Open Access. © 2018 A. A. Pisano and P. Fuschi, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License Curved and Layer. Struct. 2018; 5:136ś145 Research Article Aurora Angela Pisano* and Paolo Fuschi Stress evaluation in displacement-based 2D nonlocal fnite element method https://doi.org/10.1515/cls-2018-0010 Received Apr 06, 2018; accepted Apr 09, 2018 Abstract: The evaluation of the stress feld within a non- local version of the displacement-based fnite element method is addressed. With the aid of two numerical ex- amples it is shown as some spurious oscillations of the computed nonlocal stresses arise at sections (or zones) of macroscopic inhomogeneity of the examined structures. It is also shown how the above drawback, which renders the stress numerical solution unreliable, can be viewed as the so-called locking in FEM, a subject debated in the early seventies. It is proved that a well known remedy for lock- ing, i.e. the reduced integration technique, can be success- fully applied also in the nonlocal elasticity context. Keywords: Nonlocal displacement-based fnite element method; 8-nodes Serendipity quadrilaterals; Nonlocal stress locking; Reduced integration technique 1 Premises and motivations Nonlocal approaches in the feld of continuum mechan- ics are nowadays widely used in a number of engineer- ing applications (see e.g. [1ś3]). The common feature of such approaches relies on the need to describe at a macro- scopic level phenomena arising at a microscopic level, i.e. within the microstructure of the constituent material, that, indeed, play a signifcant role in the right description of the material/structural behavior. Typical examples of such cir- cumstance can be traced back to the early seventies when dealing with the description of the stresses at a tip of a crack (see e.g. [4]) till more recent problems facing the analysis of structural elements made of materials contain- ing nano-particles [5]. Wave dispersion, strain softening, *Corresponding Author: Aurora Angela Pisano: Department PAU - via Melissari, I-89124 Reggio Calabria, Italy, University Mediter- ranea of Reggio Calabria; Email: aurora.pisano@unirc.it Paolo Fuschi: Department PAU - via Melissari, I-89124 Reggio Cal- abria, Italy, University Mediterranea of Reggio Calabria concomitant size efects are, among others, well known ex- amples (see e.g. [6ś8] and references therein). It is also well known that classical continuum me- chanics fails in describing the above problems for the ab- sence in the constitutive laws of any information coming from the microstructure. Indeed, the above referred non- local continuum approaches succeed in this challenge by introducing in the classical modeling an internal length material scale while keeping the hypothesis of continu- ity. There are diferent constitutive hypotheses that give rise to diferent nonlocal continuum theories and an huge amount of approaches and related numerical techniques proposed to face real problems (see e.g. [6] and [2] just to quote two milestone works). The list of quotable contribu- tions is very long and out of the scope of the present pa- per. The nonlocal approach hereafter referred is the one known as nonlocal integral approach of Eringen and co- workers (see e.g. [9, 10]). It is applied within the elastic- ity feld and in a shape proposed by the authors in [5, 11ś 14] and there named strain-diference-based nonlocal inte- gral elasticity model. The stress is expressed as the sum of two contributions: a local standard one and a nonlo- cal one given in integral form and in terms of an averaged strain diference feld. A nonlocal version of the fnite ele- ment method, named NL-FEM, was also proposed in the above quoted papers on the base of a nonlocal total po- tential energy functional whose optimality conditions yield the governing equations of the pertinent boundary-value- problem governed, besides the nonlocal stress-strain law, by standard equilibrium and compatibility conditions. The details of such formulation are given in the quoted paper and will be only briefy summarized hereafter. The main goal of the present paper is however strictly related to the already implemented NL-FEM and, as declared in the ti- tle, it concerns the stress evaluation within a displacement- based nonlocal formulation of such method. The right evaluation of (nonlocal) stresses is, with no doubts, of utmost importance within a nonlocal numerical formulation. Stress based criteria to face fracture mechan- ics problems might be consistently rephrased for example, or nonlocal limit analysis for structural elements made of nonlocal materials, such as metal matrix nanocomposites, might be felds of possible fruitful application of the NL-