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Data Descriptor

Manual for Calibrating Sound Speed and Poisson’s Ratio of (Split) Hopkinson Bar via Dispersion Correction Using Excel® and Matlab® Templates

Mechanics of Materials and Design Laboratory, Department of Materials Engineering, Gangneung-Wonju National University, 7 Jugheon-ghil, Gangneung 25457, Gangwon-do, Korea
Submission received: 7 March 2022 / Revised: 26 March 2022 / Accepted: 23 April 2022 / Published: 28 April 2022

Abstract

:
This manual presents a procedure to calibrate the one-dimensional sound speed (co) and Poisson’s ratio (ν) of a (split) Hopkinson bar using the open-source templates written in Excel® and Matlab® for dispersion correction. The Excel® template carries out the Fourier synthesis and one-time dispersion correction of a traveling elastic pulse under a given set of co and ν. The MATLAB® template performs the Fourier synthesis and iterative dispersion correction of a traveling elastic pulse for a range of co and ν sets. In the case of the iterative dispersion correction, a set of co and ν is assumed at each iteration step, and the sound speed vs. frequency (cdc vs. fdc) relationship necessary for dispersion correction is obtained under the assumed set by solving the Pochhammer–Chree equation. Subsequently, dispersion correction is carried out by using the cdc vs. fdc relationship. The co and ν values of the bar are determined in the iteration process when the dispersion-corrected pulse profiles are reasonably consistent with the measured ones at two travel distances (2103 and 4000 mm) in the bar. In the case of the experimental profile considered herein, the ν and co values were calibrated to six and four decimal places, respectively. The calibration algorithm is described with the tips for using the open-source templates, which are available online in a publicly accessible repository.
Dataset License: CC-BY-NC

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 102–104 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):
s t = A A o E o ε T t ,
e ˙ t = 2 c o L ε R t ,
e t = 2 c o L 0 t ε R t d t
where s , e ˙ , and e are the nominal stress, nominal strain rate, and nominal strain of the specimen, respectively; ε R is the reflected pulse strain recorded in the incident bar; ε T 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 ( ε R and ε R ). According to the signal-processing equations (Equations (1)–(3)), the precise measurement of the specimen quantities ( s , e ˙ , and e ) directly depends on the accuracy of the c o value of the bar ( E o = ρ c o 2 , where ρ 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 ( ν ) 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 ( c o ) and bar radius (a). Therefore, the precise calibration of c o and ν for the HB and SHB is fundamental.
However, the manufacturer-provided literature values of c o and ν 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 c o and ν 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].

2. Data Description

The open-source templates used to calibrate the c o and ν values of the bars are categorized in Table 1 depending on their usage. Output files of the templates are also listed. The role of individual template listed in Table 1 is as follows.
  • “PCE_solver_n1.m”: it solves the PCE for the ground state vibration, and writes “Cdc-Fdc.xlsx”, where the Cdc vs. Fdc relationship is available.
  • “dispersion_correction.xlsm”: it carries out Fourier synthesis and one-time dispersion correction at a given ( c o , ν ) set. Two types of information, i.e., the experimental signal and the Cdc vs. Fdc relationship (copied from the “Cdc-Fdc.xlsx” file) need to be inputted to the spreadsheet of this file. When this file is executed by clicking “Synthesize” and subsequently “Shift” icons in the spreadsheet, the constants of the Fourier-synthesized function, Fourier-synthesized profile data, and dispersion corrected profile data are written in the same spreadsheet. Last two profiles are also displayed in the figures embedded in the same spreadsheet.
  • “dispersion_correction_iteration.m”: it carries out Fourier synthesis and iterative dispersion correction for a range of ( c o , ν ) sets. It reads information (elastic pulse profile; pulse magnitude vs. time data) in “experiment.csv”, and obtains the Cdc vs. Fdc relationship by itself as it includes the PCE solver algorithm. It writes three files as outputs: “signals.xlsx”, “Ak-Bk.xlsx”, and “parameters_history.csv”. The first file (signals.xlsx) contains the finally obtained dispersion-corrected signal via the iteration process where the c o and ν values were calibrated. The second one (Ak-Bk.xlsx) the constants of the Fourier-synthesized function, and the last one (parameters_history.csv) the history of the parameters ( c o , ν , and error) in the iteration process. Once the iteration process is completed, this program (dispersion_correction_iteration.m) displays the pop-up figures illustrating the Fourier-synthesized and finally obtained dispersion-corrected wave profiles with reference to the experimental profile.
  • “experiment.csv”: the recorded wave profile data in experiment need to be placed in this file: the data of signal magnitude vs. time. This file is read by “dispersion_correction _iteration.m”.
The data of an example wave profile are available in “experiment.csv” as well as “dispersion_correction.xlsm” file. The method of obtaining the experimental profile is described in [44].

