PSI - Issue 42

Roman Vodička et al. / Procedia Structural Integrity 42 (2022) 927 – 934

931

R. Vodicˇka / Structural Integrity Procedia 00 (2019) 000–000

5

g 2 ( t )

20

S 1

S

2

S

5

100

12 . 5 12 . 5

x 2

x 1

Fig. 2. A scheme for the calculations with two inclusions and basic response of the element to the load.

t = 0 . 291 s

t = 0 . 292 s

t = 0 . 288 s

t = 0 . 306 s

Fig. 3. The stress trace [MPa] ( left ) and shear strain ( right ) distribution in the structure while the interface crack is propagating shown at some selected instants t . Displacements in strain plots are magnified 10 times.

First, an overall response of the domain to increasing loading is shown in the right graph of Fig. 2. It shows abrupt change of the sti ff ness due to the newly created crack which passes through the whole width of the matrix domain. In what follows some detailed features of the fracture process are presented. The crack is initiated at the interface by an opening crack and stops propagating along the interface when the interface normal traction is not able to reach the critical value. The used model guarantees according to Vodicˇka and Manticˇ (2017) maximum of the normal cohesive stress for the value 2 κ n G i c β 1 + β = 13 . 8MPa. Similarly, to initiate an opening crack in the material with the characteristic material length ϵ , see Tanne´ et al. (2018), chosen at 1 mm, the critical stress trace σ tr has to reach the level 3 KG c 2 ϵ = 39 . 4MPa. Four instants are chosen: the first instant t = 0 . 288 s corresponds to the first interface crack formation, the second at t = 0 . 291 s documents stress concentration and initiation of phase-field fracture, the third at t = 0 . 292 s presents the crack between the inclusions, the fourth at t = 0 . 306 s contains the situation where the crack reached the outer contour. Some of the stress-strain results are shown in Fig. 3. In the studied case the parameters that a ff ect the crack propagation were adjusted so that first there appeared an in terface crack which subsequently invoked a crack in the material of matrix. Appropriate adjusting of these parameters, however, can lead to di ff erent scenarios which might appear in a wide range of technical problems. Nevertheless, the graph shows that the crack formed between the inclusions is a ff ected by shear. This is due to the fact that say in terms of fracture energy there was no di ff erence between the opening and shearing mode in this case. The crack modes are distinguished in the next example. The described computational approach for mixed-mode fracture is tested by a standard geometrical model used also in Feng and Li (2022). Here, it also involves an inclusion. The calculations are focused on showing di ff erences using various ratios between G I c and G II c , and how the interface a ff ects the behavior of the structural element. The scheme for the problem is shown in Fig. 4. The initial cracks are represented by two grooves of finite width to avoid contact after loading. For a comparison, the domain is considered also without the inclusion. The parameters characterising sti ff ness of the material and of the interface are the same as before. The di ff erence appears for the crack characteristics. The fracture energy for the opening crack mode in the matrix domain is G I c = 10 Jm − 2 , the shear mode fracture energy G II c varies between G I c and 10 G I c . For the interface, the fracture energy is G i c = 1 Jm − 2 , or 10 G i c . The phase-field degradation function Φ ( α ) = α p α p + β (1 − α ) + δ is used where the parameters are set as follows: β = 10, p = 2, δ = 10 − 6 , see Wu (2017). At the interface, the degradation function is chosen as ϕ ( ζ ) = βζ 1 + β − ζ , with β = 0 . 001, see Vodicˇka and Manticˇ (2017).