Advertisement

Dynamic Properties of Geologic Specimens Subjected to Split-Hopkinson Pressure Bar Compression Testing at the University of Kentucky

  • Russell Lamont
  • Jhon Silva
Original Paper

Abstract

Advances in materials science have shown that material behavior varies according to the rate of load application (strain-rate sensitivity). With regards to compressive strength, materials have been observed to exhibit a strengthening, weakening, or negligible response to increasing strain-rates (Zhang and Zhao in Rock Mech Rock Eng 47(4):1411–1478, 2014). Practical experimentation to ascertain these responses has been carried out for over a century, based on the fundamental equipment design pioneered by John Hopkinson in 1872 and modified by Kolsky in 1949. A contemporary Split-Hopkinson Pressure Bar (SHPB) has been constructed at the University of Kentucky (UKY) to research the dynamic properties of various geologic materials for mining and civil engineering applications. Geologic samples are of an inconsistent nature due to inherent discontinuities and large grain size. To ensure test specimens are of adequate size to reflect this inconsistent nature, the SHPB at the UKY has been constructed with component bars of 2 in. (5.08 cm) diameter. Prior publications have discussed various considerations associated with the testing procedure and data processing of this SHPB (dispersion correction, pulse shaping, etc.) (Silva and Lamont 2017). This publication presents the results of materials testing with this SHPB. Three materials were selected: Bedford (Indiana) Limestone, Berea (Ohio) Sandstone, and Aluminum 6061-T6. Two of these are common aggregates found in the mining and construction industries, while the third is an aluminum variant often encountered in industrial applications. Dynamic compression testing of these materials at various strain rates was carried out, and the results are included. Static test results have been included for comparison, and the testing and data analysis procedure are discussed in detail.

Keywords

Hopkinson bar Compression testing Dynamic properties Geologic samples Strain response 

1 Introduction

1.1 Background

The application of compressive loading is often divided into two classifications: “static”, and “dynamic”. As the names suggest, static loading represents situations in which materials are subjected to stationary loads which change negligibly over time. A great amount of historical effort has gone into refining the static testing methodology, largely due to the relative ease of data monitoring and processing that accompanies a slow moving process. However, in practical circumstances, real-world loads often change significantly over time. These are known as dynamic loads. The static properties of most materials are still useful in indicating probable responses, even under dynamic loading, and thus are included in many engineering models. Attempts to more accurately model and predict dynamic loading scenarios have led to efforts to define dynamic properties, despite the degree of difficulty involved.

The design and procedure of dynamic compression testing is a complex task, as no single established standard for this testing has yet attained general acceptance. Instead, the science describing the physical processes which occur during dynamic testing has been explored and explained, which may then be used by researchers to establish independent but often similar test protocols. The construction and operation principles of the SHPB at the UKY followed basic guidelines which are summarized by the following characteristics:
  1. 1.

    Two equal-length, thin bars (known as the incident and transmitted bars) constructed of a high-strength material such as steel, titanium, or nickel alloy.

     
  2. 2.

    A striker bar constructed of the same material, typically designed to be the same diameter but of much shorter length than the other two bars. Using a barrel, this striker bar is directed to strike in alignment with the incident bar.

     
  3. 3.

    A compressed gas system with quick-release valve used to impart momentum to the striker bar, and a laser interruption type device used to capture striker velocity.

     
  4. 4.

    Minimum of two strain gauges bonded to the surface of the incident and transmitted bars and connected to a data acquisition system. These strain gauges are usually installed equidistant from the specimen-bar interfaces to simplify the data processing.

     
As shown in Fig. 1, the SHPB constructed at the UKY is composed of perfectly machined SAE 4340 grade steel bars of 2 in. (5.08 cm) diameter. The incident and transmitted bars are 96 in. (2.45 m) long, while the striker bar (bullet) used for this testing is 24 in. (0.61 m) in length with a mass of 20.28 lbf (9.2 kg). The Poisson’s Ratio of this steel is 0.29, and it possesses a density of 490.06 pcf (7.85 g/cc). A compressed gas system capable of sustaining charge pressures up to 700 psi (4.83 MPa) allows for control of impact velocity. Strain gauges are installed at the locations shown in Fig. 1 above, with multiple gauges at each location to allow for signal combination, thus reducing any signal variation possible from gauge installations. An MREL DataTrapII data recorder is used to capture the signals, with sampling rates of up to 10 MSa/s. More detailed discussion of the SHPB characteristics and experimental setup has been covered by Silva and Lamont (2017) (Fig. 2).
Fig. 1

Configuration and dimensions of the SHPB at the UKY (U.S. customary units)

Fig. 2

Photographs of the SHPB at the UKY and the placement of the material specimen within the apparatus

The pulse generated by the impact of the striker on the incident bar will propagate at the wave speed of the bar material \(c_{b}\) until reaching the interface of the incident bar and test specimen. At this point, a portion of the strain pulse will be reflected back and a portion will be transmitted through the specimen. This concept is depicted in Fig. 3.
Fig. 3

