Section 1.  Formulated the strain rate equation for the split Hopkinson bar (SHB) test.

• Its six utilization areas are as follows. 

    (1) Tool for understanding the varying nature of specimen strain rate.    

    (2) Tool for verifying test result.

    (3) Tool for predicting specimen strain rate and stress prior to the SHB test.

    (4) Tool for predicting maximum strain of the specimen.

    (5) Tool for achieving a nearly constant strain rate.

    (6) Tool for describing SHB operation condition.

• Details are explained  in Section 23, which is available at the end of this webpage.

Section 2.  Formulated equations to predict the stress and particle velocity generated during a striker-bar impact test.

• These equations represent the first theoretical predictions of stress and particle velocity in a  striker-bar impact test in 70 years, since Kolsky 's work on the SHB in 1949.

• The equations consider three cases: (1) Zs < Zb, (2) Zs = Zb, and (3) Zs > Zb, where Z represents the general impedance (area x density × sound speed), and the subscripts s and b denote the striker and the bar, respectively.

• This  short course includes this topic.

• https://doi.org/10.1177/0954406219877210 

Section 3. Developed design guidelines for the flange-type split Hopkinson tension bar (SHTB) to prevent spurious pulses.

• Designes geometries of flange and hollow cylinder to eliminate the spurious pulses based on the equations for the stress and particle velocity presented in Section 2.

• Discloses the origin of the spurious pulses in the flange-type SHTB.

• This topic is covered during the short course.

• https://doi.org/10.1177/0954406219869984

Section 4. Disclosed the stress transfer mechanism from compression to tension in a flange-type  split Hopkinson tension bar (SHTB).

• The tensile stress profiles monitored at the strain gauge position of the incident bar are interpreted on a qualitative basis using three types of stress waves: bar (B) waves, flange (F) waves, and a series of reverberation (Rn) waves.

• When the flange length ( Lf) is long (i.e.,  Lf > Ls, where Ls is the striker length), the B wave and first reverberation wave (R1) are fully separated in the time axis. 

• When the flange length is intermediate (~Db < Lf < Ls, where Db is the bar diameter), the B and F waves are partially superposed; the F wave is delayed, then followed by a series of Rn waves after the superposition period. 

• When the flange length is short (Lf < ~Db), the B and F waves are practically fully superposed and form a pseudo-one-step pulse, indicating the necessity of a short flange length to achieve a neat tensile pulse. 

• The magnitudes and periods of the monitored pulses are consistent with the analysis result using the one-dimensional impact theory, including a recently formulated equation for impact-induced stress when the areas of the striker and bar are different, equations for the reflection/transmission ratios of a stress wave, and an equation for pulse duration time. 

• This observation verifies the flange length-dependent stress transfer mechanism on a quantitative basis. 

• This topic is covered during the short course.

• https://doi.org/10.3390/app10217601

Section 5. Developed an open-source solver for the Pochhammer–Chree equation (PCE).

• The PCE describes wave propagation in an infinitely long circular bar with a finite diameter. Its solution characterizes the relationship among any two of the following variables: sound speed, frequency, and wavelength. 

• This is the first open-source solver in 145 years since the introduction of the PCE by Pochhammer in 1876 and Chree in 1886.

• No solvers have demonstrated the capability to obtain solutions for vibration modes up to the 20th order (n = 20).

• No solvers have demonstrated the capability to obtain solutions for c/c₀ values as high as 500.

• The solver is very robust due to its two-stage solution scheme for the first time. The first stage considers only Poisson's ratio, while the second stage incorporates the bar diameter and sound speed (c₀). The revised version of this solver is handed out during the short course.

• https://doi.org/10.1177/0954406220980509

Section 6. Established a method to calibrate the sound speed and Poisson’s ratio of a circular bar using an iterative dispersion correction technique.

• This approach is referred to as the iterative dispersion correction (IDC) method in the literature.

• This study experimentally verifies, on a quantitative basis and for the first time in 74 years, the combined Pochhammer-Chree and Fourier theories originally proposed by Davies in 1948.

• Provided a manual for the established method in the form of an open-access publication. 

• This study presents, for the first time, a precise dispersion correction that includes the tail region.

