PSI - Issue 37

O.N. Belova et al. / Procedia Structural Integrity 37 (2022) 888–899 Author name / Structural Integrity Procedia 00 (2021) 000 – 000

891

4

h

specimen thickness

The study is aimed at determination of the coefficients in the Williams series expansion considering higher order terms in a rather new specimen for linear fracture mechanics – a plate with two nonparallel cracks using digital photoelasticity method. Williams proposed an infinite series expansion of the elastic stress fields around the crack tip in the form: ( ) 2 /2 1 ( ) , 1 ( , ) m k m k k ij k m ij m k r a r f    = = − = =− =   (1) with index m associated to the fracture mode; coefficients m k a related to the geometric configuration, load and mode, angular functions ( ) , ( ) k m ij f  depending on stress components and mode. Analytical expressions for circumferential eigenfunctions ( ) , ( ) k m ij f  are available (Karihaloo and Xiao (2001)): ( ) ( ) ( ) ( ) 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            = + + − − − − −     = − − − − + − −     = − + − − + − −   (2)

(

)

( ( ( ) ( / 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            = − + − − − − − −     = − − + − − + − −     = − − − − + − −   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   = = = =− =   ) ) ( ) 2,11 ( ) 2,22 ( ) 2,12

(3)

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       = + + − − = − − − +

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

(

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

( ) ( )

g g

 

k = − + − = − + − −

(

 

m k a are the unknown mode I parameters. The SIFs can be computed from the

The amplitude coefficients

coefficients as Eqs. (2) are valid for pure crack opening loads whereas Eqs. (3) are valid for pure shear loads. The goal of this study is to determine the higher order coefficients m k a in the multi-point series expansion (1) for the double edge notched specimen. 2. Photoelastic experiments: experimental setup and experimental procedure The experimental setup used is shown in fig. 1. The experimental setup consists of a traditional polariscope setup and a digital image camera. All the specimens in this work were made by casting of polycarbonate. Experimental specimens were machined from the sheet to get the test specimens. The study used ten tension strips 120 mm long by and II K a = −  . 1 2 a is related to T-stress as 1 2 4 . a 1 o  = − 1 1 2 I K a =  2 1 2