PSI - Issue 28

R.M. Zhabbarov et al. / Procedia Structural Integrity 28 (2020) 1774–1780 Author name / Structural Integrity Procedia 00 (2019) 000–000

1778

5

Fig. 5. Fringe skeleton - fringe thinned image.

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 and 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 th data 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. The results are given in Table 1. 4. Conclusions In this study coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate are obtained by the use of the digital photoelasticity method and finite element analysis. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give more accurate estimation of stress, strain and displacement fields and to extend the domain of validity for the Williams series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. The developed tool allows us to find points that belong to isochromatic fringes with minimal light intensity. The analysis of the experimental data uses a Java application programmed for the advanced determination of the fracture characteristics: coefficients of the Williams series expansion (WE) for the stress field in the vicinity of the crack tip. The tool allows us to collect experimental points from the photoelasticity tests on the cracked specimens. An automatic routine implemented as a Java application permits to determine the values of coefficients of higher order terms of the WE that describe crack-tip fields. These values are calculated using the over deterministic method which is also applied to the results of the finite element analysis of some mode I and mixed mode test geometry. Thus, the Java application provides an analytical reconstruction of the crack-tip stress field by means of the truncated WE and enables detailed analysis of the crack-tip stress field approximation. The developed Eq. (6) is a non-linear equation in terms of the unknown parameters m