• This demonstrates that any single mechanical fluctuation, including those occurring in the tail region of the SHB, holds mechanophysical significance, as evidenced by their accurate predictions through dispersion corrections.

• This study presents, for the first time, a method to accurately calibrate sound speed and Poisson's ratio through iterative dispersion correction.

• Developed a practical yet rigorous open-source PCE solver for n = 1 and demonstrated its superior accuracy compared to Bancroft's Bancroft's solution.

• Manual for the developed method is also provided. It describes practical tips for calibrating the sound speed and Poisson's ratio of a bar using the open-source PCE solver for n = 1.  We will practice this process during the short course.

• https://doi.org/10.1115/1.4054107  - https://doi.org/10.3390/data7050055

Section 7. Developed a method to calibrate the elastic properties of a circular bar through repeated simulations.

• This approach is referred to as the iterative finite element analysis (IFEA) method in the literature.

• The bar properties—elastic modulus, density, Poisson’s ratio, and sound speed—required for analyzing bar impact tests were calibrated using signal data obtained from  a striker-bar impact experiment.

• The calibration process employs two methods—the IDC method (see Section 6)  and the IFEA method—that cross-verify each other.  

• Additionally, it describes the methods used to calibrate the measured strain and to determine the impact velocity.

• This study confirms that all mechanical vibrations—including not only the fluctuating main pulses but also the noise-like tail sections—have mechanophysical significance, as they are accurately replicated by simulations.

• Manual for the developed method is also provided. It describes practical tips for calibrating the elastic properties of the bar using a GUI-based program. The GUI program will be handed out, along with the demonstration of using it during the short course.

• https://doi.org/10.1115/1.4054107 - https://doi.org/10.3390/data8030054

Section 8. Verified that the one-dimensional sound speed is maintained in bars with diameters of up to at least 200 mm.

• The prevailing consensus in the literature regarding sound speed in a circular bar is that as the diameter increases, it deviates from the one-dimensional (1D) value, c₀.

• Contrary to this consensus, we report that the bar sound speed retains the aforementioned 1D characteristics, even for diameters of at least 200 mm. 

• Quantifies the critical frequency that allows travel at a sound speed exceeding 0.99c₀ for various bar diameters by solving the PCE using Shin's solver.

• Compares the critical frequency for a range of diameters with the results obtained using Love's approximate solution in 1927. 

• Compares the critical frequency  for a range of diameters with Davies's 1948 statement.

• High-frequency wave components travel more slowly than their lower-frequency counterparts. The fastest wave component, with zero frequency (the DC component) and the greatest magnitude (dee the left figure above), travels at speed c₀ and leads the pulse, thereby determining the overall sound speed in the bar.

• Verifies the above conclusion using the simulated signals in bars of various diameters. This topic is covered during the short course.

• Handouts for the short course;   Under review

Section 9. Developed a constitutive model that accounts for the stress upturn phenomenon.

In the current model, all of the phenomena of the strain hardening (f(ε)), rate hardening (g(ε̇)), and temperature softening (h(T)) were treated to be decoupled  and the law of each phenomenon was selected with reasonable backgrounds. The employed law for each phenomenon was as follows.

• Strain hardening law: f(ε) = A + B{1−exp(−Cε)} (Voce hardening law)

• Rate  hardening law: g(ε̇) = D ln(ε̇/ε̇o) + exp(E⋅ε̇/ε̇o)

• Temperature softening law: h(T) = (1 − T*)m where T* = [(T−Tref)/(Tm−Tref)]m

The formulation of the current model is

  σ = [A+B{1−exp(−Cε)}] [D ln(ε̇/ε̇o)+exp(E⋅ε̇/ε̇o)] [1−(T−Tref)/(Tm−Tref)]m

where  σ is the flow stress,  ε is the strain, ε ̇ is the strain rate, ε ̇o is the reference strain rate, T is the temperature, Tref is the reference temperature,  Tm is the melting temperature, and  A ,  B , C , D  , E  , and m  are the material parameters. 

• Stress upturn refers to the phenomenon of a rapid increase in flow stress beyond a specific strain rate, typically around 1,000 s⁻¹.

