Issue34

R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 59-68; DOI: 10.3221/IGF-ESIS.34.05 65 0 ( ) , c i N m c c i i l N l C K dN      such that ( , , ) i c z r ic K l K       (10) ( ) ( ) / 0 i c D N l N l   where the interface equivalent SIF is indicated as ( , , ) i z r K l      since it depends on the remote stress field and on the current debonded length for the given composite material. The current detached length can be used to quantify a debonding-related damage i D which is assumed to be equal to the ratio between the current debonded length l and the critical length c l corresponding to the condition of unstable crack propagation, that is, to the condition of complete fibre detachment from the matrix material for which the damage is complete, i.e. ( ) 1 i c D N  . On the other hand, it has been shown that ( , , ) i z r K l     is a decreasing function of l [27], i.e. the SIF decreases as the detached length increases, and the critical condition cannot be reached during crack propagation. The fibre-matrix debonding-related damage i D can also be defined as follows: 0 / 1 i f D l L    (11) and measures the effectiveness of the fibre in the bearing mechanism of the composite material. On the other hand, the detachment phenomenon could synthetically be quantified also through a so-called sliding scalar function ( ) m f s  [26] (and the interface damage can thus be measured as follows: 1 ( ) m i f D s    ). The sliding scalar function can approximately be estimated as follows: ( ) ( ) / m f f f s L l L    . By means of the current debonded fibre length determined above, the sliding function parameter ( ) / m m f f f s ε    (given by the ratio of the fibre strain to matrix strain measured in the fibre direction) can be evaluated, and the tangent elastic tensor ' eq C of the homogenized material can be obtained [27]: ( ) ' ' ' ( ) ( ) ( ) m f m m eq m f f f m f ds ε E s ε p p d dε                              C C F F (12) where ,   are the fibre and matrix volume fractions, respectively; ' , ' m f E  C are the tangent elastic tensor of the matrix material and the fibre tangent elastic modulus, respectively; ( ) p   and ( ) p   are the probability distribution functions describing the fibres arrangement in 3D space; F is the second-order tensor defined as follows:   F k k , where k is the unit vector identifying the fibre axis [27]. Such a homogenization procedure is carried out as the fibre progressively detaches due to fatigue loading. N UMERICAL SIMULATIONS ow the fatigue behaviour of a 13% glass fibre-reinforced polyamide specimen (with fibres oriented parallel, i.e. with 0º   , or inclined by an angle 30º   with respect to the load direction) under constant amplitude uniaxial cyclic stress is examined [28]. The materials constituting the specimen are characterised by the following mechanical properties: matrix Young modulus 2.2 m E GPa  , Poisson’s ratio 0.4 m   , fibres Young modulus 72.45 f E GPa  , Poisson’s ratio 0.23 f   , fibre diameter equal to 10 f m    and length 4 2 5.5 10 f L m    . The Paris constants of the interface are 9 8.7 10 i C    and 13.9 i m  ( / dl dN in mm/cycle, i K  in MPa m ), i.e. those of the matrix material, whereas the Wöhler constants are 0 10 MPa   , 6 0 2 10 N   (fatigue limit and corresponding conventional number of loading cycles), 0.133 B  . N