Impact compressive response of dry sand

Mechanics of Materials 41 (2009) 777–785

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Impact compressive response of dry sand Bo Song a,*, Weinong Chen b, Vincent Luk c a b c

Sandia National Laboratories, 7011 East Avenue, Livermore, CA 94551, USA School of Aeronautics and Astronautics and School of Materials Engineering, Purdue University, West Lafayette, IN 47907-2045, USA Sandia National Laboratories, Albuquerque, NM 87185, USA

a r t i c l e

i n f o

Article history: Received 16 December 2008

a b s t r a c t A split Hopkinson pressure bar (SHPB) was properly modified to obtain dynamic compressive stress–strain curves of dry sand at various high strain rates. Quasi-static compressive properties of the sand were obtained with a MTS810 materials test system. In both dynamic and quasi-static experiments, the 1.50  103 kg/m3 dry sand, confined with a polycarbonate tube, had the same dimensions, making the strain rate the only variable. The strain rate effects on the compressive response of the sand were determined. The sand was also prepared to specimens with a higher initial density of 1.62  103 kg/m3 for dynamic experiments to investigate the initial density (gas porosity) effects. At a given dynamic strain rate, besides using the polycarbonate confining tube, polyolefin heat shrinking tubes and 4340 steel tubes were used to confine the sand specimens to study the lateral confinement effects. The results show that the compressive response of the dry sand is not sensitive to strain rate under the loading conditions in this study, but significantly dependent on the initial density and lateral confinement level. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Sand has been widely used for construction components in both civil engineering and military applications. Structural response and damage under high-rate loading are often assessed by numerical simulations. However, the accuracy of the implemented constitutive models of the component materials dominates the accuracy of the predictions from the numerical analysis. Thus, it is necessary to understand the mechanical properties of geo-materials including sand, particularly under impact loading conditions. Quasi-static mechanical properties of the sand can be characterized by using a hydraulically driven testing frame such as MTS. However, it is desired to conduct high-rate experiments to generate a database of the sand to determine the material constants in the constitutive models. The information on the high-rate mechanical response of the sand is also potentially useful in many other applications such as mining, overburden removal, earth* Corresponding author. E-mail address: [email protected] (B. Song). 0167-6636/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2009.01.003

quake engineering, containment of underground explosion, and so on (Charlie et al., 1990). Up to date, the high-rate mechanical properties of the sand have been less investigated due to the difficulties of dynamic experimental techniques in association with the complex nature of the sand. Split Hopkinson pressure bar (SHPB) has been widely employed to obtain stress–strain response of engineering materials at high strain rates since 1949 (Kolsky, 1949). SHPB has recently been utilized to characterize dry and unsaturated sands at high strain rates (Bragov et al., 1996, 2004, 2006; Gaffney et al., 1987; Charlie et al., 1990; Felice et al., 1987a,b; Veyera, 1994; Veyera et al., 1989). Since sand is a very different material compared to most engineering materials, the regular assumptions in the SHPB experiment may need to be revisited in detail. The substantial fraction of the air-filled void volume in a sand specimen makes its wave speeds low and rate of wave attenuation very high (Gaffney et al., 1987), which have significant consequences in the results obtained from SHPB experiments. One can imagine a sand specimen with low density, low rigidity, and low wave speeds. All of these characteristics pose challenges for dynamic testing. When

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B. Song et al. / Mechanics of Materials 41 (2009) 777–785

