Data‐analysis method for hydrogen embrittlement tuning‐fork test

A new data‐analysis method is proposed for processing time‐force data obtained from a recently developed novel tuning‐fork test that has been utilized in hydrogen embrittlement studies. The analysis method is based on the application of suitable functions that are shown to provide a good fit to the data. The fitted functions allow for the calculation of first‐ and second‐order time derivatives from the initially scattered data. The negative peaks of the second‐order derivatives of the fitted functions can be associated with the more sudden changes that could be observed in the experimental data. Since these sudden changes are expected to be related to the operation of the mechanisms at the microstructural level, the present analysis provides a potentially useful method for obtaining knowledge of the operation of these mechanisms.


| INTRODUCTION
Hydrogen embrittlement (HE) is a well-recognized issue with ultrahigh-strength steels.The presence of hydrogen inside the material causes unwanted brittleness and degradation of mechanical properties, which can lead to unpredictable failures in structural applications. [1,2]In HE failure, hydrogen enters the steel in its atomic form, diffuses through metal accumulating regions of high tensile stresses and with certain critical concentration causes fracture. [3]here are various testing methods to study HE in ultrahigh-strength steels.[5][6] Other techniques are used to visualize and detect hydrogen.For example, electrochemical hydrogen permeation is used to evaluate diffusion of hydrogen through metals and thermal desorption spectroscopy provides information about the hydrogen content and binding energy of hydrogen traps. [4]9][10] TFT is a fracture test that combines constant displacement with a loadcell clamping system which allows to monitor different stages of crack propagation under cathodic hydrogen charging.From each test, a time-force (t-F) data are produced and analyzed to determine the indicator values initiationtime and time-to-fracture which are used to evaluate HE susceptibility.In addition to initiation-time and time-to-fracture, fractography of fracture surfaces reveals different crack propagation mechanisms.
The indicator values obtained from t-F data could be related to actual mechanisms occurring inside of the material during the test.The effect of PAG structure on HE behavior has been investigated in martensitic steels using TFT.For equiaxed PAG structure, crack propagation was partly intergranular, leading to faster fractures with abrupt force drops in the final stages of fracturing. [11]For elongated PAGs with transgranular quasi-cleavage crack propagation, force decrease was gradual. [10]Such mechanisms can affect the first-and second-order time derivatives of the force curve.The full quantitative theory describing the methods is, however, out of the scope of the current article, which focuses on the data-analysis method for obtaining the first and second-order derivatives.Because determination of the time derivatives is of practical importance, and they provide a possible link between quantitative theory and experiments, it is beneficial to search for a fitting technique, which allows for the calculation of the time derivatives.For this reason, in this study, a new type of numerical analysis is presented for t-F data, which yields the first and second-order time derivatives.The proposed method is also compared to standard fitting techniques using polynomial fitting and smoothing with cubic splines.
In general, data fitting methods, data analysis, and numerical time series analysis methods include polynomial regression, [12] local polynomial regression, [13] Fourier series, [14] wavelet analysis, [12] and auto-correlation analysis. [15]In a previous data analysis study, we applied a combination of Avrami type function and polynomial fitting to obtain a statistical distribution of grain sizes. [16]n another previous study, a scanning electron microscopy image gray-scale histogram was analyzed by fitting a suitable set of functions to the data. [17]In the current study, we focused on finding suitable parametric functions that have the flexibility required for describing the HE test data, but still provided a compact description with a reasonably small number of fitted parameters.Most importantly, the fitted functions enable the calculation of first-and second-order time derivatives, which are expected to be correlated to the actual physical mechanisms that operate in the HE test in future studies.2), and final fracture (3).In the beginning of each test, the force values are elevated and there is a distinct plateau region (initiation stage).After a certain period, a crack initiates at the bottom of the notch and starts propagating which can be seen as a gradual decrease in the force values (propagation stage).When the crack has propagated through the specimen, the force values are close to 0 (final fracture).

| Experimental
The t-F data are currently processed with a graphical user interface (GUI) tool to determine initiation time (t i ) and time to fracture (t f ) with a ∘ 5 limiting angle.Based on the first force value or the average of multiple first values, a start line is drawn.The angle between the start line and the following data points are compared and t i is established when the angle reaches ∘ 5 .The same principle is applied in the opposite direction to determine t f with an end line.An example of filtered t-F data (10-point moving average) with marked propagation stages, start/end lines, and determined t i and t f values are presented in Figure 2.With the limiting angle technique, t-F data analysis of different materials can be challenging since angle selection requires adjustment.Furthermore, the time derivatives can potentially be linked to a physical theory in future studies.

