1. Summary
A Hopkinson bar (HB) is used to measure a transient elastic pulse generated by the impact of a near-field blast or bullets [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12]. A split Hopkinson bar (SHB) [
13,
14,
15,
16,
17,
18,
19,
20], which is also called a Kolsky bar, is used to measure dynamic material properties such as the stress–strain and strain rate–strain curves of versatile materials at strain rates of approximately 10
2–10
4 s
−1. These curves, together with the accurately extracted quasi-static material properties [
21], are generally used to calibrate a strain-rate-dependent constitutive model [
22], which is indispensable for the simulation of the dynamic deformation behavior of solids and structures [
23,
24]. Schematics of the HB and compressive SHB are presented in
Figure 1.
In an SHB experiment, the specimen properties are determined using signal processing equations [
13,
14,
15,
16,
17,
18,
19,
20] that correlate the measured elastic wave profiles in the bar to the specimen quantities (stress, strain rate, strain):
where
,
, and
are the nominal stress, nominal strain rate, and nominal strain of the specimen, respectively;
is the reflected pulse strain recorded in the incident bar;
is the transmitted pulse strain measured in the transmitted bar;
A and
L denote the initial cross-sectional area and initial length of the specimen, respectively;
Ao, Eo, and
co denote the cross-sectional area, elastic modulus, and sound speed of the bar, respectively; and
t is the time. These notations are explained here (instead of in Nomenclature for Dispersion Correction) as they are limited to the processing of SHB signals (
and
). According to the signal-processing equations (Equations (1)–(3)), the precise measurement of the specimen quantities (
,
, and
) directly depends on the accuracy of the
value of the bar (
, where
is the bar density).
The shape of the elastic wave in the HB and SHB distends as it travels across the bar; this phenomenon is called dispersion. The wave profile is generally measured at the interim axial position of the bar. The location of interest for the HB, is at the front surface where the elastic wave enters the bar; in the case of the SHB, the location of interest is the specimen position. Therefore, the measured wave profiles in the SHB and HB need to be corrected to obtain the wave profiles at the locations of interest; this correction process is called dispersion correction [
2,
9,
10,
11,
12,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36].
To perform dispersion correction, the sound speed (
cdc) vs. frequency (
fdc) relationship of the wave components that constitute the overall elastic wave must be known in advance; this relationship can be determined by solving the Pochhammer–Chree equation (PCE) [
37,
38,
39,
40]. The PCE solver in [
41,
42] handles the PCE in terms of the normalized frequency (
F) and normalized sound speed (
C). It solves the PCE first at arbitrary
F values to obtain the (
F,
C) matrix by solely using the Poisson’s ratio (
) information. Then, it subsequently obtains the solutions (
Cdc vs.
Fdc relationship) at exact
F values (
Fdc =
a fdc/
co) necessary for dispersion correction, which are determined using the information of the one-dimensional sound speed (
) and bar radius (
a). Therefore, the precise calibration of
and
for the HB and SHB is fundamental.
However, the manufacturer-provided literature values of
and
have been more readily available than the calibrated ones by the user of the bar, although the determination method of the former has hardly been disclosed. As regards the calibration studies of the user of the bar, Reference [
43] calibrated the bar properties using a limited number of frequencies involved in wave profile. However, the calibration based on a thorough dispersion correction using all involved frequencies was scarce. Accordingly, the author recently presented a method for calibrating the
and
values via iterative and thorough dispersion correction of elastic wave [
44]. This paper subsequently presents the manual of the foregoing method, which was manifested in Excel
® and MATLAB
® templates. These templates are available in a publicly accessible repository [
45].
3. Methods
Dispersion correction consists of a series of processes: (i) target signal preparation from the experimentally measured profile, (ii) mathematical modeling of the target signal (Fourier synthesis), and (iii) phase shift (dispersion correction) of the Fourier-synthesized function using the
Cdc vs.
Fdc relationship obtained by solving the PCE [
37,
38,
39,
40]. Dispersion correction assumes that the
and
values are known for the considered bar.
