Bar Technology for High-Strain-Rate Material Properties

Contents

1. Bar Technology

2. Evolution of Specimen Strain Rate in Split Hopkinson Bar Test

3. One-Dimensional Analyses of Striker Impact on Bar

4. Design Guidelines for Split Hopkinson Tension Bar and Origin of Spurious Waves

- Followings will be available here in near future -

5. A Solution Process for Pochhammer-Chree Equation.

6. A Review on the Dispersion Correction of Hopkinson Bar Signals.

- Followings will be available here in the future -

7. Calibration of Bar Properties for Impact Applications.

8. Determination of Specimen Diameter for Achieving a Constant Strain Rate in Split Hopkinson Bar Test.

9. Friction Compensation of a Stress-Strain Curve Measured using the SHPB.

10. Calibration of a Stress-Upturn Constitutive Model at Strain Rates Beyond 10⁴s^-1.

11. An Impact Instrument for Measuring Material Properties at Strain Rates Beyond 10⁵s^-1.

12. Calibration of a Stress-Upturn Constitutive Model at Strain Rates Near 10⁶s^-1.

1. Bar Technology

(1) Hopkinson Pressure Bar (HPB)

Measurement of the dynamic transient load at near-field from high energy sources such asbullet impact and high-explosive detonation isnever a simple task. Difficulty in measurement arises from limit in robustness or sensitivity of instruments/sensors.

In 1914, B. Hopkinson [1] measured the dynamic pressure pulsefirst time using a circular bar (rod). In his experiment, the flying piece (C) was attached to the rod ( B) via the weak force of the electrical solenoid magnet. The pressure pulse was produced by the detonation of gun cotton (A) on one end of the bar (rod).

B. Hopkinson's experimental set up

"A rifle bullet is fired at, or gun-cotton is detonated near, the other end; the short piece flies off and is caught in a box suspended in a similar manner to the long rod. Suitable recording arrangements register the movement both of the long rod and of the box, and the momentum in each is calculated in the usual way as for a ballistic pendulum."

Based on the measured momenta of the flying piece (C) and rod (B), the pressure-time curve when bullet impacted the rod was constructed (Fig. 7). The pressure-time curve was also constructed when gun cotton was detonated (Fig. 13).

His pressure-time curve was constructed using only the maximum magnitude and pulse duration time, based on the momentum measurement; a rough measurement. In these rough measurements of the pressure-time curves, the influence of the shape change (dispersion) of the traveling elastic wave in the rod was insignificant.

His method was developed further by Robertson (1921) [2] and Landon & Quinney (1923) [3]. Detailed pressure profile with time could be measured 34 years later by Davies (1948) [4] by measuring electrically the variation of the pressure pulse with time t of either the longitudinal displacements of the measuring end of the bar, or, alternatively, the radial displacement of the cylindrical surface of the bar. As the detailed time profile of the pressure pulse was measured, dispersion correction was possible in the study of Davies [4].

The circular bar used asa stress transducer was called later as theHopkinson pressure bar (HPB) in honor of B. Hopkinson who used the bar first time to measure the pressure-time pulseno matter how accurately the pressure-time pulse was measured. As mentioned, his study inspired/triggered later scholars to improve accuracy in measurement itself (using an electric condenser unit, strain gage, and oscilloscope) and in analysis of the measured result (via dispersion correction).

The HPB has been used not only to measure the impact load [4–8] by bullet, striker, blast, or explosive detonation, but is also used nowadays to non-destructively assess materials and structures [9–17].

(2) Split Hopkinson Bar (SHB)

In 1949, H. Kolsky [18] splitted the HPB in two pieces and sandwiched the specimen between the two bars. Then, he appliedexplosive detonation at one end of the bar (input bar). The dynamic material property (stress-strain curve) of the specimen was measured using incident, reflected, and transmitted pulses based on the one-dimensional theory developed by him; no wave dispersion was considered. This instrument is called now the split Hopkinson pressure bar (SHPB) or Kolsky bar. It was improved later via the employment of a circular-bar striker, strain gage, oscilloscope, and dispersion correction; it was also modified to measure tensile, torsional, and triaxial properties. The fundamental theory and applications of the split Hopkinson bars (SHBs) are well documented in the literature [18-26]. SHBs have been used extensively to measure the curves of uniaxial stress–strain and strain rate–strain of versatile materials.

