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H. Askes et alii, Frattura ed Integrità Strutturale, 25 (2013) 87-93; DOI: 10.3221/IGF-ESIS.25.13 88 F UNDAMENTALS OF G RADIENT E LASTICITY ack in 1964, Mindlin published the first paper [2] addressing in a systematic way gradient enriched elasticity, where a novel continuum theory containing a number of additional constitutive parameters was proposed. Amongst the different attempts made so far to simplify the above approach by reducing the number of material parameters, certainly the model devised by Aifantis and co-workers [3-5] deserves to be mentioned explicitly. In more detail, such a model postulates that the enriched stress-strain relationship can be extended with the Laplacian of the strain as follows:   2 2 C        (1) where  and  are the stresses and strains, respectively, C is a fourth-order tensor containing the elastic moduli, and l is a length scale parameter that represents the underlying microstructure. The most interesting feature of the above formulation of gradient elasticity is that it can efficiently be implemented numerically and then used, according to the procedure summarised below, to address problems of practical interest [6-8]. In particular, initially the following standard equation has to be solved: Ku f  (2) where K is the conventional linear elastic stiffness matrix, u is the vector containing the nodal displacements, and, finally, f is the vector summarising the externally applied nodal forces. Once the displacements are obtained from Eq. (2), they can be used to determine the gradient-enriched nodal stresses  from T T 2 T N N N SN S d N B d u                         (3) In the above relation, N is the matrix summarising the shape functions used to interpolate the stresses, B is the matrix containing the derivatives of the displacement shape functions, and, finally, S is the elastic compliance matrix. To conclude, it is worth recalling here that the most important implication of directly incorporating material length scale l into the stress analysis is that, even in the presence of cracks and sharp notches, the resulting stress fields are not singular in spite of the fact that the material being designed is forced to obey a linear-elastic constitutive law [9]. T HEORY OF C RITICAL D ISTANCES AND STATIC ASSESSMENT n the presence of cracks subjected to Mode I static loading, the TCD postulates that fast fracture takes place when a critical distance based effective stress,  eff , exceeds the material inherent strength,  0 [10], i.e.: eff 0     non-propagation of the crack (4) Further, as far as the static assessment is concerned, independently from the ductility level of the material being design, the stress analysis can directly be performed by adopting a simple linear-elastic constitutive law [11-13], provided that the material inherent strength  0 is determined consistently [10-13]. The appropriate way of experimentally determining  0 will be reviewed below briefly. The above considerations should make it evident that designing cracked materials against static loading according to the TCD implies performing a bi-parametrical post-processing of the linear-elastic stress fields acting on the material in the vicinity of the crack tips, the critical distance and the inherent material strength being the two adopted design parameters. When specifically dealing with static failures, the TCD’s critical distance is recommended to be determined from: 2 Ic 0 K1 L          (5) where K Ic is the LEFM plane strain fracture toughness. The TCD’s effective stress,  eff , can then be calculated according to either the Point Method, the Line Method, or the Area Method as follows [1, 14]: B I