PSI - Issue 41

Fabrizio Greco et al. / Procedia Structural Integrity 41 (2022) 576–588 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Fig. 4. (a) A pre-cracked plated under tension affected by a dynamically growing crack; (b) Computational meshes used in numerical simulations

The second mesh ( i.e. , mesh M2) is entirely coarse with 175 triangular elements, while the third mesh ( i.e. , mesh M3) is a refined arrangement made up of 1113 triangular elements. Freund (Freund (1998)) has analyzed this problem analytically, developing the following function describing the time history of the mode-I Dynamic Stress Intensity Factors (DSIFs):



     c



1 2

c t t



2

1 

a c 

0 



  0,

d

(15)

K t



r

  r

I

1

1 2 

a c





where a  is the crack tip velocity and t c is a critical time defined as the ratio between the half-height of the plate and the dilatational wave speed ( i.e. t c =H/c d ). Noting that, Eq. (15) is the solution of the fracture problem of an infinite plate with a semi-infinite crack under a remote loading. Because the geometry is finite, this solution is valid only until the reflected stress waves at the boundaries arrive at the crack front. In other words, it is possible to compare the numerical and analytical solutions in an interval of time between 0< t <3 t c . In addition to the analytical solution described by Eq. (15), several numerical results are available in the literature. Among these, Chen et al. (Chen et al. (2019)) have investigated the fracture behavior of the plate through the 0 K H   ) as a function of normalized time t/t c obtained through the proposed method (using the mesh configurations illustrated in Fig. 4-b), the analytical solution described by Eq.(15), and numerical previsions reported in (Chen et al. (2019)). The results denote that the proposed method agrees reasonably well with both analytical and numerical solutions, regardless of the computational mesh utilized in numerical analyses. This latter aspect underlines that the proposed strategy ensures accurate results also by using coarse mesh sets. In particular, the proposed method requires fine discretization only around the crack tip region where the M -integral method is applied. The Mesh M1 serves as a reference for two additional parametric studies. The first study focused attention on the influence of the shape of the q ( 1 2 , c c x x ) function adopted in the M -integral method. More precisely, two sets of suitable q ( 1 2 , c c x x ) functions are considered, i.e. , a plateau and a pyramid-shaped function whose geometric details are summarized in Table 1. Fig. 6-a reports the normalized Mode-I DSIFs achieved by using plateau functions of variable upper base size, while Fig. 6-b shows the results achieved through pyramid-shaped functions of different bottom base size. Singular Edge-based Smoothed Finite Element Method (SE-FEM). Fig. 5-a compares the normalized Mode-I DSIFs K I /K 0 ( i.e. 0