Issue 29

S. Terravecchia et al., Frattura ed Integrità Strutturale, 29 (2014) 61-73; DOI: 10.3221/IGF-ESIS.29.07 63 T HE FUNDAMENTAL SOLUTIONS he introduction of eq.(5) in (3a) valid in infinity domain   allows to express the latter in terms of displacements only [7,8] and to determine the fundamental solution of the displacements that for 2D solids proves to be 2 2 2 2 0 2 2 2 2 4 1 = 16 (1 ) 2 2(3 4 ) [ ] 2 uu r K r r r Log r K K r                                                                                      I G r r      (6) where   0 K  and   2 K  are Bessel functions of the second kind and of order 0 and 2, respectively, and   r ξ x is the distance between effect ξ and cause x points. One can note that the singularity of the fundamental solution uu G does not depend on r . Indeed, by performing the limit  r 0 one obtains   1 ( ) 2(3 4 ) ln(2 ) 1 16 (1 ) uu                G ξ x I   (7) where  is the Eulero constant and the fundamental solution for isotropic gradient elasticity shows singularity of type ln( )  . In eq.(6) the limit of uu G for 0   gives the classic isotropic elasticity solution (Kelvin) with singularity of type ln( ) r . Table 1 : Fundamental solutions for strain gradient elasticity and relative singularities. By the fundamental solution uu G in (6), taking in account eqs.(4b,c,d) and 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 provided in Table 1. The fundamental solutions hk G of Tab. 1 are characterized by two subscripts: the first indicates the effect at ξ , i.e. displacement for h u  , traction for h t  , displacement normal derivative for h g  , double traction for h r  , corner displacement for C h u  , corner force for C h t  , stress for h   ; whereas the second subscript indicates – through a T