Section 3.1. (Dispersion Correction) explains how the above processes can be carried out under the given set of
and
. This section explains the tips for using the Excel
® template (dispersion_correction.xlsm) and PCE solver (PCE_solver_n1.m).
Section 3.2. (Iterative Dispersion Correction) explains the case where
and
values are unknown. This section explains the iterative dispersion correction process using the “dispersion_correction_iteration.m” program.
3.1. Dispersion Correction
3.1.1. Fundamental Parameters
A part of the Excel
® spreadsheet (dispersion_correction.xlsm) for carrying out (i) target signal preparation, (ii) Fourier synthesis, and (iii) phase shift (dispersion correction) is presented in
Figure 2. In
Figure 2, the quantities in blue are inputted by the user; this process is explained in
Section 3.1.1 and
Section 3.1.2. The quantities in black are calculated by the template itself using the inputted quantities. The quantities in green (
Fdc(k) and
Cdc(k)) are inputted later in the Fourier synthesis stage (
Section 3.1.3).
First, the values of the Poisson’s ratio () and one-dimensional sound speed () are inputted to cells A2 and B2, respectively. As mentioned, the process of dispersion correction using the Excel template® (dispersion_correction.xlsm) and PCE solver (PCE_solver_n1.m) herein assumes that the and values are known in advance. The value is not used in the Excel template itself (it is actually used in the PCE solver), but it appears in cell A2 because this template records the values of and for which it is used.
Second, the user must input the values of Δ
t, t0, and
a. Then, as mentioned, the Excel
® template calculates all the necessary fundamental parameters (in black) for the dispersion correction process, including the values of
Fdc_max and
Ny in cells A4 and B4. These values will be copied and pasted later to the input parameter section of the PCE solver (
Section 3.1.4).
Third, the travel distances z
1 and z
2 are inputted from the measured position of the reference (incident) pulse. This information is used in the phase-shifting stage (
Section 3.1.5).
Finally, the experimental data need to be inputted to the spreadsheet from cells A6 and B6 toward the bottom cells. The recommended procedure for inputting the experimental data to the spreadsheet shown in
Figure 2 is as follows:
- (1)
Fill in new experimental data to the cells starting from cells A6 and B6 toward the bottom (in blue) after deleting the previous data. The template assumes that the number of data in the time window (Nt) is an even number for the calculation of the maximum number of frequency components in Fourier synthesis (Ny = Nt/2).
- (2)
Suppose that the time window of the user is, for instance, 1000 μs, and the sampling time interval is 0.2 μs (Nt = 5000). Then, ensure that the time data starts from zero and ends at 999.8 μs.
- (3)
Adjust the data range of all imbedded figures in the template.
As mentioned, an example profile determined in experiment is available in the cells B6 and below of the Excel® template (dispersion_correction.xlsm). The same data are available in the “experiment.csv” file.
In
Figure 2, the cells from C6 and bellow are named as “Target for Synthesis”. The method of filling these cells will be explained in the next section (
Section 3.1.2).
Figure 3 shows one of the figures embedded in the Excel
® template (dispersion_correction.xlsm) using the data available in the same spreadsheet (
Figure 2). In
Figure 3, the quantity in the ordinate (signal magnitude) is dimensionless [
44] and a positive sign is assigned to tension. Explaining the experimentally obtained wave profile in
Figure 3, the first pulse is the pulse that is incident on the bar, which is used as the reference pulse herein. The second pulse is the wave profile after traveling 2103 mm from the position where the first pulse was measured, and the third pulse is that after traveling 4000 mm [
44].
3.1.2. Target Signal Preparation
The process of target signal preparation for Fourier synthesis begins with the determination of the start and end times of the reference pulse. The Excel
® template can assist in this step. The portion of the Excel
® spreadsheet specifying the start and end positions of the reference pulse is presented in
Figure 4. The principle of determining them is described in [
44]. Once the start and end times are inputted to cells L3 and N3, respectively, following the principle, their positions are displayed in the figure embedded in the spreadsheet (
Figure 3) up to the vertical amplitude specified in cell M3.
