1. E.M. Adulina, L.V. Stepanova, Samara State University
Nonlinear eigenvalue problems arising from nonlinear fracture mechanics analysis
E.M. Adulina, L.V. Stepanova,
Samara State University

2. J. W. Hutchinson, Singular behavior at the end of tensile crack in a hardening material, J. Mech. Phys. Solids 16 (1968)
J. R. Rice, G. F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material, J. Mech. Phys. Solids 16 (1968) 1-12.
F.G. Yuan, S. Yang, Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear, Int. J. of Fracture 69, (1994) 1-26.
G.P. Nikishkov, An algorithm and a computer program for the three-term asymptotic expansion of elastic-plastic crack tip stress and displacement fields, Engineering Fracture Mechanics 50 (1995)
B.N. Nguyen, P.R. Onck, E. Van Der Giessen, On higher-order crack-tip fields in creeping solids, Transaction of the ASME 67 (2000)
I. Jeon, S. Im, The role of higher order eigenfields in elastic-plastic cracks, J. Mech. Phys. Solids 49 (2001)
C.Y. Hui, A. Ruina, Why K? High order singularities and small scale yielding, Int. J. of Fracture 72 (1995)
L. Meng, S.B. Lee, Eigenspectra and orders of singularity at a crack tip for a power-law creeping medium, Int. J. of Fracture 92 (1998)
M. Anheuser, D. Gross, Higher order fields at crack and notch tips in power-law materials under longitudinal shear, Archive of Applied Mechanics 64 (1994)

3. Hutchinson-Rice-Rosengren solution

4. Crack tip geometry and coordinate systems

5. Mode I crack. Basic equations

6. The Airy stress potential
The asymptotic behavior of the Airy function
The asymptotic behavior of the stress field

7. Nonlinear eigenvalue problem

8. Perturbation theory approach

9. Mode III crack. Basic equations

10. The asymptotic behavior of the stress function
Nonlinear eigenvalue problem

11. The set of linear differential equations
Eigenvalues
The set of linear differential equations

12. The solvability condition
The set of boundary value problems
The solvability condition

13. Closed form analytical solution

14. Closed form analytical solution

15. Mode I crack. Nonlinear eigenvalue problem

16. The perturbation method
The undisturbed linear problem

17. The set of boundary value problems

18. The solvability condition

19. The linear differential equation for

21. The three-term asymptotic expansions of the hardening exponent

22. The three-term asymptotic expansions for the hardening exponent

23. Eigenspectra at a Mode II crack under plane strain conditions
The nine-term asymptotic expansion of the hardening exponent

24. Eigenspectra at a Mode II crack under plane stress conditions
The nine-term asymptotic expansion of the hardening exponent

25. The asymptotic study of fatigue crack growth based on continuum damage mechanics
Zhao J., Zhang X. The asymptotic study of fatigue crack growth based on damage mechanics// Engn. Fracture Mechanics V P
Li J., Recho N. Methodes asymptotiques en mecanique de la rupture. Paris: Hermes Science Publications, p.
Astafiev V.I., Radayev Y.N., Stepanova L.V. Nonlinear fracture mechanics. Samara: Samara State University, p.
Stepanova L.V. Mathematical methods of fracture mechanics. Moscow: Fizmatlit, p.
Astafjev V.I., Grigorova T.V., Pastuchov V.A. Influence of continuum damage on stress distribution near a tip of a growing crack under creep conditions/ procedings of the International Colloquium «Mechanics of creep brittle materials 2», University of Leicester, UK, P

26. The asymptotic study of fatigue crack growth based on continuum damage mechanics
Consider a fatigue growing crack lying on the x-axis with the coordinate origin located at the moving crack tip

27. The asymptotic study of fatigue crack growth based on continuum damage mechanics
The essence of continuum damage mechanics is characterized by material deterioration coupled constitutive equations. Under the assumption of linear elasticity a stiffness reduction based stress-strain relationship is applied as
The kinetic law of damage evolution
The equilibrium equations

28. The asymptotic study of fatigue crack growth based on continuum damage mechanics
The compatibility equation
The constitutive equations
The constitutive equations for plane stress conditions
The constitutive equations for plane strain conditions

29. The asymptotic study of fatigue crack growth based on continuum damage mechanics
The traditional traction free conditions on crack surfaces
The Airy stress function
The Airy stress function can be presented in the form
The Mode I stress field components at the crack tip behave as follows

30. The asymptotic study of fatigue crack growth based on continuum damage mechanics
The damage field around the crack tip can be presented as
The strain components are given as
From the compatibility equation
one can obtain (for plane stress)
one can obtain (for plane strain)

31. The asymptotic study of fatigue crack growth based on continuum damage mechanics
The damage evolution law allows to obtain
The symmetry of the stress and damage fields around the crack tip
The normalizing condition is chosen as
The regularity requirement
The traction free conditions

32. The asymptotic study of fatigue crack growth based on continuum damage mechanics
The totally damaged zone needs to be modeled in the vicinity of the crack tip
The analytical result
The stress and damage fields around the crack tip

33. The asymptotic study of fatigue crack growth based on continuum damage mechanics

34. The asymptotic study of fatigue crack growth based on continuum damage mechanics
The new analytical presentation of stress, strain and continuity fields both for plane strain and plane stress conditions is given.
The results obtained differ from Zhao and Zhang's solution where the original formulation of the problem for plane stress conditions has been proposed.
An analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained.
The perturbation technique is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law.