| Numerical fitting procedure
The data fitting was conducted in two steps, where the first step provided a good approximation with fewer parameters and sparser data, and the second step provided slightly enhanced accuracy by fitting with additional parameters to a denser data set.After the fitting, further local enhancement was achieved by additional fitting functions.The experimental data contained a large number of data points (the sampling rate was 1/s), thus there was need to make it initially more sparse to enable faster fitting.In the first step, a sparse subset was collected from the data by selecting every 100th datapoint.The function described by Equation ( 1) was applied for fitting the experimental data set for the first fitting step (1 where t is time, k is the number of terms used in the fitting, and p i j , are the fitting parameters.It may seem that the use of the two parameters p i,3 and p i,4 would be redundant, but in practice it was observed that this convention helped the fitting procedure to obtain slightly better convergence to the data.The optimized fitting parameter values were obtained by applying the Matlab [18] lsqcurvefit algorithm, which employed the trust-region-reflective algorithm. [19,20]The parameters obtained from the first fitting step were used in the second fitting step as an initial guess.In the second fitting step, denser data was used: every 10th datapoint from the full data set was used at this stage.The fitting function applied in the second fitting step is described by the below equation. (2) where the additional fitting parameter, p i,5 , was introduced in the second fitting step.Initial guess in the parameter optimization for this new parameter was zero, and the optimization then proceeded to minimize the residual by changing the value.In all fitted cases and fitting steps, it was observed that the number of terms k = 3 in Equations ( 1) and ( 2) provided a good fit.
As shown in the results section (Figure 3), the function in Equation ( 2) provided a reasonably good fit to the data.However, there was still room for locally improving the fit and an even better fit was obtained by first calculating residual R t t F t ( ) = Data( ) − ( ), where t Data( ) contained the original data and F t ( ) was the fit obtained from Equation (2).Then, additional fit was made to the residual data R t ( ).For this fitting, we employed the function described by the below equation.where w i j , are the fitting parameters and u the number of terms included in the fit.After the parameters were fitted to the residual data R t ( ), the combined function F t G t ( ) + ( ) represented an enhanced fit to the experimental data.The number of terms, u, included in the fit was chosen based on the number of time segments where the residual was seen to be significant.An example of the local residual, where the introduction of the function G t ( ) was seen to enhance the fit is shown in Figure 6 in Section 3.

| RESULTS AND DISCUSSION
The functions described in Section 2.2 were fitted to the experimentally observed t-F data.A representative fit with Equation ( 2) is shown in Figure 3a.The first-order derivative of the fitted function is shown in Figure 3b and the second-order derivative is shown in Figure 3c.The peaks of the second-order derivative in Figure 3c, denoted as A-D correspond to the rate of curvature change in Figure 3a).The terms of the first-order derivative (i.e., the terms of the integrand of Equation [2] are plotted separately in Figure 3d).
We considered a method to enhance the fitting by taking into account the local twists of the data.For this aim, an additional fit using Equation (3) was made to the residual of the function and the fit is shown in Figure 3a, as described in Section 2.2, which yielded function G t ( ).The enhanced fit F t G t ( ) + ( ) is shown in Figure 4.For certain types of data, the difficulty lies in finding the turning points of the curve, since the force appears to be decreasing very smoothly.The data shown in Figure 2 in the introduction section is an example of this type of data.The data were analyzed in the same way as described above.The final fitted result is shown in Figure 5.It can be seen that the analysis provided a good fit, and points A-D where there was a more rapid drop in the derivative could be identified with negative peaks in the second-order time derivative curve b).
Figure 6 depicts the residual of the fit from Equation (2) and the correction function G t ( ) described by Equation (3), which was fitted to the residual data in the range t = 0 … 3500s.It can be seen that the introduction of the function G t ( ) locally enhances the fit by diminishing the residual.
To compare the proposed fitting method with existing standard tools, we performed fitting to the data shown in Figure 3 using some standard tools: polynomial fitting and a smoothing with cubic splines, using Matlab [18] polyfit and csaps tools.
The polynomial fitting did not produce a good local accuracy for lower-order polynomials (order less or equal to 10).Increasing the order of the polynomial order yielded a better local accuracy for certain time intervals, but it increased oscillation of the fitted function near the edges of the data range.This is the usual well-known property of polynomial fitting.
The smoothing of the data with cubic splines yielded better performance than the polynomial fitting, and it could also be used as an alternative useful method for the data analysis in the current case, and for this reason, we describe the results here in more detail, and their difference to the current method proposed in the theory section.The cubic spline smoothing includes a smoothing parameter, [18] which controls the smoothness of the splines.A test case with the smoothing parameter value 1 × 10 −7 is shown in Figure 7.
The smoothing with cubic splines appears to perform well in fitting the function, as shown in Figure 7.The first-order derivative obtained in the example case shows similar values, as obtained with the method proposed in the theory section.The second-order derivative is more noisy and has slightly different values in comparison to the proposed method.The benefit of the cubic spline smoothing is that it is readily implemented in the Matlab code base, and includes a parameter that can be used for setting the overall smoothness.The benefit of the method proposed in the theory section is that it can smoothly describe the data with relatively few fitting parameters, with the possibility of locally enhancing the fit by including additional fitting terms, as desired.
A data analysis tool for the tuning-fork test has been developed alongside the test method itself, see. [9]utomatic data reading, point average filtering and some analysis methods have been implemented into a GUI, shown in Figure 8. Tool specifics are presented in more detail in the mentioned source.For this article, the fitted data has been manually added to the tool, and in the future, the fitting method presented in this article be implemented in the GUI as well.This allows for automatization of data handling, reduces user workload, and increases data handling consistency.

