Issue 29

M.L. De Bellis et alii, Frattura ed Integrità Strutturale, 29 (2014) 37-48; DOI: 10.3221/IGF-ESIS.29.05 39 = E LU (2) where the compatibility operator is defined as 1 2 2 1 2 1 1 2 0 0 0 0 0 = . 2 0 0 0 0                                                      X X X X X X X X L (3) According to the strain driven approach, the macroscopic strain components, evaluated at , X are used as input variables for the microscopic level. The kinematic map, expressed in function of the vector , E is imposed on the UC, properly defining a BVP. At the micro-level a repetitive rectangular UC is selected, whose size is 1 2 2 2 a a  and its centre is located at the macroscopic point , X characterized by the displacement field 1 2 ={ , } T u u u , defined at each point   1 2 = , T x x x of the UC domain  . The displacement field, resulting in the UC after solving the BVP, can be represented as the superposition of the assigned field    u X, x and a perturbation field    : u X, x        =   u X, x u X, x u X, x (5) The strain vector at the microscopic level is derived by applying the kinematic operator defined for the 2D Cauchy problem and, in expanded form, it results as:   1 ,1 2 ,2 12 ,2 ,1 0 = , with and = 0                             ε lu ε l x (6) with , i  indicating the partial derivative with respect to . i x According to Eq. (5), the strain can be written as:       , = , ,   ε X x ε X x ε X x  (7) The third order polynomial map, proposed in [5 ,6] and modified in [7], is used. Different material symmetries can be considered ranging between the isotropic and the orthotropic case. In the considered orthotropic case, the kinematic map can be written in compact form as:       , =  u X x A x E X (8) with           2 2 2 3 1 2 1 1 2 2 2 1 12 1 3 1 1 2 1 2 2 2 2 3 2 1 1 1 12 2 2 1 2 3 2 1 2 2 1 1 1 0 3 2 2 1 1 0 3 2 2 x x x x x x s b x x c x x x x x x x s b x x c x                             A x (9)