the wave speed is very low, a compaction wave may propagate in the specimen, making the specimen deform progressively instead of uniformly along the axial direction (Song et al., 2006). This non-uniformity in specimen deformation violates a basic assumption of uniform deformation in SHPB experiments (Kolsky, 1949). Gas porosity in sand, which determines the initial density of the sand, is thus becoming a very important factor. Felice et al. (1987a) concluded that the initial gas porosity of the soil is a principal parameter to govern the stress– strain response. In addition, they found that, when the strain is less than the initial gas porosity, the resultant stress–strain response is insensitive to strain rate (Felice et al., 1987a). The non-uniform deformation and/or nonequilibrated stress make the obtained stress–strain response represent the structural response instead of the response of the material itself (Song et al., 2006). For sand specimens, the non-uniform stress/deformation along the axial direction may become more significant for their high initial gas porosity. This is because the higher initial gas porosity facilitates a much lower stress-wave speed, which requires much longer time to achieve the stress equilibrium in the specimen. It could be imagined that a longer specimen also needs longer time for stress equilibrium. Furthermore, sand has been recognized to be a typical sort of attenuating material. The amplitude of stress wave may significantly attenuate when propagating through the sand specimen. This attenuation of stress wave results in more severe non-uniform stress along the axial direction in the specimen. Felice et al. (1991) developed a modified Lagrangian analysis procedure, which is generally used in data reduction for plane shock wave experiments, to compute the dynamic stress–strain response for soils in SHPB experiments. However, careful consideration of specimen length could be more effective to physically minimize this attenuation effect in the sand specimen. The dimensions of the sand specimen thus need to be carefully designed to satisfy the requirement of dynamic stress equilibrium, which has been addressed by Felice et al. (1987a,b). They proposed an aspect ratio less than or equal to 0.2 for dynamic testing of the soil (Felice et al., 1987b). However, it has been demonstrated that only reducing the specimen thickness is not sufficient to satisfy the stress equilibrium particularly in soft specimens, which requires further modification to the conventional SHPB technique (Song and Chen, 2004). Pulse shaping techniques have been recently developed to generate a relatively-low rate of loading to facilitate the specimen in stress equilibrium quickly (Nemat-Nasser et al., 1991; Frew et al., 2002, 2005). In addition, the pulse shaping techniques are capable of facilitating constant strain rate deformation in the specimen by producing different shapes of incident pulses, which is always desired to study strain-rate effects (Frew et al., 2002, 2005). The sand needs to be carefully packed before it is being mechanically characterized, raising another challenge. The packing material also plays a key role in mechanical response of the confined sand. Different packing material provides different confinement in the radial direction of the specimen, resulting in different axial stress response in the specimen. Earlier numerical simulations by Bazhe-

nov et al. (2000) showed that the compliance of the confining tube and the friction between the soil specimen and the inner surface do not influence the measured characteristics of the soil, which is a little conflict with their later work. Bragov et al. (2004) found that the friction force between the soil and the inner surface affects the inner surface of the confining tube at pressures in soil above 50 MPa. A thin layer of a lubricant to the inner surface of the confining tube was thus proposed to reduce the friction (Bragov et al., 2004). A rigid confinement makes the sand specimen approximately in a uniaxial strain state, while a very soft confinement makes it in a nearly uniaxial stress state. The sand specimen is in 3-D stress and strain state when confined with most materials, which undoubtedly makes the analysis complicated (Bazhenov et al., 2000). Currently the sand specimen confined with a steel or hard aluminum alloy jacket was mostly characterized with a conventional SHPB (Bragov et al., 1996, 2006; Charlie et al., 1990; Felice et al., 1987b; Gaffney et al., 1987). The dynamic response of the sand confined with other materials has been less investigated. Furthermore, the loading conditions on the sand specimens have not been actively controlled in the previous investigations. In this research, we modified the SHPB technique to conduct dynamic characterization of the sand with an initial density of 1.50  103 kg/m3 at various strain rates. Combining the quasi-static data, the strain rate effects on the compressive response of the sand are examined. We also prepared the sand specimen to another higher density of 1.62  103 kg/m3 for dynamic characterization to study the effects of initial density on dynamic response of the sand. Dynamic experiments were also conducted with the modified SHPB on the sand specimen confined with a steel tube, a polycarbonate tube, and a polyolefin heat shrinking tube, respectively, at a certain strain rate, to study the confinement effects. 2. Experimental setups and material 2.1. The SHPB for dynamic experiments Dynamic experiments were performed at Purdue University with a modified SHPB, the schematic of which is shown in Fig. 1. Besides the standard components in a conventional SHPB (a gas gun, a striker, an incident bar, a transmission bar, a momentum absorption device, and a data acquisition system), a pulse shaper at the impact end of the incident bar, was employed in the modified SHPB (Fig. 1). The pulse shaping techniques were originally developed three decades ago to produce a non-dispersive pulse. In the past decade, a common way for the pulse shaping techniques was to attach a small ‘‘tip” material (pulse shaper) on the impact surface of the incident pulse (Fig. 1). The pulse shaper is commonly a small disk made of metal, plastic, rubber, and even paper. The selection of the pulse shapers is specimen material and strain-rate dependent, and may vary from experiment to experiment. Besides reducing the dispersion, the employment of a welldesigned pulse shaper enables to facilitate dynamic stress equilibrium and constant strain rate deformation in the