Diagram showing a general schematic of the SHPB device along with direction and nomenclature of the induced strain waves adapted from Gray III (2000)

Despite equipment design differences and varying applications, the basic science of the SHPB apparatus at the UKY remains similar to that explored by other researchers. Thus, derivation of the dynamic formulas which dictate determination of sample compressive strength is essential to understanding the experimental process. Frew et al. (2002) provide an excellent discussion of this derivation, an adaptation of which is given in this section. Note that the component steel bars of the SHPB possess properties of density \(\rho\), Young’s Modulus \(E_{b}\), bar wave speed \(c_{b}\), and cross-sectional area \(A_{b}\). When referring to the directions shown in Fig. 3, the derived equations follow sign conventions of positive stress in compression, positive strain in contraction, and positive particle velocity to the left. Subscripts 1 and 2 represent the incident-specimen and specimen-transmitted interfaces, respectively. Subscripts i, r, and t, denote the incident, reflected, and transmitted strain waves as shown in Fig. 3. If the material specimen being tested has an initial cross-sectional area \(A_{s}\) and length \(L_{s}\), then the stress at interfaces 1 and 2 may be calculated as
$$\sigma_{1} = \frac{{E_{b} A_{b} }}{{A_{s} }}\left( {\upvarepsilon_{i} +\upvarepsilon_{r} } \right)$$
(1)
$$\sigma_{2} = \frac{{E_{b} A_{b} }}{{A_{s} }}\left( {\upvarepsilon_{t} } \right)$$
(2)

As the incident strain pulse reaches the specimen, a portion of the pulse will undergo reflections within the specimen, due to the impedance mismatch which exists at interfaces 1 and 2. As these reflections take place, the specimen gradually reaches dynamic stress equilibrium. Thus, as time elapses in the experiment, the difference between the values of \(\sigma_{1}\) and \(\sigma_{2}\) should decrease and ultimately reach equivalency. In reality, due to various loading and wave phenomena it is unlikely that the two values will ever reach precisely the same value for any appreciable length of time, but in general, a level of variation may be defined as acceptable to the experimental setup. For the experiments described in this paper, that level is defined as 5%. Once this level has been achieved, the stress (\(\upsigma_{s}\)), strain rate \(\left( {\frac{{d\varepsilon_{s} }}{dt}} \right)\), and strain \(\left( {\varepsilon_{s} } \right)\) of the specimen may be calculated using either the 1-wave or 3-wave approaches (Lu and Li 2010). Equations 3 to 5 representing the 3-wave approach are commonly used when the attainment of stress equilibrium is unknown or difficult to assess.

$$\sigma_{s} \left( t \right) = \frac{{A_{b} }}{{2A_{s} }}{\text{E}}_{b} \left[ {\upvarepsilon_{i} \left( t \right) +\upvarepsilon_{r} \left( t \right) +\upvarepsilon_{t} \left( t \right)} \right]$$
(3)
$$\frac{{d\varepsilon_{s} \left( t \right)}}{dt} = \frac{{c_{b} }}{{L_{s} }}{\text{E}}_{b} \left[ {\upvarepsilon_{i} \left( t \right) -\upvarepsilon_{r} \left( t \right) -\upvarepsilon_{t} \left( t \right)} \right]$$
(4)
$$\varepsilon_{s} \left( t \right) = \frac{{c_{b} }}{{L_{s} }}\mathop \smallint \limits_{0}^{t} [\varepsilon_{i} \left( t \right) - \varepsilon_{r} \left( t \right) - \varepsilon_{t} \left( t \right)]$$
(5)

Once stress equilibrium is reached, the relationship given by \(\upvarepsilon_{i} \left( t \right) +\upvarepsilon_{r} \left( t \right) =\upvarepsilon_{t} \left( t \right)\) is true, and the equations will reduce to the 1-wave solution given in Eqs. 6 to 8. This emphasizes the importance of specimen stress equilibrium (Wu and Gorham 1997). In this research both the 1-wave and 3-wave equations were used in order to allow comparison between the two solutions.

$$\upsigma_{s} \left( t \right) = \frac{{E_{b} A_{b} }}{{A_{s} }}\varepsilon_{t} \left( t \right)$$
(6)
$$\frac{{d\varepsilon_{s} \left( t \right)}}{dt} = \frac{{ - 2c_{b} }}{{L_{s} }}\varepsilon_{r} \left( t \right)$$
(7)
$$\varepsilon_{s} \left( t \right) = \frac{{ - 2c_{b} }}{{L_{s} }}\mathop \smallint \limits_{0}^{t} \varepsilon_{r} \left( \tau \right)d\tau$$
(8)

2 Testing Procedure

2.1 Sample Materials and Data

Three common materials were selected for dynamic compression testing using the SHPB at the UKY.

