Shell and Membrane Theories in Mechanics and Biology pp 25-57

Part of the Advanced Structured Materials book series (STRUCTMAT, volume 45) | Cite as

On the Theories of Plates and Shells at the Nanoscale

Chapter

Abstract

During the last 50 years the nanotechnology is established as one of the advanced technologies manipulating matter on an atomic and molecular scale. New materials, devices or other structures possessing at least one dimension sized from 1–100 nm are developed. The question arises how structures composed of nanomaterials should be modeled. In addition, if it is necessary to perform a structural analysis what kind of theory should be used. Two approaches are suggested—theories which take into account quantum mechanical effects since they are important at the quantum-realm scale and theories which are based on the classical continuum mechanics adapted to nanoscale problems. Here the second approach will be discussed in detail. It will be shown that the classical continuum mechanics is enough for a sufficient description of the mechanical behavior of nanomaterials and-structures if surface stresses are taken into account. There are also other approaches in the literature, but they are note discussed in detail in this paper. The basic equations for nanosized plates and shells will be discussed. It is shown that for this class of objects with the help of suggested equations such effects like the size-dependent changes of the stiffness parameters can be described in a proper manner. In contrast to the results for the size-dependence of the Young’s modulus (the Young’s modulus increases when the specimen diameter becomes very thin) the plate stiffness parameters can increase or decrease when the plate thickness is in the range of several nm. Finally, the theory of plates with surface stresses will be compared with the theory of three-layered plates.