If the locations of the position bars in
Figure 3 are satisfactory in visual inspection, the cell in column A corresponding to the displayed time (e.g., 116.2 μs in L3) can be visited to determine the exact onset/end times by referring to the pulse magnitude value in column B at the considered time in column A. Note that the pulse magnitude (in column B) at the onset/end times will be zero padded later in this section. If the signal magnitudes in column B near the displayed time in L3 (116.2 μs) are examined under such notion, the onset/end times generally need to be finally tuned because the previous time (116.2 μs) was determined roughly based on the visual inspection of the pulse profile. In case the onset/end times are tuned, update the values in cells L3 and N3 with the finally tuned onset/end times. In the case of the example wave profile considered, 116.2 μs was the finally tuned value of the start time.
Figure 4 also displays the onset/end times of the error ranges at travel distances z
1 and z
2. The error of the dispersion-corrected wave profiles with reference to the experimental profiles will be calculated by the method that will be presented later in
Section 3.2 using the specified time ranges. These ranges can be determined similarly to the determination of the start/end times of the first (reference) pulse.
After the start and end times of the reference pulse are determined as described above, the target signal for Fourier synthesis can be prepared in C6 and cells below:
- (1)
Copy data from B6 and below, and paste them into cells C6 and below.
- (2)
Zero-pad values in column C from time zero to the start time of the reference pulse.
- (3)
Perform the same operation as (2) from the end time of the reference pulse to the end time.
Once the target signal data are prepared in the above manner in column C (from cell C6 toward the bottom), the prepared target signal is displayed in the figure embedded in the same spreadsheet (see
Figure 3).
3.1.3. Fourier Synthesis
Time-dependent data
can be expressed as
, where
is the magnitude constant with the dimension of
(mV herein),
is the non-dimensional shape function, and
t is the time. In this study,
M is set as 1 mV, and
is displayed in all of the figures illustrating wave profiles. Fourier synthesis refers to the mathematical modeling of the target signal using the following formula:
where
n is the index for describing time points spanning from 0 to
Nt − 1;
Nt is the number of data points in the time window with a fundamental period (
);
is the time interval of sampling;
is the fundamental frequency (=
);
k is the index for describing the Fourier series terms spanning from 1 to
K; K is the summation limit of the Fourier series, which is the Nyquist number (
Ny); and
,
, and
are the Fourier coefficients given,
Equations (5)–(7) were implemented in a macro program of the Excel
® template [
44] in Visual Basic Application (VBA) language (Alt-F11 for editing the program). Fourier synthesis of the target signal can be performed simply by clicking the “Synthesize” icon in the spreadsheet (see
Figure 2).
Once the Fourier synthesis process is completed using Equations (4)–(7), the values of
A0, Ak, and
Bk determined during the synthesis process are displayed in the same Excel
® spreadsheet (see
Figure 2). The synthesized signal is also plotted in the embedded figure of the Excel
® template (see
Figure 5).
3.1.4. Pochhammer–Chree Equation Solver
The PCE solver (PCE_solver_n1.m) solves the following PCE equation [
37,
38,
39,
40] written in physics-friendly non-dimensional variables (
C and
F) [
41]:
where
G is the value of the Pochhammer–Chree (PC) function, and
J0 and
J1 are the Bessel functions of the first kind of order zero and one, respectively.
The solution of Equation (8) gives the relationship between C and F for a given value, which is obtained by the solver (PCE_solver_n1.m) via linear extrapolation and the bisection method. It eventually derives the relationship between Cdc and Fdc via linear interpolation of the (F,C) matrix and the bisection method.
A portion of the MATLAB
® program (PCE_solver_n1.m) that specifies the required input parameters is presented in
Figure 6. The
value therein is required to derive the
C vs.
F relationship. In
Figure 6, the values of
Fdc_max and
Ny are copied from the Excel
® template (cells A4 and B4 in
Figure 2), which were calculated using information on
and
a values. These values specify the
Fdc values at which the
Cdc values are obtained by the solver.
