1. L.V. Stepanova Samara State University
An Intermediate Asymptotic Solution of the Coupled Creep-Damage Crack Problem in Similarity Variable
L.V. Stepanova
Samara State University

2. Outline
The class of self-similar solutions to coupled (creep-damage) crack problems is presented. The constitutive model is based on continuum damage mechanics. The conventional Kachanov-Rabotnov creep-damage theory is utilized to study the asymptotic behavior of damage in the region very near the crack tip.
The totally damaged zone where the damage (integrity) parameter reaches its critical value is assumed to exist in the vicinity of the crack tip. Using the similarity variable the asymptotic solutions to mode I, II and mode III crack problems are obtained. The asymptotic stress, creep strain rate and damage fields near the crack tip are analyzed by solving nonlinear eigenvalue problems resulting in a new far stress distribution. The configurations of the totally damaged zone governed by the new far stress field are found and analyzed.
The new far field stress asymptotic can be interpreted as the intermediate asymptotic valid for times and distances at which effects of initial and boundary conditions on the stress and damage distributions are lost. Higher order fields for damaged nonlinear antiplane shear and tensile crack problems are analytically derived. The higher order fields obtained permit the shape of the totally damaged zone modelled in the vicinity of the crack tip to be determined more exactly. The similarity solutions obtained can be further used in more general multiscaling models in crack tip mechanics which develops multiscale methodologies of crack tip description

3. Self-similar variable and self-similar presentation of the solution
Constitutive equations
Damage evolution law
Remote boundary condition

4. Self-similar variable and self-similar presentation of the solution
Similarity variable
Self-similar presentation of the solution

5. Mode III crack problem Equilibrium equation Compatibility equation
Kinetic law of damage evolution

6. Mode III crack problem
Traction free boundary condition on the crack surface
Symmetry condition
Asymptotic condition requiring that the stress field must approach the Hutchinson-Rice-Rosengren (HRR) field at large distances from the crack tip

7. Mode III crack problem. Asymptotic solution
Eigenfunction expansion at large distances from the crack tip
Boundary value problem for the coefficients of the first term of the stress asymptotic expansion

8. Mode III crack problem Nonlinear eigenvalue problem
Kinetic law of damage evolution

9. Mode III crack problem
Two-term asymptotic expansion for continuity (integrity) parameter
Boundary of the totally damaged zone
Two-term asymptotic expansion of the effective stress tensor components

10. Mode III crack problem
Two-term asymptotic expansion of the stress tensor components
Two-term asymptotic expansion of the effective stress and strain tensor components

11. Mode III crack problem
Boundary value problem for the coefficients of the two-term effective stress tensor asymptotic expansions

12. Mode III crack problem Kinetic law of damage evolution
Three-term asymptotic expansion of continuity parameter at large distances from the crack tip
The contour of the totally damaged zone

13. Mode III crack problem. Asymptotic expansions and higher order fields at crack tip
Three-term asymptotic expansion of the effective stress (the stress referred to the surface that really transmits the internal forces)
Three-term asymptotic expansion of stress tensor components at large distances from the crack tip
Three-term asymptotic expansion of the creep strains

14. Mode III crack problem. Asymptotic expansions and higher order fields at crack tip
The kinetic equation
Forth-order asymptotic expansion of the continuity parameter at large distances from the crack tip
The shape of the totally damaged zone is governed by

15. Mode III crack problem. The contours of the totally damaged zone (TDZ) in the vicinity of the antiplane shear crack tip
1-the contour of the TDZ given by the two-term asymptotic integrity parameter expansion
2-the contour of the TDZ given by the three-term asymptotic integrity parameter expansion

16. Mode III crack problem
To elucidate the cause of this fact one can formulate the remote boundary condition in a more general form with an unknown s which must be found as a part of the solution under the condition of convergence of the TDZ boundary to some limit contour
The self-similar variable in a more general form
The similarity solution has the form

17. Mode III crack problem
The eigenvalue s for different values of material constants

18. Mode III crack problem
The contours of the TDZ for the new stress field asymptotics
The configurations of the TDZ for the new far field stress asymptotics are shown in figures where the following notations are accepted: 1-the contour given by the two-term asymptotic expansion of the integrity parameter; 2-the contour given by the three-term asymptotic expansion of the integrity parameter; 3-the contour given by the four-term asymptotic expansion of the integrity parameter; 4-the contour given by the five-term asymptotic expansion of the integrity parameter; 5-the contour given by the six-term asymptotic expansion of the integrity parameter.