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Fig. 1. A schematic of split Hopkinson pressure bar for sand testing.

specimen. The pulse shaping techniques have been described in detail in the literature, e.g., by Song and Chen (2005). In an SHPB experiment, the specimen was sandwiched between the incident bar and the transmission bar. The impact of the striker, which is launched by the compressed gas in the gas gun, on the end of the incident bar through the pulse shaper generates an elastic wave (incident wave), which propagates through the incident bar. Due to the employment of the pulse shaper, the incident pulse may not be as trapezoidal as obtained from a conventional SHPB experiment. When the incident stress wave travels to the bar/specimen interface, part of the incident wave is reflected back into the incident bar as a reflected wave due to the mechanical impedance mismatch between the bars and the specimen. The rest is transmitted into the transmission bar as a transmitted wave which the specimen is being compressed. Both incident and reflected waves are recorded by the strain gages on the incident bar, and the transmitted wave is recorded by the strain gages on the transmission bar. The recording device is typically a highspeed digital oscilloscope with pre-amplifiers or a computer with a high-speed A/D board. During an SHPB experiment, the pressure bars must remain elastic and should be long enough to avoid the overlapping of the elastic waves. In addition, the ends of the bars in contact with the specimen must remain flat and parallel throughout the dynamic loading. Based on one dimensional wave analysis, the strain rate, strain, and stress histories in the specimen can be calculated from the recorded bar-surface strain signals with the following equations, respectively (Kolsky, 1949).

e_ ðtÞ ¼

C0 ½ei ðtÞ  er ðtÞ  et ðtÞ Ls

eðtÞ ¼

C0 Ls

rðtÞ ¼

Z

ð1Þ

t

½ei ðtÞ  er ðtÞ  et ðtÞ dt

ð2Þ

0

A0 E0 ½ei ðtÞ þ er ðtÞ þ et ðtÞ 2As

ð3Þ

where ei ðtÞ, er ðtÞ, and et ðtÞ are incident, reflected, and transmitted strain histories, respectively; A0 is the crosssectional area of the bars; E0 and C0 are Young’s modulus and elastic wave speed in the bar material, respectively;

As and Ls are initial cross-sectional area and length of the specimen, respectively. For the sands experiments in this study, the specimen diameter is the same as the bar diameter, As = A0. When the specimen is in a state of uniform stress:

ei ðtÞ þ er ðtÞ ¼ et ðtÞ

ð4Þ

Eqs. (1)–(3) can be simplified as

e_ ðtÞ ¼ 2

C0 er ðtÞ Ls

eðtÞ ¼ 2

C0 Ls

rðtÞ ¼

Z

ð5Þ

t

er ðtÞ dt

ð6Þ

0

A0 E0 et ðtÞ ¼ E0 et ðtÞ As

ð7Þ

Therefore, once the incident, reflected and transmitted signals are measured, the stress–strain data for the material under investigation can be obtained. It should be noted that Eqs. (4)–(7) are based on the assumption of dynamic stress equilibration in the specimen, which may not be satisfied automatically when the specimen material is soft, e.g., the sand specimen in this study. A conventional method to check the dynamic stress equilibrium is to use 2wave, 1-wave analysis (Eq. (4)) (Gray, 2000). The 2-wave analysis represents the difference between the incident and reflected pulses; whereas the 1-wave analysis represents the transmitted pulse only. As mentioned early, in order to achieve the stress equilibrium, the specimen cannot be too long. A longer specimen, particularly for the sand material, also makes the attenuation more significant (Gaffney et al., 1987). However, a too short specimen brings uncertainties to homogeneity for the geo-materials. In this research, the sand specimen length needs to be carefully determined. In addition, the pulse shaping technique also needs to be carefully designed to produce a relatively low rate of loading for satisfaction of dynamic stress equilibrium. A constant strain rate deformation in the specimen can also been facilitated through modifying the profile of the incident pulse with the pulse shaping technique. Collection of the accurate stress–strain curves at various constant strain rates is essential to investigate the strain-rate effects of materials and to develop strainrate-dependent material models.

