Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Acharya, A.: A model of crystal plasticity based on the theory of continuously distributed dislocations. J. Mech. Phys. Solids 49(4), 761–784 (2001)
Amar, M.B., Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids 53(10), 2284–2319 (2005)
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)
Brake, M.: An analytical elastic-perfectly plastic contact model. Int. J. Solids Struct. 49(22), 3129–3141 (2012)
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)
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)
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)
Derezin, S.V., Zubov, L.M.: Disclinations in nonlinear elasticity. Z. Angew. Math. Mech. (ZAMM) 91(6), 433–442 (2011)
do Carmo, M.: Riemannian Geometry. Mathematics: Theoryand Applications. Birkhäuser, Boston (1992). ISBN 1584883553
Doyle, T., Ericksen, J.: Nonlinear elasticity. Adv. Appl. Mech. 4, 53–115 (1956)
Ehret, A.E., Itskov, M.: Modeling of anisotropic softening phenomena: application to soft biological tissues. Int. J. Plast. 25(5), 901–919 (2009)
Epstein, M., Elzanowski, M.: Material Inhomogeneities and Their Evolution: A Geometric Approach. Springer, Berlin (2007)
Eshelby, J.: LXXXII. Edge dislocations in anisotropic materials. Lond. Edinb. Dublin Philos. Mag. J. Sci. 40(308), 903–912 (1949)
Eshelby, J., Read, W., Shockley, W.: Anisotropic elasticity with applications to dislocation theory. Acta metall. 1(3), 251–259 (1953)
Gairola, B.: Nonlinear Elastic Problems. In: Nabarro, F.R.N. (ed.) Dislocations in Solids. North-Holland Publishing Co., Amsterdam (1979)
Ghaednia, H., Marghitu, D.B.: Permanent deformation during the oblique impact with friction. Arch. Appl. Mech. 86(1–2), 121–134 (2016)
Giordano, S., Palla, P., Colombo, L.: Nonlinear elasticity of composite materials. Eur. Phys. J. B 68(1), 89–101 (2009)
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)
Golgoon, A., Yavari, A.: Nonlinear elastic inclusions in anisotropic solids. J. Elast. 130(2), 239–269 (2018)
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)
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)
Head, A.: Unstable dislocations in anisotropic crystals. Physica Status Solidi (b) 19(1), 185–192 (1967)
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)
Indenbom, V.: Dislocations and internal stresses. In: Modern Problems in Condensed Matter Sciences, Vol. 31, pp. 1–174. Elsevier (1992)
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)
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)
Katanaev, M.: Introduction to the geometric theory of defects. arXiv preprint cond-mat/0502123 (2005)
Kinoshita, N., Mura, T.: Elastic fields of inclusions in anisotropic media. Physica Status Solidi (a) 5(3), 759–768 (1971)
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)
Kondo, K.: Geometry of elastic deformation and incompatibility. Mem. Unifying Study Basic Probl. Eng. Sci. Means Geom. 1, 5–17 (1955a)
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)
Li, J.Y., Dunn, M.L.: Anisotropic coupled-field inclusion and inhomogeneity problems. Philos. Mag. A 77(5), 1341–1350 (1998)
Liu, I., et al.: On representations of anisotropic invariants. Int. J. Eng. Sci. 20(10), 1099–1109 (1982)
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)
Lu, J., Papadopoulos, P.: A covariant constitutive description of anisotropic non-linear elasticity. Z. Angew. Math. Phys. (ZAMP) 51(2), 204–217 (2000)
Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice Hall, New York (1983)
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)
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)
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)
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)
Moulton, D.E., Goriely, A.: Anticavitation and differential growth in elastic shells. J. Elast. 102(2), 117–132 (2011)
Mura, T.: Micromechanics of Defects in Solids. Martinus Nijhoff, Leiden (1982)
Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2(1), 197–226 (1958)
Noll, W.: Materially uniform simple bodies with inhomogeneities. Arch. Ration. Mech. Anal. 27(1), 1–32 (1967)
Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51(3), 032902 (2010)
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)
Pence, T.J., Tsai, H.: Swelling-induced microchannel formation in nonlinear elasticity. IMA J. Appl. Math. 70(1), 173–189 (2005)
Petersen, P.: Riemannian Geometry, vol. 171. Springer, Berlin (2006)
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)
Sadik, S., Yavari, A.: Small-on-large geometric anelasticity. Proc. R. Soc. Lond. A. Math. Phys. Eng. Sci. https://doi.org/10.1098/rspa.2016.0659 (2016)
Sadik, S., Yavari, A.: Geometric nonlinear thermoelasticity and the time evolution of thermal stresses. Math. Mech. Solids 22(7), 1546–1587 (2017)
Schaefer, H., Kronmüller, H.: Elastic interaction of point defects in isotropic and anisotropic cubic media. Physica Status Solidi (b) 67(1), 63–74 (1975)
Sozio, F., Yavari, A.: Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies. J. Mech. Phys. Solids 98, 12–48 (2017)
Spencer, A.: Part III. Theory of invariants. Contin. Phys. 1, 239–353 (1971)
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)
Stojanovic, R., Djuric, S., Vujosevic, L.: On finite thermal deformations. Arch. Mech. Stosow. 16, 103–108 (1964)
Teodosiu, C.: Elastic Models of Crystal Defects. Springer, New York (1982)
Teutonico, L.: Uniformly moving dislocations of arbitrary orientation in anisotropic media. Phys. Rev. 127(2), 413 (1962)
Triantafyllidis, N., Abeyaratne, R.: Instabilities of a finitely deformed fiber-reinforced elastic material. J. Appl. Mech. 50(1), 149–156 (1983)
Truesdell, C.: The physical components of vectors and tensors. Z. Angew. Math. Mech. (ZAMM) 33(10–11), 345–356 (1953)
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)
Volterra, V.: Sur l’équilibre des corps élastiques multiplement connexes. Annales scientifiques de l’École normale supérieure 24, 401–517 (1907)
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)
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)
Wesołowski, Z., Seeger, A.: On the screw dislocation in finite elasticity. In: Mechanics of Generalized Continua, pp. 294–297. Springer (1968)
Willis, J.: Anisotropic elastic inclusion problems. Q. J. Mech. Appl. Math. 17(2), 157–174 (1964)
Willis, J.: Stress fields produced by dislocations in anisotropic media. Philos. Mag. 21(173), 931–949 (1970)
Willis, J.R.: Second-order effects of dislocations in anisotropic crystals. Int. J. Eng. Sci. 5(2), 171–190 (1967)
Wolfram, I.: Mathematica. Version 11.0. Wolfram Research Inc., Champaign (2016)
Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20, 781–830 (2010)
Yavari, A.: On the wedge dispiration in an inhomogeneous isotropic nonlinear elastic solid. Mech. Res. Commun. 78, 55–59 (2016)
Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205, 59–118 (2012a)
Yavari, A., Goriely, A.: Weyl geometry and the nonlinear mechanics of distributed point defects. Proc. R. Soc. A 468, 3902–3922 (2012b)
Yavari, A., Goriely, A.: Nonlinear elastic inclusions in isotropic solids. Proc. R. Soc. A 469(2160), 20130415 (2013a)
Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear disclination mechanics. Math. Mech. Solids 18(1), 91–102 (2013b)
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)
Yavari, A., Marsden, J.E., Ortiz, M.: On the spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 85–112 (2006)
Yu, H.: Two-dimensional elastic defects in orthotropic bicrystals. J. Mech. Phys. Solids 49(2), 261–287 (2001)
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)
Zheng, Q.-S., Spencer, A.: Tensors which characterize anisotropies. Int. J. Eng. Sci. 31(5), 679–693 (1993)
Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies, vol. 47. Springer, Berlin (1997)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Golgoon, A., Yavari, A. Line and point defects in nonlinear anisotropic solids. Z. Angew. Math. Phys. 69, 81 (2018). https://doi.org/10.1007/s00033-018-0973-2
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-018-0973-2
Keywords
- Transversely isotropic solids
- Orthotropic solids
- Monoclinic solids
- Defects
- Disclinations
- Dislocations
- Nonlinear elasticity