogden, R.: Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int. J. Solids Struct. 40(18), 4707–4727 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Merodio, J., Ogden, R.: Tensile instabilities and ellipticity in fiber-reinforced compressible non-linearly elastic solids. Int. J. Eng. Sci. 43(8), 697–706 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Merodio, J., Ogden, R.: The influence of the invariant \({I}_8\) on the stress-deformation and ellipticity characteristics of doubly fiber-reinforced non-linearly elastic solids. Int. J. Non-Linear Mech. 41(4), 556–563 (2006)CrossRefzbMATHGoogle Scholar
  41. 41.
    Moulton, D.E., Goriely, A.: Anticavitation and differential growth in elastic shells. J. Elast. 102(2), 117–132 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Mura, T.: Micromechanics of Defects in Solids. Martinus Nijhoff, Leiden (1982)CrossRefGoogle Scholar
  43. 43.
    Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2(1), 197–226 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Noll, W.: Materially uniform simple bodies with inhomogeneities. Arch. Ration. Mech. Anal. 27(1), 1–32 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51(3), 032902 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Ozakin, A., Yavari, A.: Affine development of closed curves in Weitzenböck manifolds and the Burgers vector of dislocation mechanics. Math. Mech. Solids 19(3), 299–307 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Pence, T.J., Tsai, H.: Swelling-induced microchannel formation in nonlinear elasticity. IMA J. Appl. Math. 70(1), 173–189 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Petersen, P.: Riemannian Geometry, vol. 171. Springer, Berlin (2006)CrossRefzbMATHGoogle Scholar
  49. 49.
    Rosakis, P., Rosakis, A.J.: The screw dislocation problem in incompressible finite elastostatics: a discussion of nonlinear effects. J. Elast. 20(1), 3–40 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Sadik, S., Yavari, A.: Small-on-large geometric anelasticity. Proc. R. Soc. Lond. A. Math. Phys. Eng. Sci. (2016)
  51. 51.
    Sadik, S., Yavari, A.: Geometric nonlinear thermoelasticity and the time evolution of thermal stresses. Math. Mech. Solids 22(7), 1546–1587 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Schaefer, H., Kronmüller, H.: Elastic interaction of point defects in isotropic and anisotropic cubic media. Physica Status Solidi (b) 67(1), 63–74 (1975)CrossRefGoogle Scholar
  53. 53.
    Sozio, F., Yavari, A.: Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies. J. Mech. Phys. Solids 98, 12–48 (2017)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Spencer, A.: Part III. Theory of invariants. Contin. Phys. 1, 239–353 (1971)Google Scholar
  55. 55.
    Spencer, A.: The formulation of constitutive equation for anisotropic solids. In: Mechanical Behavior of Anisotropic Solids/Comportment Méchanique des Solides Anisotropes, pp. 3–26. Springer (1982)Google Scholar
  56. 56.
    Stojanovic, R., Djuric, S., Vujosevic, L.: On finite thermal deformations. Arch. Mech. Stosow. 16, 103–108 (1964)MathSciNetGoogle Scholar
  57. 57.
    Teodosiu, C.: Elastic Models of Crystal Defects. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  58. 58.
    Teutonico, L.: Uniformly moving dislocations of arbitrary orientation in anisotropic media. Phys. Rev. 127(2), 413 (1962)CrossRefGoogle Scholar
  59. 59.
    Triantafyllidis, N., Abeyaratne, R.: Instabilities of a finitely deformed fiber-reinforced elastic material. J. Appl. Mech. 50(1), 149–156 (1983)CrossRefzbMATHGoogle Scholar
  60. 60.
    Truesdell, C.: The physical components of vectors and tensors. Z. Angew. Math. Mech. (ZAMM) 33(10–11), 345–356 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Vergori, L., Destrade, M., McGarry, P., Ogden, R.W.: On anisotropic elasticity and questions concerning its finite element implementation. Comput. Mech. 52(5), 1185–1197 (2013)CrossRefzbMATHGoogle Scholar
  62. 62.
    Volterra, V.: Sur l’équilibre des corps élastiques multiplement connexes. Annales scientifiques de l’École normale supérieure 24, 401–517 (1907)CrossRefzbMATHGoogle Scholar
  63. 63.
    Wang, C.-C.: On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch. Ration. Mech. Anal. 27(1), 33–94 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Wang, J., Yadav, S., Hirth, J., Tomé, C., Beyerlein, I.: Pure-shuffle nucleation of deformation twins in hexagonal-close-packed metals. Mater. Res. Lett. 1(3), 126–132 (2013)CrossRefGoogle Scholar
  65. 65.
    Wesołowski, Z., Seeger, A.: On the screw dislocation in finite elasticity. In: Mechanics of Generalized Continua, pp. 294–297. Springer (1968)Google Scholar
  66. 66.
    Willis, J.: Anisotropic elastic inclusion problems. Q. J. Mech. Appl. Math. 17(2), 157–174 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Willis, J.: Stress fields produced by dislocations in anisotropic media. Philos. Mag. 21(173), 931–949 (1970)CrossRefzbMATHGoogle Scholar
  68. 68.
    Willis, J.R.: Second-order effects of dislocations in anisotropic crystals. Int. J. Eng. Sci. 5(2), 171–190 (1967)CrossRefzbMATHGoogle Scholar
  69. 69.
    Wolfram, I.: Mathematica. Version 11.0. Wolfram Research Inc., Champaign (2016)Google Scholar
  70. 70.
    Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20, 781–830 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Yavari, A.: On the wedge dispiration in an inhomogeneous isotropic nonlinear elastic solid. Mech. Res. Commun. 78, 55–59 (2016)CrossRefGoogle Scholar
  72. 72.
    Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205, 59–118 (2012a)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Yavari, A., Goriely, A.: Weyl geometry and the nonlinear mechanics of distributed point defects. Proc. R. Soc. A 468, 3902–3922 (2012b)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Yavari, A., Goriely, A.: Nonlinear elastic inclusions in isotropic solids. Proc. R. Soc. A 469(2160), 20130415 (2013a)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear disclination mechanics. Math. Mech. Solids 18(1), 91–102 (2013b)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Yavari, A., Goriely, A.: The geometry of discombinations and its applications to semi-inverse problems in anelasticity. Proc. R. Soc. A 470(2169), 20140403 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Yavari, A., Marsden, J.E., Ortiz, M.: On the spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 85–112 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Yu, H.: Two-dimensional elastic defects in orthotropic bicrystals. J. Mech. Phys. Solids 49(2), 261–287 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Yu, H., Sanday, S., Rath, B. Chang, C.: Elastic fields due to defects in transversely isotropic bimaterials. In: Proceedings of the Royal Society of London A, Vol. 449, pp. 1–30. The Royal Society (1995)Google Scholar
  80. 80.
    Zheng, Q.-S., Spencer, A.: Tensors which characterize anisotropies. Int. J. Eng. Sci. 31(5), 679–693 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies, vol. 47. Springer, Berlin (1997)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of Civil and Environmental EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.School of Civil and Environmental Engineering,The George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

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