PSI - Issue 32

O.N. Belova et al. / Procedia Structural Integrity 32 (2021) 32–41 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

38

7

3. Over-deterministic method The stress optic law relates the fringe order and the in-plane principal stresses as

1 2 / Nf h     

(4)

where f  is the material stress fringe value, N is the number of generated fringes (or fringe order) and h is the thickness of the specimen. For a plane stress problem, the stress components are related to the principal stresses as     2 2 1 2 11 22 11 22 12 , / 2 / 4             . (5) Substituting Eq. (5) in (4) one can define an error function for m thdata point       2 2 2 11 22 12 / 2 / (2 ) m m m m g Nf h             . (6) k a . Initial estimates are made for these unknown parameters and possibly the error will not be zero since the estimates are not accurate. The estimates are corrected using an iterative process based on Taylor series expansion of m g . One can arrive at the solution of the incremental change by solving a simple matrix problem. Thus, the classical over-deterministic method for the determination of the Williams series expansion is applied. Along with the conventional over-deterministic method the Broyden – Fletcher – Goldfarb – Shanno(BFGS)algorithm which is aniterative methodfor solving nonlinear optimizationproblems has been used. The optimization problem is to minimize the function m g . Table 1. Coefficients of the Williams series expansionfor the plate with the double edge notches (the experimental photoelasticity method). Coefficients of the Williams series expansion   1 1/2 1 45.498985 a Pa m   Eq. (6) is a non-linear equation in terms of the unknown parameters m

1 2 1 1 4 1

8.494524

a

Pa

 



 

1/2

3 8.675513 / a Pa m   3.546581 / a Pa m   5 1.185281 / a Pa cm    1 2 6 1.660511 / a Pa cm   1 7 1.432483 / a Pa m   9 1.051328 / a Pa m    1 4 10 0.017 / a kg cm    1 11 0.639445 / a Pa m     1 8 1 3 0.628455 / a Pa m   



3/2



5/2





7/2



9/2





1 12

5

0.015332 /

a

Pa m

 





1

11/2

13 0.423673 / a Pa m   1 14 0.007816 / a Pa m  



6





1 15

13/2

0.413682 /

a

Pa m

 