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 c o and ν 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 c o and ν . 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 c o and ν 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 ( ν ) and one-dimensional sound speed ( c o ) 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 ν and c o values are known in advance. The ν 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 ν and c o for which it is used.
Second, the user must input the values of Δt, 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 z1 and z2 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 z1 and z2. 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 X t can be expressed as X t = M x t , where M is the magnitude constant with the dimension of X (mV herein), x t is the non-dimensional shape function, and t is the time. In this study, M is set as 1 mV, and x t 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:
X n Δ t = A 0 2 k = 0 K A k cos 2 π k f 0 n Δ t + B k sin 2 π k f 0 n Δ t
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 ( t 0   = N t Δ t ); Δ t is the time interval of sampling; f 0 is the fundamental frequency (= 1 / t 0 ); 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 A 0 , A k , and B k are the Fourier coefficients given,
A 0 = 2 T 0 0 t 0 f t d t
A k = 2 T 0 0 t 0 f t cos 2 π k f 0 t d t
B k = 2 T 0 0 t 0 f t sin 2 π k f 0 t d t
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]:
G C , F ,   ν = C 2 1 + ν 1 2 Φ 2 π β F 2 1 + ν F / C 2 β C 2 1 + ν 1 C 2 1 + ν Φ 2 π 2 F 2 1 + ν F / C 2 = 0
Φ y = y J 0 y / J 1 y
J 0 y = m = 0 1 m y 2 m 2 2 m m ! 2
J 1 y = m = 0 1 m y 2 m + 1 2 2 m + 1 m + 1 ! 2
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 ν 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 ν 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 c o 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 Δ z is
X n Δ t = A 0 2 k = 0 K A k cos 2 π k f 0 n Δ t Δ z c k + B k sin 2 π k f 0 n Δ t Δ z c k
where Δ z 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 ν and c o must be assumed first. The literature values can be considered as the starting point. Once the assumed set of ν and c o 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 z1 and z2 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 ν and c o 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 f dc , k = f k = k f 0 and c dc , k = c k (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 z1 and z2 based on the formula:
Error   % = i = 1 i = N p x i dc x i exp N p x max exp × 100
In Equation (10), i is the index of the time data of the pulse at a travel distance of either z1 or z2; x is the value of the non-dimensional function x t ; superscripts dc and exp denote dispersion corrected and experiment, respectively; Np is the number of data in a given traveled pulse; and x max exp 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 z1 and z2 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 ν and c o 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 ν and c o 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 c o and ν 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 c o and ν values. At each iteration step, a set of ν and c o values is assumed and the PCE is solved for the assumed set. The c o and ν 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 ν and c o to the optimization function “fminsearch”. This function determines the values of ν and c o (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 c o information, inputs them with ν 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 c o and ν 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 ν and c o 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 ν and c o , 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 ν and c o set more quickly. Nevertheless, it was found via separate trials that the calibrator program successfully reached the optimum set of ν and c o from even fairly far initial guess values of ν and c o 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 ( N t = t 0 / Δ t ), 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 ν and c o 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 ν 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 z1 and z2.