For the first two materials, past published research was found which allowed for comparison of results. These results will be presented as verification that the SHPB construction and data analysis procedure reflect past research findings. Literature containing dynamic properties for the third material (Berea Sandstone) could not be found, and thus the results are included as a viable characterization for the strain-rate response of this material. The descriptions of the materials and their static properties can be found in several publications and are listed in Table 1.
Table 1

Listing of various static properties for the materials tested in this research.

(Hill 2017; Churcher et al. 1991; ASM International 1990; Holt 1996)

 

Limestone

Sandstone

Aluminum 6061-T6

Compressive strength

4000+ psi (27.58+ MPa)

6150–8670 psi (42.40–59.78 MPa)

35,000+ psi (241.32+ MPa)

Specific gravity

2.10–2.75

2.06

2.70

Modulus of elasticity

3300–5400 kpsi (22,753–37,232 MPa)

2010–2270 kpsi (13,858–15,651 MPa)

10,000 kpsi (68,948 Mpa)

Shear strength

900–1800 psi (5.52–12.41 MPa)

1160–5802 psi (8–40 MPa)

30,000 psi (207.84 MPa)

Ultimate tensile strength

300–715 psi (2.07–4.93 MPa)

580–3626 psi (4–25 Mpa)

45,000 psi (310.26 MPa)

Poisson’s ratio

0.18–0.33

0.21–0.38

0.33

Bedford (Indiana) Limestone Mississippian-age free-stone (homogenous texture and grade). This stone is considered chemically pure, with 97% calcium carbonate and 1.2% calcium–magnesium carbonate. Indiana Limestone has been used extensively in building construction, including the Empire State Building.

Aluminum 6061-T6 Commonly used industrial aluminum alloy. Typical applications include aircraft fittings, couplings, marine fittings and hardware, pistons, bike frames, and many others. Generally recognized to exhibit excellent joining characteristics, good acceptance of applied coatings, relatively high strength, good workability, and high resistance to corrosion.

Berea (Ohio) Sandstone Mississippian-age rock composed primarily of quartz (87–97%) with trace amounts of feldspar. Commonly referred to as “Berea Grit” by the oil and gas industry. Ohio Sandstone is a moderate to well sorted, medium-grained, quartz-rich sandstone. Matrix content (silt and clay) is minimal, making up 2% or less of the rock.

2.2 Specimen Preparation and Geometry Selection

The sample preparation and geometry selection for static compression testing was performed according to ASTM standards D4543 and D7012. A length to diameter ratio of 2:1 was selected for the specimens subjected to static tests and the results obtained from these 2 in. (5.08 cm) diameter samples are presented. The values obtained here provide a baseline reference to which other experimental results may be compared, as the dynamic testing requires a different sample geometry. These size and shape effects have been explored in depth elsewhere (Broch 1983; Forster 1983; Petrov and Selyutina 2015).

Selection of the dynamic specimen geometry involves several considerations not specified by ASTM standards, each of which has the potential to affect the results obtained during testing. First, an appropriate sample diameter must be selected. As the bars of this SHPB are constructed of 2 in. (5.08 cm) diameter steel, it is important that the diameter of the specimens be less than this dimension. Some researchers cite 80–90% of the bar diameter as an appropriate specimen diameter (Gama et al. 2004). Following this recommendation, a specimen diameter of approximately 1.75 in. (4.45 cm) was chosen for dynamic testing. Next, the length of the samples had to be selected. When dynamic compression testing using the SHPB, no definite standard exists for the selection of specimen length. Instead, some researchers state that a length should be selected such that it yields a length to diameter ratio within the range of 0.5 to 2.0 (Gray III 2000; Gama et al. 2004). According to the authors, one of the most fundamental requirements of the specimen length is that the specimen should be short enough to ensure the sample reaches stress equilibrium in a timely manner, and yet still long enough to limit axial inertial effects.

Figure 4 shows actual experimental results obtained using the SHPB at the UKY for the face stress values of \(\sigma_{1}\) and \(\sigma_{2}\) over time. The initial “ringing-up” period at the beginning of the experiment is easily identified, and consists of the minimum 3–4 pulse reflections required to reach equilibrium in the sample (Davies and Hunter 1963; Ravichandran and Subhash 1994). Once the specimen reaches equilibrium, the two values nearly approximate one another until the failure process begins to occur. A variance between the values of less than 5% may be considered acceptable (Subhash and Ravichandran 2000). Wu and Gorham (1997), and Gong et al. (1990) have published similar results.
Fig. 4

Values of \(\sigma_{1}\) and \(\sigma_{2}\) over time in an actual SHPB experiment performed on a limestone sample at the UKY

