PSI - Issue 2_A

Stepanova Larisa et al. / Procedia Structural Integrity 2 (2016) 793–800 Author name / Structural Integrity Procedia 00 (2016) 000–000

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parameter whereas the blue line shows the boundary of the CDZ obtained by the three-term asymptotic expansion of the integrity parameter. From Fig. 2 it can be seen that the boundary of the CDZ determined by the use of the 1 k + - term asymptotic expansion of continuity is very close to the boundary built by the k -term asymptotic expansion of the continuity parameter whereas the HRR stress field results in the boundary of the CDZ given by the two-term expansion which differs substantially from the boundary of the CDZ given by the three-term asymptotic expansion of the integrity parameter by the form and dimensions. 4. Summary Asymptotic crack-tip fields in damaged materials are developed for a stationary plane stress crack under mixed mode loading conditions. The asymptotic solutions are obtained by the use of the similarity variable and the similarity presentation of the solution. On the basis of the self-similar representation of the solution the near crack tip stress, creep strain rate and continuity distributions are given. It is shown that meso-mechanical effect of damage accumulation near the crack tip results in new intermediate stress field asymptotic behavior and requires the solution of nonlinear eigenvalue problems. To attain eigensolutions a numerical scheme is worked out and the results obtained provide the additional eigenvalues of the HRR problem. By the use of the method proposed the whole set of eigenvalues for the mode crack in a power law material under mixed mode loading can be determined. The self similar solutions are based on the idea of the existence of the completely damaged zone near the crack tip. The higher order terms of the asymptotic expansions of stresses, creep strain rates and continuity parameter allowing to obtain the contours of the completely damaged zone in the vicinity of the crack tip are derived and investigated. References Altenbach, H., Sadowski, T., 2015. Failure and Damage Analyses of Advanced Materials, Springer, Berlin. Barenblatt, G.I., 2014. Flow, Deformation and Fracture: Lectures on Fluid Mechanics and the Mechanics of Deformable Solids for Mathematicians and Physicists, Cambridge University Press, Cambridge. Bui, H.D., 2006. Fracture Mechanics. Inverse Problems and Solutions, Springer, Dordrecht. Chousal, J.A.G., de Moura, M.F.S.F. 2013. Mixed mode I+II continuum damage model applied to fracture characterization of bonded joints. Int. J. of Adhesion and Adhesives 41, 92-97. Kuna, M., 2013. Finite Elements in Fracture Mechanics. Theory-Numerics-Applications. Springer, Dordrecht. Murakami, S., 2012. Continuum Damage Mechanics. A Continuum Mechanics Approach to the Analysis of Damage and Fracture. Springer, Dordrecht. Ochsner, A., 2016. Continuum Damage and Fracture Mechanics. Springer Science + Business Media, Singapore. Richard, H.A., Schramm, B., Schrimeisen, N.-H., 2014. Cracks on Mixed Mode loading – Theories, experiments, simulations. International Journal of Fatigue 62, 93 – 103. Riedel, H., 1987. Fracture at High Temperature, Springer – Verlag, Berlin. Soyarslan, C, Richter, H., Bargmann, S., 2016. Variants of Lemaitre damage model and their use in formability prediction of metallic materials. Mechanics of Materials 92, 58-79. Stepanova, L.V., Adylina, E.M. 2014. Stress-strain state in the vicinity of a crack under mixed loading. Journal of Applied Mechanics and Technical Physics 55(5), 885-895. Stepanova, L.V., Igonin, S.A., 2014. Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium. Applied Mathematical Modelling 38(14), 3436-3455. Stepanova, L.V., 2008. Eigenspectra and orders of stress singularity at a mode I crack tip for a power-law medium. Comptes Rendus. Mecanique 336 (1-2), 232-237. Stepanova, L.V., 2009. Eigenvalue analysis for a crack in a power-law material. Computational Mathematics and Mathematical Physics 49 (8), 1332-1347. Stepanova, L.V., Yakovleva, E.M., 2014. Mixed-mode loading of the cracked plate under plane stress conditions. PNRPU Mechanics Bulletin 3, 129-162. Torabi, A.R., Abedinasab, S.M., 2015. Brittle fracture in key-hole notches under mixed mode loading. Experimental study and theoretical predictions. Engineering Fracture Mechanics 134, 35 – 53. Tumanov, A.V., Shlyannikov, V.N., Chandra Kishen, J.M., (2015). An automatic algorithm for mixed mode crack growth rate based on drop potential method. International Journal of Fatigue 81, 227 – 237. Voyiadis, G.Z., Kattan, P.I., 2012. Damage Mechanics with Finite Elements: Practical Applications with Computer Tools. Springer, Berlin. Voyiadis, G.Z., 2015. Handbook of Damage Mechanics, Springer – Verlag, New York. Wei, R.P., 2014. Fracture Mechanics. Integration of Mechanics, Materials Science and Chemistry, Cambridge University Press, Cambridge. Zhang, W., Cai, Y., 2010. Continuum Damage Mechanics and Numerical Applications. Springer Science & Business Media, Heidelberg.