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 z1 and z2, 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 ν and c o 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 ν and c o 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 ν and c o values used in the final step. The example of the calibration result displayed when ν = 0.335050 and c o = 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 ν and c o 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 ν = 0.30 and c o = 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 ( ν , c o ) 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 ( ν , c o ) 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 ( ν , c o ) 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 ν = 0.25 and c o = 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 ( ν , c o ) set from fairly broad ranges of initial guess values.
As can be observed in Table 2 and Table 3, multiple ( ν , c o ) sets can yield the same error values of 1.292531%. For the case of the considered experimental profile (experiment.csv), the ν 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 ν value can be calibrated as ν = 0.335050.
For the case of the c o 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 c o value for the considered experimental profile was calibrated herein as c o = 4588.2335 m/s.
For each iteration, the “fminsearch” function varies values of ν and c o 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 ν and c o . In Table 2 and Table 3, each run with an initial ( ν , c o ) set resulted in more than one hundred iterations. In separate trials, more than ten sets of initial ( ν , c o ) 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 ν and c o 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 c o and ν 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 ( ρ ) [46,47], from which the one-dimensional sound speed is suitably obtained via the relationship, c o = E o / ρ . The c o values of Maraging steel C350 determined in this way using information in Reference [46] (Eo = 200 GPa, ρ = 8082.5 kg/m3) and [47] (Eo = 200 GPa, ρ = 8080 kg/m3) are 4974 and 4975 m/s, respectively. The ν value available in Reference [47] is 0.3. The foregoing values of c o and ν 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.

4. Conclusions

The process of calibrating the c o and ν values of a (split) Hopkinson bar using the open-source templates written in Excel® and Matlab® for dispersion correction of elastic wave was presented. The Excel® template (dispersion_correction.xlsm) and the PCE solver in MATLAB® (PCE_solver_n1.m) are used for the Fourier synthesis and one-time dispersion correction of elastic pulse under a given set of c o and ν . The MATLAB® template named “dispersion_correction_iteration.m” is used for the Fourier synthesis and iterative dispersion correction of elastic pulse for a range of c o and ν sets. At each iteration of dispersion correction (using the last template), a set of c o and ν is assumed, and the sound speed vs. frequency (Cdc vs. Fdc) relationship is obtained under the assumed set using the PCE solver imbedded in the MATLAB® template (dispersion_correction_iteration.m). The (Cdc vs. Fdc) matrix is subsequently converted to (cdc vs. fdc) matrix, and finally each phase of the constituting waves of the overall elastic pulse is shifted using the cdcfdc relationship. The c o and ν values of the bar are determined in the iteration process when the dispersion-corrected pulse profiles are reasonably consistent with the measured ones at the two travel distances (2103 and 4000 mm) in the bar. For the experimental profile considered, the ν and c o values were calibrated to six and four decimal places, respectively.

Funding

This work was financially supported by a National Research Foundation of Korea grant under contract No. 2020R1A2C2009083, funded by the Ministry of Science and Technology (Korea).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data available in a publicly accessible repository [45].

Acknowledgments

The author appreciates Jae Eon Kim, Jun Moo Lee, and Sung Bin Kim, for their technical assistance.

Conflicts of Interest

The author declares no conflict of interest.

