PSI - Issue 42

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

932

6

R. Vodicˇka / Structural Integrity Procedia 00 (2019) 000–000 g 2 ( t )

g 1 ( t ) g 2 ( t )

g

v 1 t 0 v 2 t 0

25

S

40

g 1 ( t )

t

0

2 t 0

3 t 0

t 0

x 2

99 2 99

x 1

Fig. 4. Description of the computational domain ( left ), and prescribed hard-device loading ( right ).

Fig. 5. Total forces at the top face for the case without the inclusion ( left ) and for the case with the inclusion ( right ).

The domain is loaded by displacement loading in a way shown in Fig. 4: first, horizontal load g 1 is applied, and afterwards under constant horizontal load the vertical one g 2 is gradually increased. The parameters of the load are: v 1 = v 2 = 1 mm s − 1 , t 0 = 0 . 1 s. This load is applied incrementally in time steps refined to 1 ms. The global response of the material elements is shown in Fig. 5. It shows the evolution of the total vertical force F applied at the top face of the domain in time. The prescribed displacement is closely related to the time variable, though it is not proportional in the whole time range. Actually, it is seen in the first part which corresponds to the lateral pressure causing compression at the top face. Three ratios between fracture energies in shearing and opening mode ( G II c and G I c , respectively) were set in the case without the interface. If there is no di ff erence between them ( G II c / G I c = 1) a shear mode crack is prevailing which means that the domain is ruptured relatively early due to lateral pressure. In the case of large di ff erence between them ( G II c / G I c = 10) rupture in the shearing mode is suppressed and the crack in the domain waits for a substantially large vertical load. The influence of the inclusion is presented in the other graph of Fig. 5. For material fracture energies based on the option G II c = 10 G I c , two cases of the interface fracture energy G i c were chosen according to the picture. In the case G iI c / G I c = 10, there was no interface damage as it will be seen below. However, decreasing the interface fracture energy leads to an interface crack which appears in a mixed mode though initiated by the lateral loading causing shearing stress along the interface. The total force causing fracture is smaller. The only case with G II c = G I c documents that a crack appeared due to shear as decrease of sti ff ness appeared during the first phase of loading. The comments above are supported by the next graphs which show stress distribution in the domain, and forming of the cracks in terms of the phase-field damage variable. First, let us check the domain without the inclusion in Fig. 6. In the case G II c = 10 G I c shear does not contribute much to fracture thus the crack initiates in the direction of maximal principle stress and as the load increases in vertical orientation the crack turns to pertinent direction. With increasing vertical loading, the crack is attracted in accordance with observations and calculations of Feng and Li (2022). Compression applied from the left side then inhibits the crack progression. For G II c = G I c , shearing plays an important role in fracture during the first phase of loading, and when vertical loading starts the material breaks in the central part due to tension. Next, the case with the inclusion is discussed. Setting the ratio between fracture energies in material to G II c = 10 G I c , the influence of the magnitude of the interface fracture energy is studied. To this end, stresses, shear strain and resulting crack in terms of the phase-field parameter are shown in Figs. 7,8. All these graphs are snapshots of selected instants corresponding to initiation of damage in materials or at interface, and formation of an interface or a material cracks. In the stress plot it can be seen that in the case of small interface fracture energy the interface crack is formed at t = 0 . 165 s, because the stress at such a contour decreases. The interface crack can be observed also at the strain plot which also includes magnified deformations, at the instants t = 0 . 172 s and t = 0 . 200 s the crack is clearly visible. No