Issue 29

C. Maruccio et alii, Frattura ed Integrità Strutturale, 29 (2014) 49-60; DOI: 10.3221/IGF-ESIS.29.06 54 In eq. 14 the membrane strain components   and the change of curvature   read: , , 0, 0, 1 ( ) 2           ψ ψ ψ ψ (15) 0 0 , , , , 0, , 0, , 1 ( ) 2 2 2                   h h ψ d ψ d ψ g ψ g (16) where comma indicates partial derivation, Greek indices take the values 1, 2; 0 , ψ ψ are respectively the current and initial position vectors of the shell middle surface, g is the initial shell director and 0 h is the initial shell thickness. Moreover we have introduced the quantity: 0 1 2 1 2 ( , ) ( , ) 2       h d a (17) where 1 2 ,   are the natural coordinates of the shell middle surface,   1 2 0 ,     h h is the thickness stretch, h is the current shell thickness and a is the current normal. Furthermore in eq.14, the shear strain components  take the form: 0 , 0, 2               h ψ d ψ g (18) and the electric components  E read:        E (19) where  is the electric potential. Finally 0 1 33 33 ,   are the constant and linear components of the thickness strain and 0 1 3 3 , E E represent the constant and linear parts of the electric field along the thickness direction. We now introduce the transformation matrix A between the generalized Green Lagrange strain vector of the solid g E and the generalized strain column vector of the shell s E such as  g s E AE with: 3 3 3 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                                   A (20) With some algebra and after integration on the shell thickness the final constitutive equation of the homogenized shell can be recast in the following form:  macro shell s L D E (21) with