The data obtained during the “ring-up” period of wave reflections should not be regarded as valid, and if failure occurs during this stage the test results should be discarded. Although in this particular figure \(\sigma_{1}\) shows as initially negative, this is not always the case, as shown in Fig. 11. The data obtained during this stage is erroneous due in large part to impedance mismatch, inertial effects, and the physical impossibility of ensuring a seamless connection between the bars and specimen. The time required to achieve equilibrium (\(\tau_{m} )\) may be easily calculated using the length of the specimen \(L_{s}\), the elastic wave speed in the specimen \(c_{s}\), and the minimum number of wave reflections required. According to literature, the recommended wave reflections are between 3 and 4 (Davies and Hunter 1963), and possibly as much as 6–8 (Ravichandran and Subhash 1994). The travel time \(\left( {t_{o} } \right)\) from one bar-specimen interface to the other is given by Chen et al. (1999):
$$t_{o} = \frac{{L_{s} }}{{c_{s} }}$$
(9)
Once this travel time has been determined, the number of stress wave reflections \(\left( n \right)\) necessary to ensure dynamic equilibrium can be calculated using the expression given by:
$$n = \left[ {\frac{{\tau_{m} }}{{2t_{0} }}} \right]$$
(10)
where \((\tau_{m} )\) is the minimum required loading time. This yields that the required rise time of an incident pulse must be greater than \(\tau_{m}\). For the dynamic SHPB experiments presented in this paper, a length to diameter ratio of 1:1 was selected. This meant the limestone and sandstone specimens were ground to a length of approximately 1.75 in. (4.45 cm). Using the knowledge that the Berea Sandstone is the most porous of the materials tested and hence has the lowest wave velocity of 7808 ft/s (2380 m/s) (Shankland et al. 1993), it is possible to find the maximum required loading time for the experiments carried out with this specimen length. To guarantee dynamic equilibrium, it was assumed that 8 reflections are required. Thus, the wave must travel a distance of 1.75 in. × 8 = 14 in. (35.56 cm). The time required to do so is 149 microseconds. Comparing this time to the pulse-shaped waveform shown in Fig. 7, it appears that theoretically even the “worst case” requirements (defined as requiring the greatest span of time to achieve equilibrium) should still be met by the constant-rate loading portion of the shaped pulse.
In contrast to the rock specimens, the aluminum samples were obtained from a round bar already pre-machined to the required diameter. The bar was then cut at intervals to produce a 1:1 length to diameter ratio. Figure 5 below provides a visual representation of the rock and aluminum samples, along with a side-by-side comparison of the size difference between the samples prepared for traditional static testing and those prepared for dynamic SHPB testing.
Fig. 5

Photographs showing the rock and aluminum test specimens, at the geometries selected for both static and dynamic testing

2.3 Static Testing Procedure

Static testing of the various materials was carried out using a SATEC C600B Compressive Testing System with Instron controller. Testing was performed on samples with the typical 2:1 length to diameter ratio, as well as samples having a 1:1 ratio, to isolate the effect of the length to diameter ratio variance. This effect is shown and discussed in the results portion of this paper. The photographs in Fig. 6 depict the static testing of 2:1 ratio rock samples, 1:1 ratio rock samples, and the before and after condition of the 1:1 aluminum samples (notice the barreling-type failure as opposed to the brittle failures of the rocks).
Fig. 6

Photographs showing the static testing carried out on both geologic and aluminum samples of 1:1 and 2:1 geometric ratios

2.4 Dynamic Testing Procedure

The dynamic testing procedure involving the UKY-SHPB was carried out on the sandstone, limestone, and aluminum samples with 1:1 length to diameter ratios. A Ranger high-speed camera was used to capture the events visually at frame rates of up to 16,000 fps, as the event duration for each test may be measured in microseconds. A planned test matrix was designed in which compression tests for each of the three materials would be carried out at system charge pressures of 20, 40, 60, 120, 240, and 400 psi (138, 275, 413, 827, 1654, 2758 kPa). It was anticipated that this change in charge pressure would provide some variation in the experimental strain rate, which could be used to determine the material’s strain rate response. Strain gauges manufactured by Micro-Measurements with resistances of 120 ohms in the CEA series were selected with grid dimension constraints allowing for data recording at rates exceeding the maximum recording rate of the capture device (5 MS/s). To limit the effects of radial inertia, a silicone-based lubricant was applied to the specimen-bar interfaces. The data was recorded at sample rates of 5 Ms/s, and a 0.75 in. (1.91 cm) diameter by 16 GA thick copper pulse shaper was used for each test. A pulse shaper is a small disc, often composed of a soft metal, which through deformation lengthens the pulse load time. It is placed on the end of the incident bar impacted by the striker, and upon impact will deform, slowing the transfer of energy. This is often necessary to ensure the constant-rate loading phase of the pulse is adequate to achieve stress equilibrium. Prior testing of a wide range of disc materials and geometries yielded the conclusion that this particular pulse shaper generates the most desirable pulse results. The pulse generated by using this particular shaper with no test specimen between the incident and transmitted bars is shown in Fig. 7 (the thick dashed line). The unshaped pulse (thin solid line) is also shown for comparison. Note the much longer rise time with a relatively constant slope, the length of which exceeds the “worst-case” requirement of 149 microseconds. Thus, the shaped pulses used for this testing are adequate to allow achievement of stress equilibrium of the specimens in even the most extreme situations. Other benefits of using the pulse shaper include the reduction of oscillations in the loading phase and dampening of dispersion effects.
Fig. 7

