PSI - Issue 39

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

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ScienceDirect

Procedia Structural Integrity 39 (2022) 735–747

© 2021 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 CP 2021 – Guest Editors Abstract Nonlinear eigenvalue boundary value problems occurring 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 are discussed. The study is aimed at analytical determination and approximate analytical expressions of eigenfunctions and eigenvalues 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 analytical approach 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. © 2021 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 CP 2021 – Guest Editors 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)). Thus, the asymptotic presentation of the © 2 h 7th International Conference on Crack Paths Nonlinear Eigenvalues Problems Engendered by 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 CP 2021 – Guest Editors

2452-3216 © 2021 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 CP 2021 – Guest Editors 10.1016/j.prostr.2022.03.148