References

  1. 1.
    Altenbach, H., Eremeyev, V.: Direct approach-based analysis of plates composed of functionally graded materials. Arch. Appl. Mech. 78, 775–794 (2008a)CrossRefMATHGoogle Scholar
  2. 2.
    Altenbach, H., Eremeyev, V.A.: Analysis of the viscoelastic behavior of plates made of functionally graded materials. ZAMM 88(5), 332–341 (2008b)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011a)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Altenbach, H., Eremeyev, V.A. (eds.): Shell-like structures–non-classical theories and applications. Advanced Structured Materials, pp. 549–560. Springer, Berlin (2011b)Google Scholar
  5. 5.
    Altenbach, H., Eremeyev, V.A.: Cosserat-type shells. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications, pp. 131–178. Springer, Vienna (2013). CISM International Centre for Mechanical Sciences No. 541CrossRefGoogle Scholar
  6. 6.
    Altenbach, H., Morozov, N.F. (eds.): Surface Effects in Solid Mechanics. Springer, Heidelberg (2013)Google Scholar
  7. 7.
    Altenbach, H., Zhilin, P.A.: A general theory of elastic simple shells (in Russ.). Uspekhi Mekhaniki 11(4), 107–148 (1988)MathSciNetGoogle Scholar
  8. 8.
    Altenbach, H., Altenbach, J., Kissing, W.: Mechanics of Composite Structural Elements. Springer, Berlin (2004)CrossRefGoogle Scholar
  9. 9.
    Altenbach, H., Eremeev, V.A., Morozov, N.F.: On equations of the linear theory of shells with surface stresses taken into account. Mech. Solids 45(3), 331–342 (2010a)CrossRefGoogle Scholar
  10. 10.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale. Int. J. Eng. Sci. 59, 83–89 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Mechanical properties of materials considering surface effects. In: IUTAM Symposium on Surface Effects in the Mechanics of Nanomaterials and Heterostructures, pp. 105–115, Springer, Heidelberg (2013)Google Scholar
  12. 12.
    Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells—a short review and bibliography. Arch. Appl. Mech. 80, 73–92 (2010b)CrossRefMATHGoogle Scholar
  13. 13.
    Ambarcumyan, S.A.: Theory of Anisotropic Plates: Strength, Stability, and Vibrations. Hemispher Publishing, Washington (1991)Google Scholar
  14. 14.
    Ashoori Movassagh, A., Mahmoodi, M.: A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eur. J. Mech. A. Solids 40, 50–59 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Buzea, C., Pacheco, I., Robbie, K.: Nanomaterials and nanoparticles: Sources and toxicity. Biointerphases 4(2), MR17–MR71 (2007)Google Scholar
  16. 16.
    Capriz, G.: Continua with Microstructure. Springer, New York (1989)CrossRefMATHGoogle Scholar
  17. 17.
    Cauchy, A.L.: Sur l’équilibre et le mouvement d’une plaque solide. Exercises Mathématiques 3, 328–355 (1828)Google Scholar
  18. 18.
    Challamel, N., Ameur, M.: Out-of-plane buckling of microstructured beams: Gradient elasticity approach. J. Eng. Mechanics 139(8), 1036–1046 (2013)CrossRefGoogle Scholar
  19. 19.
    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statics and Dynamics of Multyfolded Shells. Wydawnictwo IPPT PAN, Warszawa, Nonlinear Theory and Finite Elelement Method (in Polish) (2004)Google Scholar
  20. 20.
    Cosserat, E., Cosserat, F.: Sur la théorie de l’élasticité. Ann Toulouse 10, 1–116 (1886)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Herman, Paris (1909)Google Scholar
  22. 22.
    Donnell, L.H.: Beams, Plates, and Shells. McGraw-Hill, New York (1976)MATHGoogle Scholar
  23. 23.
    Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Advances in Applied Mechanics, pp. 1–68. Elsevier, San Diego (2008)Google Scholar
  24. 24.
    Eremeyev, V.A., Altenbach, H., Morozov, N.F.: The influence of surface tension on the effective stiffness of nanosize plates. Doklady Phy. 54(2), 98–100 (2009)CrossRefMATHGoogle Scholar
  25. 25.
    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer-Briefs in Applied Sciences and Technologies. Springer, Heidelberg (2013). )Google Scholar
  26. 26.
    Eringen, A.C.: Microcontinuum Field Theory I: Foundations and Solids. Springer, New York (1999)CrossRefGoogle Scholar
  27. 27.
    Eringen, A.C.: Microcontinuum Field Theory II: Fluent Media. Springer, New York (2001)Google Scholar
  28. 28.
    Goldenweiser, A.L.: Formulation of approximative theory of shells with the help of the asymptotic integration of the equations of the theory of elasticity (in Russ.). Prikl Mat i Mekh 26(4), 668–686 (1962)Google Scholar
  29. 29.
    Günther, W.: Analoge Systeme von Schalengleichungen. Ing-Arch 30, 160–188 (1961)Google Scholar
  30. 30.
    Grigolyuk, E.I., Kogan, A.F.: Present state of the theory of multilayered shells (in Russ.). Prikl Mekh 8(6), 3–17 (1972)Google Scholar
  31. 31.
    Grigolyuk, E.I., Selezov, I.T.: Nonclassical theories of vibration of beams, plates and shells (in Russ.). In: Itogi nauki i tekhniki, Mekhanika tverdogo deformiruemogo tela, vol 5, VINITI, Moskva (1973)Google Scholar
  32. 32.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Huang, D.W.: Size-dependent response of ultra-thin films with surface effects. Int. J. Solids Struct. 45(2), 568–579 (2008)CrossRefMATHGoogle Scholar
