INTRODUCTION

The development of universal models for calculating the properties of promising materials of complex composition, including porous models, requires reliable experimental data for verification under shock compression. An example of a substance that is relatively simple and suitable for modeling is spheroplastics, which are lightweight polymer-based composite materials containing hollow microsized spheres. In addition, these substances widely applied [1]. Due to the variety of polymer matrices, microspheres, their coatings, and concentrations in the sample, the properties of the materials obtained on their basis are very different. The structural and deformation properties of spheroplastics are well studied under low stress conditions [2]. There are much less data on their properties under shock compression. It is known that, under relatively low pressures, the front of a compression wave propagating in spheroplastics has a complex structure [3, 4], in which the primary and secondary waves are distinguished. The primer is associated with the spread of elastic compression, and the latter is related to the transfer of plastic stresses as a result of impact leading to the fracture of microspheres. In this regard, the issues related to the pressure range in which a complex wave structure is observed and the effect of changes in the concentration of microspheres on the nature of shock compression waves are relevant for experimental research.

EXPERIMENT

Results of experiments

Etal-370 epoxy resin and Etal-45M hardener based samples are under study investigated. Glass microspheres (MSs) with an average diameter of 83 \(\mu\)m and a wall thickness of the order of 1 \(\mu\)m are selected as filler. The epoxy resin is mixed with the hardener in a volume ratio of 2 : 1 and with microspheres until a homogeneous medium is obtained. Then, the sample solidifies at room temperature for more than 24 h. Particular attention is paid to the uniform distribution of microspheres in the volume. The volume fractions of the microspheres are \(\varphi\) = 0.27 and 0.55, the densities of the samples are 0.90 and 0.64 g/cm3, respectively, and the density of the samples not containing any MSs is 1.15 g/cm3. The characteristics of the samples and experimental parameters are presented in the table.

The sample is placed in the assembly. The assembly consists of a screen, a sensor recording the time at which a wave enters the sample, the sample itself, a foil reflecting the laser beam, and a water window. The screen is made of a D16T aluminum alloy, with its thickness in all experiments being 1.80–2.00 mm. The thickness of the samples is \(h_s\) = 3.5–5.3 mm, and their diameter is 40 mm. The assembly is mounted on a mounting ring which serves as a support when placed in the measurement chamber. The chamber is evacuated to a residual air pressure of less than 0.1 torr. This makes it possible to virtually exclude the influence of the head shock wave arising on the surface of the impactor during movement in the atmosphere on the measured velocity profile of the sample surface. The experiments are carried out at room temperature.

In each experiment, the measured parameters are those that make it possible to determine the impactor velocity, the propagation velocity of the shock compression wave in the sample, and the velocity profile of the sample surface. There are tests in which one of the parameters could not be measured for technical reasons.

To accelerate 10-mm thick flat impactors made of the D16T aluminum alloy to velocities of 320–780 m/s, a projector with a caliber of 57 mm was used. The velocity is determined using two pairs of contact needle sensors, which the impactor closes sequentially during movement to the assembly screen. The sensors are placed around the perimeter of the mounting ring in which the assembly is placed.

The velocity profiles of the sample/water and sample/vacuum boundaries are obtained using a VISAR inteferometer [5].

The propagation velocity of the compression wave in the sample is determined using the measured time interval from the instance at which the wave enters the sample to the instance at which it exits the sample and the known thickness of the sample. The instance at which wave enters the sample is determined by a signal from an aluminum foil sensor located between the screen and the sample. The instance at which the wave arrives at the studied surface is determined by the appearance of a signal in the VISAR measurement channels. The general experimental design and assembly composition are shown in Fig. 1.

Fig. 1
figure 1

Experimental design: impactor (1), mounting ring (2), sensors (3), screen (4), sensor for measuring the time it takes the wave to enter the sample ( 5), sample (6), reflecting foil (7), water window (8), window cover made of PMMA (9), and VISAR beam (10).

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Fig. 2
figure 2

Velocity profiles of the sample/water window boundary for the samples with a microsphere content of 0.27 (a) and 0.55 (b); the numbers near the curves refer to the test numbers in the table.

