Strain gradient elasticity within the symmetric BEM formulation 
    Terravecchia S.1,a* , Panzeca T2,b  and Polizzotto C3,c
       1,2,3 DICAM, Viale delle Scienze, 90128 Palermo, Italy
a silvioterravecchia@gmail.com, b teotista.panzeca@unipa.it, c castrenze.polizzotto@unipa.it
Keywords: Strain gradient elasticity, Symmetric Galerkin BEM. 
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 moment tractions. The fundamental solutions - featuring a fourth order partial differential equations (PDEs) system - exhibit singularities which in 2D may be of the order . 
This research communication shows a part of the results being elaborated within a more general paper [1].
After 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, 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 functions 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]. 
The symmetric formulation is motivated by the high efficiency achieved  within classical elasticity by the method in [9] with regard to the techniques used to eliminate the singularities of the fundamental solutions, the evaluation of the coefficients of the solving system and the computational procedures characterized by great implementation simplicity. This has already led to the birth of the computer code Karnak sGbem [10] operating in the classical elasticity.
The objective of this paper is to experiment new techniques and procedures that, applied in the context of strain gradient elastic materials, may permit one to obtain the related solving system.
The class of strain gradient elastic materials herein considered is featured by the following strain elastic energy
 	(1)
where  is the internal length,  is the classical strain and is the classic isotropic elasticity tensor
Eq.(1) provides the "primitive" stresses
	(2)
The constitutive equation relating the total stress  to the strain is 
	(3)
where  is the classical strain.
In [7,8] the fundamental solution of the displacements for 2D solids is obtained through which, using the well-known procedure given in [11], it is possible by exploiting the known properties of symmetry of the fundamental solutions to derive the entire tableau of the fundamental solutions.
Computational techniques have been pursued in order to eliminate the singularities of order  , ,  in the blocks of the coefficients related to the corners of the solid and new techniques based on the rigid motion strategy have been introduced  in order to test the coefficients of the blocks of the solving system. The displacement and internal deformation fields were obtained.
Numerical techniques in order to remove the singularity of higher order like   and   are in advanced study.
References.
[1]	C. Polizzotto, T. Panzeca, S. Terravecchia, A symmetric Galerkin BEM formulation for a class of gradient elastic materials of Mindlin type. Part I: Theory, (2014) in preparation.
[2]	R.D.Mindlin, Second gradient of strain and surface tension in linear elasticity, Int. J. Solids Struct. 1 (1965) 417-438.
[3]	R.D. Mindlin, N.N. Eshel, On first strain-gradient theories in linear elasticity, Int. J. Solids Struct. 28 (1968) 845-858.
[4]	C.H. Wu, Cohesive elasticity and surface phenomena, Quart. Appl. Math. L(1) (1992) 73-103.
[5]	E.C. Aifantis, On the role of gradients in the localization of deformation and fracture,. Int. J. Eng. Sci. 30 (1992) 1279-1299.
[6]	H. Askes, E.C. Aifantis, Gradient elasticity in statics and dynamics:  An overview of formulations, length scale identification procedures, finite  element implementations and new results,  Int. J. Solids Struct. 48 (2011) 1962-1990.
[7]	D. Polyzos, K.G. Tsepoura, S.V. Tsinopoulos, D.E. Beskos, A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part.I: integral formulation, Comput. Meth. Appl. Mech. Engng. 192 (2003) 2845-2873. 
[8]	G. F. Karlis,  A. Charalambopoulos, D. Polyzos, An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain gradient theory of elasticity, Int. J. Numer. Meth. Engng. 83 (2010) 1407-1427. 
[9]	T. Panzeca, F. Cucco, S. Terravecchia, Symmetric Boundary Element Method versus Finite Element Method, Comp. Meth. Appl. Mech. Engrg. 191 (2002) 3347-3367.
[10]	F. Cucco, T. Panzeca, S. Terravecchia, The program Karnak.sGbem Release 2.1, Palermo University (2002).
[11]	C. Polizzotto, An energy approach to the boundary element method. Part.I: elastic solids, Comput. Meth. Appl. Mech. Engng. 69 (1988) 167-184.