• Most existing strain-rate-dependent constitutive models fail to adequately capture this behavior. 

• This topic is covered during the short course.

• https://doi.org/10.1115/1.4000225

Section 10. Reviewed twelve constitutive models that depend on strain rate.

• Analyzes and visualizes the characteristics of strain hardening, strain rate hardening, and thermal softening models.

• This study identifies realities of several well-known models, including the Zerilli-Armstrong, Preston-Tonks-Wallace,  Steinberg–Cochran–Guinan–Lund, Mechanical Threshold Model, Khan-Huang-Liang, and Voyiadjis-Abed models.

• Points out our current position in developing constitutive models and outlines the future direction to be pursued.

• It should be noted that until the proposed type of the constitutive model becomes available, the Johnson-Cook model and the Shin-Kim stress-upturn model remain the most practical alternatives.

• This study proposes the Voce-Polynomial and Ludwik-Polynomial models for materials exhibiting strain softening following necking.

• This topic is covered during the short course.

• https://doi.org/10.3390/technologies10020052

Section 11. Developed an algorithm to accurately extract the stress-strain curve, including the post-necking region, during a tensile test without relying on any constitutive models.

• The stress-strain curve obtained from a standard tensile test, especially in the post-necking region, deviates significantly from the actual material behavior.

• A slightly tapered geometry is employed around the specimen’s center to reliably initiate necking at that location in both the experiment and the simulation.   

• The algorithm iteratively traces the experimentally measured force–elongation curve using finite element analysis (FEA) without assuming any constitutive model.

• This method employs a novel stress update algorithm for subsequent finite element analysis (FEA), eliminating the need for a constitutive model during the extraction process. It is compatible with general-purpose finite element solvers, making it accessible to a broad range of users.

• Points out that necking simulations using geometrically perfect finite element (FE) models in existing studies rely on numerical errors in finite element analysis (FEA) to initiate necking, which may lead to inaccuracies in the resulting curves.

• This study presents the first comparison between the simulated deformed shape after testing and the corresponding experimental results. This comparison is enabled by accurately initiating necking through the incorporation of geometric imperfections in both the simulation and the experiment. 

• The results of the mesh sensitivity test on the final extracted stress-strain curve  provide guidance on which stress-strain curve should be used in finite element analysis (FEA) when employing a large mesh size (approximately 15 mm) to simulate a large structure. This topic is covered during the short course.

• An accurately extracted stress-strain curve at a quasistatic strain rate should be used as the reference for comparison with the curve obtained at high strain rates using the Split Hopkinson Bar (SHB) method.

• https://doi.org/10.1115/1.4064372 - https://doi.org/10.3795/KSME-A.2024.48.8.553

Section 12. Developed a method for calibrating a constitutive model using a compression-to-tension conversion (CTC) specimen in an SHPB test.

• The conventional method for calibrating a strain rate-dependent constitutive model involves measuring stress-strain curves using the SHB at various strain rates and temperatures, followed by  nonlinear curve fitting. This approach is both costly and often inaccurate, particularly due to the challenges associated with high-temperature testing.

• The method requires prior calibration of the bar properties using either the 1D method (as described above in Section 6) or the 3D method (as described above in Section 7).

• An accurate prior measurement of the specimen's tensile stress-strain curve, including the post-necking regime, is also required, using the method described above in Section 11.

• It employs a compression-to-tension conversion (CTC) structure that transforms compressive stress into tensile stress during the SHPB test. 

• It overcomes the limitations of both the SHPB and SHTB by leveraging the advantages of each through the use of the CTC specimen.  

• The algorithm iteratively traces the experimentally measured bar signals using FEA by assuming a set of constitutive parameters. This process continues until the simulated bar signals closely match the experimental data, at which point the input set of constitutive parameters is identified as the calibrated values.

• It determines the temperature parameter of a constitutive model from an SHPB test conducted at ambient temperature by utilizing the adiabatic heating phenomenon resulting from the specimen's plastic deformation.

• It is highly cost-efficient because it requires only one quasistatic tensile test and one SHPB test, while also being significantly more accurate than the conventional method, which cannot reliably simulate the bar signals using the calibrated parameter set. This topic is covered during the short course.

