# Application of the complex cohesive zone model to the edge mi xed-mode crack problem for orthotropic media

 1Selivanov, MF1Chornoivan, YO1S.P. Timoshenko Institute of Mechanics of the NAS of Ukraine, Kyiv Dopov. Nac. akad. nauk Ukr. 2019, 11:31-40 https://doi.org/10.15407/dopovidi2019.11.031 Section: Mechanics Language: Ukrainian Abstract:  The elasticity problem for a semiinfinite orthotropic body with an edge crack that is normal to the boundary is studied. The crack is aligned with one of the orthotropic material axes, and a biaxial stress field is applied at infinity. The multiple cohesive zone model developed to study the mixed-mode fracture is used to determine the critical stress parameters, as well as the cohesive stress field. The difference of mode I and mode II cohesive lengths is inherent to the model. The problem is formulated in terms of integral equations for the unknown displacement discontinuity along the crack line. The collocation method, which is applied to two singular integral equations with generalized Cauchy kernel, gives the equations for critical loads and respective discrete cohesive tractions and separations. These equations are non-linear, as a dependence of cohesive stresses on the crack opening is taken according to the potential-based traction—separation law. An algorithm for determining the parameters of limiting equilibrium of a mixed-mode crack is constructed. An iterative procedure is used to enforce the smooth crack closure for the two pure fracture modes, and two cohesive lengths can be determined with the precision of a mesh size. An example of calculating the critical state parameters and the corresponding cohesive stress field is given. The numerical example is built for the traction–separation law based on the trapezoidal laws of pure modes that are coupled without parameters of mode-mixity. Keywords: crack in an orthotropic body, mixed-mode fracture, multiple cohesive zone model, traction—separation law for a mixed-mode crack
References:

1. Dimitri, R., Trullo, M., Zavarise, G. & De Lorenzis, L. (2014). A consistency assessment of coupled cohesive zone models for mixed-mode debonding problems. Frattura ed Integrità Strutturale, 8, No. 29, pp. 266-283. doi: https://doi.org/10.3221/IGF-ESIS.29.23
2. Högberg, J. L. (2006). Mixed mode cohesive law. Int. J. Fract., 141, No. 3-4, pp. 549-559. doi: https://doi.org/10.1007/s10704-006-9014-9
3. Jensen, S. M., Martos, M. J., Bak, B. L. V. & Lindgaard, E. (2019). Formulation of a mixed-mode multilinear cohesive zone law in an interface finite element for modelling delamination with r-curve effects. Composite Struct., 216, pp. 477-486. doi: https://doi.org/10.1016/j.compstruct.2019.02.029
4. de Moura, M., Gonçalves, J. & Silva, F. (2016). A new energy based mixed-mode cohesive zone model. Int. J. Solids and Struct., 102-103, pp. 112-119. doi: https://doi.org/10.1016/j.ijsolstr.2016.10.012
5. Li, S., Thouless, M. D., Waas, A. M., Schroeder, J. A. & Zavattieri, P. D. (2006). Mixed-mode cohesive-zone models for fracture of an adhesively bonded polymer–matrix composite. Eng. Fract. Mech., 73, No. 1, pp. 64-78. doi: https://doi.org/10.1016/j.engfracmech.2005.07.004
6. Tvergaard, V. & Hutchinson, J. W. (1992). The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J. Mech. Phys. Solids., 40, No. 6, pp. 1377-1397. doi: https://doi.org/10.1016/0022-5096(92)90020-3
7. Tvergaard, V. (2004). Predictions of mixed mode interface crack growth using a cohesive zone model for ductile fracture. J. Mech. Phys. Solids., 52, No. 4, pp. 925-940. doi: https://doi.org/10.1016/S0022-5096(03)00115-7
8. Park, K. & Paulino, G. H. (). Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces. Appl. Mech. Reviews, 2013. 64, No. 6, 060802-060802-20. doi: https://doi.org/10.1115/1.4023110
9. Park, K., Paulino, G. H. & Roesler, J. R. (2009). A unified potential-based cohesive model of mixed-mode fracture. J. Mech. Phys. Solids., 57, No. 6, pp. 891-908. doi: https://doi.org/10.1016/j.jmps.2008.10.003
10. Selivanov, M. F. (2019). Subcritical and critical states of a crack with failure zones. Appl. Math. Model., 72, pp. 104-128. doi: https://doi.org/10.1016/j.apm.2019.03.013
11. Selivanov, M. F. & Chornoivan, Y. O. (2018). A semi-analytical solution method for problems of cohesive fracture and some of its applications. Int. J. Fract., 212, No. 1, pp. 113-121. doi: https://doi.org/10.1007/s10704-018-0295-6
12. Selivanov, M. F., Chornoivan, Y. O. & Kononchuk, O. P. (2018). Determination of crack opening displacement and critical load parameter within a cohesive zone model. Continuum Mech. Thermodyn., 31, No. 2, pp. 569-586. doi: https://doi.org/10.1007/s00161-018-0712-0
13. Selivanov, M. F. (2019). An edge crack with cohesive zone in orthotropic body. Dopov. Nac. acad. nauk Ukr., No. 6, pp. 25-35 (in Ukrainian). doi: https://doi.org/10.15407/dopovidi2019.06.025
14. Selivanov, M. F. (2019). An edge crack with cohesive zone. Dopov. Nac. acad. nauk Ukr., No. 3, pp. 46-54 (in Ukrainian). doi: https://doi.org/10.15407/dopovidi2019.03.046