Issue 29

S. Terravecchia et al., Frattura ed Integrità Strutturale, 29(2014) 61-73; DOI: 10.3221/IGF-ESIS.29.07 62 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. N OTATION s a rule, a compact notation is used, with vectors and tensors denoted by bold-face symbols. The dot  , the colon products : and :: indicate the simple and higher index contraction. The following notations are also utilized: the gradient operator: grad ( ) ( )     ; the divergence operator : div ( ) ( )      ; the Laplacian operator : 2 2 2 ( ) ( ) ( ) ( ) x y               ; the component of the normal derivation of u is denoted as n k k n     g u u ; the quantities with bar superscripts indicate know quantities. Other symbols will be specified in the text at their first appearance. B ASIC RELATIONS IN 2D he class of strain gradient elastic materials herein considered is featured by the following strain elastic energy 2 1 : : :: ( ) 2 2 T W         ε E ε E ε ε  (1) that is a function of the symmetric 2nd order strains (0) ( ) S    ε ε u and of the strain gradient (1)   ε ε , where  is the internal length which relates the microstructure with macrostructure. Eq.(1) provides the stress tensors work-coniugate of (0) ε and (1) ε respectively (0) (1) 2 2 (0) : ;    σ E ε σ σ  (2a,b) where E is the classic isotropic elasticity tensor. By the principle of the virtual work, the indefinite equilibrium equation and the following boundary conditions prove to be in Ω     σ b 0 , on Γ  t t , on Γ  r r , on the corner C C  t t (3a,b,c,d) where the total stresses σ , the tractions t , the double tractions r and the force at the corner P are so defined (0) 2 2 (0)    σ σ σ  , ( ) (1) ( ) ( ) s K        t n σ n n σ , (1) :  r nn σ , (1) : C  t sn σ       (4a,b,c,d) In eq.(4b), the symbol ( ) ( ) s     I nn denotes the tangent gradient on the boundary and ( ) s K    n is the curvature on the boundary having normal n ; in eq.(4d) s denotes a unit vector tangent to the two boundary portions convergent in the corner C and the brackets    indicate that the enclosed quantity is the difference between the related values taken on two sides of the corner C. For a more exhaustive understanding of the jump    to see [2]. The constitutive equation relating the total stress σ to the strain is 2 2 )    σ E ε ε  : ( . (5) A T