Abstract
The method of Lamé’s strain potential for solving analytically a certain class of problems of classical elasticity is extended here to plane gradient elasticity of the simple Mindlin’s type with just one constant (internal length) in addition to the two classical elastic moduli. According to this method, the strains are expressed as second-order derivatives of a scalar function (Lamé’s strain potential). Thus, combining compatibility, equilibrium and stress–strain equations under plane stress or plane strain conditions and zero body forces, one can prove that this strain potential satisfies an equation of the fourth order instead of the second one for the classical case. General solutions of this equation in Cartesian and polar coordinates are provided. Four examples, two in Cartesian and two in axisymmetric polar coordinates, are presented to illustrate the method and demonstrate its advantages and limitations.
Similar content being viewed by others
References
Mindlin R.D.: Micro-structure in linear elasticity. Arch. Ratio. Mech. Anal. 16, 51–78 (1964)
Eringen C.A.: Linear theory of micropolar elasticity. J. Mathe. Mech. 15, 909–923 (1966)
Toupin R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
Koiter, W.T.: Couple stresses in the theory of elasticity, I & II. In: Proceedings of the Royal Netherlands Academy of Sciences (K. Ned. Akad. Wet.), B67, pp. 17–44 (1964)
Tiersten H.F., Bleustein J.L.: General elastic continua. In: Hermann , G. (eds) R.D. Mindlin and Applied Mechanics, pp. 67–103. Pergamon Press, New York (1974)
Lakes R.: Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Mühlhaus, H.B. (eds) Continuum Models for Materials with Microstructure, pp. 1–25. Wiley, Chichester (1995)
Vardoulakis I., Sulem J.: Bifurcation Analysis in Geomechanics. Chapman and Hall, London (1995)
Exadactylos G.E., Vardoulakis I.: Microstructure in linear elasticity and scale effects: a reconsideration of basic rock mechanics and rock fracture mechanics. Tectonophysics 335, 81–109 (2001)
Altan B.S., Aifantis E.C.: On the structure of the mode-III crack-tip in gradient elasticity. Scripta Metallurgica et Materialia 26, 319–324 (1992)
Ru C.Q., Aifantis E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mechanica 101, 59–68 (1993)
Vardoulakis I., Exadactylos G., Kourkoulis S.K.: Bending of marble with intrinsic length scales: a gradient theory with surface energy and size effects. J. de Physique IV 8, 399–406 (1998)
Tsepoura K.G., Papargyri-Beskou S., Polyzos D., Beskos D.E.: Static and dynamic analysis of gradient elastic bars in tension. Arch. Appl. Mech. 72, 483–497 (2002)
Papargyri-Beskou S., Tsepoura K.G., Polyzos D., Beskos D.E.: Bending and stability analysis of gradient elastic beams. Int. J. Solids Struct. 40, 385–400 (2003)
Yang F., Chong A.C.M., Lam D.C.C., Tong P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)
Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)
Georgiadis H.G.: The mode-III crack problem in microstructured solids governed by dipolar gradient elasticity: static and dynamic analysis. ASME J. Appl. Mech. 70, 517–530 (2003)
Lazar M., Maugin G.A.: Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184 (2005)
Lazar M., Maugin G.A., Aifantis E.C.: Dislocations in second strain gradient elasticity. Int. J. Solids Struct. 43, 1787–1817 (2006)
Vardoulakis I., Giannakopoulos A.E.: An example of double forces taken from structural analysis. Int. J. Solids Struct. 43, 4047–4062 (2006)
Giannakopoulos A.E., Stamoulis K.: Structural analysis of gradient elastic components. Int. J. Solids Struct. 44, 3440–3451 (2007)
Papargyri-Beskou S., Beskos D.E.: Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates. Arch. Appl. Mech. 78, 625–635 (2008)
Papargyri-Beskou S., Beskos D.E.: Stability analysis of gradient elastic circular cylindrical shells. Int. J. Eng. Sci. 47, 1379–1385 (2009)