The solver first obtains the
C vs.
F curve at
F intervals of 0.001; the result is shown in
Figure 7a. It subsequently obtains the
Cdc vs.
Fdc relationship via linear interpolation of the
C vs.
F curve, followed by carrying out the bisection method. The obtained the
Cdc vs.
Fdc curve is shown in
Figure 7b, where the
Fdc interval (
dFdc =
Fdc_max/
Ny) is 0.001466 for the experimental wave profile considered. To emphasize the difference of data interval (0.001 vs. 0.001466), a solid curve is used in
Figure 7a while open circles in
Figure 7b, respectively. The
Fdc vs.
Cdc relationship illustrated in
Figure 7b is used for the phase shifting of the wave components that constitute the overall elastic pulse. Once the program terminates, the
Fdc vs.
Cdc data used to plot
Figure 7b are written in the “Fdc_Cdc.xlsx” file. After opening this file, the cell values from and below A2 and B2 therein need to be copied, and then pasted into the cells from and below A6 and B6 (
Fdc(k) and
Cdc(k)) in the “dispersion_correction.xlsm” file (
Figure 2).
3.1.5. Dispersion Correction
As mentioned, dispersion correction is the process of predicting the wave profile at a given travel distance. The Fourier series expression for the elastic wave after traveling a distance
is
where
is positive for forward travel and negative for backward travel. Dispersion correction using Equation (9) was implemented in the Excel
® template in VBA.
For carrying out dispersion correction, the values of and must be assumed first. The literature values can be considered as the starting point. Once the assumed set of and are inputted to the respective cells in the Excel® template (A2 and B2, respectively) together with the quantities in blue, the maximum limit of the normalized frequency in dispersion correction (Fdc_max) and the number of Fdc values in dispersion correction (Ny = Fdc_max/dFdc) are displayed in cells A4 and B4, respectively.
As mentioned, the range of error calculation (see
Figure 8) must be specified in the spreadsheet before carrying out dispersion correction. The principle of determining the error calculation range for the traveled pulses at z
1 and z
2 is described in [
44]. The determination of the error calculation range for each traveled pulse can be assisted by using the Excel
® template (dispersion_correction.xlsm), similarly to the determination process of the range of the first (reference) pulse (
Section 3.1.2 and
Figure 4).
Once the set of
and
together with the error calculation ranges are specified in the spreadsheet (
Figure 2), dispersion correction can be carried out using the Excel
® template following the procedure:
- (1)
Open the “Fdc-Cdc.xlsx” file, copy the highlighted portion (PCE solutions) therein, and paste it into the Fdc(k) and Cdc(k) columns (green colored) starting from cells H6 and I6 toward the bottom of the spreadsheet.
- (2)
Close the “Fdc-Cdc.xlsx” file for use in the PCE solver.
- (3)
Click the “Shift” icon in the Excel
® spreadsheet (see
Figure 2) to carry out the phase shift (dispersion correction).
Clicking the “Shift” icon renders the macro program embedded in the Excel® template to calculate the ck vs. fk relationship from the Ck vs. Fk relationship using information on a and co. The determined values of ck at fk are eventually used for dispersion correction using Equation (9), where and (k = 1, 2, 3…).
Once dispersion correction is completed, the Excel
® template calculates the error of the predicted (dispersion-corrected, i.e., phase-shifted) wave profiles with reference to the measured profiles at travel distances of at z
1 and z
2 based on the formula:
In Equation (10),
i is the index of the time data of the pulse at a travel distance of either z
1 or z
2;
x is the value of the non-dimensional function
; superscripts dc and exp denote dispersion corrected and experiment, respectively;
Np is the number of data in a given traveled pulse; and
is the maximum magnitude (positive) of the measured pulse in a given pulse. These notations are not listed in Nomenclature for Dispersion Correction but explained here for readability; Equation (10) is limited to the error calculation. Once the macro program completes the error calculation, the dispersion-corrected (phase-shifted) wave profiles at travel distances z
1 and z
2 are displayed in the figures embedded in the Excel
® template (see
Figure 8).
