Abstract:
A method is proposed to calculate the eigenvalues of the class of nonlinear eigenvalue problems resulting from the problem of determining the stress-strain state in the vicinity of a crack tip in power-law materials over the entire range of mixed modes of deformation, from the opening mode to pure shear. The proposed approach was used to found eigenvalues of the problem that differ from the well-known eigenvalue corresponding to the Hutchinson–Rice–Rosengren solution. The resulting asymptotic form of the stress field is a self-similar intermediate asymptotic solution of the problem of a crack in a damaged medium under mixed loading. Using the new asymptotic form of the stress field and introducing a self-similar variable, we obtained an asymptotic solution of the problem of a crack in a damaged medium and constructed the regions of dispersed material near the crack.
Keywords:
mixed deformation, self-similar representation of solution, self-similarity of the second kind, intermediate asymptotic form.
Citation:
L. V. Stepanova, E. M. Adylina, “Stress-strain state in the vicinity of a crack tip under mixed loading”, Prikl. Mekh. Tekh. Fiz., 55:5 (2014), 181–194; J. Appl. Mech. Tech. Phys., 55:5 (2014), 885–895
\Bibitem{SteAdy14}
\by L.~V.~Stepanova, E.~M.~Adylina
\paper Stress-strain state in the vicinity of a crack tip under mixed loading
\jour Prikl. Mekh. Tekh. Fiz.
\yr 2014
\vol 55
\issue 5
\pages 181--194
\mathnet{http://mi.mathnet.ru/pmtf1038}
\elib{https://elibrary.ru/item.asp?id=22318154}
\transl
\jour J. Appl. Mech. Tech. Phys.
\yr 2014
\vol 55
\issue 5
\pages 885--895
\crossref{https://doi.org/10.1134/S0021894414050186}
Linking options:
https://www.mathnet.ru/eng/pmtf1038
https://www.mathnet.ru/eng/pmtf/v55/i5/p181
This publication is cited in the following 26 articles:
Yu. N. Bakhareva, “Multiparametric presentation of the crack-tip fields in the vicinity of longitudinal shear crack”, Vestnik of Samara University. Natural Science Series, 30:4 (2025), 18
Lyudmila Frishter, Lecture Notes in Networks and Systems, 706, Fundamental and Applied Scientific Research in the Development of Agriculture in the Far East (AFE-2022), 2023, 375
L.V. Stepanova, E.M. Yakovleva, “Nonlinear EigenProblems Arising From Nonlinear Fracture Mechanics Boundary Value Problems”, Procedia Structural Integrity, 37 (2022), 908
Larisa V. Stepanova, “Methods of perturbation theory and their applications in nonlinear fracture mechanics: From the pioneering studies by Hutchinson, Rice and Rosengren until today”, Mat Design & Process Comms, 3:6 (2021)
S. A. Sokolov, D. E. Tulin, “Modeling of Elastoplastic Stress States in Crack Tip
Regions”, Phys Mesomech, 24:3 (2021), 237
A N Kosygin, L N Kosygina, V N Usacheva, “One example of image digital processing of experimental data obtained by the method of photoelasticity”, J. Phys.: Conf. Ser., 1479:1 (2020), 012144
Larisa Stepanova, Sergey Bronnikov, “A computational study of the mixed–mode crack behavior by molecular dynamics method and the multi – Parameter crack field description of classical fracture mechanics”, Theoretical and Applied Fracture Mechanics, 109 (2020), 102691
L. V. Stepanova, “Influence of the higher order terms in Williams' series expansion of the stress field on the stress-strain state in the vicinity of the crack tip. Part II”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 25:1 (2019), 80–96
Xu Li, Xiaotao Li, Shihao Cao, Xiaoyu Jiang, “Analysis of different damage configurations at a macro-crack tip”, Arch Appl Mech, 89:7 (2019), 1259
L. V. Stepanova, “Influence of the higher order terms in Williams' series expansion of the stress field on the stress-strain state in the vicinity of the crack tip. Part I”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 25:1 (2019), 63–79
L. V. Stepanova, “Asymptotic analysis of the crack tip stress field (consideration of higher order terms)”, Num. Anal. Appl., 12:3 (2019), 284–296
A. N. Kosygin, L. N. Kosygina, “Digital processing of interferograms obtained by the photoelasticity method”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 25:2 (2019), 75–91
LV Stepanova, O N Belova, “Estimation of crack propagation direction angles under mixed mode loading in linear elastic isotropic materials by generalized fracture mechanics criteria and by molecular dynamics method”, J. Phys.: Conf. Ser., 1096 (2018), 012060
L. N. Kosygina, “Asymptotic representation of the stress field near the crack tip of an infinite plate with two semi-infinite symmetrical edge notches: theoretical study and computational experiment”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 24:2 (2018), 55–66
Stepanova Larisa, “Intermediate self-similar asymptotic presentation of stress and damage fields in the vicinity of mixed mode crack tip under creep regime”, Procedia Structural Integrity, 13 (2018), 255
L Stepanova, “Asymptotic methods and their applications in nonlinear fracture mechanics: a review”, J. Phys.: Conf. Ser., 1096 (2018), 012058
Larisa Stepanova, Sergej Bronnikov, “Mathematical modeling of the crack growth in linear elastic isotropic materials by conventional fracture mechanics approaches and by molecular dynamics method: crack propagation direction angle under mixed mode loading”, J. Phys.: Conf. Ser., 973 (2018), 012046
L V Stepanova, V S Dolgikh, “Interference-optical methods in mechanics for the multi-parameter description of the stress fields in the vicinity of the crack tip”, J. Phys.: Conf. Ser., 1096 (2018), 012117
L. V. Stepanova, AIP Conference Proceedings, 1915, 2017, 040059
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