  34. 34.
    Ieşan, D.: Deformation of thin chiral plates in strain gradient elasticity. Euro. J. Mech. A. Solids 44, 212–221 (2014)CrossRefGoogle Scholar
  35. 35.
    Jaiani, G., Podio-Guidugli, P. (eds.): Relations of Shell, Plate, Beam, and 3D Models. Springer, Berlin (2008)MATHGoogle Scholar
  36. 36.
    Javili, A., McBride, A., Steinmann, P.: Thermomechanics of solids with lower-dimensional energetics: On the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Appl. Mech. Rev. 65, 010,802–1-31 (2012)Google Scholar
  37. 37.
    Kafadar, C.B., Eringen, A.C.: Polar field theories. In: Eringen, A.C. (ed.) Continuum Physics. Academic Press, New York (1976)Google Scholar
  38. 38.
    von Kármán, T.: Festigkeitsprobleme im Maschinenbau. In: Encyk. d. Math. Wiss., vol. IV, pp. 311–385, Teubner, Leipzig (1910)Google Scholar
  39. 39.
    Kienzler, R.: Erweiterung der klassichen schalentheorie; der einfluß von dickenverzerrung und querschnittverwölbungen. Ingenieur-Archiv 52, 311–322 (1982)CrossRefMATHGoogle Scholar
  40. 40.
    Kienzler, R.: On consistent plate theories. Arch. Appl. Mech. 72, 229–247 (2002)CrossRefMATHGoogle Scholar
  41. 41.
    Kienzler, R., Altenbach, H., Ott, I. (eds.): Theories of Plates and Shells: Critical Review and New Applications. Springer, Berlin (2004)Google Scholar
  42. 42.
    Kirchhoff, G.R.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Crelles Journal für die reine und angewandte Mathematik 40, 51–88 (1850)CrossRefMATHGoogle Scholar
  43. 43.
    Kraatz, A.: Anwendung der Invariantentheorie zur Berechnung des dreidimensionalen Versagens- und Kriechverhaltens von geschlossenzelligen Schaumstoffen unter Einbeziehung der Mikrostruktur. PhD thesis, Zentrum für Ingenieurwissenschaften, Halle (2007)Google Scholar
  44. 44.
    Krenk, S.: Theories for elastic plates via orthogonal polynomials. Trans. ASME J. Appl. Mech. 48(4), 900–904 (1981)CrossRefMATHGoogle Scholar
  45. 45.
    Lazopoulos, K.: On the gradient strain elasticity theory of plates. Eur. J. Mech. A. Solids 23(5), 843–852 (2004)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Lazopoulos, K.: On bending of strain gradient elastic micro-plates. Mech. Res. Commun. 36(7), 777–783 (2009)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Lazopoulos, K., Lazopoulos, A.: Nonlinear strain gradient elastic thin shallow shells. Eur. J. Mech. A. Solids 30(3), 286–292 (2011)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Lazopoulos, K., Alnefaie, K., Abu-Hamdeh, N., Aifantis, E.: The GRADELA plates and shells. In: Pietraszkiewicz, W., Górski, J. (eds.) Shell Structures: Theory and Applications, vol. 3, pp. 121–124. CRC Press, London (2014)Google Scholar
  49. 49.
    Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New York (2010)CrossRefMATHGoogle Scholar
  50. 50.
    Levinson, M.: An accurate, simple theory of the statics and dynamics of elastic plates. Mech. Res. Comm. 7(6), 343–350 (1980)CrossRefMATHGoogle Scholar
  51. 51.
    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)CrossRefMATHGoogle Scholar
  52. 52.
    Lo, K.H., Christensen, R.M., Wu, E.M.: A high-order theory of plate deformation. Part I: Homogeneous plates. Trans. ASME J. Appl. Mech. 44(4), 663–668 (1977)CrossRefMATHGoogle Scholar
  53. 53.
    Lu, P., He, L.H., Lee, H.P., Lu, C.: Thin plate theory including surface effects. Int. J. Solids Struct. 43(16), 4631–4647 (2006)CrossRefMATHGoogle Scholar
  54. 54.
    Lurie, A.I.: Theory of Elasticity. Foundations of Engineering Mechanics. Springer, Berlin (2005)Google Scholar
  55. 55.
    Meenen, J., Altenbach, H.: A consistent deduction of von kármán-type plate theories from threedimensional non-linear continuum mechanics. Acta Mech. 147, 1–17 (2001)CrossRefMATHGoogle Scholar
  56. 56.
    Mikhasev, G.: On localized modes of free vibrations of single-walled carbon nanotubes embedded in nonhomogeneous elastic medium. ZAMM 94(1–2), 130–141 (2014)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Trans. ASME J. App. Mech. 18, 31–38 (1951)MATHGoogle Scholar
  58. 58.
    Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)CrossRefGoogle Scholar
  59. 59.
    Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109–124 (1968)CrossRefMATHGoogle Scholar
  60. 60.
    Mushtari, K., Galimov, K.: Nonlinear theory of thin elastic shells. NSF-NASA, Washington (1961)Google Scholar
  61. 61.
    Naghdi, P.M.: The theory of shells and plates. In: Flügge, S. (ed.) Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Heidelberg (1972)Google Scholar
  62. 62.
    Nami, M., Janghorban, M.: Resonance behavior of FG rectangular micro/nano plate based on nonlocal elasticity theory and strain gradient theory with one gradient constant. Compos. Struct. 111(1), 349–353 (2014)CrossRefGoogle Scholar
  63. 63.
    Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon, Oxford (1986)MATHGoogle Scholar
  64. 64.
    Nye, J.: Physical Properties of Crystals. Oxford Science Publications, Oxford (2000)Google Scholar