As the error between the dispersion-corrected and measured signals decreases, the precise calibration of the
and
set is impractical if solely the Excel
® template (dispersion_correction.xlsm) and PCE solver (PCE_solver_n1.m) are used as it would require an overly large number of trials in assuming
and
sets. The process of precise calibration can be facilitated greatly by utilizing the calibrator program (explained in
Section 3.2.) and the foregoing Excel
® template (dispersion_ correction.xlsm) as the preprocessor.
3.2. Iterative Dispersion Correction
As mentioned, iterative dispersion correction is carried out when and values are unknown. In such a case, once the Fourier synthesis is completed, the phase shift (dispersion correction) of the Fourier-synthesized function is carried out iteratively for a range of and values. At each iteration step, a set of and values is assumed and the PCE is solved for the assumed set. The and values are determined as the calibrated values when the dispersion-corrected (predicted; phase-shifted) wave profiles after traveling certain distances in the bar are in reasonable agreement with the experimental profiles. This section explains the iterative dispersion correction process using the calibrator program (dispersion_correction_iteration.m).
3.2.1. Skeleton of Calibrator Program
The algorithm of the calibrator program that carries out iterative dispersion correction is presented in
Figure 9a. The algorithm consists of the main part, an optimization function (fminsearch), and two subroutines. The feedbacks among the main part and subroutines are illustrated in
Figure 9b.
Briefing the algorithm and feedbacks, the main part of the solver synthesizes the Fourier signal and inputs the initial values of and to the optimization function “fminsearch”. This function determines the values of and (except for the first run) and inputs them to the “executor” subroutine, and receives the error value from the “executor” subroutine. This subroutine (executor) calculates the Fdc_max and dFdc values using information, inputs them with to the “PCE solver” subroutine, and receives the (Cdc, Fdc) matrix outputted from it. The “executor” subroutine subsequently calculates the (cdc, fdc) matrix from the (Cdc, Fdc) matrix, carries out dispersion correction, and calculates the error. Once the error value is less than the prescribed value (variable named “error_limit” in the calibrator program) or the number of iterations reaches its prescribed limit (variable named “counr_limit”), the program terminates and outputs the current and values to the user as the calibrated result. Otherwise, the “executor” subroutine reports the error value to the “fminsearch” function for the determination of the next set of and for a new iteration.
3.2.2. Preprocessing and Input Parameters
The Excel
® template (dispersion_correction.xlsm) and PCE solver (PCE_solver_n1.m) can be used to obtain a good initial guess set of
and
, which can prevent the optimization algorithm from reaching any local minimum in the iterative optimization process. A good initial guess set also allows the calibrator program to reach the optimized
and
set more quickly. Nevertheless, it was found via separate trials that the calibrator program successfully reached the optimum set of
and
from even fairly far initial guess values of
and
from the optimized values, which will be presented later (
Section 3.2.7).
The input parameters sections I and II in the calibrator program are shown in
Figure 10. Before executing this program, the “experiment.csv” file needs to be prepared in a 1 ×
Nt matrix (
), which contains the wave profile data measured in the experiment. Then, the input parameters sections I and II (
Figure 10) need to be inputted. The variables in the input parameters section II include the start and end times of the reference (incident) pulse, which, as mentioned, can be suitably determined using the Excel
® template (
Figure 2 and
Figure 4).
3.2.3. First Time Run
The calibrator program can be executed once the initial guess values of
and
together with the parameters in
Figure 10 are inputted. The algorithm reads the experimental signal in the “experiment.csv” file, prepares the target signal with zero-padded portions, and carries out the Fourier synthesis of the target signal using Equations (4)–(7). The algorithm then calculates the
Fdc_max and
Ny values, and transfers them together with the
value to the PCE solver subroutine; the calibrator program (dispersion_correction_iteration.m) includes the PCE solver (PCE_solver_n1.m) as a subroutine. Once the algorithm receives the solutions (
Cdc and
Fdc matrix) from the PCE solver subroutine, it converts the (
Cdc, Fdc) to (
cdc, fdc) matrix, and subsequently performs dispersion correction using Equation (9) to predict the wave profiles at travel distances of z
1 and z
2.