Papargyri-Beskou S., Beskos D.E.: Static analysis of gradient elastic bars, beams, plates and shells. Open Mech. J. 4, 65–73 (2010)
Georgiadis H.G., Anagnostou D.S.: Problems of Flamant-Boussinesq and Kelvin type in dipolar gradient elasticity. J. Elast. 90, 71–98 (2008)
Gourgiotis P.A., Georgiadis H.G.: Plane-strain crack problems in microstructure solids governed by dipolar gradient elasticity. J. Mech. Phys. Solids 57, 1898–1920 (2009)
Gao X.L., Ma H.M.: Green’s function and Eshelby’s tensor based on a simplified strain gradient elasticity theory. Acta Mechanica 207, 163–181 (2009)
Gao X.L., Ma H.M.: Strain gradient solution for Eshelby’s ellipsoidal inclusion problem. Proc. R. Soc. Lond. A 466, 2425–2446 (2010)
Papargyri-Beskou, S., Charalambakis, N.: Airy’s stress function approach in plane gradient elasticity. In: Papanastasiou, P., et al. (eds.) CD-ROM Proceedings of 9th HSTAM International Congress on Mechanics, Limassol, Cyprus (12–14 July 2010)
Papargyri-Beskou S., Giannakopoulos A.E., Beskos D.E.: Variational analysis of gradient elastic flexural plates under static loading. Int. J. Solids Struct. 47, 2755–2766 (2010)
Aravas N.: Plane strain problems for a class of gradient elasticity models-A stress function approach. J. Elast. 104, 45–70 (2011)
Amanatidou E., Aravas N.: Mixed finite element formulations of strain-gradient elasticity problems. Comput. Methods Appl. Mech. Eng. 191, 1723–1751 (2002)
Giannakopoulos A.E., Amanatidou E., Aravas N.: A reciprocity theorem in linear gradient elasticity and the corresponding Saint–Venant principle. Int. J. Solids Struct. 43, 3875–3894 (2006)
Askes H., Gutierrez M.A.: Implicit gradient elasticity. Int. J. Numer. Methods Eng. 67, 400–416 (2006)
Askes H., Morata I., Aifantis E.C.: Finite element analysis with staggered gradient elasticity. Comput. Struct. 86, 1266–1279 (2008)
Zervos A.: Finite elements for elasticity with microstructure and gradient elasticity. Int. J. Numer. Methods Eng. 73, 564–595 (2008)
Zervos A., Papanicolopulos S.A., Vardoulakis I.: Two finite-element discretizations for gradient elasticity. J. Eng. Mech. ASCE 135, 203–213 (2009)
Markolefas S.I., Tsouvalas D.A., Tsamasfyros G.I.: Mixed finite element formulation for the general anti-plane shear problem, including mode III crack computations, in the framework of dipolar linear gradient elasticity. Comput. Mech. 43, 715–730 (2009)
Tsepoura K.G., Papargyri-Beskou S., Polyzos D.: A boundary element method for solving 3D static gradient elastic problems with surface energy. Comput. Mech. 29, 361–381 (2002)
Polyzos D., Tsepoura K.G., Tsinopoulos S.V., Beskos D.E.: A boundary element method for solving 2-D and 3-D static gradient elastic problems, part I: integral formulation. Comput. Methods Appl. Mech. Eng. 192, 2845–2873 (2003)
Tsepoura K.G., Tsinopoulos S.V., Polyzos D., Beskos D.E.: A boundary element method for solving 2-D and 3-D static gradient elastic problems, part II: numerical implementation. Comput. Methods Appl. Mech. Eng. 192, 2875–2907 (2003)
Karlis G., Tsinopoulos S.V., Polyzos D., Beskos D.E.: Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity. Comput. Methods Appl. Mech. Eng. 196, 5092–5103 (2007)
Karlis G.F., Charalambopoulos A., Polyzos D.: An advanced boundary element method for solving 2D and 3D static problems in Mindlin’s strain-gradient theory of elasticity. Int. J. Numer. Methods Eng. 83, 1407–1427 (2010)
Barber J.R.: Elasticity. 2nd edn. Kluwer, Dordrecht (2002)
Timoshenko S.P., Goodier J.N.: Theory of Elasticity. 3rd edn. McGraw-Hill, New York (1970)
Mathematica: Version 4. Wolfram Research Inc., Champaign (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Papargyri-Beskou, S., Tsinopoulos, S. Lamé’s strain potential method for plane gradient elasticity problems. Arch Appl Mech 85, 1399–1419 (2015). https://doi.org/10.1007/s00419-014-0964-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-014-0964-5