Skip to content
BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access January 13, 2017

Structural symmetry within nonlocal integral elasticity: theoretical issues and computational strategies

  • Aurora Angela Pisano EMAIL logo and Paolo Fuschi


The structural symmetry and the appropriate definition of a reduced (symmetric) mechanical/ numerical model is discussed within a nonlocal elasticity context. In particular, reference is made to an integral model of Eringen-type. The paper highlights how the classical, i.e. local, concepts of structural symmetry have to be rephrased through the definition of an enlarged symmetric model of the analyzed structure. This enlarged model, endowed with apposite nonlocal boundary conditions enforced in an iterative fashion, is proved to be able to recover the nonlocal effects that the neglected portion of the structure exerts on the portion chosen for the analysis. It is shown how the mirrored symmetric solution exactly matches the complete one. Theoretical issues and computational strategies referred to a nonlocal version of the finite element method are discussed with reference to the analysis of a case-study.


[1] Bažant Z.P., Cedolin L. [2010] Stability of structures: elastic, inelastic, fracture and damage theories, World Scientific Publishing Company Ed.10.1142/7828Search in Google Scholar

[2] Eringen A.C. [1999] Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag New York.10.1007/978-1-4612-0555-5Search in Google Scholar

[3] Rogula D. [1982] Introduction to nonlocal theory of material media., In: Nonlocal theory of material media, D. Rogula (Ed.), Springer-Verlag, Berlin. 125-222.10.1007/978-3-7091-2890-9_3Search in Google Scholar

[4] Silling S.A. [2000] “Reformulation of elasticity theory for discontinuities and long-range forces," Journal of Mechanics and Physics of Solids 48, (1) 175-209.10.1016/S0022-5096(99)00029-0Search in Google Scholar

[5] Andrianov I.V., Awrejcewicz J., and Weichert D. [2010] “Review Article. Improved Continuous Models for Discrete Media," Hindawi Publishing Corporation Mathematical Problems in Engineering Article ID 986242.10.1155/2010/986242Search in Google Scholar

[6] Eringen A.C. [2002] Nonlocal continuum field theories, Springer- Verlag New York.Search in Google Scholar

[7] Askes H., Aifantis E.C. [2011] “Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results," International Journal of Solids and Structures 48, 1962-1990.10.1016/j.ijsolstr.2011.03.006Search in Google Scholar

[8] Polizzotto C., Fuschi P. and Pisano A.A. [2006] “A nonhomogeneous nonlocal elasticity model," European Journal of Mechanics A/Solids 25, 308-333.10.1016/j.euromechsol.2005.09.007Search in Google Scholar

[9] Fuschi P., Pisano A.A. and De Domenico D. [2015] “Plane stress problems in nonlocal elasticity: finite element solutions with a strain-difference-based formulation," Journal of Mathematical Analysis and Applications 431, 714-736.10.1016/j.jmaa.2015.06.005Search in Google Scholar

[10] Fuschi P., Pisano A.A. [2016] “Symmetric structures made of a nonlocal elastic material" International Journal of Applied Mechanics 8, 1650052(17 pages).10.1142/S1758825116500526Search in Google Scholar

[11] Borino G., Failla B., Parrinello F. [2003] “A symmetric nonlocal damage theory" International Journal of Solids and Structures 40, 3621-3645.10.1016/S0020-7683(03)00144-6Search in Google Scholar

[12] Fleck N.A., Muller G.M., Ashby M.F. and Hutchinson J.W. [1994] “Strain gradient plasticity: theory and experiments," Acta Metall. Mater. 42, 475-487.10.1016/0956-7151(94)90502-9Search in Google Scholar

[13] Ghosh S., Kumar A., Sundararaghavan V. and Waas A.M. [2013] “Non-local modeling of epoxy using an atomistically-informed kernel," International Journal of Solids and Structures 50, 2837-2845.10.1016/j.ijsolstr.2013.04.025Search in Google Scholar

[14] Allegri G., Scarpa F.L. [2014] “On the asymptotic crack-tip stress field in nonlocal orthotropic elasticity," International Journal of Solids and Structures 51, 504-515.10.1016/j.ijsolstr.2013.10.021Search in Google Scholar

[15] Polizzotto C. [2001] “Nonlocal elasticity and related variational principles," International Journal of Solids and Structures 38, 7359-7380.10.1016/S0020-7683(01)00039-7Search in Google Scholar

Received: 2016-7-25
Accepted: 2016-8-1
Published Online: 2017-1-13
Published in Print: 2017-1-26

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Downloaded on 23.3.2023 from
Scroll Up Arrow