Issue 29

S. Terravecchia et al., Frattura ed Integrità Strutturale, 29 (2014) 61-73; DOI: 10.3221/IGF-ESIS.29.07

T HE FUNDAMENTAL SOLUTIONS

T

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

     

      

    

r K  

  

2 2 2 4 

      I

 





   

2

2

r







1

(6)

=

G

uu

 

16 (1 )   

r               

 

2

r       

2

  

 



   

 

[ ] Log r K 





 r r 

2(3 4 ) 

K

2

0

2

2

r

 

  0 K  and

 

where 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

63