B. Song et al. / Mechanics of Materials 41 (2009) 777–785

In this study, the bars in the modified SHPB with pulse shaping technique had a common diameter of 19.05 mm. We used the steel and aluminum bars, respectively, to obtain the compressive response of the sand under different confining levels. The purpose of using the aluminum bar for testing of sand confined with the soft polyolefin heat shrinking tube is to obtain more sensitive transmitted signal because of its much lower Young’s modulus than steel.

28

24

20 Volume %

780

16

12

2.2. The MTS for quasi-static experiments

8

Quasi-static compressive experiments of the sand were conducted with an MTS 810 materials test system in the mode of displacement control. The schematic of the testing section is shown in Fig. 2. Two 19.05-mm-diameter steel bars are held and guided with an alignment frame to keep a precise alignment during compression. Besides the same specimen specimens, the configuration in the MTS experiments is the same as in the SHPB experiments such that the strain rate is the only variable from the quasi-static to dynamic experiments.

4

2.3. Sand material and specimens The material studied in this research is silica based fine grain sand, kiln dried and poorly graded (ASTM, 2001). Fig. 3 shows the sand particle size distribution, which indicates that most sand particles are in the diameter range from 150 to 450 lm. The minimum and maximum densities of the dry sand material are 1.40 and 1.63  103 kg/ m3, respectively. In this study, the dry sand specimens were made to an initiate density of 1.50  103 kg/m3 to study the strain-rate and lateral confinement effects. At the same strain rate and confining pressure, another higher density of the sand material, 1.62  103 kg/m3, was selected to investigate the initial density effect. The sand specimens were confined with hardened 4340 steel, polycarbonate, and polyolefin heat shrinking tubes, the schematics of which are shown in Fig. 4. As shown in Fig. 4, the sand specimen was determined to be 9.30 mm

Fig. 2. A schematic of gripping system in MTS810 for sand testing.

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 Particle Size (µ µm)

Fig. 3. Particle size distribution in the sand sample.

long. This specimen length has been demonstrated to satisfy the requirement of dynamic stress equilibrium through preliminary experiments and confirmed with high-speed images of specimen deformation. A more quantitative confirmation of dynamic stress equilibrium will be carried out with Eq. (4), as presented later. In the following, we are presenting detailed sand specimen preparation procedures because it is very important to influence the experimental results. 2.3.1. Confined with a polycarbonate tube (Fig. 4(a)) The commercially purchased polycarbonate tube had an outer diameter of 25.4 mm, an inner diameter of 19.1 mm, and a length of 50.8 mm. A 6.35-mm-thick, 19.05-mmdiameter steel platen was placed inside the polycarbonate tube, aligning with a pair of drilled small holes with 180° apart. The tube was then fitted into the Hopkinson bar by placing the incident and transmission bars inside the tube from two ends, respectively. One bar was held stationary while the other bar was moved towards the stationary bar to make the incident bar, steel platen, and transmission bar aligned. Two small set screws were installed into the holes to hold the platen before the tube was taken off from the Hopkinson bar and placed vertically on the top of a 19.05-mm-diamter short rod. 4.0-g of dry sand was poured into the tube, which was then tapped slightly to make the top surface even. Another steel platen was placed on the top surface of sand, which is then slightly pressed with a small rod to make the specimen length at 9.30 mm (corresponding to 1.50  103 kg/m3 density). It is noted that, when we prepared the dry sand specimens at a higher density (1.62  103 kg/m3), the dry sands were pre-compressed to 8.60 mm for the same amount of sand (4.0 g). Similar to the bottom ones, two set screws were installed to hold the top platen if necessary. The assembled tube specimen was then installed to the Hopkinson bar. A drop of super glue was applied on the end surface of the incident bar to firmly attach the top platen on it. This is to prevent the platen from flying off when a reflected tensile wave arrives. The transmission bars was then moved into the tube in contact with the other platen. Double check the specimen thickness and make necessary adjustment. All the positioning screws were taken off before impact loading.

