PSI - Issue 28

Domenico Ammendolea et al. / Procedia Structural Integrity 28 (2020) 1981–1991 Ammendolea et Al./ Structural Integrity Procedia 00 (2019) 000–000

1985

5





1 I I II II K K K K 1 2 2 aux

aux

E 

M



'

(8)

where, E E   for plane stress and  1 1 , I II K K can be evaluated by means of two integration integrals evaluated by superimposing the actual state (1) with a pure mode I (2a) and pure mode II (2b) auxiliary fields, i.e. (1,2 ) a M and (1,2 ) b M . Eq.(8) provides simplified expressions of the actual SIFs as follows:   2 1 E E     for plane strain. The SIFs of the actual problem 



 



 

1,2

1,2

a

b

' E M

' E M





1

2   a

1

2   b

aux K K

aux K K

0

0

K

K





I

II

II

II

I

I

2

2

2

2

a

b

K

K

(9)

I

II

3 Numerical implementation The ALE formulation and the M-integral have been implemented in COMSOL Multiphysics (COMSOL (2018)), a commercially available software, that provides an effective FE environment to model any kind of structural problem (see for instance Lonetti and Pascuzzo (2016), Lonetti et al. (2016), Lonetti and Pascuzzo (2020)). Also, it provides easy tools for expanding standard capabilities. Among these, the LiveLink for MatLab platform permits the integration between COMSOL and MatLab (Lonetti and Pascuzzo (2014), Bruno et al. (2016), Lonetti et al. (2019)). This tool has been used to develop a user-made script that manages the propagation process automatically, thus creating a unified framework between the structural problem, the ALE method, and the M-integral approach. The key steps of the script are reported in Table 1 and a schematic of the propagation process between two consecutive steps is sketched in Fig. 3. The first step consists of setting the geometry and the boundary conditions of the problem together with an initial mesh configuration. Subsequently, the external loads start to increase and, for each increment, the SIFs are evaluated through the M-integral (Eq.s (7)-(9)).

Fig. 2. J-integral in the equivalent domain integral form and a schematic of the arbitrary function q( X 1 ,X 2 )

Table 1. Description of the steps involved in the propagation process START 1. Set the input data of the problem: geometry, material, boundary conditions, and initial mesh configuration 2. Loading process: evaluate SIFs (M-integral); check crack onset conditions and evaluate the direction of crack propagation 3. Propagation process: solve the structural problem for increasing external loads, compute the ALE problem, evaluate SIFs, and direction of crack propagation. Check mesh quality. 4. Check tolerance conditions for the angle variation STOP