• https://doi.org/10.1080/15376494.2025.2457125

Section 13. Developed a method for simultaneously calibrating a constitutive model and a fracture model using the Type II compression-to-tension conversion (CTC Type II) specimen in an SHPB test.

• The first type of CTC structure described in Section 12  induces necking but does not cause fracture during the stress pulse duration. 

• This study presents the second type of CTC structure (Type II CTC), which can induce fractures during the stress pulse duration in an SHPB test.

• This demonstrates the simultaneous calibration of strain rate and temperature parameters for the Johnson-Cook constitutive model and the Johnson-Cook fracture model using a  single 'SHPB' test with a Type II CTC structure. This topic is covered during the short course.

• Handouts for the short course;  https://doi.org/10.2495/SUSI250141 ;  Under review

Section 14. Developed a coupling scheme for materials exhibiting non-zero fracture strength.

• The coupling scheme for damage models of materials, commonly used in the literature and in general-purpose commercial finite element software, assumes that the material's stress reduces to zero when the damage variable (D) reaches one at fracture. 

• This scheme does not accurately reflect reality, as materials generally exhibit considerable fracture strength. Additionally, it causes numerical instability in finite element analysis (FEA) when  D approaches one.

• Here we present for the first time a coupling scheme for materials exhibiting non-zero fracture strength.

• This coupling scheme was integrated with the uncoupled Johnson–Cook damage model to create a simple coupled damage model. When calibrated using tensile test data from an aluminum alloy, this coupled model accurately describes the stress–strain behavior of the material under investigation, outperforming the coupled model that employs the conventional coupling scheme.

• In FEA, the damage model utilizing the proposed coupling scheme precisely replicates the experimentally measured load–displacement curve from the tensile test and maintains numerical stability, even as the damage parameter approaches unity. 

• The calibration process for this coupled damage model is provided in an Excel template included in the Supplementary Data, along with a detailed explanation in the Supplementary Material. 

• This topic is covered during the short course.

• Handouts for the short course;   Under review

Section 15. Developed a method to calibrate the fracture strain model that accounts for the varying triaxiality throughout the loading process.

• Johnson-Cook fracture model employs a triaxiality-dependent fracture strain curve characterized by three fitting constants: D1, D2, and D3. 

• The majority of studies including the pioneering studies of Wierzbicki group have overlooked the varying nature of  triaxiality during loading. Specifically, they have used the average (or final) triaxiality value of a given fracture specimen and applied this value across a series of fracture specimens with different average (or final) triaxiality values.

• In contrast to most existing studies, this research accurately accounts for the variability of the triaxiality value during loading. In other words, it effectively accounts for the non-proportional nature of the loading process.

• The fracture strain curves obtained using the developed method are presented for various types of fracture specimens, including shear, tensile, and round-notched specimens. 

• It also compares the  fracture strain curves obtained using the newly developed method with those obtained by the conventional method.

• It also presents a specimen type that can reliably exhibit negative triaxiality during loading.

• The new calibration process is provided in an Excel template included in the Class Note of the Short Course, along with a detailed explanation. 

• Through exercises conducted during the short course, we will analyze and observe the  fracture strain curves of a series of fracture specimens, which accurately reflect the varying triaxiality characteristics throughout the loading process.

• Handouts for the short course ;  Under review

Section 16. Developed a damage model that can be coupled with constitutive behavior prior to necking.

• Can be coupled with constitutive behavior prior to necking.

• Employs Banerjee’s damage accumulation scheme, the Johnson–Cook fracture strain curve, and the Kachanov–Rabotnov coupling scheme.

• Four types of sheet specimens were prepared using SGAFC780 steel for tensile fracture tests. These specimens include center-holed, round-notched, U-shaped notched, and shear specimens, designed to encompass a wide range of triaxiality levels.

• The parameters of both the proposed damage model and the constitutive model were inversely identified by tracking the experimentally measured load–displacement curves through repeated finite element analyses using an optimization algorithm.

• The identified parameters demonstrated coupling from the pre-necking stage and accurately replicated the measured load–displacement curves for not only the four types of specimens used to calibrate the models but also for the specimen subjected to the standard tensile test.

