Issue 29

S. Terravecchia et al., Frattura ed Integrità Strutturale, 29(2014) 61-73; DOI: 10.3221/IGF-ESIS.29.07 64 work-conjugate rule – the cause applied at x , i.e. a unit concentrated force for k u  , a unit surface relative displacement for k t  , a unit concentrated double force for k g  , a unit higher order surface distortion for k r  , a unit corner force for C k u  , a unit corner displacement for C k t  , a unit imposed strain for k   . To consider the way in which the fundamental solutions included in the single    and double      brackets to see [2]. While the fundamental solutions represent the response in a point ξ of the unlimited domain caused by unit kinematical or mechanical action imposed at x , the response to the distributed actions is provided by the SIs that in [1] are obtained by a generalization of the Betti theorem for the gradient elastic materials. A CASE STUDY s a premise, it should be noted that the theory of strain gradient should be applied to those bodies where the ratio between surface and volume in 3D and between boundary and area in 2D is high. In these cases, the body should be considered as composed of a nucleus, a crust and its boundary. The example we want to meet will show that on the basis of this theory the behavior of the cortex has different characteristics from the classical behavior of the nucleus, and this depends on the presence of new variables such as double traction r and normal derivative g of the displacements. Obviously, the body will suffer the effects due to the presence of these new variables. From computational point of view, the fundamental solutions present in the Tab. 1 show various orders of singularity, up to 4 1/ r in the column related to  u , up to 3 1/ r in that related to C  u ; furthermore the singularities present in the column C  u present an order lower than that of  u , relatively to the same effect. In order to investigate the techniques useful to remove the singularities from the blocks of coefficients, a simple application in two-dimensional space is shown, but at this stage involving the study of fundamental solutions having the maximum order of singularity equal to 2 1/ r . It has to be remembered that in the symmetric BEM the coefficients are obtained by a double integration, the first (inner) regarding the modeling of the cause through appropriate shape functions and the second (external) regarding the effect weighing through shape functions, but dual in the energetic sense. x y 1 1 1 1 a b c d e f g 1 2 3 4 5 6 7 8 x y h a) b) Figure 1 : Plate: a) geometry and constraints; b) discretization into boundary elements and linear modelling of the boundary quantities. For this purpose, the example shown concerns a plate completely constrained on the boundary, subjected to the simple distortions  u , C  u and distortions of higher order / n    g u . This example is also used to develop the techniques of rigid motion that could be used in other applications to compute the coefficient blocks in which the techniques available are not sufficient to suppress the residual singularities. For the plate of Fig.1a the boundary conditions are  u u on u   g g on g  with u g    (8a,b) Let us proceed to write the SIs on the boundary A