Issue 29

C. Maruccio et alii, Frattura ed Integrità Strutturale, 29 (2014) 49-60; DOI: 10.3221/IGF-ESIS.29.06 53         4 3 1 2 4 1 4 3 1 2 4 1 1 4 2 3 2 1 1 4 2 3 2 1                             u u u u u u u u (11) where p u and  p , p = 1, 2, 4 are the position vectors and the electric potential of the corner nodes 1, 2, and 4 in the undeformed state and ( 4 3 1 2 1 4 2 3 , , ,     u u u u ), ( 4 3 1 2 1 4 2 3 , , ,         ) are position vectors and electric potential for every respective pair of nodes on the top–bottom and right–left boundaries of the RVE. Once the RVE problem is solved, the macroscopic material properties are determined from the homogenization procedure and the final constitutive equations read:                        T C e T S D E e  (12) where the macroscopic (overall, effective) mechanical moduli C , piezoelectric moduli e , and dielectric moduli   , are introduced. The overall material behaviour resulting from the homogenization procedure is nonlinear due to the electromechanical frictionless contact conditions at the microscale. Therefore, the coefficients in Eq. (12) are to be considered secant values determined for a given load increment. For the next steps we define the constitutive matrix macro solid D and the generalized Green Lagrange strain vector g E of the homogenized solid as:            macro solid T C e D e         g S E E (13) Within this approach only constitutive equations at the RVE scale are required. M ACROSCALE FORMULATION ased on the analysis of the shell kinematics, see Fig.3, the Green-Lagrange strains and the electric field components in convective coordinates can be arranged in a generalized strain column vector: 0 1 0 1 11 22 12 11 22 12 1 2 1 2 33 33 3 3 , , 2 , , , 2 , , , , , , , ,                T s E E E E E (14) Figure 3 : Shell kinematics. B