• https://doi.org/10.1016/j.jmrt.2025.05.245

Section 17. Disclosed the minimum distance required for the stress state in the transmitted bar to become uniform—that is, the minimum distance necessary for placing the strain gauge on the transmitted bar.

• The minimum required distance was determined when the strain–time profiles at r = 0, 0.5Ro and 1.0Ro, were coincident (r is the radial position and Ro is the radius of the bar.).

• Here, epsilonT represents the transmitted strain, z stands is the axial distance in the transmitted bar, zc denotes the critical axial distance where the stress state becomes uniform, r indicates the radial position, Ro and Do signify the radius and diameter of the bar, and D is the specimen diameter.

• The determined minimum required distance, f(x), is presented as a function of the relative specimen diameter to that of the bar (x = D/Do): f(x) = – 0.9385x3 + 0.6624x2 – 0.7459x + 1.4478 (x = 0.1 - 0.9).

• This result demonstrates the Saint-Venant’s principle of rapid dissipation of localized stress in transient loading.

• The result will also allow one to avoid unnecessarily remote strain gage position from the specimen.

• This topic is covered during the short course.

• https://doi.org/10.1051/matecconf/202030804005 

Normalized values (epsilonT(z)/epsilonT(zc)) at (a) r = 0, (b) r =0.5Ro, and (c) r = 1.0 Ro for a range of D/Do ratios. (d) Minimum required distance (zc) for the gauge position in the transmitted bar. 

Section 18. Disclosed which friction-correction model has to be used in a compression test.

• Disclosed that the Schroeder-webster model for a sliding surface is the most effective among the compared models in correcting the influence of friction included in the results of an axial compression test.

• The above result was based on an examination of four friction correction models: Schroeder-Webster, Hill, Rand, and Cha et al.

• It also examined the reasons behind the effectiveness of the Schroeder-Webster model for a sliding surface.

• This study additionally investigated the Schroeder-Webster model in greater detail, focusing on the three surface types considered within the model: the sliding surface, the mixed surface, and the sticky surface.

• This topic is covered during the short course.

•  https://doi.org/10.1115/1.4044131 

The solid curves in the four other diagrams represent sigmao, which was defined as the input material property for the numerical experiment conducted using finite element analysis. The dashed curves correspond to the measured results from the numerical experiment. Note that Ho/Do = 0.1, indicating a very thin disk specimen, and the value of mu is typically 0.1 in an SHPB test.

Section 19. Developed a method for measuring a nearly friction-free stress-strain curve of rubber in compression testing.

• Rubber is commonly used for impact energy absorption via compression, but measuring  its compressive stress-strain curve is challenging due to friction during testing.

• The contact area was measured by applying stamp ink to the specimen's sidewall to identify the optimal L/D ratio that produces a stress–strain curve closest to that of a friction-free specimen. 

• Ink traces on the platen after the compression test indicate that rollover occurred.

• When the L/D ratio is below 1.0, the contact area remains smaller than that of a frictionless specimen, despite a slight supplement from rollover.

• When the L/D ratio increases up to 1.0, the contact area increases toward that of the ideal specimen that deforms uniformly under the friction-free condition; the stress–strain curve of the specimen with the L/D ratio of 1.0 can be regarded as the nearly friction-free property of silicone rubber.

• When the L/D ratio reaches 1.0, the contact area approximates that of an ideal specimen deforming uniformly without friction. The stress–strain curve at this ratio reflects the nearly frictionless behavior of silicone rubber.

• The above knowledge can be applied to SHPB testing for rubber. The measured stress-strain curve may be corrected from the knowledge of ink traces on the bar surface.  

• This topic is covered during the short course.

• https://doi.org/10.1007/s11340-018-0426-z

Section 20. Disclosed the origin of anomalously long transmitted pulse from  soft materials, such as rubber, in SHPB tests.

• The soft specimen is in a compressive state even after the passage of the pulse because it cannot push the bars, such that a fairly long transmitted pulse is monitored at the gauge position of the output bar. 