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Fig. 4. Specimen configurations with different confinement materials. (a) Polycarbonate tube; (b) 4340 steel tube; and (c) polyolefin heat shrinking tube.

2.3.2. Confined with a 4340 steel tube (Fig. 4(b)) The 4340 steel tube had the same dimensions as the polycarbonate tube. Unlike the transparent polycarbonate tube, the sand specimen is invisible after confined with the 4340 steel tube, which makes another challenge to prepare the sand specimen possessing the right initial density (or thickness). Two screwed holes with 180° apart were drilled at the locations of 14 mm from one end. A 19.05mm-diameter supporting rod that is marked at the position of 14 mm from the top end was placed inside the steel tube, the edge of which was aligned with the marked line. A 6.35-mm-thick, 19.05-mm-diameter steel platen that is the same material as the bar material is placed on the top of the supporting rod inside the tube. The steel platen is positioned by two screws through both screwed holes. After the 4.0-g dry sand was poured into the tube evenly, another steel platen was placed on the top surface of sands. Another rod with a marked line at the position of 14.8 mm from the bottom end was used to slightly press the steel platen until the top surface of the steel tube was aligned with the marked line on the top rod. Therefore, the specimen length is 50.8  14.0  14.8  6.35  2 = 9.3 mm (corresponding to 1.50  103 kg/m3 density), as shown in Fig. 4(b). Similarly two lines at the positions of 14 and 14.8 mm from the specimen end were marked on the inci-

dent and transmission bars, respectively. The assembled steel tube was then installed on the Hopkinson bar in the right position by aligning both marked lines to the steel surfaces. It is noted that the super glue is necessary to attach the steel platen to the end of the incident bar. 2.3.3. Confined with a polyolefin heat shrinking tube (Fig. 4(c)) The polyolefin heat shrinking tube had an actual inner diameter of 20.6 mm, a wall thickness of 0.3 mm, and a length of 52 mm (Fig. 4(c)). The specimen preparation is very similar to the steel confinement. The difference is that the screws were not applicable. We used a hair dryer to heat the tube until the tube is shrunk to snug-fit the platens, giving the platens sufficient support. It should be noted that the aluminum platens were used here because the sand confined with the polyolefin heat shrinking tube will be characterized with aluminum SHPB. Using the same platen material as the bar material avoids the wave disturbance due to the mechanical impedance mismatch. 3. Experimental results The detailed pulse shaping design and the corresponding striking speed at each loading condition are tabulated

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Table 1 Dynamic testing conditions for sand experiments. Striker length (mm)

Striker speed (m/s)

Pulse shaper(s)

686 686 686

5.3 10.1 16.2

686 610 686

6.2 3.3 5.8

U 6.35  0.5 mm C11000 annealed copper disk U 7.14  0.8 mm C11000 annealed copper disk U 9.65  4.6 mm M-2 steel stacked with a U 3.97  0.5 mm C11000 annealed copper disk U 6.35  0.8 mm C11000 annealed copper disk U 4.76  0.05 mm C11000 annealed copper disk U 7.14  0.8 mm C11000 annealed copper disk

in Table 1. Fig. 5 shows a typical set of oscilloscope records of incident, reflected, and transmitted pulses from an SHPB experiment. In this experiment, a steel SHPB was used to characterize the 1.5  103 kg/m3 dry sand confined with a polycarbonate tube. The incident pulse in Fig. 5 is different from that obtained in a conventional SHPB experiment because of the usage of pulse shaping technique. The rise time of the incident pulse, which is approximately 10 ls in a conventional SHPB experiment, has been increased to 150 ls due to the employment of the pulse shaping technique. The long rise time in the modified incident pulse makes the stress in the specimen in equilibration. Dynamic stress equilibrium was checked by ‘‘2-wave”, ‘‘1-wave” method (Eq. (4)), the result of which is shown in Fig. 6. The nearly overlapped stress histories at both ends of the specimen demonstrate that the stress in the specimen was uniform (Fig. 6). The pulse shaping technique also modified the profile of the incident pulse (Fig. 5) for the purpose of achieving constant strain rate deformation in the specimen. Under dynamic stress equilibrium, the reflected pulse represents the strain-rate history in specimen (Eq. (5)). Fig. 7 shows the specimen strain-rate history calculated from the reflected pulse, and its integration, the strain history in the specimen. As shown in Fig. 7, the strain rate was nearly a constant (470/s) from 150 to 325 ls that correspond to the strains from 0.03 to 0.11. Hence, the specimen was deformed at

