PSI - Issue 39

Riccardo Cappello et al. / Procedia Structural Integrity 39 (2022) 179–193 Author name / Structural Integrity Procedia 00 (2019) 000–000

182

4

        

         

... ...

...

...

1  ∆   ∆    ×    ∆   2 3 .. . A A A

...           ∆   . .. i

2

4 6

co

s(

/ 2) ...

r

i θ

i

T

T = −  

o th K

r

(4)

i

... ...

...

...

N  

 



m

w

m N ×

w

Given the high number of experimental points that can be used from the IR maps, eq. (4) represents an over deterministic system of linear equations that can be solved using a suitable software such as Matlab. The first and second terms of the series respectively carry the information related to the Stress Intensity Factor and the T-stress. In particular, they are linked by these simple equations: ( ) 1 2 2 4 I K A T stress A π ∆ ∆ = − = ∆ − ∆ (5) The points used to solve the Over Deterministic Least Squares Fitting (ODLSF) are collected from a pac-man shaped area close to the crack tip, as shown in Figure 1. The annular area is defined by an external radius, r max , an internal radius, r min , and a fixed angular opening from 22.5° to 337.5°. An accurate definition of the data input collection area is required to perform reliable evaluations of the fracture parameters: points too close to the crack tip do not belong to the LEFM singularity dominated zone, hence also the thermoelastic law is no longer valid; points too distant from the crack tip may see a diminishing influence of the singularity [26]; the chosen angular opening extent allows to avoid points too close to the wake of the crack, where the signal might be corrupted or very low. In this work, the coefficient of determination R 2 is used to evaluate the quality of the fitting, comparing the experimental data with the results obtained by fitting the Williams’ model [26]. The coefficient of determination ranges between 0 and 1 (0 < R 2 < 1), where the value R 2 = 1 indicates that the model perfectly fit the experimental data.

T [°C]

0.26

0.24

0.22

0.2

0.18

r

max

0.16

0.14

0.12

r

0.1

min

0.08

Data input

0.06

0.04

0.02

0

Figure 1 – Data collection input area for the least squares fitting of the Williams’ series

One key aspect is the accurate determination of the crack tip location, since the coordinates of the experimental points represent an input of the model. Hence, an inaccurate crack tip location would yield inaccuracies in the SIF evaluation. In this work, the identification of the crack tip has been performed starting from a user-defined initial guess point, and the crack tip coordinates are then optimized iteratively by using a pattern-search algorithm through the built-in Matlab function, that allows the solution of non-linear optimization problems without requiring the evaluation of derivatives and requires short computational times [27]. The optimization procedure aimed at finding the crack tip coordinates that yields the best fitting of the Williams’ stress formulation, associated with the higher R 2 value.