Abstract
In the framework of the general nonlinear plate theory we consider a buckling problem for an elastic plate with incompatible plane strains generated by continuous distributions of edge dislocations and wedge disclinations as well as other sources of residual stress (non-elastic growth or plasticity). In contrast to the Föppl-von Kármán model the plane strains are not supposed to be small. To explore buckling transition of such kind of structures, the problem is reduced to a system of nonlinear partial differential equations with respect to the transverse deflection of the plate and the embedded metrics coefficients, which naturally leads to the non-trivial plate shapes that are seen even in the absence of any external forces. In the case of very thin plate (membrane) that doesn’t resist bending we present several exact solutions for the axially-symmetric domains.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Seung and Nelson [3] deduced this equation in the inextensional limit.
References
Eshelby, J.D., Stroh, A.N.: Dislocations in thin plates. Phil. Mag. (Ser. 7) 42, 1401–1405 (1951)
Mitchell, L.H., Head, A.K.: The buckling of a dislocated plate. J. Mech. Phys. Solids 9, 131–139 (1961)
Seung, H.S., Nelson, D.R.: Defects in flexible membranes with crystalline order. Phys. Rev. A 38(2), 1005–1018 (1988)
Föppl, A.: Vorlesungen über technische Mechanik. Bd. 5. Teubner, Leipzig (1907)
von Kármán, T.: Festigkeitsprobleme im Maschinenbau / Encyclopädie der Mathematischen Wissenschaften, vol. 4/4C. Teubner, Leipzig (1910)
Stojanovitch, R., Vujoshevitch, L.: Couple stress in non Euclidean continua. Publ. Inst. Math. (Beograd) (N.S.) 2(16), 71–74 (1962)
Stojanovitch, R.: Equilibrium conditions for internal stresses in non-Euclidean continua and stress spaces. Int. J. Eng. Sci. 1(3), 323–327 (1963)
Ben-Abraham, S.I.: Generalized stress and non-Riemannian geometry. In: Simmons, J.A., de Wit, R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory, vol. 2, pp. 943–962. National Bureau of Standards Special Publication 317, Washington (1970)
Clayton, J.D.: On anholonomic deformation, geometry, and differentiation. Math. Mech. Solids 17(7), 702–735 (2012)
Lewicka, M., Mahadevan, L., Pakzad, M.R.: The Föppl-von Kármán equations for plates with incompatible strains. Proc. R. Soc. A 467, 402–426 (2011)
Dervaux, J., Ciarletta, P., Ben Amar, M.: Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Föppl-von Karman limit. J. Mech. Phys. Solids 57, 458–471 (2009)
Kücken, M., Newell, A.C.: A model for fingerprint formation. Europhys. Lett. 68(1), 141–146 (2004)
Chen, S., Chrzan, D.C.: Continuum theory of dislocations and buckling in graphene. Phys. Rev. B 84, 214103 (2011)
Kochetov, E.A., Osipov, V.A., Pincak, R.: Electronic properties of disclinated flexible membrane beyond the inextentional limit: application to graphene. J. Phys.: Condens. Matter 22, 395502 (2010)
Li, K., Yan, S.P., Ni, Y., Liang, H.Y., He, L.H.: Controllable buckling of an elastic disc with actuation strain. Europhys. Lett. (EPL) 92, 16003 (2010)
Qiao, D.-Y., Yuan, W.-Z., Yu, Y.-T., Liang, Q., Ma, Z.-B., Li, X.-Y.: The residual stress-induced buckling of annular thin plates and its application in residual stress measurement of thin films. Sens. Actuators A 143, 409–414 (2008)
Derezin, S.V., Zubov, L.M.: Equations of a nonlinear elastic medium with continuously distributed dislocations and disclinations. Dokl. Phys. 44(6), 391–394 (1999)
Derezin, S.V., Zubov, L.M.: Disclinations in nonlinear elasticity. Z. Angew. Math. Mech. 91(6), 433–442 (2011)