Fig. 5. A typical set of oscilloscope records in a modified SHPB experiment on the sand.

Testing condition Specimen density (kg/m3)

Confinement

Bar material

Strain rate (/s)

1.50  103 1.50  103 1.50  103

Polycarbonate (PC) tube Polycarbonate (PC) tube Polycarbonate (PC) tube

Steel Steel Steel

450 900 1400

1.62  103 1.50  103 1.50  103

Polycarbonate (PC) tube Polyolefin Heat Shrinking tube 4340 steel tube

Steel Aluminum Steel

520 500 500

Fig. 6. Dynamic stress equilibrium in the sand specimen.

the nearly constant rate of strain under dynamic stress equilibrium within the most loading duration, validating the resultant stress–strain curve. In order to check the specimen repeatability, we repeated 23 experiments under the same loading condition. Fig. 8 summarizes all 23 stress–strain curves of the 1.50  103 kg/m3 dry sands confined with the polycarbonate tubes at the very close strain rates (470/s). All resulting stress–strain curves of the sand under the same loading condition exhibit a nearly linear compressive behavior when the strain is smaller than 0.11 except for some kind of oscillation. This oscillation may come from either vibration of sand particles or friction between the confining tube and the bar. Fig. 9 presents the variation of stresses at the strains of 5% and 10% obtained from all the 23 experiments, the results of which indicate reasonable repeatability. Following the similar procedure, we conducted three dynamic strain-rate experiments (470/s, 900/s, and 1450/ s) as well as three quasi-static strain-rate experiments (0.01/s, 1/s, and 20/s) using a MTS810 materials test machine. For each strain rate three to five experiments were repeated and only the mean curves with error bars are shown in Fig. 10. The error bars shown in Fig. 10 represent the maximum and minimum values of stress at a certain strain and strain rate, which clearly indicate the scattering of experimental results obtained under the same loading condition. The results in Fig. 10 exhibit reasonable repeatability for the sand testing. When the specimen deforms

B. Song et al. / Mechanics of Materials 41 (2009) 777–785

783

Fig. 7. Strain-rate and strain histories in the sand specimen.

Fig. 8. Stress–strain curves of the sand under the identical loading condition.

Fig. 9. Comparison of stresses at 0.05 and 0.10 strain obtained from identical experiments.

within a small strain, e.g., <0.1, the stress–strain response obtained from quasi-static to dynamic testing exhibit a nearly linear behavior. However, a hardening behavior becomes notable to the sand specimen when it deforms to a larger strain. The most important characteristic is that the sand specimen under the polycarbonate confinement exhibits little strain rate effects, which confirms the conclusion by Felice et al. (1987a). At the similar strain rates (470/s), we also conducted dynamic experiments on the dry sand with a higher initial density of 1.62  103 kg/m3 to investigate the effects of specimen initial density. Fig. 11 compares the stress–strain curve for the 1.62  103 kg/m3 sand to that for the previous 1.50  103 kg/m3 sand under the same loading condition. By contrast to the strain-rate effect, the dry sand exhibits significant density effect. The sand with a higher density behaves stiffer than the lower-density sand though the shapes of stress–strain curves are very similar. More quan-

Fig. 10. Stress–strain curves of the sand at various strain rates.