H. Kolsky's experimental set up

(3) High-Speed (High-Strain-Rate) Events

High-strain-rate events such as, crashes of the high-speed transportation means (airplanes, express trains, and automobiles), high-speed machining, rock blasting, building blasting, explosive welding, projectile penetration, explosion, blast impact, meteorite impact, and earthquake/tzunami impact need an in-depth understanding of physics behind the events to achieve a reliable design of solids and structures associated with the events. Relying solely on experimental approaches for the design/understanding of the high-strain-rate behaviour of solids and structures may not be practical because of the burden of cost and time; however, an experimental verification of the design result is inevitable. Modelling and simulation (M & S) helps one to understand the evolution of the high-strain-rate phenomena on a step-by-step basis and is a time- and cost-efficient design approach. Therefore, it is advantageous to combine the M & S-based design with experimental verification of the mechanical behaviour of solids and structures exposed to high-strain-rate events.

Airplane crash

Express train crash (1)

Express train crash (2)

Auto-Train crash

Automobile crash (1)

Automobile crash (2)

High-speed machining (1)

High-speed machining (2)

High-speed machining (3)

Earthquake impact (1)

Earthquake impact (2)

Earthquake impact (3)

Earthquake impact (4)

Tzunami impact

Rock blasting(1)

Rock blasting (2)

Building blasting (1)

Building blasting (2)

Blast impact

Penetration (1)

Penetration (2)

Explosive welding

Meteorite impact (1)

Meteorite impact (2)

Fig. 1. High-strain-rate events.

(4) Split Hopkinson bar nowadays

For the M & S-based design/understanding of the high-strain-rate behaviour of solids and structures, a strain-rate-dependent constitutive model is indispensable (together with the fracture model and equation of state, if necessary). A number of uniaxial stress–strain curves measured precisely at a wide range of strain rates are required for the calibration of the strain rate-dependent constitutive model.

The split Hopkinson bar (SHB) [1–2], which is also called the Kolsky bar, measures the uniaxial stress–strain curves at the strain rates from approximately 10² to 10⁴ s^-1. The SHB system is schematically illustrated in Fig. 2. Good sources for the fundamental theory of SHB are listed in references [3–15]. They describe the classic (one-dimensional) theory in detail; it is not repeated here.

Fig. 2.

Compression-mode SHB.

The example of the measured signals in the SHB test is illustrated in Fig. 3(a). In this figure, the incident, reflected, and transmitted waves were measured using strain gages attached in the incident and transmit bars. The mili-volt signal in Fig. 3(a) is converted to strain values using appropriate conversion factor of the electronic measuring system (gage factor of the wheat stone bridge and the amplification factor of the amplifier): the incident (ε_I), reflected (ε_R), and transmitted (ε_T) pulses are recorded.

The strain valuesrecorded at the bars (ε_I, ε_R, and ε_T) are then used to determine the engineering stress (s), engineering strain rate (é ), and engineering strain (e) of thespecimen via a one-wave analysis of the SHB theory, which is illustrated in Fig. 3(b). In Fig. 3(b) ,ε_I,ε_R, andε_T denote engineering strain records from the strain gage for incident (ε_I), reflected (ε_R), and transmitted pulses (ε_T), respectively;A andL are the initial cross- sectional area and initial length of the specimen, respectively;A_o,E_o, andC_o are the cross-sectional area, elastic modulus, and sound speed of the bar, respectively;t is time.

Once the curves of engineering stress–strain and engineering rate–strain are determined by treating the bar signals using the equations in Fig. 3(b), they are converted to curves of true stress–strain and true strain rate–strain. In the conversion process from engineering properties to true properties, it is assumed that the specimen is incompressible (at Poisson’s ratio of 0.5) and deforms uniformly although, in the elastic regime, the specimen is compressible according to the Poisson’s ratio of approximately 0.3 for metallic materials. This limitation exists only in the elastic part of deformation because the plastic deformation of a metallic specimen is incompressible. An example of the finally obtained curves of true stress–strain and true strain rate–strain are illustrated in Fig. 3(c).The curves illustrated in Fig. 3(c) are used for the calibration of the strain rate-dependent constitutive models.

