Asymptotics of eigenvalues of the nonlinear eigenvalue problem arising from the near mixed-mode crack-tip stress-strain field problems

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• 作者：L. V. Stepanova ; E. M. Yakovleva
• 刊名：Numerical Analysis and Applications
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：9
• 期：2
• 页码：159-170
• 全文大小：1,317 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：1995-4247
• 卷排序：9

文摘

In the present paper, approximate analytical and numerical solutions to nonlinear eigenvalue problems arising in nonlinear fracture mechanics in studying stress-strain fields near a crack tip under mixed-mode loading are presented. Asymptotic solutions are obtained by the perturbation method (the artificial small parameter method). The artificial small parameter is the difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear “undisturbed” problem. It is shown that the perturbation technique is an effective method of solving nonlinear eigenvalue problems in nonlinear fracture mechanics. A comparison of numerical and asymptotic results for different values of the mixity parameter and hardening exponent shows good agreement. Thus, the perturbation theory technique for studying nonlinear eigenvalue problems is offered and applied to eigenvalue problems arising in fracture mechanics analysis in the case of mixed-mode loading.Keywordsnonlinear eigenvalue problemperturbation theory (small parameter method)asymptotics of stress and strain fields in the vicinity of the mixed-mode crackmixed-mode loadingconstitutive power laweigenspectrum