Nomenclature for Dispersion Correction

aBar radius (m)
CNormalized sound speed (=c/co)
CdcC value used in dispersion correction
cSound speed of a wave with frequency f (m/s) and wavelength Λ (m)
c dc c value used in dispersion correction (m/s)
c o One-dimensional sound speed (m/s). c at f = 0 (Λ = or a = 0)
dFF interval
dFdcF interval in dispersion correction
dff interval
dfdcf interval in dispersion correction
Δ t Sampling time interval (s)
Δ z Travel distance (m)
FNormalized frequency (=af/co)
FmaxMaximum F value
FdcF value used in dispersion correction
Fdc_maxMaximum F value in dispersion correction
fFrequency (Hz)
f dc f value used in dispersion correction (Hz)
f 0 Fundamental frequency ( = 1 / t 0 ; Hz)
f N y Nyquist frequency ( = f s / 2 ; Hz)
f s Sampling frequency (= 1 / Δ t ; Hz)
K Summation limit in Fourier synthesis ( = N dc = N y )
N dc Number of F (f) components in dispersion correction (=Ny)
N p Number of data in a traveled pulse in the experiment
N t Number of data in the time window ( = t 0 / Δ t )
N y Nyquist number (= f N y / d f = Ndc)
ν Poisson’s ratio
tTime (s)
t 0 Fundamental time period (s)

