PSI - Issue 59

Bernadett Spisák et al. / Procedia Structural Integrity 59 (2024) 3–10 B. Spisa ´ k et al. / Structural Integrity Procedia 00 (2019) 000 – 000

8

6

sets. To do this, the so- called ”fit” funct ion was used, where nonlinear regression was chosen. Since the toolbox library did not contain the desired parametric equation, a custom equation was created, which, based on equation (1), took the following form:

W a

2



(2)

   

   

2

3

4

P

W a

W a

W a

W a

  

  

  

   

  

   

  

  

,

K

 b c

d

e

f



1

1/ 2

3/ 2

BW

W a

  

  

1



from which the parameters b , c , d , e and f are to be determined. The least squares method was used to fit the data. The fitting requires a parametric model that relates the response data to the predictor data by one or more coefficients. The result of the fitting process is an estimate of the coefficients of the model, in which the least squares method minimises the sum of squares of the residuals. Further subtypes of least squares fitting can be distinguished, of which, the nonlinear least squares method was applied, since the equation which was chosen is nonlinear. Nonlinear models are more difficult to fit than linear models because the coefficients cannot be estimated using simple matrix techniques. Instead, an iterative approach is needed, following the steps below.  Initial estimation of each coefficient: for some nonlinear models, a heuristic approach can be given that yields reasonable initial values. For other models, random values within the interval 0 and 1 are required.  Construct a curve fitted to the actual coefficients.  Adjust coefficients and determine whether the fit improves. The direction and magnitude of the adjustment depends on the fitting algorithm. The toolbox provides the followings. – Reliability Region – it is the default algorithm and can be used when coefficient constraints are specified. It can solve difficult nonlinear problems more efficiently than the other algorithms. – Levenberg-Marquardt - it has been used for many years and is the most proven method for a wide range of nonlinear models and initial values. If the confidence region algorithm does not produce a reasonable fit and there are no coefficient constraints, then it is worth using the Levenberg-Marquardt algorithm.  Repeat the process, returning to the second step, until the fit reaches the specified convergence criteria. Taking this knowledge into account, the surface fit was performed on the curves defined for the hybrid 1T specimen, as illustrated in Figure 5. 3.4. Formula for the determination of the stress intensity factor for the hybrid specimen The method described in the previous subsection was also applied to the other specimen sizes (1T, 0.25T and 0.16T). Then, the minimum and maximum values for each parameter ( b , c , d , e , f ) and their average were determined for each dimension. These values are presented in Table 2. Finally, in order to use the formula to determine the stress intensity factor for all specimen sizes, it was necessary to unify the parameters. For this purpose, the maximum value was chosen from the minimum values of the parameters and the minimum value from the maximum values, ensuring that the formula could be applied to all specimen sizes, and the average of these two values was determined. Thus, by transforming equation (2), the stress intensity factor can be written as a function of the load force and the a/W ratio as follows:

W a

(3)

2



   

   

2

3

4

P

W a

W a

W a

W a

  

   

  

   

  

   

  

  

0.127 1.286

3.209

3.316

1.233

,

K



 

1

1/ 2

3/ 2

BW

W a

  

  

1