  65. 65.
    Palmov, V.A.: Fundamental equations of the theory of asymmetric elasticity (in Russ.). Prikl. Mat. Mekh. 28(3), 401–408 (1964)MathSciNetGoogle Scholar
  66. 66.
    Paufler, P.: Physikalische Kristallographie. Akademie, Berlin (1986)Google Scholar
  67. 67.
    Pietraszkiewicz, W., Gorski, J. (eds.): Shell Structures: Theory and Application, vol. 3. CRC Press, Boca Raton (2014)Google Scholar
  68. 68.
    Podstrigach, Y.S., Povstenko, Y.Z.: Introduction to mechanics of surface phenomena in deformable solids (in Russ.). Naukova Dumka, Kiev (1985)Google Scholar
  69. 69.
    Povstenko, Y.: Mathematical modeling of phenomena caused by surface stresses in solids. In: Altenbach, H., Morozov, N.F. (eds.) Surface Effects in Solid Mechanics, pp. 135–153. Springer, Heidlberg (2013)CrossRefGoogle Scholar
  70. 70.
    Preußer, G.: Erweiterung der klassichen Schalentheorie; der Einfluß von Dickenverzerrung und Querschnittverwölbungen. Ingenieur-Archiv 54, 51–61 (1981)CrossRefGoogle Scholar
  71. 71.
    Ramezani, S.: A shear deformation micro-plate model based on the most general form of strain gradient elasticity. Int. J. Mech. Sci. 57(1), 34–42 (2012)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Ramezani, S.: Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory. Nonlinear Dyn. 73(3), 1399–1421 (2013)MathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Reddy, J., Srinivasa, A., Arbind, A., Khodabakhshi, P.: On gradient elasticity and discrete peridynamics with applications to beams and plates. Adv. Mater. Res. 745, 145–154 (2013)CrossRefGoogle Scholar
  74. 74.
    Reddy, J.N.: A simple higher-order theory for laminated composite plates. Trans. ASME J. Appl. Mech. 51, 745–752 (1984)CrossRefMATHGoogle Scholar
  75. 75.
    Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2007)Google Scholar
  76. 76.
    Reissner, E.: On the theory of bending of elastic plates. J. Math. Phys. 23, 184–194 (1944)MathSciNetMATHGoogle Scholar
  77. 77.
    Reissner, E.: Reflections on the theory of elastic plates. Appl. Mech. Rev. 38(11), 1453–1464 (1985)CrossRefGoogle Scholar
  78. 78.
    Rothert, H., Zastrau, B.: Herleitung einer Direktortheorie für Kontinua mit lokalen Krümmungseigenschaften. ZAMM 61, 567–581 (1981)CrossRefMATHGoogle Scholar
  79. 79.
    Rubin, M.B.: Cosserat Theories: Shells, Rods and Points, Solid Mechanics and Its Applications, vol. 79. Springer, Berlin (2000)Google Scholar
  80. 80.
    Sahmani, S., Ansari, R.: On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory. Compos. Struct. 95, 430–442 (2013)CrossRefGoogle Scholar
  81. 81.
    Saito, M., Kukula, S., Kataoka, Y., Miyata, T.: Practical use of statistically modified laminate model for injection moldings. Mater. Sci. Eng., A 285(1–2), 280–287 (2000)CrossRefGoogle Scholar
  82. 82.
    Schaefer, H.: Das Cosserat-Kontinuum. ZAMM 47, 485–498 (2006)CrossRefGoogle Scholar
  83. 83.
    Serpilli, M., Krasucki, F., Geymonat, G.: An asymptotic strain gradient Reissner-Mindlin plate model. Meccanica 48(8), 2007–2018 (2013)MathSciNetCrossRefMATHGoogle Scholar
  84. 84.
    Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. R. Soc. Lond. A 455(1982), 437–474 (1999)MathSciNetCrossRefMATHGoogle Scholar
  85. 85.
    Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, New York (1985)Google Scholar
  86. 86.
    Touratier, M.: An effcient standard plate theory. Int. Eng. Sci. 29(8), 901–916 (1991)CrossRefMATHGoogle Scholar
  87. 87.
    Tsiatas, G.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46(13), 2757–2764 (2009)CrossRefMATHGoogle Scholar
  88. 88.
    Vekua, I.N.: Shell Theory: General Methods of Construction. Pitman, Boston (1985)Google Scholar
  89. 89.
    Wang, B., Zhou, S., Zhao, J., Chen, X.: A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory. Eur. J. Mech. A. Solids 30(4), 517–524 (2011)CrossRefGoogle Scholar
  90. 90.
    Wang, J., Duan, H.L., Huang, Z.P., Karihaloo, B.L.: A scaling law for properties of nano-structured materials. Proc. R. Soc. A 462(2069), 1355–1363 (2006)CrossRefMATHGoogle Scholar
  91. 91.
    Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., Wang, T.: Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solida Sinica 24, 52–82 (2011)CrossRefGoogle Scholar
  92. 92.
    Wilson, E.B.: Vector Analysis. Yale University Press, New Haven (1901). (Founded upon the Lectures of J. W. Gibbs)MATHGoogle Scholar
  93. 93.
    Wlassow, W.S.: Allgemeine Schalentheorie und ihre Anwendung in der Technik. Akademie, Berlin (1958)MATHGoogle Scholar
  94. 94.
    Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976)MathSciNetCrossRefGoogle Scholar
  95. 95.
    Zhilin, P.A.: Applied mechanics. Foundations of the theory of shells (in Russ.). St. Petersburg State Polytechnical University, St. Petersburg (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät für MaschinenbauOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
  2. 2.Southern Scientific Center of Russian Academy of Science and Southern Federal UniversityRostov on DonRussia

Personalised recommendations