• The arrival of the second pulse (tensile) from the right-end surface of the output bar releases the first transmitted signal (compressive) coming from the compressed state of the specimen, which puts an end to the fairly long transmitted pulse. 

• The exact duration of the transmitted pulse is dependent on the distance between the strain gage and the end surface of the transmitted bar: it is instrument-dependent.

• Because the observation of a prolonged transmitted pulse is a natural phenomenon, the prolonged transmitted pulse should not annoy researchers in the process of confirming the validity of experiments on soft specimens. 

• Our concern in verifying the validity of experiments can be focused to other aspects such as the process of measurement and the amplification of weak transmitted signals coming from soft specimens. 

• https://link.springer.com/article/10.1007/s12541-016-0026-8

In the left diagram, u represents displacement. Subscripts 1 and 2 denote the end surfaces of the incident bar and the transmitted bar, respectively; both are in contact with the specimen, which has a variable elastic modulus. Explanations for the middle and right diagrams are provided below. 

Stress profiles of elastic specimens with a varying elastic modulus.

Section 21. Demonstrated a calibration method for hyperelastic and hyperfoam constitutive models of rigid polyurethane foam.

• Hyperelastic and hyperfoam constitutive models are calibrated for rigid polyurethane (PU) foam exhibiting the characteristics of both soft foam and rubber.

• Test data from uniaxial and volumetric compression are used for calibration of the models with the goal of simulating a compressive-loading event (a hemispherical indentation).

• The applicability and limitations of the models for describing the indentation behavior of the rigid PU foam are discussed.

• This topic is covered during the short course.

• https://doi.org/10.1016/j.compositesb.2018.11.045

(first) Schematic illustration of the experimental setup for determination of the curve of pressure vs. volume ratio of the specimen using a thick-walled cylinder (TWC). (second) 2D axisymmetric finite element model for the indentation test. Least-square-fitted curves of various hyperelastic constitutive models for data from (third) the uniaxial compression test and (fourth) the TWC test for the pressure vs. volume ratio.

Section 22. Gave explanations for some failure models for geomaterials with compressible character and developed an EOS measurement method for them.

Section 23. Six Usage Areas of the Rate Equation Presented in Section 1.

• https://doi.org/10.1177/0954406218813386

• This paper on the Strain Rate Equation is quite lengthy. The following summary will help the general audience save time while grasping its key points.

• The  short course includes an exercise on applying this equation.

• The six areas of utilizing the Strain Rate Equation are summarized below.

Unless the first and second terms are balanced, the specimen strain rate generally varies (decreases or increases) with strain (with specimen deformation), which is the physical origin of the varying nature of the specimen strain rate in the SHB test. The competition between the rate-increasing first term and the rate-decreasing second term is discussed here using the relative area (A/Ao) of the specimen and the stress of the specimen (σ), which appear only in the second term. When the rate-decreasing second term is more dominant (when the values of σ and A/Ao are considerably large), the strain rate decreases (with strain) in the plastic deformation regime. When the rate-decreasing second term is negligible (when the values of σ  or A/Ao are overly diminished) and thus the rate-increasing first term is dominant, the strain rate increases (with strain) in the plastic deformation regime. The term D/Do (the diameter ratio) is used hereafter instead of A/Ao for convenience.

The above mentioned varying nature of the specimen strain rate with deformation (strain) depending on the magnitude of D/Do  is  illustrated in Fig. 1 where oxygen-free copper was considered as the specimen material. In Fig. 1, curve R is the measured rate–strain curve using the bar signals (via the one-wave signal processing equation of the classic SHB theory), which was obtained via an explicit finite element analysis. Curve R* is the predicted rate–strain curve using the rate equation. The consistency of curve R* and curve R verifies the formulated rate equation. In Fig. 1(a) where the D/Do value is considerable (0.9), the slope of the rate–strain curve in the plastic deformation regime is negative because the rate-decreasing second term is more dominant than the rate-increasing first term. In contrast, In Fig. 1(b) where the D/Do value is very small (0.15), the slope of the rate–strain curve in the plastic regime is positive because the rate-decreasing second term is overly diminished compared with the rate-increasing first term.

