PSI - Issue 39

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L.V. Stepanova et al. / Procedia Structural Integrity 39 (2022) 735–747 Author name / Structural Integrity Procedia 00 (2019) 000–000

The numerical method to solve the nonlinear eigenvalue problems arising in nonlinear fracture mechanics is proposed. The method is aimed at computational determination of the whole eigenvalue spectrum of the nonlinear eigenvalue problems following from the near crack-tip field determination in hardening and creeping materials. The stress leading stress singularities orders and associated stress eigenfunctions were obtained. Acknowledgements The work is supported by the Russian Science Foundation (project 21-11-00346). Anheuser, M., Gross. D., 1994. Higher order fields at crack and notch tips in power-law materials under longitudinal shear. Archive of Applied Mechanics 64, 509 – 518. Carpinteri, A., Paggi, M., 2009. Asymptotic analysis in Linear Elasticity: From the pioneering studies by Wieghardt and Irwin until today. Engineering Fracture Mechanics 76, 1771-1784. Dai, Y., Liu, Y., Qin, F., Chao, Y.J., 2019. A unified method to solve higher order asymptotic crack-tip fields of mode I, mode II and mixed mode I/II crack in power-law creeping solids. Engineering Fracture Mechanics, 218, 106610. Dai, Y., Liu, Y., Qin, F., Chao, Y.J., Berto, F., 2019. Estimation of stress field for sharp V-notch in power-law creeping solids: An asymptotic viewpoint. International Journal of Solids and Structures, 180-181, 189-2054. Dai, Y., Qin, F., Liu, Y., Chao, Y.J., 2021. On the second order term asymptotic solution for sharp V-notch tip field in elasto-viscoplastic solids. International Journal of Solids and Structures, 2170218, 106-122. Hutchinson, J.W., 1968a. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 16, 13-31. Hutchinson, J.W., 1968b. Plastic stress and strain fields at a crack tip. J Mech Phys Solids 16, 337-347. Loghin, A., Joseph, P.F., 2020. Mixed mode fracture in power law hardening materials for plane stress. Journal of the Mechanics and Physics of Solids 139, 103890. Loghin, A., Joseph, P.F., 2001. Asymptotic solutions for mixed mode loading of cracks and wedges in power law hardening materials. Engineering Fracture Mechanics, 68(14), 1511-1534. Meng, L., Lee, S.B., 1998. Eigenspectra and orders of singularity at a crack tip for a power-law creeping medium. International Journal of Fracture 92, 55-70. Murakami, S., Hirano. T., Liu, Y., 2000. Asymptotic fields of stress and damage of a mode I creep crack in steady -- state growth. International Journal of Solids and Structures 37, 6203-6220. Niu, Z., Li, C., Ge. R., Hu, Z., Hu, B., 2019. Analysis of plastic stress singularities of cracks and wedges under plane stress conditions. Engineering Fracture Mechanics 208, 72-89. Rice, J. R., Rosengren, G. F., 1968. Plain strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 16, 1-12. Sotiropoulou, A., Panayotounakou, N., Panayotounakos, D., 2006. Analytic parametric solutions for the HRR nonlinear elastic fields with low hardening exponents. Acta Mecahnica 183, 209-230. 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, 885-895. Stepanova, L.V,, Yakovleva, E.M., 2015. Asymptotic stress field in the vicinity of a mixed-mode crack under plane stress conditions for a power law hardening material. Journal of Mechanics of Materials and Structures 10(3), 367-393. Stepanova, L.V., Yakovleva, E.M., 2016. Stress-strain state near the crack tip under mixed-mode loading: Asymptotic approach and numerical solutions of nonlinear eigenvalue problems. AIP Conference Proceedings 1785, 030030. 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., Peksheva, A.A., 2021. Mixed mode fracture in perfect plastic materials for plane stress conditions. Journal of Physics: Conference series 1745(1), 012072. Stepanova, L., 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. Zhao, J., Zhang, X., 1995. The asymptotic study of fatigue crack growth based on damage mechanics. Engineering Fracture Mechanics 50(1,) 131 – 141. Zhao, J., Zhang, Z., 2001. On the process zone of a quasi - static growing tensile crack with power-law elastic-plastic damage. International Journal of Fracture 108, 383-395. References