Issue34

R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 59-68; DOI: 10.3221/IGF-ESIS.34.05 61 In the present paper, the damage taking place in the matrix and at the fibre-matrix interface is assumed to be non interacting, i.e. each of them can be assessed independently of the other. Moreover, the effect of the cyclic loading on the fibres is completely neglected. Finally, as far as the matrix damage is concerned, the hypothesis of no crack formation and propagation in the bulk material is adopted, since the matrix is assumed to be ductile. The multiaxial fatigue strength determination in the case of biaxial tension-torsion cyclic stress state is typically tackled by empirical approaches. In particular, for normal and shear cyclic stresses (with amplitudes a  and a  , respectively), a suitable relationship to identify the fatigue limit is [13]: 2 2 , 1 , 1 ( / ) ( / ) 1 a af a af         (1) that can be determined once different values of biaxiality ratio, / a a r    , and phase shift angles have been fixed. Typically the stress amplitudes a  , a  are stresses measured in a particular plane, called critical plane. In the case of a reinforced material, the relevant cyclic stress amplitudes for the material interface can be assumed to be equal to , r a  and , z a  , i.e. the radial and the axial one. In Fig. 2a, the effective radial and axial (in phase) cyclic stresses amplitudes against time are shown. As can be noted, they are evaluated by neglecting the compressive portion of the cyclic stress diagrams, having assumed no damage when the material is compressed. Equation (1) can be rewritten as follows (Fig.1a, b): 2 2 , , 1 , , 1 ( / ) ( / ) 1 z a af r a af         (2) which represents a circle in the , , z a r a    plane (Fig. 3a) related to a given number of loading cycles to failure, f N ( f N is the number of loading cycles to failure for an uniaxial fatigue stress with the amplitude , z a  or , r a  ). Then, the normalized amplitudes are defined as follows: , , 0, , , 0, ' / , ' / z a z a ref r a r a ref         (3) with 0, ref  = reference remote applied stress amplitude. The fatigue domain (2) can be rewritten through the above dimensionless stress amplitudes: 2 2 , 0, , 0, , 1 , 1 ' ' 1 z a ref r a ref af af                             (4) Such a biaxial fatigue stress domain can conveniently be applied to assess the damage degradation of the fibre-matrix interface layer, as is discussed later. In Fig. 2b, the radial and axial stresses for a generic fibre are represented in the case of a reinforced body under remote uniaxial cyclic stress. A multiaxial fatigue-related parameter can be defined as follows: , ( ) / ( ) f n f i D N N N S  (5) where N is the current number of cycles characterising the stress state A and , ( ) f n N A is the number of cycles to failure under the uniaxial cyclic stress with amplitude 2 2 1/2 , , ( ) , 1, 2, 3,.... n r a z a S n      , while its dimensionless amplitude is 2 2 1/2 , , , 1 1 ' ( ) , 1, 2, 3,.... n r a z a af S n        (Fig. 3a). The Wöhler curve (Fig. 3b) is generally a continuously decreasing function of the number of loading cycles, since a well defined horizontal asymptote cannot easily be identified from experiments for many materials. In this context, the so-called fatigue limit loses its original meaning because experimental results classified as ‘run out’ simply represent the achievement of a number of cycles beyond that of interest for the applications being considered.