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\psz9\linex0\headery709\footery284\colsx709\pgbrdropt32\sectdefaultcl\sectrsid2429172\sftnbj {\rtlch\fcs1 \af0 \ltrch\fcs0 \cs40\cf17\insrsid7941742\charrsid7941742 . \par }\pard\plain \ltrpar\s19\qj \li0\ri0\widctlpar\wrapdefault\faauto\adjustright\rin0\lin0\itap0\pararsid7941742 \rtlch\fcs1 \ab\af0\afs24\alang1025 \ltrch\fcs0 \b\fs24\lang1033\langfe1033\cgrid\langnp1033\langfenp1033 {\rtlch\fcs1 \af0 \ltrch\fcs0 \b0\insrsid6254288 This }{\rtlch\fcs1 \af0 \ltrch\fcs0 \b0\insrsid7941742\charrsid7941742 research communication }{\rtlch\fcs1 \af0 \ltrch\fcs0 \b0\insrsid6254288 shows a part of the results}{\rtlch\fcs1 \af0 \ltrch\fcs0 \b0\insrsid7941742\charrsid7941742 being elaborated within a more}{\rtlch\fcs1 \af0 \ltrch\fcs0 \b0\insrsid6254288 general paper [1]}{\rtlch\fcs1 \af0 \ltrch\fcs0 \b0\insrsid7941742\charrsid7941742 . \par }\pard\plain \ltrpar\s20\qj \li0\ri0\widctlpar\wrapdefault\faauto\adjustright\rin0\lin0\itap0\pararsid5447388 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0 \fs24\lang1033\langfe1033\cgrid\langnp1033\langfenp1033 {\rtlch\fcs1 \af0 \ltrch\fcs0 \cs40\cf17\insrsid5447388\charrsid5447388 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. \par 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 }{\rtlch\fcs1 \af0 \ltrch\fcs0 \cs40\insrsid5447388\charrsid5447388 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 a l. [6] for historical details about the latter formulations and its applications). \par }\pard\plain \ltrpar\s21\qj \li0\ri0\widctlpar\wrapdefault\faauto\adjustright\rin0\lin0\itap0\pararsid5447388 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0 \fs24\lang1033\langfe1033\cgrid\langnp1033\langfenp1033 {\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid5447388\charrsid5447388 Formulations in the boundary element method based on the strain gradient elasticity were pioneered by }{\rtlch\fcs1 \af0 \ltrch\fcs0 \cs40\cf17\insrsid5447388\charrsid5447388 Polyzos et al. [7], Karlis et al. [8] who provide a collocation BEM form ulation 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.}{\rtlch\fcs1 \af0 \ltrch\fcs0 \cs40\insrsid5447388\charrsid5447388 \par }\pard\plain \ltrpar\s20\qj \li0\ri0\widctlpar\wrapdefault\faauto\adjustright\rin0\lin0\itap0\pararsid5447388 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0 \fs24\lang1033\langfe1033\cgrid\langnp1033\langfenp1033 {\rtlch\fcs1 \af0 \ltrch\fcs0 \cs40\cf17\insrsid5447388\charrsid5447388 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]. \par The symmetric formulation is motivated by the high efficiency achieved within c lassical 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 imple mentation simplicity. This has already led to the birth of the computer code Karnak sGbem [10] operating in the classical elasticity.}{\rtlch\fcs1 \af0 \ltrch\fcs0 \cf17\insrsid5447388\charrsid5447388 \par }{\rtlch\fcs1 \af0 \ltrch\fcs0 \cs40\cf17\insrsid5447388\charrsid5447388 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.}{\rtlch\fcs1 \af0 \ltrch\fcs0 \cs40\cf17\insrsid5447388 \par }\pard \ltrpar\s20\qj \li0\ri0\widctlpar\wrapdefault\faauto\adjustright\rin0\lin0\itap0\pararsid16123085 {\rtlch\fcs1 \af0\afs22 \ltrch\fcs0 \fs22\insrsid16123085 The class of strain gradient elastic materials herein considered is featured by the following strain elastic energy \par }\pard \ltrpar\s20\qj \li0\ri0\sb120\sa120\widctlpar\tqr\tx9525\wrapdefault\faauto\adjustright\rin0\lin0\itap0\pararsid16123085 {\pard\plain \ltrpar\s20\qj \li0\ri0\sb120\sa120\widctlpar 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\lang1033\langfe1033\langnp1033\langfenp1033\insrsid5447388\charrsid2826952 \hich\af0\dbch\af0\loch\f0 [1]\tab}}\pard\plain \ltrpar\qj \fi-454\li454\ri0\sa120\sl288\slmult0\widctlpar \jclisttab\tx454\wrapdefault\aspalpha\aspnum\faauto\ls2\adjustright\rin0\lin454\itap0\pararsid13374019 \rtlch\fcs1 \af0\afs20\alang1025 \ltrch\fcs0 \fs20\lang1031\langfe1033\cgrid\langnp1031\langfenp1033 {\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid2826952 C. Polizzotto, T. Panzeca, S. Terravecchia, }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid5447388 A symmetric Galerkin BEM formulation for a class of gradient elastic materials }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388 of Mindlin type. Part I: Theory, (2014) i}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid5447388 n preparation.}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid5447388 \hich\af0\dbch\af0\loch\f0 [2]\tab}}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388 R.D.}{ \rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 Mindlin, Second gradient of strain and surface tension in linear elasticity}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388 ,}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 Int. J. Solids Struct}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388 .}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 1}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388 (1965) }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 417-438. \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid5447388\charrsid5447388 \hich\af0\dbch\af0\loch\f0 [3]\tab}R.}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388 D.}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 Mindlin}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388 , N.N. }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 Eshel, On first strain-gradien}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388 t theories in linear elasticity, }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 Int. J. Solids Struct}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \i\fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 .