Graph showing the triangular pulse obtained by using a 0.75 in. (1.91 cm) by 16 GA copper pulse shaper, compared to the unshaped waveform at the UKY–SHPB

Still frames captured by the hi-speed camera are shown in the photographs of Fig. 8, both pre- and post-failure. Note the “shattering” type failure experienced by the brittle rock specimens, as opposed to the relatively indiscernible shape variation of the aluminum specimens (some barreling effect was observable under close inspection).
Fig. 8

Photographs showing the pulse wave effects on the rock and aluminum samples

3 Results

3.1 Static Testing Results and Discussion of Data Processing

Static testing was performed for each of the three test materials at both the 2:1 and 1:1 geometry ratios. Up to 8 individual tests were run for each of 6 possible experimental scenarios, and the average results are depicted in the three graphs below (Fig. 9). Note that in all scenarios, the stress–strain curve shifts to the right when reducing the ratio from 2:1 to 1:1. This may be translated as a reduction in the Young’s Modulus of the material (represented by the slope of the linear portions of the curves). However, only negligible differences were observed in the yield strength. This may be attributed to the fact that rock types exhibiting extremely homogenous and uniform structure were selected for testing, and thus any effect of geologic discontinuities may be insignificant. Uniformity of the aluminum is believed to be the main explanatory factor behind the negligible change in failure strength as well.
Fig. 9

Graphs showing the quasi-static stress–strain average curves for the three test materials, at geometric ratios of 2:1 and 1:1

Valuable information obtained from this testing included the average failure strengths of the 1:1 samples, which provide the starting point (origin) for strain response curves which were later completed using the data obtained during dynamic testing. These average failure points are represented in Table 2.
Table 2

Average quasi-static failure strengths of the test materials, at 2:1 and 1:1 geometric ratios

Static failure strengths

Material

L:D ratio

σ (psi)

Indiana limestone

2:1

6562

Indiana limestone

1:1

6246

Ohio sandstone

2:1

7774

Ohio sandstone

1:1

8038

Aluminum 6061-T6

2:1

45,821

Aluminum 6061-T6

1:1

46,105

3.2 Discussion of Data Processing

The dynamic testing procedure is much more complex than the static, largely due to the lack of standardized equipment and data processing tools. To demonstrate the data processing procedure followed in this research, Figs. 10, 11, 12, 13 and 14 were derived and included from a test of Ohio Sandstone at a charge pressure of 20 psi (138 kPa). Although not shown in Fig. 10, the first step in the data reduction process is the averaging of signals from strain gauges installed equidistant on the bars (i.e. 2 gauges were installed at the midpoint of both the incident and transmitted bars, and thus the signals shown in Fig. 10 are the average of these two signals at each midpoint).
Fig. 10

Graph showing the incident (positive), transmitted and reflected (negative) waves, the time-shifted incident wave is included

Fig. 11

Calculated stress levels on the faces of an Ohio Sandstone test specimen, subjected to a 20 psi (138 kPa) charge impact

Fig. 12

a Graph showing the stress–strain curve for a test of Ohio Sandstone subjected to a 20 psi (138 kPa) charge, and b enlargement of a portion of the same curve important to failure determination

Fig. 13

a Graph showing the first derivative (slope) of the stress–strain curve from the Ohio Sandstone 20 psi (138 kPa) test, and b graph showing second derivative (rate of change of the slope) of the same stress–strain curve

Fig. 14

Graph showing the stress values for the 1 and 3-wave solutions, as well as the strain rate of the experiment

Once these average signals have been obtained, it is necessary to time-shift the incident wave in such a way that it is superimposed over the transmitted and reflected waveforms, with a relevant common time origin. To perform this action, an initial starting point of the incident wave must be defined. For the analyses performed in this paper, the starting points of the incident waves were defined by finding the slope of data in “windows”, containing 200 individual data points. These slopes were then analyzed to find the coefficient of variation for these windows, using Eq. 11:
$$CoV = \frac{\sigma }{\mu } *100\%$$
(11)

A CoV limit of 50 was deemed to be most representative of the graphically verifiable starting point. Thus, once the recorded signal enters an area where the slope is consistently trending positively as opposed to the positive–negative variation which may be attributed to random “static”, the limit of \(CoV\) = 50 provides a standardized means of designating the start of the waveform. This method and limit was applied to each test run.

