PSI - Issue 39

O.N. Belova et al. / Procedia Structural Integrity 39 (2022) 761–769 Author name / Structural Integrity Procedia 00 (2021) 000–000

763 3

G

shear modulus

f σ

the material stress fringe value (stress-optical constant)

t

specimen thickness The present analysis is aimed at determination of the coefficients in the Williams series expansion considering higher order terms in the classical specimen for linear elastic fracture mechanics – a plate with two nonparallel cracks using holographic interferometry method. Williams proposed an infinite series expansion of the elastic stress fields around the crack tip in the form: ( ) 2 / 2 1 ( ) , 1 ( , ) l k l k k ij k l ij l k r a r f σ θ θ = =∞ − = =−∞ = ∑ ∑ (3)

with index l is associated to the loading mode; coefficients l

k a are related to the geometric configuration, load and

( ) , ( ) k l ij f θ depending on stress components and mode. Analytical expressions for

mode, circumferential functions circumferential eigenfunctions

( ) , ( ) k l ij f θ are available (Karihaloo and Xiao (2001), Stepanova (2021)): ( ) ( ) ( ) ( ) 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) 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   = − + − − − − − −     = − − + − − + − −     = − − − − + − −   The displacement fields around the crack tip can be described via the Williams expansion as ( ) ( ) 2 / 2 ( ) , 1 ( , ) / , l k l k k i k l i l k u r a G r g θ θ = =∞ = =−∞ = ∑ ∑ ( ) ( ( ) ) ( ) 2,11 ( ) 2,22 ( ) 2,12

(4)

(5)

where in above equations the following notations are adopted ( ) ( ) ( ) ( ) ( ) 1,1 ( ) 1,2 ( ) ( ) k k k k g k k k g k k k θ κ θ θ κ θ = + + − − = − − − +

/ 2 ( 1) cos / 2 ( / 2) cos( / 2 2) , / 2 ( 1) sin / 2 ( / 2)sin( / 2 2) , θ θ − − / 2 ( 1) sin / 2 ( / 2)sin( / 2 2) , / 2 ( 1) cos / 2 ( / 2) cos( / 2 2) . k k k k k k k θ θ θ θ + − + − ) ( ) ) ( )

(

( ) 2,1 ( ) 2,2 k k

( ) ( )

g g

θ κ

k = − + − = − + − −

(

θ κ

l k a are the unknown mode I and mode II parameters. The SIFs can be computed from

The amplitude coefficients

1 1 2

2 1 2

I K a =

II K a = −

1 2 a is related to T-stress as

1 2 4 . a

π

π

σ = −

and

.

The goal of this

the coefficients as

1

o

study is to determine the higher order coefficients l

k a in the multi-point series expansion (1) for the double edge

notched specimen.