Fig. 3(a).

Example of signals measured in the bars [3].

Fig. 3(b).

Signal processing equations.

Fig. 3(c).

Example curves of true stress–strain and true strain rate–strain [3].

Since the pioneering studies of Hopkinson in 1913 [1] and Kolsky in 1949 [2], much advancement in the SHB test technology has been achieved, which made the SHB to be used extensively nowadays for measuring dynamic stress–strain curves of various materials: not only metals but also non-metallic materials such as ceramics, concrete, rocks, soil (sand), plastics, rubber, foam, honeycombs, wood, and various types of composites. It is an ‘open’ tool to researchers. While the original inception of the SHB was based on the compression mode, much modification has been made thereafter including tensile, torsional, and triaxial SHBs.

However, there is no international standard for the SHB test. Each mechanical bar system for generating elastic pulses and each electronic system for signal acquisition/filtering have their own particular characteristics. Furthermore, test conditions including the specimen geometry, bar properties, bar geometry, and the signal processing after the test for versatile materials depend on the choice of researchers. Not only the instrument characteristics but also the test conditions need to be specified with in-depth understanding of physics behind the specifications to reduce the uncertainty of the test result using the SHB instrument, which will foster the establishment of a standard for the SHB test.

[1] Hopkinson B. A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets.Philos Trans R Soc Lond Series A 1914; 213: 437–456.

[2] Robertson, R. Some properties of explosives.Trans Chem Soc 1921; 119: l–29.

[3] Landon JW and Quinney H. Experiments with the Hopkinson pressure bar.Proc R Soc A 1923; 103: 622–643.

[4] Davies RM. A critical study of the Hopkinson pressure bar.Philos Trans R Soc London Ser A 1948; 240~821: 375–457.

[5] Hsieh DY, Kolsky H. An Experimental study of pulse propagation in elastic cylinder.Proc Phys Soc 1958; 71(4): 608–612.

[6] Nagy PB. Longitudinal guided wave propagation in a transversely isotropic rod immersed in fluid. The Journal of the Acoustical Society of America 1995; 98: 454.

[7] Clarke S, Fay S, Warren J et al. A large scale experimental approach to the measurement of spatially and temporally localized loading from the detonation of shallow-buried explosives.Meas Sci Technol 2015; 26(1): 015001.

[8] Fay S, Rigby S, Tyas A et al. Displacement timer pins: an experimental method for measuring the dynamic deformation of explosively loaded plates.Int J Impact Eng 2015; 86: 124–130.

[9] Rigby S, Tyas A, Clarke S et al. Observations from preliminary experiments on spatial and temporal pressure measurements from near-field free air explosions.Int J Protective Struct 2015; 6(2): 175–190.

[10] Rigby S, Fay S, Clarke S et al. Measuring spatial pressure distribution from explosives buried in dry Leighton Buzzard sand.Int J Impact Eng 2016; 96: 89–104.

[11] Pavlakovic B, Lowe M and Cawley P. High-frequency low-loss ultrasonic modes in imbedded bars.J Appl Mech 2001; 68(1): 67–75.

[12] Hayashi T, Song WJ and Rose J. Guided wave dispersion curves for a bar with an arbitrary cross-section, a rod and rail example.Ultrasonics 2003; 41(3): 175–183.

[13] Cui H and Zhang B. Excitation mechanisms and dispersion characteristics of guided waves in multilayered cylindrical solid media.J Acoustical Soc Am 131, 2048 (2012)

[14] Subhani M, Li J and Samali B. A comparative study of guided wave propagation in timber poles with isotropic and transversely isotropic material models.J Civil Struct Health Monitoring 2013; 3(2): 65–79.

[15] Farhidzadeh A and Salamone S. Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves.Ultrasonics 2015; 57: 198–208.

[16] Leinov E, Lowe MJ and Cawley P. Ultrasonic isolation of buried pipes.J Sound Vibration 2016; 363: 225–239.

[17] Crespo BH, Courtney CRP, and Engineer B. Calculation of guided wave dispersion characteristics using a three-transducer measurement system.Appl Sci 2018; 8:1253.