3.2.4. Error Calculation
To calculate the error of the predicted (phase-shifted, i.e., dispersion-corrected) wave profiles with reference to the measured ones at two travel distances, the onset and end points of each traveled pulse in the experiment need to be identified. As mentioned, this identification process can be assisted if the Excel
® template (dispersion_correction.xlsm) is used. The determined error calculation ranges for the two traveled pulses considered are marked in
Figure 8. The end point of the traveled pulse was selected to include the tail part of the traveled pulse, which reason is explained in Reference [
44]. The calibrator program refers to the onset and end times of the two traveled pulses at distances z
1 and z
2, respectively, and calculates the average error value of the two pulses based on Equation (10) for each pulse.
3.2.5. Termination Conditions of the Iteration Loop
The calibrator program checks the user-specified termination conditions at each iteration step. Two user-specified termination conditions are set in the calibrator program: “error limit” of the dispersion corrected wave profiles and “count limit” of the iteration number. If neither of these user-specified termination conditions is met, the program (“fminsearch” function available in MATLAB®) determines a new set of and by reflecting the error value (the average error of the two dispersion-corrected pulses at travel distances of z1 and z2) at the current iteration step. The program terminates if the number of iterations reaches the predefined count limit or if the calculated error value reaches the predefined error limit. If the algorithm exits the iteration loop by one of the user-specified termination conditions, the program plots the dispersion-corrected result in the final iteration step. The dispersion-corrected signals in the final iteration stage are also written in the “signals.xlsx” file.
In addition to the foregoing user-specified termination conditions, the calibrator program sets the “fminsearch” function to check the termination condition additionally via the TolX option, which is imbedded in the “fminsearch” function. The TolX option also terminates the iteration loop if the and values do not change further in a number of iteration steps. The tolerance limit for this termination option (TolX) was set as 1 10−6 in the current calibrator program, while the default value is 1 10−4. Other “fminsearch”-governed termination options are also available as can be observed in the calibrator program near the TolX option. The exit of the iteration loop caused by the termination conditions of the “fminsearch” function itself (e.g., the TolX option) also leads to the plots of the dispersion-corrected signals together with the “signals.xlsx” file, like in the case of the terminations by the user-specified conditions.
3.2.6. Calibration Result
Unless terminated by the user-specified conditions, the calibrator program exits the iteration loop by the termination options of the “fminsearch” itself when both
ν and
co values with six decimal places do not vary appreciably in the iterations. As mentioned, when the calibrator program is terminated, the dispersion corrected wave profiles obtained at the final iteration step are plotted together with the display of the
and
values used in the final step. The example of the calibration result displayed when
= 0.335050 and
= 4588.233496 m/s is illustrated in
Figure 11.
3.2.7. Dependency of the Calibrated Values on Initial Guess Values
This section investigates whether the calibrated values of and are dependent on the initially guessed values of them. If the Excel template (dispersion_correction.xlsm) and PCE solver (PCE_solver_n1.m) are used as the preprocessors using the considered experimental profiles, an initial guess set of = 0.30 and = 4600 m/s can be obtained suitably after a few trials. When these values were inputted to the current calibrator program together with the current experimental profile (experiment.csv), it took approximately 14 min and 15 s for a personal computer with a 4.0 GHz CPU to complete the Fourier synthesis, to perform 120 iterations, and to write the iteration history (parameters_history.csv) and optimized wave profiles (signals.csv).