In the literature, there has been no theory describing the varying nature of the specimen strain rate with deformation. Only the maximum limit of the specimen strain rate was described by the empirical relationship:

ε̇   <  Vo /L

where Vo is the impact velocity and L is the initial length of the specimen. The maximum rate limit predicted using the above relationship is indicated in Fig. 1(a) by the dashed horizontal line. This line is drawn only up to a limited strain to avoid complexity. By purely judging from the result of Fig. 1(a), where the strain rate decreases with strain because the rate-decreasing second term is dominant, the above relationship seems to predict the upper bound of the strain rate while the rate equation (curve R*) predicts a detailed change in the strain rate within the bound. However, when the rate-decreasing second term is diminished (when the values of σ and/or D/Do are diminished), as can be observed in Fig. 1(b), the above relationship turns out to be invalid while the rate equation (curve R*) still predicts the strain rate reasonably even in such a case. In overall, the rate equation is indeed an informative tool which allows one to understand the varying nature of the specimen strain rate in the SHB test.

Similarly, the converted stress–strain curve from the measured rate–strain curve (R) is named S** in this study. If the experiment is to be valid (if the curves S and R were measured reliably in experiment), the curves S and S** should be coincident. The correlation of the measured curves of stress–strain (S) and rate–strain (R) can also be checked from the coincidence of the curves S and S**. 

Curve R in Fig. 2(a) was also applied into the rate equation. The converted curve is shown in Fig. 2(b) as curve S**. Included in Fig. 5(b) is the measured curve S. As can be observed in Fig. 2(b), the fluctuating curves of S and S** coincide remarkably. Therefore, the measured stress–strain curve (S) is verified to be correlated with the measured rate–strain curve (R) via curve S**. 

As described above, the rate equation indicates that the measured curves of the stress–strain (S) and rate–strain (R) are correlated. This point was verified both numerically and experimentally in the original paper by demonstrating that the curves S and S** are coincident as well as the curves R and R**. Therefore, for the experiment and the bar-signal processing to be valid, the curves of S and R should reasonably coincide with S** and R**, respectively, as illustrated in Figs. 1-2. In this regard, the rate equation is a rigorous tool to verify the measured rate–strain curve simultaneously with the measured stress–strain curve, i.e., the reliability of the experiment. If the coincidence is not confirmed, it is necessary to check the experimental procedure or calibration of the SHB. The correlation method presented here can be used for the calibration of the instrument system as well.

The curves of rate–strain and stress–strain, which should be measured (manifested) in the SHB test, can be predicted before carrying out the SHB test using the rate equation provided the constitutive parameters are available. Fig. 3 shows the examples of the predicted curves of stress–strain (S*) and rate–strain (R*). The respective measured curves (S and R) using the bar signals are also included in Fig. 3.  In reality, the constitutive parameters are unknown for the specimen to be tested in the SHB. However, the constitutive parameters of the specimen can be reasonably estimated as follows.

 In general, the quasi-static test of a specimen is carried out before the SHB test. If two or more stress–strain curves are measured at two or more different strain rates in the quasi-static test (e.g., 10-5 and 10-2 s-1), the values of a, b, n, and c of the Johnson-Cook (JC) model can be determined suitably via non-linear curve fitting of the measured stress–strain curves. As for the thermal softening parameter (m), it is noted that its value does not vary significantly for similar types of material. For instance, the values of m for 1006 steel, 4340 steel, and S-7 tool steel are 1.00, 1.03, and 1.00, respectively. The values for aluminium alloy 2024-T351 and aluminium alloy 7039 are 1.00 and 1.00, respectively. The values for Armco® iron and Carpenter® electrical iron are 0.55 and 0.55, respectively. Therefore, the value of m for similar types of material to the current specimen can be reasonably obtained from the literature; the selected value of m in this way should not be far away from that of the current specimen. 