[18] Kolsky H. An investigation of the mechanical properties of materials at very high rates of loading.Proc Phys Soc Lond Sect B 1949; 62(11): 676–700.

[19] Meyers MA.Dynamic behavior of materials. John Wiley & Sons, Inc, New York, 1994.

[20] Field J, Walley S, Bourne N, Huntley J. Experimental methods at high rates of strain.J de Physique IV 1994; 04(C8): C8-3–C8-22.

[21] Al-Mousawi MM, Reid SR and Deans WF. The use of the split Hopkinson pressure bar techniques in high strain rate materials testing.Proc Inst Mech Engr Part C: J Mechanical Engineering Science 1997; 211: 273–292.

[22] Chen W and Song B.Split Hopkinson (Kolsky) bar - Design, testing, and Applications. Springer Science+Business Media, LLC, New York, 2011.

[23] Othman R.The Kolsky-Hopkinson bar machine. Springer International Publishing, 2018.

[24] Shin H and Kim J-B. Evolution of specimen strain rate in split Hopkinson bar test.Proc IMechE, Part C: J Mechanical Engineering Science 2019. https://doi.org/10.1177/0954406218813386

[25] Shin H, Lee J-H, Kim J-B, et al. Design guidelines for the striker and transfer flange of a split Hopkinson tension bar and the origin of spurious waves.Proc IMechE, Part C: J Mechanical Engineering Science 2019.

[26] Shin H and Kim D. One-dimensional analyses of striker impact on bar with different general impedance. 2019.

2. Evolution of Specimen Strain Rate in Split Hopkinson Bar Test

(1) Evolution of Specimen Strain Rate

(a) Evolution of specimen strain rate

One of the important phenomena in the SHB test that needs to be understood is thatthe magnitude of the reflected pulse monitored in the input bar often varies 'significantly' with time (with specimen deformation). A typical example of the varying magnitude of the reflected pulse (wave) with time is illustrated in Fig. 3(a). Even though the fluctuating characteristics of the elastic wave in the bar is admitted, the varying nature of the magnitude of the reflected pulse is undoubtedly clear. According to the fundamental (one-dimensional) theory of SHB, the reflected pulse monitored in the input bar represents the specimen strain rate (see the second equation in Fig. 3(b)).

For the reliable test using the SHB,it is necessary to understand the varying nature (evolution) of the specimen strain rate in the SHB test from the viewpoint of mechanical science. Such fundamental understanding may lead to finding practical means of solving further issues to be cleared in the SHB test, which will be described below (section 4: Utilisation of the Rate Equation). Here at this site, a strain rate equation is introduced, which equation describes the evolution of the specimen strain rate as a function of strain (specimen deformation).

The term ‘rate’ means the ‘strain rate’ hereafter. The term ‘rate–strain curve’ will be used preferably to the term ‘strain rate–strain curve’ for improving readability. The term ‘rate equation’ is used interchangeably with the term ‘strain rate equation’.

(b) Assumptions

The classic theory of SHB is based on the following assumptions:

(i) The velocity of the elastic wave in the bars is described by a slender rod (one-dimensional) approximation (C=(E/ρ )^1/2, whereC is the velocity of sound,E is the elastic modulus, andρ is the density).

(ii) The specimen expands freely along the radial direction (no friction between the specimen and the bars) while it deforms axially.

(iii) There is no inertia in the specimen and bars.

(iv) The specimen deforms uniformly along the axial direction; the stress and strain of the specimen at the front surface of the specimen are the same as those at the rear surface of the specimen; this stress uniformity of the specimen results in the force equilibrium between the ends of the bars, which are in contact with the specimen.

The same assumptions as the classic theory of SHB (as described above) are employed in the derivation process of the strain rate equation [16].

(c) Strain rate equation

The strain rate equation formulated in the study of Shin and Kim [16] is presented here (right side). In the presented equation, a positive sign is assigned to the compressive property. The strain rate equation when a positive sign was assigned to the tensile property was also derived in reference [16]. In the presented form of the rate equation, the exponential function is located at the denomenator to keep the same fashion as the expressions for the rate equations under different sign conventions (see reference [16]).

