Issue 29

M. Marino et alii, Frattura ed Integrità Strutturale, 29 (2014) 241-250; DOI: 10.3221/IGF-ESIS.29.21 243 where f v is the fiber volume fraction, 1 m f v v   , [ ] I is the identity matrix, [ ] c c ij S s      the compliance matrix for the constituent c , and   [ ] ij A a  is the bridging matrix. In the case of a plane-stress state [ ] S , [ ] c S and [ ] A reduce to 3 x 3 matrices, with [ ] A defined component-wise by [14]: 11 m A f E a E  , 22 0.5 1 m T f E a E          , 33 0.5 1 m AT f G a G          , 21 31 32 0 a a a    (2)     12 12 11 22 12 11 11 f m f m s s a a a s s     , 2 11 1 21 13 11 22 12 21 d d a          , 1 22 2 12 23 11 22 12 21 d d a          , (3) where   1 13 11 33 m d s a a   ,         2 23 11 22 33 13 33 12 m m f m f m d s v v a a a s v v a a      (4) 11 12 12 m f s s    , 12 11 11 m f s s    ,     22 22 12 12 m f f m v v a s s     (5)         21 12 12 12 11 22 22 f m f m m f m v s s a v v a s s       (6) Referring to composite laminated plates comprising mono-directional fiber-reinforced layers, stress analysis is conducted by employing the classical laminate theory [12], where the compliance matrix   k S of each k -th composite layer is obtained from Eq. (1), and it is suitably expressed by passing from the local coordinate system (aligned with the fiber direction) to a global reference one. When necessary, a local stress measure for each layer’s constituent can be recovered starting from the homogenized stress field within the layer and by considering as localization matrices [ ] B and [ ][ ] A B for fiber and matrix, respectively, where the 3 x 3 matrix [ ] A is defined in Eqs. (2-6), and the 3 x 3 matrix [ ] B is defined (in components) as:    11 22 33 / f m f m b v v a v v a     ,     12 12 33 / m f m b v a v v a     , 21 31 32 0 b b b    (8)        13 12 23 22 13 [ / m m f m m b v a v a v v a v a     ,    22 11 33 / f m f m b v v a v v a     (9)     23 23 11 / m f m b v a v v a     ,    33 22 11 / f m f m b v v a v v a    (10) where     11 22 33 f m f m f m v v a v v a v v a      . In detail, referring to an incremental approach, and denoting with k d  the increment of the homogenized stress vector within the k -th layer, the corresponding stress increments in fiber and matrix result in:   f k d B d    ,    m k d A B d    (11) Failure analysis Failure analysis has been based on local criteria. In detail, two approaches differently accounting for micro-mechanical features have been employed to predict degradation of the constituents’ material properties. The first criterion herein considered, provided by Rotem [13], operates on the stress state k  in the k -th laminate layer, whose increment results from   1 k k d S d     , where   k S is computed by Eq. (1) and where d  is the homogenized strain increment in the laminate. This latter is obtained as the solution of the incremental problem and it is assumed to be constant along the laminate thickness. This criterion is based on the basic assumptions that: a) the failure onset is a localized phenomenon, b) only in-plane stresses are effective (that is, interlaminar stresses do not cause failure), c) the matrix material is weaker and softer than the fibers. In agreement with this approach, it is also assumed that micro- buckling does not occur. On the basis of these assumptions the failure criterion actually combines two separate criteria, a failure criterion along the fibers direction and a failure criterion along the tansversal-to-fibers direction. The fibers, being