Some authors have recommended using the point at which the incident wave crosses the x-axis as the start of the reflected wave (Kaiser 1998). However, in the course of testing at the UKY it was found that trials using high impact energy levels (high strain rates), presented a “stress bleed-off”, causing the trailing edge of the incident signal to be affected. This condition significantly altered the point at which the incident wave signal crosses the x-axis (this effect will be discussed in more detail later in this section). Due to this stress bleed-off condition, the reflected wave starting point was defined using the speed of pulse travel in the steel bars (found in previously published experiments to be 5148.6 m/s or 16,891.7 in./s), to avoid dependence of the reflected wave starting point on the incident wave ending point. Coupling this velocity with the knowledge that the strain gauges are placed at the midpoints of the bars 48 in. (1.22 m), the elapsed time required for the signal to travel the 96 in. (2.44 m) from the strain gauge to the specimen and back was found. Thus the incident wave may be superimposed over the reflected and transmitted waves by adding this time (found to be 473.60 μs). In this way, the time-shifted incident wave in Fig. 10 was generated and placed on the x-axis.

In Fig. 11, a representation of the stress on each of the specimen faces over the elapsed time of the experimental event is shown. The values graphed here are obtained by application of Eqs. 1 and 2 to the recorded signals. Note that for the portion of the experiment which is pertinent (before specimen failure), the stresses on the two faces are roughly equivalent. Thus, we may conclude that the specimen was in dynamic stress equilibrium when failure occurred. During the data analysis process, it was found that for high energy impacts (those with charge pressures of 120 psi (827 kPa) and over), it was difficult if not impossible to achieve this stress equilibrium. This is likely due to the specimen length being too great to allow achievement of stress equilibrium before failure occurs. Future testing with shorter specimens will explore this hypothesis.

Using the superimposed incident, transmitted, and reflected waves, the 1-wave and/or 3-wave solution equations (Eqs. 3 to 8) may be used to determine the average stress, strain, and strain rate at any point in time. Once this data is obtained, the familiar stress–strain curve may be generated, as shown in Fig. 12. Note that there is some level of “static”, or abrupt variation between each data point. To reduce this unwanted variation while still maintaining data accuracy, a 10 point averaging function was carried out, as shown in Fig. 12a. This is a necessary first step in finding the precise failure point in the data. Visual inspection of the graph allows easy determination of which part of the signal is pertinent to failure analysis, and an enlarged portion of the averaged waveform in Fig. 12a is shown in Fig. 12b.

One technique for finding the precise failure point is finding the point of inflection. This means finding the point at which the slope of the stress–strain curve changes most rapidly (in the negative direction). Figure 13a is the graphical representation of the first derivation of the enlarged portion of the stress–strain curve in Fig. 12b. As expected, this line trends downward over time, as the slope of the stress–strain curve in this area changes from positive to negative. Figure 13b is the graphical representation of the second derivation, which is the rate of change of the slope of the stress–strain curve. Failure point determination corresponds with that part of the curve which changes most rapidly in the negative direction. The most negative point of the second derivative is visible in this figure. The strain value at which this occurs may be transferred to the time domain, allowing for simple correlation to the stress value at which failure occurs.

Another verification of the specimen stress equilibrium is a comparison of the solutions given by the 1-wave and 3-wave methodologies. Figure 14 depicts the stress value solutions of these two methods. Note the nearly equivalent values. Also shown on this graph is the curve of the strain rate solution. The plateau portion of this curve is used to determine the strain rate of the experiment.

Carrying out this analysis procedure for each of the experimental trials yielded results that appeared to be meaningful, up to the 60 psi (414 kPa) charge pressure. For the trials using pressures of 120, 240, and 400 psi (827, 1654, 2758 kPa), a stress “bleed-off” tail appeared on the trailing edge of the incident wave. This significantly impacts the values of the reflected wave, and hence the results were rendered invalid and discarded. This stress bleed-off is visible in Fig. 15a. Note how the trailing edge of the incident signal bleeds over the transmitted wave, and appears to affect the leading edge of the reflected wave. This figure comes from an experiment run on Indiana Limestone at a 400 psi (2758 kPa) charge pressure. Additionally, Fig. 15b shows how for the same test, the specimen face stresses never approach equivalency. Similar effects have been reported by another researcher when high-velocity tests were carried out (Wu and Gorham 1997). Future testing will address this issue by finding ways to control the stress bleed-off, such as using shorter length striker bars, etc.
Fig. 15

a Graph showing the incident (positive phase), reflected (negative phase), transmitted, and time-shifted incident waves for a test conducted on Indiana Limestone at 400 psi (2758 kPa), and b graph showing that the specimen face stresses of this test conducted on Indiana Limestone at 400 psi (2758 kPa) never approach one another (dynamic equilibrium never achieved)

4 Dynamic Testing Results and Comparisons

4.1 Bedford (Indiana) Limestone

Results from dynamic testing of the Bedford Limestone are shown in Figs. 16 and 17. The average result from the static testing is included in each figure for comparison. A substantial degree of scatter is evident in the results, but in general a strong trend is manifested of higher strain rate leading to higher failure strength. The individual stress–strain curves of each limestone test are shown in Fig. 16. Figure 17 is a plot of the relationship between the failure strength and strain rate of each test. Note that the overall positive trend is highly significant for the range of strain rates tested.
Fig. 16

Graph of the stress–strain curves for each of the tests conducted on Bedford (Indiana) Limestone

Fig. 17

a Plot of the Bedford (Indiana) Limestone failure strength versus the strain rate of each test., and b graph showing the results of dynamic testing on Bedford (Indiana) Limestone performed by and figure taken from Frew et al. (2001)

Dynamic testing of the Bedford (Indiana) Limestone was previously carried out by Frew et al. (2001) and the results are shown in Fig. 17b. The results of this testing largely confirm the results obtained by the SHPB at UK. Interestingly, those authors also tested specimens exhibiting length to diameter ratios from 1 to 2, and reported no significant effect. This serves as confirmation in the dynamic range of the static range geometric ratio testing carried out at the UKY.