(2) Utilisation of the Rate Equation

Reference [16] is comprised of (1) a fairly long article with 21 double-column pages, (2) a Supplemental Material, and (3) an Excel® program wherein the rate equation is implemented. Utilisation areas of the rate equation described below may serve as an executive summary of the paper.

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

Theprimary, and probably the most important contribution of the rate equation is allowing one to understand the physical origin of the varying nature of specimen strain rate in the SHB test. According to the formulated rate equation, the specimen strain rate is found out to be controlled by the stress and strain of the specimen during deformation, geometry (the length and diameter) of specimen, impedance of bar, and impact velocity. To reveal the varying nature (evolution) of the strain rate with strain (specimen deformation), the formulated rate equation is analysed as follows.

The rate equation is composed of two terms. The first term in the right side of the rate equation increases the magnitude of the rate with strain. On the other hand, the second term decreases the magnitude of the rate with strain. Therefore, the specimen strain rate evolves as a result of the competition between the rate-increasing first term and rate-decreasing second term.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/A_o) 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 ofσ andA/A_o 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 ofσ orA/A_o are overly diminished) and thus the rate-increasing first term is dominant, the strain rate increases (with strain) in the plastic deformation regime. The termD/D_o (the diameter ratio) is used hereafter instead ofA/A_o for convenience.

The above mentioned varying nature of the specimen strain rate with deformation (strain) depending on the magnitude ofD/D_o is illustrated in Fig. 4 where oxygen-free copper was considered as the specimen material. In Fig. 4, 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. 4(a) where theD/D_ovalue 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. 4(b) where theD/D_ovalue 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 [6, 8]:

ε̇ <V_o /L

whereV_ois the impact velocity andL is the initial length of the specimen. The maximum rate limit predicted using the above relationship is indicated in Fig. 4(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. 4(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 ofσ and/orD/D_o are diminished), as can be observed in Fig. 4(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.

Fig. 4(a).

Rate–strain curve when the magnitude of the rate-decreasing second term is considerable (D/D_o=0.9).

Fig. 4(b).

Rate–strain curve when the magnitude of the rate-decreasing second term is overly diminished (D/D_o=0.15).

2) Tool for verifying test result

In the literature, the reliability of the measured stress–strain curve has been verified by checking the coincidence of stresses at the front and back surfaces of the specimen (stress equilibrium) [17–26]. However, as for the reliability of the strain rate–strain curve of specimen, which is also required for the calibration of the constitutive model, there has been no direct tool to verify the measured result. If a method for verifying the measured strain rate–strain curve is available, it can also be verified, improving the reliability of the SHB test. In this regard, thesecond application area of the rate equation is using it as a tool to verify the measured strain rate–strain curve simultaneously with the measured stress–strain curve using the formulated rate equation based on the correlation of the strain rate–strain curve with the stress–strain curve; the rate equation describes the relationship between the two curves.

Themeasured curves of stress–strain (S) and rate–strain (R) using the bar signals can also be converted to each other via the rate equation. In this study, the converted rate–strain curve from the measured stress–strain curve (S) is named R**. According to the rate equation, if the experiment is to be valid (if the curves S and R were measured reliably in experiment), the curves R and R** should be coincident. The correlation of the measured curves of stress–strain (S) and rate–strain (R) can be checked from the coincidence of the curves R and R**.

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

Fig. 5(a).

Coincidence of curves R and R** .

Fig. 5(b).

Coincidence of curves S and S** .

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. 5(a) was also applied into the rate equation. The converted curve is shown in Fig. 5(b) as curve S**. Included in Fig. 5(b) is the measured curve S. As can be observed in Fig. 5(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 themeasured curves of the stress–strain (S) and rate–strain (R) are correlated. This point was verified both numerically and experimentally in reference [16] 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. 4-5. 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.

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

In the SHB test, the unknown stress–strain curve of the specimen is determined at a target strain rate. The issue is that the specimen strain rate is also unknown. The state-of-the-art technology to obtain the target strain rate in the SHB test relies on trials or previous experience for specimens with similar property and geometry to those of the current specimen. The actually manifested specimen strain rate in the SHB test can be revealed only after the experiment is finished. In the literature, it is difficult to find a method for predicting the specimen strain rate before carrying out the SHB test. If a method for predicting the specimen strain rate is available, it should be useful for the design of the experiment. In this regard, thethird application area of the rate equation is using it as a tool to practically predict the specimen strain rate before carrying out the SHB test: the method simultaneously solves the formulated strain rate equation and a reasonably estimated constitutive equation of the specimen, which results in the anticipated curves of strain rate–strain and stress–strain in the SHB test.An Excel® program for solving the two equations is provided in Reference [16].

