PSI - Issue 13

Danilo D’ Angela et al. / Procedia Structural Integrity 13 (2018) 939–946 Danilo D’Angela and Marianna Ercolino / Structural Integrity Procedia 00 ( 2018) 000 – 000

941

= 2 ′ ; ′ = { (1− 2 )

3

(1)

Irwin (1957, 1948) extended Griffith’s th eory to metals. He defined the stress intensity factor ( K ) which describes the state of stress close to the crack tip (Stephens and Fuchs, 2001). The formulation of K for the Mode I ( K I ) for flawed plates is shown in Eq. 2, where Y is a geometrical factor. The theoretical relationship between G and K is shown in Eq. 3 ( E’ is defined in Eq. 2). = √ (2) = 2 ′ (3) Paris and Erdogan (1963) investigated the experimental correlation between the fatigue crack growth ( da/dN , i.e., the slope of the crack length a over the cycles N ) and the stress intensity factor range ( ΔK , i.e., the difference between the maximum and the minimum K over the single cycles). They demonstrated that the relationship between da / dN and the ΔK essentially depends on material and environmental conditions; the Paris law was assumed for Stage II (Eq. 4), where c and m are material and environmental constants. = (4) Fatigue fracture tests are often performed by means of compact tension (CT) samples, i.e., notched plates in compliance to the standards (ASTM International, 2015; ISO, 2012). The ideal response related to constant load amplitude testing of a pre-cracked CT (i.e., a 0 is the pre-crack length) is shown in Fig. 2. The fracture stages are essentially defined by ΔK th and K C , which are the upper bound ΔK threshold and the fracture toughness, respectively.

(a) (b) Fig. 2. Idealised response of constant load-amplitude fatigue testing of CT sample: (a) crack length ( a ) vs. number of cycles ( N ) and (b) crack growth rate ( da/dN ) vs. stress intensity factor ( ΔK ). 1.2. eXtended Finite Element Method and Low-Cycle Fatigue approach The eXtended Finite Element Method (XFEM) is an advanced FE modelling technique for fracture mechanics. It overcomes the main issues of traditional FE analysis of fracture (Belytschko and Black, 1999). Existing simulation approaches, such as cohesive behaviour, do not allow to implement arbitrary fracture in solid materials because the potential cracking surface should be known a priori . Moreover, cracking can only affect the pre-defined mesh and nodes (Hedayati and Vahedi, 2014); automatic re-meshing could be a solution but this procedure often causes convergence problems and requires high computational costs (Ashari and Mohammadi, 2010; Hedayati and Vahedi, 2014; Hulton and Cavallaro, 2016). XFEM technology is based on the partition of unity concept; the cracking process is implemented by the enrichment of the finite elements (Kucharski et al., 2016; Melson, 2014). The strength of the XFEM is given by the distance functions (i.e., Ψ and Φ ), which quantitatively characterize the crack surface. Recent works demonstrated the accuracy of XFEM for fracture simulations (Hulton and Cavallaro, 2016; Melson, 2014; Pathak et al., 2013; Singh et al., 2012).