4.2 Aluminum 6061 T6

Results from testing carried out on Aluminum 6061 T-6 are shown in Figs. 18 and 19a. Note that although it is clear that the Young’s Modulus shifted to the left for each test, no discernible effect on failure strength was observed. Also, it appears that the test conducted with a 20 psi (138 kPa) impact charge (51 s−1 strain rate) may not have possessed enough energy to induce failure in the aluminum sample.
Fig. 18

Graph showing the stress strain curves from testing carried out on Aluminum 6061 T-6

Fig. 19

a Graph showing the plot of strain rate versus failure strength for tests carried out on Aluminum 6061 T-6, and b graph showing the results of dynamic testing on Aluminum 6061-T6 performed by and figure obtained from Manes et al. (2011)

Dynamic testing of Aluminum 6061 T-6 was previously carried out by Manes et al. (2011) over a wide range of strain rates, from quasi-static up to 104 s−1. The results of this testing are shown in Fig. 19b. Conclusions from this research were that until reaching a strain rate of 103 s−1, this aluminum alloy does not exhibit strain rate sensitivity. Because the range of values tested at UK was at a maximum of approximately 150 s−1, this explains why no significant change in yield strength was observed. If future testing were carried out a higher strain rates, confirmation of the higher range results obtained by Manes et al. (2011) could be obtained.

4.3 Berea (Ohio) Sandstone

Figures 20 and 21a show the results of testing performed on the Berea (Ohio) Sandstone. Although it was clear from testing that all dynamic testing results yielded failure strengths above the static testing average, it is not apparent that within the range of dynamic strain rates tested that increasing the strain rate had any effect on the strength. This is evidenced by the random scatter of the stress–strain curves shown in Fig. 20, and by the weak positive correlation plotted in Fig. 21a.
Fig. 20

Graph showing the stress–strain curves for dynamic testing carried out on sandstone specimens

Fig. 21

a Graph showing the plot of strain rate versus failure strength for each sandstone sample, and the overall weak positive trend, and b results from dynamic testing of sandstone taken from China Qinling, Taibai Mountain, Shaanxi Province performed by and figure taken from Liu et al. (2011)

Although no published results from dynamic testing of Berea (Ohio) Sandstone could be found, results from another sandstone were found and included here for the sake of providing a reference to what may be expected from a sandstone type of rock. The results of this testing, performed by Liu et al. (2011), are shown in Fig. 21b. This testing was carried out using a 3.94 in. (100 mm) SHPB, on samples possessing diameter 3.82 in. (97 mm) and length 1.69 in. (43 mm). Note that this is a length to diameter ratio of approximately 0.44, compared to 1.0 used in this research.

5 Discussion and Conclusions

In this work testing of three materials was carried out using an SHPB of 2 in. (5.08 cm) diameter constructed at the University of Kentucky. The materials tested were Bedford (Indiana) Limestone, Berea (Ohio) Sandstone, and Aluminum 6061-T6. The tests were carried out using samples with 1:1 length to diameter ratios, at strain rates ranging from quasi-static to 210 s−1.

Conclusions of this work are as follows.
  1. 1.

    The limestone samples exhibit a strong positive correlation between strain rate and failure strength. This is confirmed by work previously carried out by Frew et al. (2001).

     
  2. 2.

    The aluminum samples exhibit no significant strain rate sensitivity for the range tested. This is confirmed by work previously carried out by Manes et al. (2011). Future testing at higher strain rates may allow for confirmation of the positive correlation at high rates shown in that research.

     
  3. 3.

    The sandstone samples exhibit a weak positive correlation between strain rate and failure strength. Without other published research for this specific rock, this is difficult to confirm. However, work performed by Liu et al. (2011) on another sandstone did show a strong positive correlation. Additional testing is recommended to confirm the results from this publication.

     
  4. 4.

    Future testing should be performed with attempts to control the “stress bleed-off” phenomena which caused the higher velocity (and higher strain-rate) experimental data to be discarded. This may include the use of shorter striker bars, and changing the pulse shaper. Thinner samples with a lower L:D ratio may also be required to assist in achieving dynamic equilibrium.

