PSI - Issue 39

O.N. Belova et al. / Procedia Structural Integrity 39 (2022) 770–785 Author name / Structural Integrity Procedia 00 (2021) 000–000

772

3

with index m associated to the fracture mode; coefficients m

k a related to the geometric configuration, load and mode,

( ) , ( ) k m ij f θ depending on stress components and mode. Analytical expressions for circumferential , ( ) k m ij f θ are available (Karihaloo and Xiao (2001), Hello et al. (2012), Hello (2018)): ( ) ( ) ( ) ( ) 1,11 ( ) 1,22 ( ) 1,12 ( ) ( / 2) 2 / 2 ( 1) cos( / 2 1) ( / 2 1) cos( / 2 3) , ( ) ( / 2) 2 / 2 ( 1) cos( / 2 1) ( / 2 1) cos( / 2 3) , ( ) ( / 2) / 2 ( 1) sin( / 2 1) ( / 2 1)sin( / 2 3) , k k k k k k f k k k k k f k k k k k f k k k k k θ θ θ θ θ θ θ θ θ   = + + − − − − −     = − − − − + − −     = − + − − + − −   (2) ( )

angular functions

eigenfunctions

(

)

( ( ( ) ( / 2) 2 / 2 ( 1) sin( / 2 1) ( / 2 1)sin( / 2 3) , ( ) ( / 2) 2 / 2 ( 1) sin( / 2 1) ( / 2 1)sin( / 2 3) , ( ) ( / 2) / 2 ( 1) cos( / 2 1) ( / 2 1) cos( / 2 3) . k k k k k k f k k k k k f k k k k k f k k k k k θ θ θ θ θ θ θ θ θ   = − + − − − − − −     = − − + − − + − −     = − − − − + − −   ) ) ( ) 2,11 ( ) 2,22 ( ) 2,12

(3)

Nomenclature ij σ

stress tensor components around the crack tip

, r θ

polar coordinates of the system with its origin at the crack tip

N

fringe order

m

k a

coefficients of the terms of the Williams series expansion for mode I and mode II of loading

, I II K K , ( ) k m ij f θ and mode ( ) , ( ) k m i g θ ( )

mode-I and mode II stress intensity factors

known angular functions included in stress distribution related to the geometric configuration, load

known angular functions included in displacement distribution related to the geometric

configuration, load and mode m

index associated to the fracture mode

1 2 , σ σ

principal stresses Young’s modulus Poisson’s ration

E ν G

shear modulus

f σ

the material stress fringe value (stress-optical constant)

h

specimen thickness

The displacement fields around the crack tip can be described via the Williams expansion as ( ) ( ) 2 / 2 ( ) , 1 ( , ) / , m k m k k i k m i m k u r a G r g θ θ = =∞ = =−∞ = ∑ ∑

where in above equations the following notations are adopted