Issue 37

R. Sepe et alii, Frattura ed Integrità Strutturale, 37 (2016) 369-381; DOI: 10.3221/IGF-ESIS.37.48 371 according to the relation:   0 3 2 b R e    (1) As we can see, the bigger the initial radius of the particle the larger the cavity will be, for a given strain. In analogy, if X 0 is the initial average distance between the particles, the current value of the average distance, X , can be written as: 2 0 X X e    (2) this equation states that the applied load leads to a reduction of the distance between the particles, which is added to the increasing effect of the cavity dimensions. As a consequence of those assumptions, high-localised stresses are produced into the surrounding material, which can lead to a coalescence phenomenon of near cavities if a critical value of strain (  f ) is reached. This critical value is strictly related to the strain hardening coefficient of the material, n : high values of n mean that the strain peaks are transferred to less strained surrounding zones; consequently, the contraction of the material between the cavities becomes less localized, and the coalescence of the cavities delayed. Once the critical strain is reached, the ratio b/X exhibits a limit value that can be expressed as follows:       1 1 3 0 3 2 2 0 0 4 / 3 2 3 2 3 f f f f cr R b X e e f e e X                 (3) which, rearranging the terms, leads to an equation, which let us evaluate the critical strain:   1 1 3 /2 3 0 4 3 2 3 f f cr b e e f X                   (4) In the previous equation the critical strain, f  is implicitly showed as dependent on the initial volume fraction of the particles, f 0 . Rice and Johonson [20], assuming that the crack tip is located at distance Xo from the nearest cavity with radius R 0 , formulated a bi-dimensional model for the localised strain and the coalescence process of cavities. The crack tip opening displacement, at the conditions for which the coalescence between the growing cavity and the plastically deformed crack tip occurs, represents the CTOD i value at which the crack starts to growth. Intuitively, for a fixed space between the particles, the CTOD i is bigger if the volumetric ratio is smaller. Experimentally, it has been found a relation like CTOD i / X 0 = cost · ε f and, according to some detailed finite element analyses, as well as experimental tests, that constant can be assumed to be function, g , of the square of the hardening coefficient of the material, n [21]. Therefore, the following equation can be considered: 2 0 ( ) i f CTOD X g n     (5) According to this expression, the value of CTOD i could be converted in an equivalent value of fracture toughness K Ic , by means of the well-known relationship: 2 0.5 I c i K CTOD E   (6) where  represents the yielding stress of the material; from the Eqs. (4) and (6) follows: