PSI - Issue 37

Available online at www.sciencedirect.com Structural Integrity Procedia 00 (2019) 000 – 000 Available online at www.sciencedirect.com ScienceDirect

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Procedia Structural Integrity 37 (2022) 908–919

© 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Pedro Miguel Guimaraes Pires Moreira Abstract We discuss nonlinear eigenvalue boundary value problems arising in nonlinear fracture mechanics due to determination of the stress-strain state in the vicinity of the crack tip in power law hardening or creep power law materials. The study is aimed at analytical determination of eigenfunctions and eigen values of the nonlinear eigenvalue problems following from the problem of determining the crack tip fields at crack in power law materials under Mode III, Mode I, Mode II and Mixed Mode loadings. The study is based on the perturbation theory technique (small artificial parameter method) allowing us to find the analytical solution for the eigenvalues in the approximate form. The method of analytical determination of eigenfunctions of the problem is presented. © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Pedro Miguel Guimaraes Pires Moreira Keywords: Nonlinear eigenvalue problems, artificial small parameter method; HRR solution; new eigenvalues. 1. Introduction. Fundamental asymptotic solution of Hutchinson, Rice and Rosengren Eigenvalue problems arise in several areas of linear elastic and plastic fracture mechanics (Loghin and Joseph (2020), Dai at al. (2019), Loghin and Joseph (2001), Dai et al. (2017), Niu et al. (2019), Dai et al. (2021), Hutchinson (1968), Rice and Rosengren (1968), Anheuser and Gross (1994), Sotiropoulou (2006), Stepanova and Adylina (2014), Murakami et al. (2000), Zhao and Zhang (1995), Zhao and Zhang (2001)). For instance, the determination of the near ICSI 2021 The 4th International Conference on Structural Integrity Nonlinear EigenProblems Arising From Nonlinear Fracture Mechanics Boundary Value Problems L.V. Stepanova a , E.M. Yakovleva a * a Samara Nationa Research University, Mosckovskoye shosse, 34, Samara 443086, Russian Federation

* Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: Stepanova.lv@ssau.ru

2452-3216 © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Pedro Miguel Guimaraes Pires Moreira

2452-3216 © 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Pedro Miguel Guimaraes Pires Moreira 10.1016/j.prostr.2022.02.025