The optimization results for the above case are summarized in Run No. A1 of
Table 2. The optimized (
,
) result listed in Run No. A1 was used as the initial guess values for the next run (Run No. A2 in
Table 2). The optimized (
,
) set from this second run (Run No. A2) is also listed in
Table 2. In this way, three post runs in total (runs from A2 to A4) were carried out. The results (calibrated values and average error) are listed in
Table 2. The post runs stopped after run No. A4 because the optimized (
,
) set in this run was the same as the optimized set in run No. A2, while the average values were the same (1.292531%); further runs would have repeated the results obtained in run Nos. A2 and A3.
The capability of the current calibrator program for the current experimental profile was also tested by inputting another initial guess set of
= 0.25 and
= 5000 m/s, which is fairly far from the optimized result in
Table 2. The calibration result obtained using this set of initial values is presented in run No. B1 of
Table 3. As before, the optimized result from run No. B1 was used as the initial guess values for run No. B2. In this series of runs, the calibration result of run No. B3 (in
Table 3) is the same as that of run No. A3 in
Table 2. Therefore, the post runs were carried out only up to run No. B3; further post runs would have repeated the results obtained in run Nos. A2 and A3. This observation indicates that the current solver is capable of finding the optimized (
,
) set from fairly broad ranges of initial guess values.
As can be observed in
Table 2 and
Table 3, multiple (
,
) sets can yield the same error values of 1.292531%. For the case of the considered experimental profile (experiment.csv), the
values in
Table 2 and
Table 3 are the same down to the sixth decimal place for six runs in total. Based on this observation, the
value can be calibrated as
= 0.335050.
For the case of the
values in
Table 2 and
Table 3, the fourth and higher decimals of it varies depending on the set of the initial guess values, while the same error value of 1.292531% is yielded. Based on this observation in
Table 2 and
Table 3, the
value for the considered experimental profile was calibrated herein as
= 4588.2335 m/s.
For each iteration, the “fminsearch” function varies values of
and
down to six decimal places. The algorithm of the calibrator program consequently requests the PCE solver to provide the solutions for such values of of
and
. In
Table 2 and
Table 3, each run with an initial (
,
) set resulted in more than one hundred iterations. In separate trials, more than ten sets of initial (
,
) values were tested, which resulted in more than thousand iterations in total. These iterations were carried out successfully to yield the same calibrated values as above. The current PCE solver reliably conformed with the massive and fastidious requirements of the optimization algorithm by providing the PCE solutions for a wide range of
and
values with six decimal places.
As can be tested using the current calibrator program, once the Fourier synthesis is completed, the rate-limiting step in the iteration process is the phase shifting (dispersion correction) stage, which utilizes Equation (9) and PCE solutions, that is, the (Fdc vs. Cdc) relationship. The process of obtaining the PCE solution itself (Fdc vs. Cdc) is never the rate-limiting step, which indicates the prompt nature of the current solver for n = 1. The version of the PCE solver embedded in the calibrator program herein is faster than the separate PCE solver (PCE_solver_n1.m) because the latter requires some time for plotting the solutions (F vs. C and Fdc vs. Cdc) and writing the Fdc vs. Cdc relationship in the Excel® file (Cdc-Fdc.xlsx); this operation is not carried out in the embedded version. The calibrator program herein (dispersion_correction_ iteration.m) may be used for the calibration of the and values of the versatile bars used in contemporary bar technology, including (S)HB applications.
3.2.8. Further Discussion
Manufacturers of the bar usually provide Poisson’s ratio, elastic modulus (
E) and density (
) [
46,
47], from which the one-dimensional sound speed is suitably obtained via the relationship,
. The
values of Maraging steel C350 determined in this way using information in Reference [
46] (
Eo = 200 GPa,
= 8082.5 kg/m
3) and [
47] (
Eo = 200 GPa,
= 8080 kg/m
3) are 4974 and 4975 m/s, respectively. The
value available in Reference [
47] is 0.3. The foregoing values of
and
are called the manufacturer-provided values herein. As mentioned, these values are more readily available than the calibrated ones by the user of the bar. However, notable differences are observed between the manufacturer-provided and user-calibrated values herein for the bar introduced to the author’s laboratory under the premise of the material specification of Maraging steel C350.