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Fig. 3
figure 3

Velocity profile of the sample/water window boundary for the epoxy resin sample containing no glass microspheres (test No. 15).

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Fig. 4
figure 4

Velocity profiles of the free surface of the samples with no glass window (\(\varphi\) = 0.27).

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RESULTS

The resulting profiles of the surface velocity \(v\) are shown in Figs. 2 and 3 (profile numbers refer to test numbers in the table). It is clearly seen that the surface velocity profiles of spheroplastic samples have a two-wave structure in contrast to a profile in the epoxy resin samples, with the exception of test 7 in which the projectile velocity is the smallest (Fig. 3). The profile begins with a sharp increase in velocity, called the first wave or a precursor. It is followed by a smooth section separating the first wave from the second one, called the main wave. The profiles 1, 3–6, and11–14 have another velocity jump (marked by arrows) that follows the arrival of the main wave at the sample/water window boundary, associated with the reflection of the shock wave from the sample/window boundary into the sample, which reflects from the screen and then returns to the boundary [4]. In the tests 9 and 10 with no water window and with a PMMA cover, no similar picture is observed (Fig. 4). In these profiles, the stepwise character of the increasing amplitude is clearly distinguished when the main wave exits with a pronounced step at\(v\) = 300–350 m/s. A similar structure is associated with the circulation of the precursor between the free surface of the sample and the front of the main wave [4].

The precursor velocity amplitude reaches 80–140 m/s and 30–80 m/s for samples whose microspheres have volume fractions of 0.27 and 0.55, respectively. In the test 7, the following situation is observed: at a shock compression pressure \(p\) = 0.29 GPa, the main plastic wave is poorly distinguishable against the precursor background. The oscillations of this profile can be associated with damage to the reflecting foil by microspheres. These circumstances make it possible to measure the velocity of the main wave only approximately and do not allow for estimating the maximum strain rate in it within this experiment. In the test 12, the precursor is almost completely attenuated starting from a value of 50 to 20 m/s, and its total duration was 3.7 ms.

Fig. 5
figure 5

Shock adiabats of the spheroplastics, precursor (2 and 4), and the main wave (1 and 3) for \(\varphi\) = 0.27 (1 and 2) and  0.55 (3 and 4).

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Based on measuring the propagation velocity of the precursor and the main wave, shock adiabats are constructed in \(u\)–\(D\) coordinates [the mass velocity–shock wave velocity (in km/s)]. The mass velocity is calculated by the reflection method using the known characteristics of the impactor material and the screen [7]. The results are presented in Fig. 5. It is noteworthy that, for the purpose of clarity, the precursor velocities correspond to the mass velocities in the main wave rather than in it. It is clearly seen that the shock adiabat of the main wave in samples with a lower MS content lies significantly higher and has a greater angle of slope than the adiabat of samples with a high MS content. The dependences of the precursor wave velocity have a negative angle of slope, and, if extrapolated onto the range of higher compression pressures, they touch the adiabats of their main waves at \(u>1\) km/s, which corresponds to \(p>2\) GPa for samples with \(\varphi\) = 0.27 and \(p>1\) GPa for samples with \(\varphi\) = 0.55. The data on the velocity of the main shock wave are approximated by the linear dependences

$$D=0.21+2.35u\mbox{ [km/s],}$$
(1)
$$D=0.01+1.60u\mbox{ [km/s],}$$
(2)

for samples with \(\varphi\) = 0.27 and 0.55, respectively. The adiabat for samples \(\varphi\) = 0.27 reasonably corresponds to the results of the similar study [3].

The shock compression rate of the samples is estimated using the expression

$$\dot{\varepsilon}_x = \dot{u}_p/D\mbox{ [s$^{-1}$]},$$
(3)

where \(\dot{u}_p\)[m/s2] denotes the maximum velocity gradient in the region where the amplitude of the main wave increases, and this gradient is calculated from the profile taking into account the difference in the dynamic impedances of the materials of which the sample and the window are made.