| CONCLUSIONS AND OUTLOOK
Fitting functions were tested for describing the timeforce data obtained from the recently developed tuningfork test for HE of steels.A suitable set of equations and fitting procedure was found, which was capable of adequately fitting the data and provided the calculation of smooth first and second-order time derivatives of the time-force curve.In addition, comparison to usual fitting procedures using polynomial fitting and smoothing the data with cubic splines was examined.The cubic splines method was found to be a good alternative method for analyzing the data.
In future studies, the fitting procedure could be automated and included to the previously developed graphical user interface, which would allow for easy use of the fitting procedures.
The article describes the methods that are capable for identifying the more sudden changes in the time-force data by analyzing the first and second-order derivatives.This information can be used in the future to identify the time intervals associated with the changes.Future research work could focus on explaining the observed time derivatives based on the mechanisms operating on the microstructural level during the test.

− 1 . 2 V
A direct-quenched (DQ) 500 HBW martensitic steel (0.25C-0.1Si-0.25Mnwt.%) was utilized in tuning-fork tests.Notched tuning-fork specimens were wire electrical discharge machined (WEDM) from 150 × 50 × 5 mm billets with longest side longitudinal to the rolling direction.During testing, specimens were elastically stressed at 1000 MPa and in situ charged until failure (0.1 M H SO + Hg Hg2SO4 ).Tuning-fork geometry and Von Mises stress distribution at the bottom of the notch are presented in Figure1.Stressing of the specimens was conducted with a loadcell clamp, which monitors the change in force values that is, clamp relaxation during hydrogen charging.The changes in force values are correlated to different crack propagation stages: crack initiation (1), propagation ( Tuning-fork testing (TFT) geometry with dimensions in millimeters.(b) Von Mises stress distribution at the notch.[Color figure can be viewed at wileyonlinelibrary.com]

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I G U R E 2 Time-force (t-F) data example.[Color figure can be viewed at wileyonlinelibrary.com]F I G U R E 3 (a) The function F t ( ) (red line) described by Equation (2) fitted to the data (black markers).(b) The first-time derivative of the fitted function.(c) The second time derivative of the fitted function.(d) The terms of the first-order derivative plotted separately.[Color figure can be viewed at wileyonlinelibrary.com]

F I G U R E 4
The function F t G t ( ) + ( ) compared to the experimental data, where the function G t ( ) is described by Equation (3).G t ( ) was fitted to the residual t F t Data( ) − ( ).The inset shows detail in the region contained within the rectangle.[Color figure can be viewed at wileyonlinelibrary.com]F I G U R E 5 (a) The function F t G t ( ) + ( ) compared to the experimental data for the experimental case shown in the introduction section, Figure 2. (b) The second order time derivatives of the function, showing negative peaks at points A-D.[Color figure can be viewed at wileyonlinelibrary.com]POHJONEN ET AL.

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I G U R E 6 The function G t ( ) fitted to the residual data R t ( ) for the experimental data are shown in Figure 5. [Color figure can be viewed at wileyonlinelibrary.com]F I G U R E 7 For comparison, a smooth fitting with cubic splines was performed to the data.(a) Fitted function (red solid line) and the data (black markers), (b) the time derivative of the fitted function, (c) the second-order time derivative of the fitted function.[Color figure can be viewed at wileyonlinelibrary.com]

F I G U R E 8
Graphical user interface (GUI) example for the tuning-fork testing (TFT) force reading tool.[Color figure can be viewed at wileyonlinelibrary.com]