19. Mode III crack problem
The contours of the TDZ for the new stress field asymptotics
The configurations of the TDZ for the new far field stress asymptotics are shown in figures where the following notations are accepted: 1-the contour given by the two-term asymptotic expansion of the integrity parameter; 2-the contour given by the three-term asymptotic expansion of the integrity parameter; 3-the contour given by the four-term asymptotic expansion of the integrity parameter; 4-the contour given by the five-term asymptotic expansion of the integrity parameter; 5-the contour given by the six-term asymptotic expansion of the integrity parameter.

20. Finite difference solution of mode III crack problem in copled formulation
To justify the asymptotic solution obtained one can address to the direct numerical integration of mode III crack problem equations formulated in terms of the similarity variable.
Stresses and integrity distributions in the vicinity of the antiplane shear crack

21. Finite difference solution of mode III crack problem in coupled formulation
It is seen that there are two rectilinear parts: one linear region corresponds to the Rice’s asymptotics while the order linear part corresponds to the new intermediate asymptotic solution
Logarithmic plot of the effective stress

22. Mode I crack. Fundamental equations and asymptotic solution
The creep power-law constitutive equations in the coupled formulation
The equilibrium equations are written in the form
The compatibility equation
The kinetic evolution law of damage

23. Mode I crack. Fundamental equations and asymptotic solution
The creep power-law constitutive equations in the coupled formulation
The traction-free conditions on the crack surfaces
The remote boundary conditions

24. Mode I crack problem. Statement of the problem in terms of the similarity variable
The equilibrium equations
The compatibility equation
The kinetic law of damage evolution
Constitutive equations (plane strain conditions)

25. Mode I crack problem. Statement of the problem in terms of the similarity variable
Constitutive equations (plane strain conditions)
Traction-free boundary conditions
Symmetry requirements
The asymptotic remote boundary conditions

26. Asymptotic solution The Airy stress function
Asymptotic expansions for the Airy stress function and for the continuity parameter
The three-term stress tensor asymptotic expansion

27. Nonlinear eigenvalue problem
Eigenvalus s. Numerical results

28. The boundary of the TDZ for different values of materials constants (plane strain conditions)
1 - the contour given by the two-term asymptotic expansion of the continuity parameter; 2 - the contour given by the three-term asymptotic expansion of the continuity parameter; 3- the contour given by the four-term asymptotic expansion of the continuity parameter;

29. The boundary of the TDZ for different values of materials constants (plane strain conditions)
1 - the contour given by the two-term asymptotic expansion of the continuity parameter; 2 - the contour given by the three-term asymptotic expansion of the continuity parameter; 3- the contour given by the four-term asymptotic expansion of the continuity parameter;

30. The boundary of the TDZ for different values of materials constants (plane stress conditions)
1 - the contour given by the two-term asymptotic expansion of the continuity parameter; 2 - the contour given by the three-term asymptotic expansion of the continuity parameter; 3- the contour given by the four-term asymptotic expansion of the continuity parameter;

31. Rate of growth of the totally damage zone
One can estimate the size of the TDZ and find a law according to which the TDZ evolutes
The rate of the TDZ

32. Conclusions
The effects of material damage on the asymptotic stress and creep strain rate fields of mode I and mode III cracks were analysed on the basis of continuum damage mechanics by postulating power-law creep damage theory.
Based on the similarity variable a stress analysis is carried out for the mode I crack under plane stress and plane strain conditions and for the mode III crack and assuming the existence of a totally damaged zone near the crack tip. It is found that the Hutchinson-Rice-Rosengren solution can’t be used as the remote boundary condition and the actual far field stress is obtained. The shape of the totally damaged zone is given and analysed.
It is shown that the new far field stress asymptotics can be interpreted as the intermediate asymptotic valid for times and distances at which effects of initial and boundary conditions on the stress and damage distributions are lost.
Higher order fields for damaged nonlinear antiplane shear and tensile crack problems are analytically derived. The higher order fields obtained permit the shape of the totally damaged zone modelled in the vicinity of the crack tip to be determined more exactly.