Issue 29

S. Terravecchia et al., Frattura ed Integrità Strutturale, 29 (2014) 61-73; DOI: 10.3221/IGF-ESIS.29.07 61 Focussed on: Computational Mechanics and Mechanics of Materials in Italy Strain gradient elasticity within the symmetric BEM formulation S. Terravecchia, T. Panzeca, C. Polizzotto University of Palermo silvioterravecchia@gmail.com , teotista.panzeca@unipa.it , castrenze.polizzotto@unipa.it A BSTRACT . The symmetric Galerkin Boundary Element Method is used to address a class of strain gradient elastic materials featured by a free energy function of the (classical) strain and of its (first) gradient. With respect to the classical elasticity, additional response variables intervene, such as the normal derivative of the displacements on the boundary, and the work-coniugate double tractions. The fundamental solutions - featuring a fourth order partial differential equations (PDEs) system - exhibit singularities which in 2D may be of the order 4 1/ r . New techniques are developed, which allow the elimination of most of the latter singularities. The present paper has to be intended as a research communication wherein some results, being elaborated within a more general paper [1], are reported. K EYWORDS . Strain gradient elasticity; Symmetric Galerkin BEM. I NTRODUCTION fter the pioneering work of Mindlin [2], theories of strain gradient elasticity have become very popular, particularly within the domain of nano-technologies, that is, for problems where the ratio surface/volume tends to become very large and there is a need to introduce at least one internal length. However the model introduced by Mindlin and then improved by Mindlin et al. [3] and Wu [4] leads to an excessive number of material coefficients, which at the best for isotropic materials reduce to the number of seven. In the early 1990’s, Aifantis [5] introduced a signified material model of strain gradient elasticity which requires only three material coefficients, that is, the Lame' constants and one length scale parameter. The latter model was then developed further following the so-called Form II format given by Mindlin et al. [3], that is a theory centered on the existence of a free energy function of the (classical) strain and of its first gradient, which leads to the generation of symmetric stress fields (see Askes et al. [6] for historical details about the latter formulations and its applications). Formulations in the boundary element method based on the strain gradient elasticity were pioneered by Polyzos et al. [7], Karlis et al. [8], who provide a collocation BEM formulation where the simplified constitutive equation by Aifantis [5], has been adopted. In latter papers only the fundamental solutions used in the collocation approach to BEM are provided. In the case of the symmetric formulation of the BEM, Somigliana Identities (SIs) for the tractions and for the double tractions are also needed. These new SIs are necessary in order to get, through the process of modeling and weighing, a solving equation system having symmetric operators. In Polizzotto et al. [1] all the set of fundamental solutions is derived starting from the displacement fundamental solution given in [7,8]. A