PSI - Issue 37

L.V. Stepanova et al. / Procedia Structural Integrity 37 (2022) 908–919 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

918

11

Function 0 ( ) f  is shown by light blue color. The numerical solution of the problem (42), (43) is shown by red line. The two-term asymptotic solution 0 1 ( ) ( ) ( ) f f f     = + is shown by green color, the three-term asymptotic expansion 2 0 1 2 ( ) ( ) ( ) ( ) f f f f       = + + is shown by black color and the fourth-term asymptotic expansion 2 3 0 1 2 3 ( ) ( ) ( ) ( ) ( ) f f f f f         = + + + is shown by blue color. One can see that the angular distributions tend to approach the numerical solution as the number of retained terms in the asymptotic expansion of the function ( ) f  increases. It is seen from Figs. 9 – 11 that it is sufficient to retain four terms in the asymptotic expansion if the small parameter method is used, because the angular distribution of the function ( ) f  obtained with the use of the four-term asymptotic expansion is close to the limiting numerical solution. Conclusions The numerical method to solve the nonlinear eigenvalue problems arising in nonlinear fracture mechanics is presented. The method is aimed at computational determination of the whole eigenvalue spectrum of the nonlinear eigenvalue problems following from the near crack-tip field determination in hardening and creeping materials. The stress leading stress singularities orders and associated stress eigenfunctions were obtained. Acknowledgements The work is supported by the Russian Science Foundation (project 21-11-00346). Anheuser, M., Gross. D., 1994. Higher order fields at crack and notch tips in power-law materials under longitudinal shear. Archive of Applied Mechanics 64, 509 – 518. Carpinteri, A., Paggi, M., 2009. Asymptotic analysis in Linear Elasticity: From the pioneering studies by Wieghardt and Irwin until today. Engineering Fracture Mechanics 76, 1771-1784. Dai, Y., Liu, Y., Qin, F., Chao, Y.J., 2019. A unified method to solve higher order asymptotic crack-tip fields of mode I, mode II and mixed mode I/II crack in power-law creeping solids. Engineering Fracture Mechanics, 218, 106610. Dai, Y., Liu, Y., Qin, F., Chao, Y.J., Berto, F., 2019. Estimation of stress field for sharp V-notch in power-law creeping solids: An asymptotic viewpoint. International Journal of Solids and Structures, 180-181, 189-2054. Dai, Y., Qin, F., Liu, Y., Chao, Y.J., 2021. On the second order term asymptotic solution for sharp V-notch tip field in elasto-viscoplastic solids. International Journal of Solids and Structures, 2170218, 106-122. Hutchinson, J.W., 1968a. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 16, 13-31. Hutchinson, J.W., 1968b. Plastic stress and strain fields at a crack tip. J Mech Phys Solids 16, 337-347. Loghin, A., Joseph, P.F., 2020. Mixed mode fracture in power law hardening materials for plane stress. Journal of the Mechanics and Physics of Solids 139, 103890. Loghin, A., Joseph, P.F., 2001. Asymptotic solutions for mixed mode loading of cracks and wedges in power law hardening materials. Engineering Fracture Mechanics, 68(14), 1511-1534. Meng, L., Lee, S.B., 1998. Eigenspectra and orders of singularity at a crack tip for a power-law creeping medium. International Journal of Fracture 92, 55-70. Murakami, S., Hirano. T., Liu, Y., 2000. Asymptotic fields of stress and damage of a mode I creep crack in steady -- state growth. International Journal of Solids and Structures 37, 6203-6220. Niu, Z., Li, C., Ge. R., Hu, Z., Hu, B., 2019. Analysis of plastic stress singularities of cracks and wedges under plane stress conditions. Engineering Fracture Mechanics 208, 72-89. Rice, J. R., Rosengren, G. F., 1968. Plain strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 16, 1-12. Sotiropoulou, A., Panayotounakou, N., Panayotounakos, D., 2006. Analytic parametric solutions for the HRR nonlinear elastic fields with low hardening exponents. Acta Mecahnica 183, 209-230. Stepanova, L.V., Adylina, E.M., 2014. Stress-strain state in the vicinity of a crack under mixed loading. Journal of Applied Mechanics and Technical Physics 55, 885-895. References