Abstract
Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with the configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we showthat two dimensional defectswith trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in a Euclidean space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aharoni, H., Abraham, Y., Elbaum, R., Sharon, E., Kupferman, R.: Emergence of spontaneous twist and curvature in non-Euclidean rods: application to erodium plant cells. Phys. Rev. Lett. 108, 238, 106 (2012)
Armon S., Efrati E., Sharon E., Kupferman R.: Geometry and mechanics of chiral pod opening.. Science 333, 1726–1730 (2011)
Benzécri J.P.: Sur les variétés localement affines et localement projectives. Bull. de la S.M.F. 88, 229–332 (1960)
Bilby B., Bullough R., Smith E.: Continuous distributions of dislocations: a new application of the methods of Non-Riemannian geometry. Proc. R. Soc. A 231, 263–273 (1955)
Bilby B., Smith E.: Continuous distributions of dislocations. III. Proc. R. Soc. Edin. A 236, 481–505 (1956)
Dawson C., Vincent J., Rocca A.M.: How pine cones open. Nature 390, 668 (1997)
Derezin S., Zubov L.: Disclinations in nonlinear elasticity. Z. Angew. Math. Mech. 91, 433–442 (2011)
Dervaux, J., Ben Amar, M.: Morphogenesis of growing soft tissues. Phys. Rev. Lett. 101, 068,101 (2008)
Do Carmo, M.: Riemannian Geometry. Birkhauser, Boston, 1992
Farb, B., Margalit, D.: A primer on mapping class groups. Princeton University Press, Princeton, 2011
Forterre Y., Skotheim J., Dumais J., Mahadevan L.: How the Venus flytrap snaps. Nature 433, 421–425 (2005)
Fried D., Goldman W., Hirsch M.: Affine manifolds with nilpotent holonomy. Comment. Math. Helvetici 56, 487–523 (1981)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge, 2002
Katanaev M., Volovich I.: Theory of defects in solid and three-dimensional gravity. Ann. Phys. 216, 1–28 (1992)
Klein Y., Efrati E., Sharon E.: Shaping of elastic sheets by prescription of non-Euclidean metrics.. Science 315, 1116–1120 (2007)
Kondo, K.: Geometry of elastic deformation and incompatibility. Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, Vol. 1 (Eds. K. Kondo) pp. 5–17, 1955
Kroner, E.: The physics of defects. Les Houches Summer School Proceedings (Eds. R. Balian, M. Kleman, J.P. Poirier). North-Holland, Amsterdam, 1981
Kroner E.: The internal mechanical state of solids with defects. Int. J. Solids Struct. 29, 1849–1852 (1992)
Liang, H., Mahadevan, L.: The shape of a long leaf. Proc. Natl. Acad. Sci. USA (2009)
Marder M., Papanicolaou N.: Geometry and elasticity of strips and flowers. J. Stat. Phys. 125, 1069–1092 (2006)
Miri M., Rivier N.: Continuum elasticity with topological defects, including dislocations and extra matter.. J. Phys. A Math. Gen. 35, 1727–1739 (2002)
Noll W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Rational Mech. Anal. 2, 197–226 (1958)
Ozakin A., Yavari A.: Affine development of closed curves in Weitzenböck manifolds and the burgers vector of dislocation mechanics. Math. Mech. Solids 19, 299–307 (2014)
Romanov A.: Mechanics and physics of disclinations in solids.. Europ. J. Mech. A/Solids 22, 727–741 (2003)
Seung H., Nelson D.: Defects in flexible membranes with crystalline order. Phys. Rev. A 38, 1005–1018 (1988)
Volterra, V.: Sur l’équilibre des corps élastiques multiplement connexes. Ann. Sci. Ecole Norm. Sup. Paris 1907 24, 401–518 (1907)
Wang C.C.: On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch. Rational Mech. Anal. 27, 33–93 (1967)
Wu, Z., Moshe, M., Greener, J., Therien-Aubin, H., Nie, Z., Sharon, E., Kumacheva, E.: Three-dimensional shape transformations of hydrogel sheets induced by small-scale modulation of internal stresses. Nat. Commun. 4, 1586 (2013). doi:10.1038/ncomms2549
Yavari A., Goriely A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Rational Mech. Anal. 205, 59–118 (2012)
Yavari A., Goriely A.: Weyl geometry and the nonlinear mechanics of distributed point defects. Proc. R. Soc. A 468, 3902–3922 (2012)
Yavari A., Goriely A.: Riemann–Cartan geometry of nonlinear disclination mechanics. Math. Mech. Solids 18, 91–102 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Ortiz
Rights and permissions
About this article
Cite this article
Kupferman, R., Moshe, M. & Solomon, J.P. Metric Description of Singular Defects in Isotropic Materials. Arch Rational Mech Anal 216, 1009–1047 (2015). https://doi.org/10.1007/s00205-014-0825-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-014-0825-y