PSI - Issue 37

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

909

2

crack-tip stress and strain fields 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 solution 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.

Nomenclature ij 

stress tensor components strain tensor components

ij 

displacement vector components

i u

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



n

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

0 

0 



polar coordinates mixity parameter

, r  p M

eigenvalue corresponding to the nonlinear problem



( ) f  eigenfunction 0 

eigenvalue corresponding to the linear problem

It is well-known that in (Hutchinson (1968), Rice and Rosengren (1968)) crack-tip stress and strain singularities for pure power law material response are investigated. Power law material response is described by the formula ( ) 0 0 / / , n      = (1) where  is a material constant, 0  is the reference yield strength, n is the strain hardening exponent, 0 0 / E   = is the reference yield strain. With this assumptions the crack tip fields can be derived in the form 1/ ( 1) n + / ( 1) n n +

 

  

 

  

J

J

(2)

( , ) r   

( , ), n

( , ) r   

( , ), n

= 

= 

ij  

ij  

0

0

ij

ij

n I r

n I r

0 0  

0 0  

/ ( 1) +

n n

 

  

J

(3)

1/ ( 1) n r u n  +

( , )

( , ),

u r

= 

 

0

i

i

I

0 0  

n

where J is the path-independent integral, n I is the dimensionless J -integral (an integration constant that depends on n ). The asymptotic fields (2) – (3) are referred to as the HRR fields in the vicinity of the crack tip in power-law materials.