The duration of the sections of the profiles taken to determine the maximum velocity gradients in all tests is larger than the resolution limit of the VISAR interferometer, which is equal to 1 ns. For pressures higher than those in this study, a modified technique is applied with a transition to stepwise shock compression, which is necessary because the method for estimating the maximum compression rate described above becomes incorrect [8].

Fig. 6
figure 6

Compression rate versus the shock compression pressure.

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The generalizing dependence of the measured shock compression rates on the pressure is calculated only for tests with the water window (Fig. 6). The resulting data can be approximated by the power dependences for samples with \(\varphi\) = 0.27 and 0.55, respectively:

$$\dot{\varepsilon}_x = 1.58\cdot 10^7p^{3.2}\mbox{ [s$^{-1}$]},$$
(4)
$$\dot{\varepsilon}_x = 4.46\cdot 10^7p^{1.5}\mbox{ [s$^{-1}$]}.$$
(5)

Here the dimensionality of the numerical coefficient in front of the pressure is [1/(s\(\, \cdot\,\)GPa\(^{n})\)].

RESULTS AND DISCUSSION

According to the data of [3], an elastic precursor in spheroplastics arises due to the ability of the walls of MSs to deform elastically. Upon reaching the elastic limit corresponding to the maximum amplitude of the precursor on the velocity profile, the MSs begin to fracture. A plastic deformation wave propagates through a substance with partially or completely fractured MSs.

For the spheroplastics under study, the fracture threshold of MSs is 0.10 \(\pm\)0.04 GPa (\(\varphi\) = 0.27) and 0.04 \(\pm\)0.02 GPa (\(\varphi\) = 0.55). Thus, the fracture threshold under shock compression of samples with \(\varphi\) = 0.27 turns out to be an order of magnitude higher than under a static load [9]. For samples with \(\varphi\) = 0.55, it is several times higher. The observed decrease in the fracture threshold of MSs with an increase in their concentration in the sample is consistent with the Hermann \(p\)–\(\alpha\) model [10]. The dependence for solid metals, which is similar to Eqs. (4) and (5), has a fourth degree [11]. For a fluid, the degree of 2.1 is obtained using glycerol as an example [8]. In [12], it is shown that the degree of dependence drops from 1.5 to 1.1 along with the initial temperature from 18.5 to 6.5 for commercial glycerol, and the degree of 3.2 is obtained for water, which turns out to be close to this degree for metals. Apparently, the compression process during high-velocity impact in samples containing hollow MSs changes its character as their concentration becomes higher. Samples with a smaller volume fraction of MSs are compressed in a manner similar to solid metals, and, with a larger fraction of microspheres, they are close in their properties to fluids. Moreover, the higher the MS content, the higher compression rate. The differences obtained are due to the fact that the number of defects (plastic deformation carriers) arising at the boundary between the matrix and MS fragments is greater at a higher initial concentration of MSs. It should be noted that, in spheroplastics, as the MS concentration increases to a certain value, a quasicontinuous volumetric framework of microspheres can form [2]. However, for the filler and microspheres studied in this work, this value of concentration is not investigated. This does not allow us to unambiguously relate the possible qualitative difference in the samples before shock compression with the experimental results.

CONCLUSIONS

1. The study performed shows that the complex structure of the shock compression wave in the samples containing glass microspheres is observed in a finite pressure range. Accurately determining its upper bound and how it depends on the volumetric fraction of microspheres in the compound is possible in tests under high compression pressures. Extrapolating the resulting data may allow one to estimate the bound as that lying above a pressure of 2 GPa for samples with \(\varphi\) = 0.27 and 1 GPa in the case \(\varphi\) = 0.55. The analysis of the dependences between the shock compression velocity and the pressure behind the wave front shows that, as the volumetric fraction of microspheres in the samples becomes larger, their properties during shock compression become closer to those of viscous fluids. In this case, the strain rate of shock compression is higher in samples with a larger fraction of microspheres, which is generally typical of substances with multiple defects.

2. With an increase in the concentration of microspheres of the same type in the epoxy matrix, the threshold for the onset of their fracture under shock compression decreases, which is consistent with the model concepts.