Fig. 6(a).

Examples of predicted curves of stress–strain (S*) using the rate equation. Curve S is the measured curve.

Fig. 6(b).

Examples of predicted curves of rate–strain (R*) using the rate equation. Curve R is the measured curve.

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 equationprovided the constitutive parameters are available. Fig. 6 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. 6. 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 ofa, b, n, andc of the Johnson-Cook (JC) model [27] 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 [9]. For instance, the values ofm 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 ofm for similar types of material to the current specimen can be reasonably obtained from the literature; the selected value ofm 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 [26] 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 [28] would be more desirable than the JC model. A stress-upturn constitutive model [28] is introduced in the subpage of this site, titled asStress-Upturn Constitutive Models. When the phenomena of rate-hardening and temperature-softening are coupled [29] or when strain-hardening and rate-hardening are coupled [30], models developed for such cases [29, 30] 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® program. 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 [28] will be released soon after publication.

4) Tool for predicting maximum strain of the specimen

Neither the rate equation nor constitutive equation has a limit in strain (there is no strain limit in drawing curves S* and R*) as far as the strain rate is positive. The specimen strain measured in the SHB test can be limited by the pulse duration time and fracture of the specimen. This study considers the case where the specimen strain is limited by the pulse duration time (t_p=2L_o/C_o); it is assumed that specimen fracture does not occur meanwhile the stress pulse passes the specimen. The maximum specimen strain (ε_max) in the SHB test is determined by the pulse duration (passage) time (t_p) and the strain rate that the specimen experiences: ε_max=t_p ε̇. Because the pulse duration time is fixed, a higher rate of the specimen results in a higher maximum strain.

In the current technology, the maximum strain that a specimen experiences in the SHB test is revealed only after the test is finished. However, once the rate–strain curve is available prior to the SHB test, the maximum strain can be predicted by combining the rate–strain curve with the pulse duration time (thefourth application area of the rate equation). The method calculates first the incremental deformation time (dt) at each strain step (dε):dt=dε/ε̇ . Then, the value of the strain, at the moment when the cumulative deformation time over the strain steps reaches the pulse duration time (t_p), is determined to be the maximum strain.The algorithm to predict the maximum strain in this way is included in the Excel® program. It considers the condition that the strain rate is positive. The predicted maximum strains using the Excel® program are reasonably consistent with the results of the numerical simulations in Ref. [16].

5) Tool for achieving a nearly constant strain rate

As mentioned, the specimen strain rate usually varies during the SHB test. From the viewpoint of measuring the material properties or investigating a dynamic phenomenon at a given strain rate, it is necessary to control the specimen strain rate in the SHB test to achieve a constant strain rate. In this regard, researchers employed the pulse shaping techniques [12,31–40] which utilises a conical striker, dummy specimen, or tip material to obtain a nearly constant strain rate. The achieved nearly constant strain rate using these techniques can be confirmed from the result of numerical analyses. However, a theory-based understanding of the achieved result using the mentioned techniques is limited because no analytical expressions describing the reason for achieving a nearly constant strain rate are available at the present time. The process of achieving a nearly constant strain rate in these techniques is an iterative process based on an open loop control [12], which means that the process does not need the feedback of the previous result. The conditions for achieving a nearly constant strain rate depend on the unknown properties of the specimen to be tested such as the dynamic stress–strain curve. The conditions of the constant strain rate also depend on the impact velocity and specimen geometry. The reason for such dependencies of the constant-strain-rate conditions in the mentioned techniques is currently unavailable either. If a theory-based method for achieving a nearly constant strain rate in the standard SHB test (without the aid of the pulse shaper or conical striker) is available, the method will allow researchers to readily understand why a nearly constant strain rate was achieved in their test and may serve as an informative method for achieving a nearly constant strain rate. In this regard, thefifth application area of the rate equation is using it as a tool to tailor the slope of the rate–strain curve by simply controlling the specimen diameter. Two practical methods to determine the specimen diameter for achieving a nearly constant strain rate are presented in detail in reference [16], which drastically facilitates the convergence of a closed loop control process of the handy method for achieving the optimal specimen diameter. They are briefed below.