The above paragraph described the procedure for reasonably estimating the parameters of the JC model employed in this study. This model probably has been used and calibrated most extensively for simulating many high-strain-rate events of materials and structures. However, there are indeed numerous types of constitutive models, which were developed for capturing various aspects of complicated constitutive behaviours of versatile materials. For instance, when the specimen material exhibits the phenomenon of stress upturn and the material is anticipated to be used in the strain rate regime where the stress upturn takes place, the use and calibration of a stress upturn model would be more desirable than the JC model. A stress-upturn constitutive model is introduced in the subpage of this site, titled as Stress-Upturn Constitutive Models. When the phenomena of rate-hardening and temperature-softening are coupled or when strain-hardening and rate-hardening are coupled, models developed for such cases would be more appropriate. A similar procedure to the JC model described in the above paragraph can be employed for the reasonable estimation of the parameters of such other constitutive models. 

As mentioned, according to the rate equation, the specimen strain rate is controlled by the stress of the deforming specimen, geometry (the length and diameter) of specimen, impedance of bar, and impact velocity. One can suitably explore the effects of these variables by inputting appropriate values in the spread sheet cells in the provided Excel file, and the resulting curves of rate–strain and stress–strain are updated immediately after running the program. For the design of experiment, prediction of the rate–strain curve in this way before carrying out the SHB test should be more desirable than determining the manifested rate–strain curve after the experiment is finished. 

Once the constitutive parameters are reasonably estimated as above, the curves of rate–strain and stress–strain which are anticipated to be manifested (measured) in the SHB test can be predicted simultaneously using the provided Excel software. While this study illustrates the usage of the rate equation by combining it with the JC model, the provided program can be modified suitably for different constitutive models. An Excel program combining the rate equation and a stress-upturn constitutive model will be released soon after publication.

According to the diameter equation, because the specimen parameters vary with strain, there is no single (D/Do)c value for the perfectly constant strain rate as the strain increases. In other words, the diameter equation explains the theoretical reason why only a nearly constant strain rate is achieved in the SHB test even when the D/Do value is appropriately tuned. Therefore, in essence, the optimal D/Do value for achieving a nearly constant strain rate has to be determined experimentally by trials. However, the first trial value of D/Do can be reasonably estimated as described in the following paragraphs. 

To determine the first trial value of D/Do for achieving a nearly constant rate–strain curve, this study presents two practical methods as follows. The first practical method to determine the first trial value of D/Do is using the diameter equation. The diameter equation indicates that if a given stress–strain curve is applied into the equation, a (D/Do)c vs. strain plot is obtained. The original paper numerically verifies that the plot of (D/Do)c vs. strain is very useful to obtain the optimal  D/Do value that yields a nearly constant strain rate at the plastic deformation regime. The input stress–strain curve to the diameter equation can be obtained by multiplying the rate factor (the second bracket in the JC model) and temperature factor (the third bracket in the JC model) to a stress–strain curve measured at a quasi-static strain rate and temperature, which is considered as the  reference curve (measured at ε ̇o and Tref). The methods of obtaining the parameters c and m in the rate and temperature factors, respectively, were described previously. The resultant (D/Do)c vs. strain plot constructed by applying the input stress–strain curve into the diameter equation can be used to determine the first trial value of D/Do. 

The second practical method to find the first trial value of D/Do is using the rate equation itself, instead of using its strain derivative, i.e., the diameter equation. As described before, the constitutive parameters can be reasonably estimated from a couple of quasi-static tests and by referring to the value of m in the literature for similar types of materials. Then, the numerical solution using the provided Excel program will readily produce the rate–strain curves such as the ones shown in Fig. 4(a) for a range of D/Do values, which will allow one to determine the D/Do value for the first trial to obtain a nearly constant strain rate (Fig. 4(b)). Actually, constructing the anticipated rate–strain curve in this way is desirable before carrying out the SHB test. The optimum value of (D/Do)c shown in Fig. 4 is limited to the considered specimen, bar, and impact condition. 

(6) Tool for Describing SHB Operating Conditions

Predicting the SHB operation condition is important especially when high-strength materials are tested. In order to achieve a positive specimen strain rate in the SHB test, the rate equation should be positive:

 Vo - 2Aσ exp(ε)/(AoρoCo) > 0

 According to the above  inequality, an overly high specimen stress (σ), an overly high A/Ao ratio, and/or an overly low impact velocity (Vo) yields a negative value of specimen strain rate; the SHB does not operate. The Excel program of the original paper allows one to screen such an impact condition. 

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