Abstract
The strain gradient elasticity is enriched with fractional derivatives of the strain, contributing for a more accurate description of the non-local stress–strain response, introducing a better description of the influence of microstructure (local and non-local), since in some respects the fractal texture of the material is also introduced. Using that theory, the uniaxial tension problem is discussed. The one-dimensional variational problem is presented including not only the strain but also the strain gradient and fractional derivatives of the strain and the strain gradient. The solution to that uniaxial problem is found by the finite element numerical method, suitably revisited for the fractional strain gradient elastic problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agrawal O.P.: A general finite element formulation for fractional variational problems. J. Math. Anal. Appl. 337, 1–12 (2008)
Aifantis E.C.: Update on a class of gradient theories. Mech. Mater. 35, 259–280 (2003)
Aifantis E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)
Altan, B.S., Aifantis, E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater 8, 231–282 (1997) [see also: Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 3]
Atanackovic T.M., Stankovic B.: Stability of an elastic rod on a fractional derivative of foundation. J. Sound Vib. 227, 149–161 (2004)
Atanackovic T.M., Spacic D.T.: On the impact of elastic bodies with adhesive forces. Mechanica 34, 367–377 (1999)
Carpinteri A., Cornetti P.: A fractional calculus approach to the description of stress and strain localization on fractal media. Chaos, solitons fractals 13, 85–94 (2002)
Carpinteri A., Sapora A.: Diffusion problems in fractal media defined on Cantor sets. ZAMM 90(3), 203–210 (2010)
Das S.: Functional fractional calculus for system identification and controls. Springer, Berlin (2007)
Fleck N.A., Hutchinson J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)
Gorenflo R., Mainardi F.: Fractional calculus: integral and differential equations of fractional order. Springer, Wien (1997)
Hilfer R.: Applications of fractional calculus in physics. World Scientific, Singapore (2000)
Kilbas, A.A., Srivastava, H.M., Trujillo, I.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204 (2006)
Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P.: Experiments and theory in gradient strain elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)
Lazar M., Maugin G.A.: A note on line forces in gradient elasticity. Mech. Res. Commun. 33, 674–680 (2006)
Lazopoulos K.A., Lazopoulos A.K.: Bending and buckling of thin strain gradient elastic beams. Eur. J. Mech.-A/Solids 29(5), 837–843 (2010)
Lazopoulos K.A.: Nonlocal continuum mechanics and fractional calculus. Mech. Res. Commun. 33, 753–757 (2006)
Liang Y.S., Su W.Y.: The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus. Chaos, solitons fractals 34, 682–692 (2007)
Liang Y.S., Su W.Y.: Connection between the order of fractional calculus and fractional dimensions of a type of fractional functions. Anal. Theory Appl. 23(4), 354–362 (2007)
Liouville J.: Sur le calcul des differentielles a indices quelconques. J. Ic. Polytech. 44 13, 71 (1832)
Magin R.L.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)
Mindlin R.D.: Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Sruct. 1, 417–438 (1965)
Oldham K.B., Spanier J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York and London (1974)
Papargyri-Beskou S., Beskos D.E.: Response of gradient-viscoelastic bar to static and dynamic axial load. Acta Mech. 170, 199–212 (2004)
Peterson L.J., Rajfur Z., Maddox A.S., Freel C.D., Chen Y., Edlund M., Otey C., Burridge K.: Simultaneous stretching and contraction of stress fibers in vivo. Mol. Biol. Cell 15(7), 3497–3508 (2004)
Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198 (1999)
Podlubny I.: Fractional Differential Equations (An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications). Academic Press, San Diego-Boston-New York-London-Tokyo-Toronto (1999)
Riemann B.: Versuch einer allgemeinen Auffassung der Integration and Differentiation. Gesammelte Werke 62, 331–334 (1876)
Ru C.Q., Aifantis E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993)
Sabatier J., Agrawal O.P., Tenreiro Machado J.A.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Berlin (2007)
Samko S., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London (1993)
Stankovic B., Atanackovic T.M.: On a model of a viscoelastic rod. Fract. Calculus Appl. Anal. 4, 501–522 (2001)
Tsepoura K.G., Papargyri-Beskou S., Polyzos D., Beskos D.E.: Static and dynamic analysis of a gradient-elastic bar in tension. Arch. Appl. Mech. 72, 483–497 (2002)
Truesdell C.: A First Course in Rational Continuum Mechanics vol. 1. Academic Press, New York, San Francisco, London (1977)
Vardoulakis I.G., Sulem J.: Bifurcation Analysis in Geomechanics. Blackie/Chapman and Hall, London (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lazopoulos, K.A., Lazopoulos, A.K. Fractional derivatives and strain gradient elasticity. Acta Mech 227, 823–835 (2016). https://doi.org/10.1007/s00707-015-1489-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-015-1489-x