Issue 37

R. Sepe et alii, Frattura ed Integrità Strutturale, 37 (2016) 369-381; DOI: 10.3221/IGF-ESIS.37.48 372 1 1 6 6 0 0 0 4 ( ) 2 ( ) 2 3 Ic f f K g n E X g n E R f                 (7) The above equation states that in the limit state preceding the coalescence of voids and, therefore, the crack growth, the fracture toughness K Ic of the material depends on the power (–1/6) of the initial volumetric void fraction, f 0 , and on the basic material parameters ( E ,  , n ) as well as on the initial dimension of the particles, R 0 . Based on an equivalent approach, the GT model provides a constitutive relation for the increasing of the volumetric void fraction up to the starting of the coalescence phenomenon. The coalescence phenomenon is not accounted for in the model, as the basic assumption of the model theory is the homogeneity of the strain field, and therefore it does not allow reliable prediction in the case of localised strain phenomena. This limit of the Gurson model can be overcome if a critical value of the cavities volume fraction is introduced; the coalescence phenomenon starts when the actual volume fraction of cavities reaches this critical value, as explained below. Fortunately, it was possible to find a strong dependency of this critical volume fraction, f c , from its initial value f 0 and therefore from the cavity radius, R 0 , and from the distance from the crack tip, X 0 . The original Gurson model considers a rigid sphere of perfectly plastic material, encapsulating a spherical cavity under a homogenous strain. The derived model that can also consider material hardening, called the Gurson-Tvergaard model [10], can follow the growth of the volume fraction of existing cavities from f 0 to f c and can be written as:     2 2 2 1 3 2 3 , , , 2 cosh 1 0 2 m m q f q f q f                    (8) where q 1 , q 2 and q 3 are correction parameters introduced by Tvergaard [9] (in order to take into account the hardening behaviour of material), σ m is the average normal stress, σ is the equivalent Von Mises stress; with such model it is possible to predict the cavity growth ratio in the plastic field. For real components, where many inclusions are to be found, the GT model considers that the volume fraction of voids increases over an increment of load because of both the continuing growth of existing voids and the nucleation of new voids; this is taken into account by increasing the void growth rate linked to the actual plastic strain flow at crack tip, (1 ) p growth df f d    , by a quantity, ( ) nucleation df A d    , where: 2 1 ( ) exp 2 2 N N N N f A S S                      (9) Therefore, the introduced nucleation parameters are:  f N : particles volume fraction from which the nucleation of cavities can take place;   N and S N : mean value and standard deviation of the supposedly normal distribution of the critical strain value at which the nucleation takes place. As it is possible to observe, the nucleation acceleration, driven by the parameter A , is limited to an internal strain condition in which the actual equivalent strain  is quite close to N  . Anyway, the nucleation phenomenology is of complex understanding and, even though some results from metallographic analyses about particle distribution and concentration are available, it is not sure if voids nucleate from all of them; this implies that the particles total volume fraction obtained by the metallographic analyses cannot be directly used as parameter in the GT model [22]. Within this work, the values of the physical parameters of the GT model related to the nucleation mechanism were calibrated by adapting numerical and available experimental results by means of an iterative process that is described in the following pages. This crack growth mechanism leads to a numerical model characterised by cubic elements parallel to the crack plane; all those elements, which are equal in dimensions, exhibit material properties according to the GT model. Each element represents an elementary cell of material containing an initial cavity volume fraction, f o ; the cell dimension, D o , is comparable with the distance between the largest inclusions detected by microscopic analyses at the interior of the bulk material. According to the GT model, the volumetric fraction of the cavities increases with the stress-strain state; because