PSI - Issue 37

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

895

8

Fig. 9. The isoclinic data

Fig. 10. The isoclinic data

3. The over-deterministic method and the Broyden – Fletcher – Goldfarb – Shanno algorithm 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 ,

/ 2

/ 4

 −

11  

+



(5)

.

22

12

Substituting Eq. (5) in (4) one can define an error function for m th data point ( ) ( ) ( ) 2 2 2 11 22 12 / 2 / (2 ) m m m m g Nf h     =  −  + −   .

(6)

Eq. (6) is a non-linear equation in terms of the unknown parameters m k a . Initial estimates should be 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. It should be noted that here the over-deterministic method in the form based on the stress field is used. According to numerous computational experiments performed one can conclude that this form relying on the stress field has indisputable advantages compared to the classical over-