784

B. Song et al. / Mechanics of Materials 41 (2009) 777–785 50 Dry Sand (1.50 g/cc) @ 470/s

Engineering Stress (MPa)

Dry Sand (1.62 g/cc) @ 520/s

40

30

20

10

0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Engineering Strain

Fig. 11. Comparison of dynamic stress–strain curves of the sand with initial densities of 1.50 and 1.62  103 kg/m3.

titatively, the 8% increase in initial density produces approximately 30% increase in stiffness. This is because the less gas porosity in a higher-density sand specimen motivates the increase of the equivalent stiffness. It is noted that the above results were obtained for the sand confined with polycarbonate tubes. More dynamic experiments were carried out at the similar strain rate for the dry sand but confined with the polyolefin heat shrinking tubes and 4340 steel tubes to study the stressstate effects of the sand. The stress–strain curves at the similar strain rates under the three confinements are compared in Fig. 12. Compared to the sand specimen confined with the polycarbonate tube, the stresses at certain strains drastically decrease when confined with the polyolefin heat shrinking tube and approximately 20% increase when confined with the 4340 steel tube. In addition, the profiles of stress–strain curves for the sands confined with the polycarbonate and steel tubes are similar but different from that under the heat shrinking tube confinement. The thin-wall polyolefin heat shrinking tube produces less confining force to the sand specimen along the radial direction. This makes the stress–strain curve of the confined sand a nearly perfectly plastic behavior: the axial stress changes little with the increasing strain. Compared to the nearly uniaxial strain state when confined with the steel tube, the stress state in the sand specimen confined with 40 Steel Tube Confinement

Engineering Stress (MPa)

35

Polycarbonate Tube Confinement Polyolefin Tube Confinement

30 25 20 15 10

the polyolefin heat shrinking tube is close to the state of uniaxial stress. Hence, the results in Fig. 12 demonstrate significant effects of stress state on the dynamic compressive response of the sand. In this study, we found that the strain rate does not significantly influence the compressive stress–strain response of the sand material. However, the initial density and stress-state are becoming dominant parameters to dynamic compressive response for the sand studied in this research. It could be understood that the grain size and distribution of the sand material are the other important parameters, from the point of view of material, to affect the mechanical properties. It has to be emphasized that the size effect has not been investigated in this study, even though it might affect the mechanical response of the sand (Felice et al., 1987b). All these parameters make difficult to compare the mechanical properties, particularly the strength and stiffness, of the sand with others documented in the literature. This research, however, provides view points of the effects of strain rate, density, and stress state on the compressive stress–strain response of the sand material. 4. Conclusions We modified the SHPB to conduct dynamic characterization of sand with the pulse shaping technique. This modification ensured that the specimen deformed at a nearly constant strain rate under dynamic stress equilibrium such that the stress–strain curves were obtained reliably. By using this technique, we obtained dynamic stress–strain curves of the sands ranged from quasi-static to dynamic strain rates under polycarbonate tube confinement. Under the polycarbonate tube confinement, the dry sand with an initial density of 1.50  103 kg/m3 exhibited little strainrate effects in the strain-rate and strain ranges we studied. At the same strain rate and under the same polycarbonate tube confinement, we studied the effects of initial density on dynamic response of the sand. At certain strains, the stresses significantly increased when the initial density increases from 1.50 to 1.62  103 kg/m3. In addition to being confined with a polycarbonate tube, the 1.5  103 kg/m3 dry sand was characterized under confinements with a polyolefin heat shrinking tube and a 4340 steel tube, to examine the effects of stress-state in the specimen. Compared to the polycarbonate confinement, the stiffness increased 20% when a more rigid steel tube was used to confine the sand. However, the stress drastically decreased when a polyolefin heat shrinking tube was used. Moreover, the stress–strain curve of the sand confined with the polyolefin heat shrinking tube exhibits nearly perfectly plastic behavior, which is different from the nearly linear behavior of the sand under the steel and the polycarbonate tubes. Acknowledgments

5 0 0

0.02

0.04

0.06 0.08 Engineering Strain

0.1

0.12

0.14

Fig. 12. Comparison of dynamic stress–strain curves of the sand confined with the polycarbonate, 4340 steel and polyolefin heat shrinking tubes.

The authors would like to thank Bradley Martin at US Eglin Air Force Base for providing the material information of the sand. The U.S. Department of Energy and the Joint DoD/DOE Munitions Technology Development Program provided funding for this work.

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