Geodesic
Asymptotics of the stress field near a crack tip under mixed-mode loading: small parameter method
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 10 (2015), pp. 77-90 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the present paper approximate analytical and numeric solutions to nonlinear eigenvalue problems arising in nonlinear fracture mechanics in analysis of stress — strain fields near a crack tip under mixed mode loading are presented. Asymptotic solutions are obtained via perturbation method technique (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 gives an effective method of solving nonlinear eigenvalue problems in nonlinear fracture mechanics. Comparison results of numeric and asymptotic results for different value of the mixity parameter and hardening exponent shows good agreement. Thus the perturbation theory technique for study of nonlinear eigenvalue problems is offered and applied for eigenvalue problems arising from fracture mechanics analysis in the case of mixed mode loading.
Keywords: mode II crack under plane stress conditions, power-law material, nonlinear eigenvalue problem, eigenspectrum and orders of stress singularity
Mots-clés : perturbation technique. 
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L. V. Stepanova; E. M. Yakovleva. Asymptotics of the stress field near a crack tip under mixed-mode loading: small parameter method. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 10 (2015), pp. 77-90. http://geodesic.mathdoc.fr/item/VSGU_2015_10_a6/

[1] Andrianova I. V., Awrejcewicz J., Methods of asymptotic analysis and synthesis in nonlinear dynamics and solid mechanics, Institut komp'iuternykh issledovanii, M.–Izevsk, 2013, 276 pp. (in Russian)

[2] Wei R. P., Fracture Mechanics. Integration of Mechanics, Materials Science and Chemistry, Cambridge University Press, Cambridge, 2014, 232 pp. (in English)

[3] Kuna M., Finite Elements in Fracture Mechanics. Theory-Numerics-Applications, Springer, Dordrecht, 2013, 336 pp. (in English) | MR | Zbl

[4] A. Ehrlacher, H. Markenscoff (eds.), Duality, Symmetry and symmetry lost in solid mechanics, Selected works of H. D. Bui, Presses des Ponts, Paris, 2011, 396 pp. (in English)

[5] Williams M. L., “Stress singularities resulting from various boundary conditions in angular corners of plates in extention”, J. Appl. Mech., 19 (1952), 287–298 (in English) | MR

[6] Stepanova L. V., Mathematical Methods of Fracture Mechanics, Samarskii universitet, Samara, 2006, 231 pp. (in Russian)

[7] Astafiev V. I., Stepanova L. V., Shesterikov S. A., “Asymptotics of stress-strain state in the vicinity of the crack tip under creep conditions”, Vestnik of Samara State University, 1995, no. S, 59–64 (in Russian) | MR

[8] Sapora A., Carpinteri A., “A Finite Fracture Mechanics approach to V-notched element subjected to mixed-mode loading”, Engineering Fracture mechamics, 97 (2013), 216–226 (in English) | DOI

[9] Weibgraeber P., Becker W., “Finite Fracture Mechanics model for mixed mode fracture in adhesive joints”, International Journal of Solids and Structures, 50:14 (2013), 2383–2394 (in English) | DOI

[10] Stepanova L. V., “Eigenvalue analysis for a crack in a power-law material”, Computational Mathematics and Mathematical Physics, 49:8 (2009), 1332–1347 (in English) | DOI | MR | Zbl

[11] Natarajan S., Song C., Belouettar S., “Numerical evaluation of stress intensity factors and T-stress for interfacial cracks and cracks terminating at the interface without asymptotic enrichment”, Computer methods in applied mechanics and engineering, 279 (2014), 86–112 (in English) | DOI | MR

[12] Stepanova L. V., “Refined study of stress-strain state near the crack tip under cyclic loading in a damaged medium”, Vestnik of Samara State University, 2011, no. 2(83), 105–115 (in Russian)

[13] Rice J. R., Rosengren G. F., “Plane strain deformation near a crack tip in a power-law hardening material”, J. Mech. Phys. Solids, 16 (1968), 1–12 (in English) | DOI | Zbl

[14] Hutchinson J. W., “Singular behavior at the end of tensile crack in a hardening material”, J. Mech. Phys. Solids, 16 (1968), 13–31 (in English) | DOI | Zbl

[15] Hutchinson J. W., “Plastic stress and strain fields at a crack tip”, J. Mech. Phys. Solids, 16 (1968), 337–349 (in English) | DOI

[16] Carroll J., Daly S., Fatigue, Failure and Damage Evolution, Springer, Berlin, 2015, 252 pp. (in English)

[17] Rahman S., Mohammad E., Effects of mixed-mode overloading on the mixed-mode I+II fatigue crack growth, Springer, Berlin, 2013, 987–1000 (in English)

[18] Anheuser M., Gross D., “Higher order fields at crack and notch tips in power-law materials under longitudinal shear”, Archive of Applied Mechanics, 64 (1994), 509–518 (in English) | DOI | Zbl

[19] Adylina E. M., Igonin S. A., Stepanova L. V., “About a non-linear task on eigenvalues incurring from the analysis of tensions at the fatigue crack tip”, Vestnik of Samara State University, 2012, no. 3/1(94), 83–102 (in Russian)

[20] Adulina E. M., Stepanova L. V., “On development of multiscale fracture models”, Vestnik of Samara State University, 2012, no. 9(100), 70–83 (in Russian)

[21] Stepanova L. V., Adylina E. M., “Stress-strain state in the vicinity of a crack tip under mixed mode loading”, Applied Mechanics and Technical Physics, 55:5(327) (2014), 181–194 (in Russian) | MR | Zbl