}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 28}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \b\fs24\lang1033\langfe1033\langnp1033\insrsid5447388 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 (1968}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388 ) }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 845-858. \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid5447388\charrsid5447388 \hich\af0\dbch\af0\loch\f0 [4]\tab} C.H. Wu, Cohesive elasticity and surface phenomena, Quart. Appl. Math. L(1) (1992) 73-103. \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid5447388\charrsid5447388 \hich\af0\dbch\af0\loch\f0 [5]\tab}E.C. Aifantis, On the role of gradients in the localization of deformation and fra cture,. Int. J. Eng. Sci. 30 (1992) 1279-1299. \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid5447388\charrsid5447388 \hich\af0\dbch\af0\loch\f0 [6]\tab}H. Askes, E.C. Aifantis, }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1040\langnp1033\langfenp1040\insrsid5447388\charrsid5447388 Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, }{ \rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 Int. J. Solids Struct. 48 (2011) 1962-1990. \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid5447388\charrsid5447388 \hich\af0\dbch\af0\loch\f0 [7]\tab} 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. \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid5447388\charrsid5447388 \hich\af0\dbch\af0\loch\f0 [8]\tab} 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, }{\rtlch\fcs1 \ai\af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1040\langnp1033\langfenp1040\insrsid5447388\charrsid5447388 Int. J. Numer. Meth. Engng.}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1040\langnp1033\langfenp1040\insrsid5447388\charrsid5447388 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid5447388 83 (2010) 1407-1427. \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid13374019\charrsid2826952 \hich\af0\dbch\af0\loch\f0 [9]\tab}}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid13374019\charrsid2826952 T. }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid2826952 Panzeca, }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid13374019\charrsid2826952 F. }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid2826952 Cucco, }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid13374019\charrsid2826952 S. }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid2826952 Terravecchia}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid13374019\charrsid2826952 ,}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid2826952 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid13374019 Symmetric Boundary Element Method versus Finite Element Method}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid13374019 ,}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid13374019 Comp. Meth. Appl. Mech. Engrg.}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid13374019 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid13374019 191}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid13374019 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019 (2002)}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid13374019 3347-3367.}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid13374019 \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1040\langfe1033\langfenp1033\insrsid13374019 \hich\af0\dbch\af0\loch\f0 [10]\tab}}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019 F.}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019\charrsid5447388 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid5447388\charrsid5447388 Cucco}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019 ,}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid5447388\charrsid5447388 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019 T.}{ \rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019\charrsid5447388 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid5447388\charrsid5447388 Panzeca}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019 ,}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid5447388\charrsid5447388 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019\charrsid5447388 S. }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid5447388\charrsid5447388 Terravecchia}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019 ,}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid5447388\charrsid5447388 }{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid5447388 The program Karnak.sGbem Release 2.1}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid13374019 ,}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid5447388 Palermo University}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid13374019 }{ \rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019\charrsid5447388 (2002}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1040\langfe1033\langnp1040\insrsid13374019 )}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\insrsid5447388\charrsid5447388 .}{\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid5447388\charrsid11276684 \par {\listtext\pard\plain\ltrpar \rtlch\fcs1 \af0 \ltrch\fcs0 \lang1033\langfe1033\langnp1033\langfenp1033\insrsid11276684\charrsid11276684 \hich\af0\dbch\af0\loch\f0 [11]\tab}}\pard \ltrpar\qj \fi-454\li454\ri0\sa120\sl288\slmult0\widctlpar \jclisttab\tx454\wrapdefault\aspalpha\aspnum\faauto\ls2\adjustright\rin0\lin454\itap0\pararsid11276684 {\rtlch\fcs1 \af0\afs24 \ltrch\fcs0 \fs24\lang1033\langfe1033\langnp1033\insrsid11276684\charrsid11276684 C. Polizzotto, An energy approach to the boundary element method. Part.I: elastic solids, Comput. Meth. Appl. Mech. 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