PSI - Issue 40

O.N. Belova et al. / Procedia Structural Integrity 40 (2022) 46–60 O.N. Belova, L.V. Stepanova / Structural Integrity Procedia 00 (2022) 000 – 000

49

4

f 

the material stress fringe value (stress-optical constant)

h

specimen thickness

The current 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 inclined parallel cracks using digital photoelasticity method. The Williams solution uses the mathematical form of an infinite series to describe the crack tip stresses according to which asymptotic expressions for the stress field in a plane medium with a traction-free crack submitted to mode I and mode II loading are presented as:   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 coefficients as

The coefficients

1 1 2 and 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. II K a    . 1 2 a is related to T-stress as 1 2 4 . a 1 o    2 1 2 I K a  