Lurie, A.I.: Nonlinear Theory of Elasticity. North-Holland, Amsterdam (1990)
de Wit, R.: Linear theory of static disclinations. In: Simmons, J.A., de Wit, R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory, vol. 1, pp. 651–673. National Bureau of Standards Special Publication 317, Washington (1970)
Volterra, V.: Sur l’équilibre des corps élastiques multiplement connexes. Ann. Ecole Norm. Super. (Ser. 3) 24, 401–517 (1907)
Kondo, K.: Geometry of Elastic Deformation and Incompatibility: RAAG Memories, vol. 1, Division C. Gakujutsu Bunken Fukyu-kai, Tokyo (1955)
Bilby, B.A., Bullough, R., Smith, E.: Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry. Proc. R. Soc. London A 231, 263–273 (1955)
Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Rat. Mech. Anal. 4, 273–334 (1960)
Le, K., Stumpf, H.: Nonlinear contimuum theory of dislocations. Int. J. Eng. Sci. 34(3), 339–358 (1996)
Steinmann, P.: Views on multiplicative elastoplasticity and the continuum theory of dislocations. Int. J. Eng. Sci. 34(15), 1717–1735 (1996)
Forest, S., Cailletaud, G., Sievert, S.: A Cosserat theory for elastoviscoplastic single crystals at finite deformation. Arch. Mech. 49(4), 705–736 (1997)
Anthony, K.-H.: Die Theorie der Disklinationen. Arch. Rat. Mech. Anal. 39, 43–88 (1970)
Acharya, A., Fressengeas, C.: Coupled phase transformations and plasticity as a field theory of deformation incompatibility. Int. J. Fract. 174, 87–94 (2012)
Norden, A.P.: Affinely Connected Spaces. Nauka, Moscow (1976). (in Russian)
Cartan, E.: Sur les variétés à connexion affine et la théorie de la relativité généralisée. Ann. Sci. École Norm. Super. (Ser. 3) 40, 325–412 (1923)
Mura, T.: Micromechanics of Defects in Solids. Kluwer Academic Publishers, Boston (1987)
Zubov, L.M.: Large deformation of elastic shells with distributed dislocations. Dokl. Phys. 57(6), 254–257 (2012)
Zubov, L.M.: Von Kármán equations for an elastic plate with dislocations and disclinations. Dokl. Phys. 52(1), 67–70 (2007)
Pogorelov, A.V.: Multidimensional Monge-Ampére Equation. Cambridge Scientific Publishers, Cambridge (2008)
Karyakin, M.I.: Equilibrium and stability of a nonlinear-elastic plate with a tapered disclination. Appl. Mech. Tech. Phys. 33(3), 464–470 (1992)
Zubov, L.M., Pham, T.H.: Strong deflections of circular plate with continuously distributed disclinations (in Russian). Izv. VUZov, Sev.-Kav. Reg. Issue 4, 28–33 (2010)
Lazopoulos, K.A.: On the gradient strain elasticity theory of plates. Euro. J. Mech. A. Solids 23, 843–852 (2004)
Altan, B.S., Aifantis, E.C.: On the structure of the mode III crack-tip in gradient elasticity. Scr. Metall. 26, 319–324 (1992)
Acknowledgments
The author thanks Prof. L.M. Zubov for fruitful discussions.
This work was supported by the Russian Foundation for Basic Research (via the grants 12-01-00038 and 12-01-91270) and the Federal target programme “Research and Pedagogical Cadre for Innovative Russia” for 2009–2013 years (state contract N P596).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Derezin, S. (2013). Buckling of Nonlinearly Elastic Plates with Microstructure. In: Altenbach, H., Forest, S., Krivtsov, A. (eds) Generalized Continua as Models for Materials. Advanced Structured Materials, vol 22. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36394-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-36394-8_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36393-1
Online ISBN: 978-3-642-36394-8
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)