PSI - Issue 39

L.V. Stepanova et al. / Procedia Structural Integrity 39 (2022) 735–747 Author name / Structural Integrity Procedia 00 (2019) 000–000

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near crack-tip stress -strain state in power law materials by the eigenfunction expansion method results in classical nonlinear eigenvalue problems (Hutchinson (1968), Rice and Rosengren (1968), Anheuser and Gross (1994), Stepanova and Adylina (2014)) the solutions of which is of significant interest. The eigenfunction expansion method based on the perturbation theory approach is widely used in fracture mechanics (Carpinteri and Paggi (2009)). Singular fields and higher order fields in the vicinity of the crack in a power-law material are investigated in many works (Hutchinson (1968), Rice and Rosengren (1968)). Hutchinson, Rice and Rosengren (Hutchinson (1968), Rice and Rosengren (1968)) proposed the very neat approach singular fields near a sharp notch in a power-law hardening material. They solved the governing nonlinear differential equations for the stress function (describing an eigenvalue problem) by a numerical procedure. They found the eigenvalue of the corresponding to the problem. In (Anheuser and Gross (1994)) using the perturbation theory method the whole set of eigenvalues is determined. A closed form solution for the eigenvalues, determining the asymptotic behavior of the fields is analytically derived by applying the perturbation method. However, there are not such solutions for Mode I, Mode II and mixed mode crack problems. Nowadays along with the well-known eigenvalue of the HRR solution for Mode I and Mode II crack problems it is important to know the new eigenvalues different from the HRR problem and the eigenfunctions corresponding to the new eigenvalues derived. In the present paper the approach developed by Anheuser and Gross (Anheuser and Gross (1994)) is generalized for Mode I, Mode II and Mixed Mode loadings. In (Stepanova and Peksheva (2021)) the close class of problems is considered. The asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for the full range of the mode mixities are presented. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities for plane strain conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions all values of the mixity parameter to within 0.1 were considered and analysed. The asymptotic solution to the statically determinate problem is obtained by the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. Thus, some questions remains and many problems require still thorough research.

Nomenclature ij σ

stress tensor components strain tensor components

ij ε

displacement vector components

i u

material constant of the constitutive power law hardening exponent or creep exponent

α

n

the reference yield strength the reference yield strain the Airy stress potential

0 σ

0 ε

χ

polar coordinates mixity parameter

, r θ p M

eigenvalue corresponding to the nonlinear problem

λ

( ) f θ eigenfunction 0 λ

eigenvalue corresponding to the linear “undisturbed” problem

Power law material response is described by the formula ( ) 0 0 / / , n n B ε ε α σ σ σ = =

(1)