The analytical expression of theD/D_o value for achieving a constant specimen strain rate, (D/D_o)_c, can be obtained by differentiating the rate equation with respect to ε, i.e., from the condition of dε̇ /dε = 0. The obtained result is presented below, which is named as the diameter equation for the constant strain rate in this study.

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

To determine the first trial value ofD/D_o 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 ofD/D_o is using the diameter equation. The diameter equation indicates that if a given stress–strain curve is applied into the equation, a (D/D_o)_c vs. strain plot is obtained. Reference [16] numerically verifies that the plot of (D/D_o)_c vs. strain is very useful to obtain the optimalD/D_o 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 atε ̇_o andT_ref). The methods of obtaining the parametersc andm in the rate and temperature factors, respectively, were described previously in subsection (3). The resultant (D/D_o)_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 ofD/D_o.

The second practical method to find the first trial value ofD/D_o is using the rate equation itself, instead of using its strain derivative, i.e., the diameter equation. As described in subsection (3), the constitutive parameters can be reasonably estimated from a couple of quasi-static tests and by referring to the value ofm 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. 7(a) for a range ofD/D_o values, which will allow one to determine theD/D_o value for the first trial to obtain a nearly constant strain rate (Fig. 7(b)). Actually, constructing the anticipated rate–strain curve in this way is desirable before carrying out the SHB test. The optimum value of (D/D_o)_c shown in Fig. 7 is limited to the considered specimen, bar, and impact condition.

Fig. 7(a).

Predicted rate-strain curves for a range ofD/D_o values using the Excel program provided in reference [16].

Fig. 7(b).

Predicted curves of rate-strain (R*), demonstrating a nearly constant strain rate during specimen deformation.

In the SHB test, the use of the specimen with the first trial value ofD/D_o based on either of the methods presented above should be more appropriate than using the specimen with an arbitraryD/D_ovalue because the latter may yield a significantly varying strain rate of the deforming specimen (such as curve R in Fig. 6(b) when the initial temperature is 25 degree Celsius). Even the first trial value ofD/D_o determined by the presented methods may not be far away from the optimal value. TheD/D_o value can be tuned further via the second-trial test only when a better nearly-constant strain rate than the first-trial test result is necessary. As can be observed in Fig. 7(a), if the slope of the rate–strain curve in the second trial test needs to be increased, a smallerD/D_ovalue (which reduces the magnitude of the second term of the rate equation) is required and vice versa to decrease the slope of the rate–strain curve in the second trial test.

The rate equation not only provides a theory-based handy method for achieving a nearly constant strain rate by controlling the specimen diameter but also drastically speeds up the convergence process towards the optimumD/D_o value by providing the first trial value, which is anticipated to be reasonably close to the optimum value. Even when the second or more trial tests are needed, the progress of convergence to the optimumD/D_o value can be monitored suitably and systematically because the process of determining the optimumD/D_o value (described in the above paragraph) is a closed loop process that utilises the feedback of the previous test result. Being able to monitor such a systematic convergence is a big advantage of the closed loop control process.

A succeeding experimental study on the determination of the optimum diameter of the specimen in the SHB test is currently carried out following the above-mentioned methods. The result will be published elsewhere and reference information for the article will be available at this site.

6) Tool for describing SHB operation condition

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:

V_o - 2Aσ exp(ε)/(A_oρ_oC_o) > 0

According to the above inequality, an overly high specimen stress (σ), an overly highA/A_o ratio, and/or an overly low impact velocity (V_o) yields a negative value of specimen strain rate; the SHB does not operate. The Excel program of Reference [16] allows one to screen such an impact condition.

(3) Relationship: Rate Equation and Classic SHB Theory

Reference [16] formulated the strain rate equation, and presented some application areas. As the same assumptions as the classic (one-dimensional) theory of SHB were employed in the derivation process of the rate equation,the rate equation does not contradict the SHB theory but is within its boundary. The rate equation allows one to investigate an unexplored area in the classic SHB theory:the evolution of the specimen strain rate with strain (specimen deformation).

