PSI - Issue 2_A

Mikhail Perelmuter et al. / Procedia Structural Integrity 2 (2016) 2030–2037 M. Perelmuter / Structural Integrity Procedia 00 (2016) 000–000

2037

8

from the regime of bonds rupture alone to the regime of quasistatic crack growth occurs at the following fracture characteristics: 1) c 0 = 0 . 1: σ cr ≈ 25 . 1 MPa, G bond ( d cr , ℓ ) ≈ 10 . 1 J / m 2 ; 2) c 0 = 0 . 25: d cr ≈ 0 . 105 ℓ , σ cr ≈ 16 . 1 MPa, G bond ( d cr , ℓ ) ≈ 4 . 1 J / m 2 ; 3) c 0 = 0 . 50: d cr ≈ 0 . 225 ℓ , σ cr ≈ 11 . 7 MPa, G bond ( d cr , ℓ ) ≈ 2 . 1 J / m 2 . The points, which correspond with these values of d cr , are shown also in Fig. 5 as the gray circles for appropriated values of relative bonds sti ff ness. Under quasistatic crack growth conditions there is an significant decrease of the critical external loading when the crack length extends (Fig. 6), but in this case the crack bridged zone critical length d cr and the adhesion power characteristic G cr = G bond ( d cr , ℓ ), are changing slightly for rather long cracks. The main features of the nonlocal criterion for quasistatic growth of a bridged crack (4) are taking into account the energy consumed by bonds during cracks advance and the analysis of non self-similar cracks growth as has been clearly demonstrated in Fig. 5 and Fig. 6. This criterion was implemented within the bridged crack model with assumption of a singularity at the crack tip, but it includes the cohesive model as the special case for long cracks, the details in (Perelmuter, 2015b). The nonlocal criterion (4) consists of two conditions and the appropriate equations include by several physical mechanical parameters or functions, the dependence on bonds deformation law shape which can be defined exper imentally or obtained starting from the micromechanical modelling (Goldstein et al., 1997; Budiansky et al., 1995; Zhu et al., 2009). In the simplest cases as considered in this paper (linear-elastic bonds) or in (Perelmuter, 2015b) the criterion (4) includes only two experimentally defined parameters (in the case of linear-elastic bonds it is the maximal bonds stress and the critical crack opening at the trailing edge of crack bridged zone) and it can be regarded as the two-parametric fracture criterion. Acknowledgements This work was partly supported by Russian Foundation for the Basic Research, the projects number are 14-08 01163 and 14-01-00869. Barenblatt, G.I., 1959. Concerning equilibrium cracks forming during brittle fracture. The stability of isolated cracks. Relationships with energetic theories. Journal of Applied Math. and Mech. (PMM) 23, 1273 – 1282. Budiansky, B., Amazigo, J.C., Evans, A.G., 1988. Small-scale crack bridging and the fracture toughness of particulate-reinforced ceramics. Journal of the Mechanics and Physics of Solids 36, 167–187. Budiansky, B., Evans, A., Hutchinson, J., 1995. Fiber-matrix debonding e ff ects oncracking in aligned fiber ceramic composites. International Journal of Solids and Structures 32, 315 – 328. Dugdale, D.S., 1960. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 8, 100–104. Goldstein, R., Bakirov, V., Perelmuter, M., 1997. Modeling of the adhesion strength and fracture kinetics of the microelectronic package polymer polymer joints. Proceedings of the Institute of Physics and Technology of the Russian Academy of Sciences 13, 115–125. Goldstein, R., Perelmuter, M., 1999. Modeling of bonding at an interface crack. International Journal of Fracture 99, 53–79. Goldstein, R.V., Perelmuter, M.N., 2012. Kinetics of crack formation and growth on the material interface. Mechanics of Solids 47, 400–414. Panasyuk, V.V., 1971. Limiting Equilibrium of Brittle Solids with Fractures. Michigan Information Service. Perelmuter, M., 2007. A criterion for the growth of cracks with bonds in the end zone. Journal of Appl. Math. and Mech. (PMM) 71, 137 – 153. Perelmuter, M., 2011a. Integro-di ff erential equations for stress analysis in the bridged zone of interface cracks, in: Constanda, C., Harris, P.J. (Eds.), Integral Methods in Science and Engineering. Birkhauster Boston, pp. 287–298. Perelmuter, M., 2011b. An interface crack with non-linear bonds in a bridged zone. Journal of Appl. Math. and Mech. (PMM) 75, 106 – 118. Perelmuter, M., 2013. Boundary element analysis of structures with bridged interfacial cracks. Computational Mechanics 51, 523–534. Perelmuter, M., 2015a. BEM analysis of structures with bridged cracks and weak interfaces, in: Proceedings of the 10th UK Conference on Boundary Integral Methods, University of Brighton, UK, Brighton, UK. pp. 131–141. Perelmuter, M., 2015b. Nonlocal criterion of bridged cracks growth: analytical analysis. Acta Mechanica 226, 397–418. Salganik, R., 1963. The brittle fracture of cemented bodies. Journal of Appl. Math. and Mech. (PMM) 27, 1468 – 1478. Zhu, Y., Liechti, K.M., Ravi-Chandar, K., 2009. Direct extraction of rate-dependent traction-separation laws for polyurea / steel interfaces. Interna tional Journal of Solids and Structures 46, 31 – 51. Zhurkov, S., 1965. Kinetic concept of the strength of solids. International Journal of Fracture Mechanics 1, 311–323. References 4. Conclusion