Issue34

R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 59-68; DOI: 10.3221/IGF-ESIS.34.05 63 plane with an interface crack [23, 24]. According to the above assumption, the debonded zone corresponds to a 3D cylindrical crack lying between two different materials [21] (Fig. 1b). Typically, the fibre-matrix stress interaction is analysed through the well-known shear lag model [25, 26], even if such an approach presents some limitations due to the inability of examining a complex stress state around the fibre. The generic fibre, embedded in an elastic matrix (Fig 1b) under remote axial ( z   ) and radial ( r   ) stresses, is characterised by an energetically equivalent SIF (along the circular crack front) defined as follows [27]: 2 2 ( ) ( ) ( ) 0 ( ) 0 I r II r II z r i II z r K K K K K                           (7) The generic dimensionless Mode M SIF ( , M I II  ), due to the remote stress w   ( , w r z  ), can be defined as * ( ) / Mw M w w K K l       with ( ) Mw M w K K    . All the above SIFs , ( ), ( ), ( ) i I r II r II z K K K K       are independent of the angular co-ordinate  due to the axial symmetry of the examined configuration. An application of the above fracture mechanics problem concerns the dimensionless Mode II SIFs, * IIz K and * IIr K , due to remote longitudinal ( z   ) and radial ( r   ) stresses. In Fig. 4, such quantities are represented for different values of the material Young modulus ratio, f m E E , and two values of the relative fibre detached length, / f l L   . As can be noted, the SIF is an increasing function of the modulus ratio, while its value decreases by increasing the aspect ratio of the fibre, 2 f f L   , i.e. the SIF is much more severe for short fibre-matrix detached length. The remote axial stress approximately produces only a Mode II SIF, whereas the remote radial positive stress is mainly responsible for both Mode I and mainly Mode II SIFs. The last cited stress-intensity factor arises due to the different elastic properties of the two joined materials. In Fig. 5, the dimensionless Mode I SIF due to a remote radial stress is displayed against the relative debonded length (Fig. 5a) and against the fibre aspect ratio (Fig. 5b). Such a SIF decreases by increasing the detached fibre length and by increasing the fibre aspect ratio, i.e. it is lower for longer fibres in the case of a given relative detached length. The equivalent SIF at the fibre-matrix interface crack front is suitable to define the condition of unstable crack propagation (leading to a complete fibre-matrix separation), according to the energy-based Griffith approach: 2 plane stress plane strain 1 i i ic i i E K K E            ic ic G G (8) where ic G is the critical interface fracture energy, ic K is the corresponding fracture toughness, and i E and i  are the Young modulus and the Poisson ratio at the interface, respectively. Multiaxial fibre-matrix interface damage under cyclic loading The fracture mechanics approach to examine the fibre detachment has the benefit to allow us the use of the well-known crack growth approach to fatigue. As a matter of fact, if the total fatigue life is assumed to be the sum of the initiation and the propagation number of loading cycles, and the propagation stage prevails over the first one as in pre-cracked components or in presence of high stress concentration effects, the crack growth evaluation can lead to a proper expected fatigue life. In the present case, the pre-existing crack can be considered to be always present due to the unavoidable fibre-matrix detachment at the fibre extremities. Moreover, irrespective of the stress state in the material surrounding the fibre, the crack path is always well defined, since it corresponds to the outer fibre surface. The crack propagation assessment can be performed through the debonding length rate law, or the crack growth velocity cg v quantified with respect to the number of loading cycles, evaluated through standard power laws such as the classical Paris law: