Finite element solutions for nonhomogeneous nonlocal elastic problems Aurora A.Pisano * , Alba Sofi,Paolo Fuschi Dipartimento DASTEC, Università Mediterranea di Reggio Calabria, via Melissari,89124 Reggio Calabria, Italy a r t i c l e i n f o Article history: Received 21 January 2009 Received in revised form 13 May 2009 Available online 24 June 2009 Dedicated to Prof.Castrenze Polizzotto on the occasion of his 85th birthday. Keywords: Nonhomogeneous nonlocal elasticity Nonlocal finite elements 2D mechanical problems a b s t r a c t A finite element procedure for analysing nonhomogeneous nonlocal elastic 2D problems is presented and discussed. The procedure grounds on a variationally consistent approach known, in the relevant literature,as NonlocalFinite Element Method. The latter is recast making use of a recently theorized phenomenological strain-difference-based nonhomogene- ous nonlocal elastic model. The peculiarities of the numerical procedure together with the pertinent nonlocal operators are expounded and discussed. Two simple numerical 2D examples close the paper. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction The key idea of the nonlocal elastic approaches resides on the concept that within a nonlocal elastic medium the particle influence one another not simply by contact forces and heat diffusion (as it occurs in local materials), but also by long range cohesive forces which imply some long distance energy interchanges (see e.g. Kröner, 1967; Edelen, 1976). Such assumptio allows one to overcome within the context of continuum theories the inability of the local theory to describe physical phe- nomena affected by events arising at microstructure or, even,at atomic level. The singular stress field predicted at a sharp crack-tip in a continuum fracture mechanics problem is a typical example. A very extensive list of contributions is traceable in the relevant literature, see e.g. Eringen and Kim (1974) or Zhou and Wang (2005) just to quote one of the early contribu- tion and one of the more recent one. An effective nonlocal continuum approach for solving problems involving (spontaneou formation of discontinuities, so including fracture mechanics problems, is the one known as peridynamic model proposed by Silling (2000); see also the recent contributions of Silling et al. (2003) and Emmrich and Weckner (2007). The application of nonlocal elastic continuum approaches to nanomaterials, where size effects often become prominent and have to be ac- counted for, is another current example. The list of contributions is also in this case quite extensive, starting with the work of Peddieson et al.(2003) who developed a nonlocal Euler–Bernulli beam model, among others,some of the more recent papers are those of Zhang et al. (2005); Ece and Aydogdu (2007); Heireche et al. (2008) addressing problems of vibration, buckling and wave propagation in carbon nanotubes on the base of a nonlocal Timoshenko beam theory. Variational prin- ciples for multi-walled carbon nanotubes have recently been presented by Adali (2008) who assumed a continuum model- ling which takes into account small scale effects via the nonlocal theory of elasticity. Till the very recent and remarkable contribution of Aifantis (2009), related to gradient elasticity, but showing that continuum elasticity can indeed describe a variety of problems at micro/nano regime if long range or nonlocal material point interactions and surface effects are taken into account.The list of the above quoted references is far to be exhaustive; to this concern reference can be made to the 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.06.003 * Corresponding author.Tel.: +39 965 3223141; fax: +39 965 21158. E-mail address: aurora.pisano@unirc.it (A.A. Pisano). Mechanics Research Communications 36 (2009) 755–761 Contents lists available at ScienceDirect Mechanics Research Communications j o u r n a l h o me pa ge : w w w . e l s e v i e r . c o m / l o c a t e / m e c h r e s c o m