PSI - Issue 39

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

737

3

where α is a material constant, 0 / E ε σ = is the reference yield strain. For these materials the first-order asymptotic field is called as the HRR singularity field defined as (Hutchinson (1968), Rice and Rosengren (1968)) 1/ ( 1) n + / ( 1) n n + 0 σ is the reference yield strength, n is the strain hardening exponent, 0

 

  

 

  

J

J

(2)

( , ) r σ θ σ

( , ), n

( , ) r ε θ αε

( , ), n



ij σ θ 

= 

= 

ij ε θ

0

0

ij

ij

n I r

n I r

0 0 ασ ε

0 0 ασ ε

/ ( 1) +

n n

 

  

J

(3)

1/ ( 1) n r u n θ + 

( , )

( , ),

u r

= 

θ αε

0

i

i

I

0 0 ασ ε

n

where J is the path-independent integral, n I is the dimensionless J -integral (an integration constant that depends on n ). 2. Fundamental equations In either generalized plane stress or plane strain, equilibrium is ensured for all stresses derived from a stress function by 2 2 (4) where cylindrical coordinates , r θ are centered at the right end of the crack. According to (Hutchinson (1968), Rice and Rosengren (1968)) the non-dimensional stress function and the radial coordinate are given in terms of the dimensional quantities by ( ) 2 0 / , L χ χ σ = / , r r L = where L is the half length of the crack. Condition of compatibility is written in the form ( ) 2 2 2 0. r rr rr r r r r r r r r θθ θ ε ε ε ε θ θ  ∂  ∂ ∂ ∂ ∂ ∂   + − − =     ∂ ∂ ∂ ∂ ∂ ∂     (5) The power-law constitutive relations for the plane strain conditions are described by expressions ( ) 1 1 3 3 , , 4 2 n n rr e rr r e r θθ θθ θ θ ε ε ασ σ σ ε ασ σ − − = − = − = (6) where the von Mises equivalent stress is expressed by ( ) 2 2 2 3 / 4 3 e rr r θθ θ σ σ σ σ = − + . For plane stress conditions the constitutive equations have the form ( ) ( ) 1 1 1 1 1 3 2 , 2 , , 2 2 2 n n n rr e rr e rr r e r θθ θθ θθ θ θ ε ασ σ σ ε ασ σ σ ε ασ σ − − − = − = − = (7) where the von Mises equivalent stress is expressed by 2 2 2 3 . e rr rr r θθ θθ θ σ σ σ σ σ σ = + − + An asymptotic expansion of the solution is attempted in the separable form ( ) 1 1 1 2 , ( ) ( ) ... r r f r f λ γ χ θ θ θ + + = + + (8) where, if the first term is to be singled out as the dominant one, . λ γ < In (Hutchinson (1968), Rice and Rosengren (1968)) one can see the two-term asymptotic expansion. However, the attention is restricted to only the dominant (leading) term in the asymptotic series expansion ( ) 1 , ( ). r r f λ χ θ θ + = (9) The resulting ordinary differential equation following from (5) is homogeneous in ( ) f θ and is associated with homogeneous boundary conditions has the form of the nonlinear eigenvalue equation for λ : 2 2 2 1 1 1 ∂  ∂  ( , ) r σ θ , ( , ) r σ θ , ( , ) r σ θ , rr r r r ∂ r r ∂ ∂  r r θθ θ χ χ χ χ θ θ ∂ ∂ ∂ ∂ = + = = −    ∂