(4) Conclusion

To reveal the physical origin of the varying nature of specimen strain rate during deformation in the SHB test, the strain rate has been formulated as a function of strain (specimen deformation) based on a one-dimensional assumption. According to the formulated rate equation, the specimen strain rate is governed by the stress and strain of specimen during deformation, geometry (the length and diameter) of specimen, impedance of bar, and impact velocity. The rate equation is composed of two terms: the rate-increasing first term (V_oexp(ε)/L) and the rate-decreasing second term (−2Aσ exp(2ε)/(A_oρ_oC_oL)). Therefore, the specimen strain rate evolves as a result of the competition between the rate-increasing first term and rate-decreasing second term. Unless these two terms are balanced, the specimen strain rate generally varies (decreases or increases) with strain (specimen deformation), which is the physical origin of the varying nature of the specimen strain rate in the SHB test. The increase in specimen stress during deformation (e.g., work hardening), appearing in the second term, plays a role in decreasing the slope of the rate–strain curve in the plastic deformation regime.

According to the formulated strain rate equation, the measured curves of stress–strain and rate–strain are mutually correlated. The rate equation can be used as a tool to verify the measured rate–strain curve simultaneously with the measured stress–strain curve, i.e., to verify the reliability of the experiment. If the experimentally measured curves of stress–strain (S) and rate–strain (R) are superposed to the curves of S** and R**, which were converted from curves of R and S, respectively, using the rate equation, the curves of S and S** should be coincident as well as the curves of R and R**. Otherwise, it is be necessary to check the experimental procedure and instrument calibration. The method of correlation can also be used as a tool for the calibration of the instrument. The intersection points between curves R and R** (S and S**) can be used to extract a less-fluctuating rate–strain (stress–strain) curve, which can be used as representative curves of stress–strain and rate–strain for the calibration of a constitutive model.

It has been numerically demonstrated that the rate–strain curve and stress–strain curve measured in the SHB test can be predicted before carrying out the test by simultaneously solving the rate equation and a constitutive equation. The program for solving the two equations is implemented in the Excel® software, which is available in the Supplementary Material. For practical prediction of the stress–strain curve and rate–strain curve, the constitutive parameters of the Johnson–Cook (JC) model can be reasonably estimated from a couple of quasi-static tests together with referring to the thermal softening parameter of a similar type of material to the current specimen in the literature. The parameters of other constitutive models can also be reasonably estimated similarly.

Once the rate–strain curve is available, the maximum specimen strain in the SHB test can be predicted by combining the strain rate–strain curve with the pulse duration time. Such an algorithm is also included in the Excel® program.

It has also been demonstrated both numerically and experimentally that the slope of the rate–strain curve in the plastic regime can be tailored by controlling the diameter of the specimen. Two practical methods to determine the value of the relative diameter (D/D_o) for achieving a nearly constant strain rate are presented. The first method is using the (D/D_o)_c vs. strain plot which can be constructed by applying an input stress–strain curve into the diameter equation; the input curve can be obtained by multiplying the rate and temperature factors of the JC model to a quasi-static stress–strain curve. The second method is simultaneously solving the rate equation and a reasonably estimated constitutive equation (using the Excel® program) for a range ofD/D_o values.

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. An overly high specimen stress (σ), an overly highA/A_o ratio, and/or an overly low impact velocity (V_o) yields a negative specimen strain rate; the SHB does not operate. The Excel program allows one to screen such an impact condition.

As the same assumptions as the fundamental theory of SHB were employed in the derivation process of the rate equation, the rate equation is within the boundary of the SHB theory. The rate equation allows one to investigate an unexplored area in the classic (one-dimensional) SHB theory: the evolution of the specimen strain rate with strain (specimen deformation).

(5) Download the Paper (Ref. [16])

The pdf file of the article as well as the Supplemental Material (the Excel file) is openly accessible at the publisher's website (Click here).

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2. One-Dimensional Analyses of Striker Impact on Bar

3. Design Guidelines for Split Hopkinson Tension Bar and Origin of Spurious Waves