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Figure 1. Schematics of the (a) Hopkinson bar (HB) and (b) compressive split Hopkinson bar (SHB).
Figure 1. Schematics of the (a) Hopkinson bar (HB) and (b) compressive split Hopkinson bar (SHB).
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Figure 2. Portion of the Excel® spreadsheet (dispersion_correction.xlsm) for carrying out (i) target signal preparation, (ii) Fourier synthesis, and (iii) phase shift (dispersion correction).
Figure 2. Portion of the Excel® spreadsheet (dispersion_correction.xlsm) for carrying out (i) target signal preparation, (ii) Fourier synthesis, and (iii) phase shift (dispersion correction).
Data 07 00055 g002
Figure 3. Embedded figure in the Excel® template (dispersion_correction.xlsm) for the experimental and target signals for Fourier synthesis.
Figure 3. Embedded figure in the Excel® template (dispersion_correction.xlsm) for the experimental and target signals for Fourier synthesis.
Data 07 00055 g003
Figure 4. Portion of the Excel® spreadsheet (dispersion_correction.xlsm) specifying the start and end positions (in time) of the reference pulse and traveled pulses at z1 and z2.
Figure 4. Portion of the Excel® spreadsheet (dispersion_correction.xlsm) specifying the start and end positions (in time) of the reference pulse and traveled pulses at z1 and z2.
Data 07 00055 g004
Figure 5. Embedded figure in the Excel® template (dispersion_correction.xlsm) for the Fourier synthesized signal in comparison with the measured signal in experiment.
Figure 5. Embedded figure in the Excel® template (dispersion_correction.xlsm) for the Fourier synthesized signal in comparison with the measured signal in experiment.
Data 07 00055 g005
Figure 6. Portion of the Matlab® program (PCE_solver_n1.m) specifying the required input parameters to run the solver.
Figure 6. Portion of the Matlab® program (PCE_solver_n1.m) specifying the required input parameters to run the solver.
Data 07 00055 g006
Figure 7. (a) C vs. F and (b) Cdc vs. Fdc relationships derived by the PCE solver (PCE_solver_n1.m).
Figure 7. (a) C vs. F and (b) Cdc vs. Fdc relationships derived by the PCE solver (PCE_solver_n1.m).
Data 07 00055 g007aData 07 00055 g007b
Figure 8. Embedded figures in the Excel® template (dispersion_correction.xlsm) for the predicted (dispersion corrected) wave profiles at travel distances of (a) 2103 and (b) 4000 mm for the case where ν = 0.335050 and c o = 4588.233496 m/s in comparison with the signal measured in the experiment.
Figure 8. Embedded figures in the Excel® template (dispersion_correction.xlsm) for the predicted (dispersion corrected) wave profiles at travel distances of (a) 2103 and (b) 4000 mm for the case where ν = 0.335050 and c o = 4588.233496 m/s in comparison with the signal measured in the experiment.
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Figure 9. (a) Flowchart for the iterative dispersion correction algorithm of the calibrator program (dispersion_correction_iteration.m) and (b) feedbacks among the main part and the subroutines.
Figure 9. (a) Flowchart for the iterative dispersion correction algorithm of the calibrator program (dispersion_correction_iteration.m) and (b) feedbacks among the main part and the subroutines.
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Figure 10. Input parameters sections I & II in the calibrator program (dispersion_correction_iteration.m).
Figure 10. Input parameters sections I & II in the calibrator program (dispersion_correction_iteration.m).
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Figure 11. Examples of plots of the calibrator program (dispersion_correction_iteration.m) when the iterative dispersion correction loop finds the optimum (calibrated) values of the ν and c o .   Dispersion-corrected signals at travel distances (a) z1 and (b) z2 in comparison with the measured experimental signal when ν = 0.335050 and c o = 4588.233496 m/s.
Figure 11. Examples of plots of the calibrator program (dispersion_correction_iteration.m) when the iterative dispersion correction loop finds the optimum (calibrated) values of the ν and c o .   Dispersion-corrected signals at travel distances (a) z1 and (b) z2 in comparison with the measured experimental signal when ν = 0.335050 and c o = 4588.233496 m/s.
Data 07 00055 g011aData 07 00055 g011b
Table 1. Open-source templates and output files used for dispersion correction and iterative dispersion correction.
Table 1. Open-source templates and output files used for dispersion correction and iterative dispersion correction.
UsageDispersion CorrectionIterative Dispersion Correction
TemplatesPCE_solver_n1.m
dispersion_correction.xlsm
dispersion_correction_iteration.m
experiment.csv
Output filesCdc-Fdc.xlsxsignals.xlsx
Ak-Bk.xlsx
parameters_history.csv
Table 2. Change in the optimized values of ν and c o in the first run (run No. A1) and post runs (run Nos. from A2 to A4) for the considered experimental profile and error calculation ranges illustrated in Figure 8. Initial guess values for run No. A1 were ν = 0.30 and c o = 4600 m/s.
Table 2. Change in the optimized values of ν and c o in the first run (run No. A1) and post runs (run Nos. from A2 to A4) for the considered experimental profile and error calculation ranges illustrated in Figure 8. Initial guess values for run No. A1 were ν = 0.30 and c o = 4600 m/s.
Run No.IterationsOptimization ResultAverage Error (%)
ν c o (m/s)
A11200.3350504588.2335421.292531
A21170.3350504588.2334761.292531
A31230.3350504588.2334911.292531
A41100.3350504588.2334761.292531
Table 3. Change in optimized values of ν and c o in the first run (run No. B1) and post runs (run Nos. B2 and B3) for the current experimental profile and error calculation ranges illustrated in Figure 8. Initial guess values for run No. B1 were ν = 0.25 and c o = 5000 m/s.
Table 3. Change in optimized values of ν and c o in the first run (run No. B1) and post runs (run Nos. B2 and B3) for the current experimental profile and error calculation ranges illustrated in Figure 8. Initial guess values for run No. B1 were ν = 0.25 and c o = 5000 m/s.
Run No.IterationsOptimization ResultAverage Error (%)
ν c o (m/s)
B11510.3350504588.2334961.292531
B21100.3350504588.2334811.292531
B31130.3350504588.2334911.292531
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Shin, H. Manual for Calibrating Sound Speed and Poisson’s Ratio of (Split) Hopkinson Bar via Dispersion Correction Using Excel® and Matlab® Templates. Data 2022, 7, 55. https://0-doi-org.brum.beds.ac.uk/10.3390/data7050055

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Shin H. Manual for Calibrating Sound Speed and Poisson’s Ratio of (Split) Hopkinson Bar via Dispersion Correction Using Excel® and Matlab® Templates. Data. 2022; 7(5):55. https://0-doi-org.brum.beds.ac.uk/10.3390/data7050055

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Shin, Hyunho. 2022. "Manual for Calibrating Sound Speed and Poisson’s Ratio of (Split) Hopkinson Bar via Dispersion Correction Using Excel® and Matlab® Templates" Data 7, no. 5: 55. https://0-doi-org.brum.beds.ac.uk/10.3390/data7050055

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