     

Ultimately, the results from the testing are confirmed by past published research. Additional testing should be carried out to solidify these findings. These materials tested in this work commonly used in construction and manufacturing and definition of dynamic properties is highly important to the engineering design process. Future testing with this SHPB will explore the properties of geologic materials which are often subject to dynamic processes in the mining and civil engineering fields, such as blasting, milling, etc.

Notes

Acknowledgements

This research was internally funded by the University of Kentucky’s Explosives Research Team.

References

  1. ASM International (1990) Metals handbook, volume 2: properties and selection: nonferrous alloys and special-purpose materials, 10 edn. ASM InternationalGoogle Scholar
  2. Broch E (1983) Estimation of strength anisotropy using the point-load test. Int J Rock Mech Min Sci Geomech Abstr 20(4):181–187CrossRefGoogle Scholar
  3. Chen W, Zhang B, Forrestal MJ (1999) A split Hopkinson bar technique for low-impedance materials. Exp Mech 39(2):81–85CrossRefGoogle Scholar
  4. Churcher PL, French PR, Shaw JC, Schramm LL (1991) Rock properties of Berea sandstone, Baker dolomite, and Indiana limestone. Soc Pet Eng.  https://doi.org/10.2118/21044-ms Google Scholar
  5. Davies EDH, Hunter SC (1963) The dynamic compression testing of solids by the method of the split Hopkinson pressure bar. J Mech Phys Solids 11(3):155–179CrossRefGoogle Scholar
  6. Forster IR (1983) The influence of core sample geometry on the axial point-load test. Int J Rock Mech Min Sci Geomech Abstr 20(6):291–295CrossRefGoogle Scholar
  7. Frew DJ, Forrestal MJ, Chen W (2001) A split Hopkinson pressure bar technique to determine compressive stress–strain data for rock materials. Exp Mech 41(1):40–46CrossRefGoogle Scholar
  8. Frew DJ, Forrestal MJ, Chen W (2002) Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar. Exp Mech 42(1):93–106CrossRefGoogle Scholar
  9. Gama BA, Lopatnikov SL, Gillespie JW Jr (2004) Hopkinson bar experimental technique: a critical review. Appl Mech Rev 57(4):223–250CrossRefGoogle Scholar
  10. Gong JC, Malvern LE, Jenkins DA (1990) Dispersion investigation in the split Hopkinson pressure bar. J Eng Mater Technol 112(3):309–314CrossRefGoogle Scholar
  11. Gray GT III (2000) Classic split Hopkinson pressure bar testing. ASM Handb Mech Test Eval 8:462–476Google Scholar
  12. Hill JR (2017) Indiana geological and water survey, “Indiana limestone”. https://igs.indiana.edu/MineralResources/limestone.cfm. Accessed 11 Mar 2018
  13. Holt JM (1996) Structural alloys handbook, Technical Ed; Ho CY (ed) CINDAS/Purdue University, West Lafayette, INGoogle Scholar
  14. Kaiser MA (1998) Advancements in the split Hopkinson bar test. Diss. Virginia TechGoogle Scholar
  15. Liu S et al (2011) SHPB experimental study of sericite-quartz schist and sandstone. Chin J Rock Mech Eng 30(9):1864–1871Google Scholar
  16. Lu YB, Li QM (2010) Appraisal of pulse-shaping technique in split Hopkinson pressure bar tests for brittle materials. Int J Prot Struct 1(3):363–390CrossRefGoogle Scholar
  17. Manes A et al (2011) Analysis of strain rate behavior of an Al 6061 T6 alloy. Proc Eng 10:3477–3482CrossRefGoogle Scholar
  18. Petrov Y, Selyutina N (2015) Scale and size effects in dynamic fracture of concretes and rocks. In: EPJ web of conferences, vol 94. EDP SciencesGoogle Scholar
  19. Ravichandran G, Subhash G (1994) Critical appraisal of limiting strain rates for compression testing of ceramics in a split Hopkinson pressure bar. J Am Ceram Soc 77(1):263–267CrossRefGoogle Scholar
  20. Shankland TJ, Johnson PA, Hopson TM (1993) Elastic wave attenuation and velocity of Berea sandstone measured in the frequency domain. Geophys Res Lett 20(5):391–394CrossRefGoogle Scholar
  21. Silva J, Lamont R (2017) Dispersion signal analysis in a Split-Hopkinson pressure bar at the University of Kentucky. Fragblast Int J Blast Fragm 11(1):7–22Google Scholar
  22. Subhash G, Ravichandran G (2000) Split-Hopkinson pressure bar testing of ceramics. ASM International, Materials Park, OH, pp 497–504Google Scholar
  23. Wu XJ, Gorham DA (1997) Stress equilibrium in the split Hopkinson pressure bar test. J Phys IV 7(C3):C3–91Google Scholar
  24. Zhang QB, Zhao J (2014) A review of dynamic experimental techniques and mechanical behaviour of rock materials. Rock Mech Rock Eng 47(4):1411–1478CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of KentuckyLexingtonUSA

Personalised recommendations