Vol. 10, No. 3, 2015

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Asymptotic stress field in the vicinity of a mixed-mode crack under plane stress conditions for a power-law hardening material

Larisa V. Stepanova and Ekaterina M. Yakovleva

Vol. 10 (2015), No. 3, 367–393
DOI: 10.2140/jomms.2015.10.367

The stress-strain state analysis near the crack tip in a power-law material under mixed-mode loading conditions is investigated. By the use of the eigenfunction expansion method the stress-strain state near the mixed-mode crack tip under plane stress conditions is found. The type of the mixed-mode loading is specified by the mixity parameter, which varies from 0 to 1. The value of the mixity parameter corresponding to a mode II crack loading is equal to 0, whereas the value corresponding to a mode I crack loading is equal to 1. It is shown that the eigenfunction expansion method results in a nonlinear eigenvalue problem. The numerical solutions of the nonlinear eigenvalue problems for all values of the mixity parameter and for all practically important values of the strain hardening (or creep) exponent are obtained. It is found that mixed-mode loading of the cracked plate gives rise to a change in the stress singularity in the vicinity of the crack tip. Mixed-mode loading of the cracked plate results in new asymptotics of the stress field, which is different from the classical Hutchinson–Rice–Rosengren (HRR) stress field. The approximate solution of the nonlinear eigenvalue problem is obtained by a perturbation theory technique (artificial small parameter method). In the framework of the perturbation theory approach, a small parameter representing the difference between the eigenvalue of the nonlinear problem and the undisturbed linear problem is introduced. The asymptotic analysis carried out shows clearly that the stress singularity in the vicinity of the crack tip changes under mixed-mode loading in the case of plane stress conditions. The angular distributions of the stress and strain components (eigenfunctions) in the full range of values of the mixity parameter are given.

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stress-strain state near the crack tip, mixed-mode loading, mixity parameter, nonlinear eigenvalue problem, perturbation technique
Received: 23 July 2014
Revised: 6 November 2014
Accepted: 25 December 2014
Published: 26 August 2015
Larisa V. Stepanova
Department of Mathematical Modelling in Mechanics
Samara State University
1 Akad. Pavlov
Ekaterina M. Yakovleva
Department of Mathematical Modelling in Mechanics
Samara State University
1 Akad. Pavlov