The Impulse Imparted upon Monolithic Metal Plates due to Blast Loading (Experimental and Applied Mechanics)

ABSTRACT

The impulse imparted onto a structure is a major component in the study of fluid-structure interactions during a blast loading event. A better understanding of the impulse transferred to a structure will lead to an improved evaluation of an object’s blast performance. However, limited experimental studies have been performed to determine the impulse imparted to a structure during a blast event. In this study, a comprehensive experimental study on the impulse imparted to free standing monolithic plates under blast loading is conducted. A series of aluminum and steel cylindrical plates were subjected to various experimental loading conditions using a shock tube apparatus. The motion of the specimens was captured using a high speed camera, Photron SA1, to determine their velocity and momentum. The relationship between the impulse transferred to the specimens and the shock wave pressures was analyzed. These results were compared with the theories developed by Xue and Hutchinson, 2004 and Kambouchev, et al, 2006, 2007. The comparisons show that the current fluid-structure model needs to be modified in air blast when the compressibility of the fluid cannot be ignored.

INTRODUCTION

The impulse imparted onto a structure is a major component in the study of fluid-structure interactions during a blast loading event. A better understanding of the impulse transferred to a structure will lead to an improved evaluation of an object’s blast performance. This in turn will help direct the design of new structures with greater blast resistance.


The momentum imparted into a structure during a blast event is typically calculated via fluid-structure interaction. This problem was first solved by Taylor [2] who calculated the solution for a one dimensional wave impinging and reflecting on a solid plate to derive the momentum transmitted into the plate. From this, the momentum transmitted to a plate during a blast loading was found to derive from the density of the fluid, the wave speed, the blast decay time and the areal density of the plate. This fluid structure interaction during a blast loading has been continuously studied for the many years [2-8] and in particular Taylor’s solution has been used to evaluate the blast resistance of sandwich composites with different core topologies [5-7]. However these researchers did not include the non-linear compressibility of the fluid in their studies. Due to the non-linear compressibility of air during blast loading events the results have been questioned [11]. In addition, Kambouchev, et al [3-4] revised Taylor’s model by including the compressibility of the air. This compressibility however, was for waves under the acoustic limit which means that they are governed by the linear wave equation with a constant wave speed. This may be valid for a low intensity air blasts but for high intensity air blasts, one for which there is a noticeable difference between the velocities of the incident and reflected shock wave [1] further considerations may be needed. Some researchers [9-10] used pendulum experiments to estimate the impulse transmitted to the structures from a blast loading. This method can only estimate the final impulse transmitted to the structures and shows neither the impulse redistribution behavior nor the imparted impulse history during the blast event.

In this paper, a series of well-designed shock wave loading experiments were conducted on free-standing monolithic flat plates to simulate the one-dimensional shock wave loading status using a shock tube apparatus.

The specimen momentum history was obtained from both the high-speed photography technique and the measured pressure-time profiles. The experimental results were carefully analyzed and compared with the theoretical prediction from the existing models.

EXPERIMENT SETUP

MATERIALS AND SPECIMENS

The metallic cylindrical plates used were 6061 T6 aluminum and 1018 steel of dimensions 77.8 mm in diameter, which is slightly larger than the inner diameter of the shock tube (76.2 mm), and 6.4 mm thick with masses of 81.5 g and 232 g respectively. Cylindrical plates were used to ensure that any loading on the specimens was limited to the area of the muzzle opening, to (as close as possible) maintain the conditions for a one dimensional uniform blast loading.


EXPERIMENTAL PROCEDURE

A shock tube apparatus was utilized in the present study to generate a controlled blast loading. It consisted of both a driver and driven section which had an overall length of 8 m. These two sections were separated by a destructible diaphragm. A picture of the shock tube can be seen in Figure 1. The driver section was pressurized with high pressure Helium gas which created a pressure difference across the diaphragm. When this difference became great enough the diaphragm ruptured quickly releasing the gas. This gas then traveled down the driven section and created a planar shock wave which imparted an impulse upon the specimen. Two pressure transducers (PCB102A) were mounted at the end of the muzzle section to record the incident and reflected pressure profiles. The first pressure sensor was mounted 20 mm away from the muzzle and the second was mounted 180 mm away (160 mm separation from the first pressure sensor). The final muzzle diameter was 76.2 mm. Figure 2 shows a typical measured pressure profile. Four incident peak pressures of the shock waves were chosen in the current study: .44, .76, 1.03, and 1.35 MPa in the present study.

Shock tube apparatus

Fig.1. Shock tube apparatus

Typical experimental pressure profile

Fig. 2 Typical experimental pressure profile

The specimen was placed on a stand created to ensure placement and a freestanding boundary condition of the specimen, while imparting minimal frictional forces upon the specimen during loading. The flat face of the specimen was set normal to the axis of the shock tube with the face completely covering the opening. A diagram of this set up can be seen in Figure 2. At least two specimens of each type were tested at each incident pressure to ensure reliability.

A high speed photography system was utilized to capture the motion of the specimens in order to determine their velocity and momentum. The lens axis of the camera was set perpendicular to the shock tube and direction of motion of the as shown in figure 3. The distance between the camera and the plate was chosen to be approximately 2 m, which is more than 20 times that of the plate’s dimension (~ 0.1 m) and 10 times that of the plate displacement in the images (~ 0.2 m), to avoid image distortion during plate propagation. In addition, the side of all of the specimens was painted white to give good contrast between the object and the background. The camera was a Photron SA1 high-speed digital camera which has the ability to capture images at a framing rate of 40,000 fps with an image resolution of 512×256 pixels for a 2 second time duration.

The muzzle setup

Fig. 2 The muzzle setup

orientation of camera to shock tube and specimen

Fig. 3 orientation of camera to shock tube and specimen

EXPERIMENTAL RESULTS AND DISCUSSION

HIGH-SPEED SIDE-VIEW IMAGES

The real time observation of the motion of the aluminum specimens at different initial overpressure is shown in figure 5. On the right side of the each image is the shock tube and the shock wave impinges upon the plate from this side. The images show that the circular plate moves linearly away from the shock tube near the time of impact and then rotates slightly as it travels away from the muzzle. This rotation may be due to the reflection of the expanded gas from the boundaries of the dump tank. From the high-speed images, it can be clearly seen that under higher incident pressure, the plate moved faster (with larger distance to the muzzle at the same time).

High-speed side-view images of aluminum plates for different shock loadings

Fig. 5 High-speed side-view images of aluminum plates for different shock loadings

MOMENTUM OF PLATE FROM HIGH SPEED IMAGES

The momentum of the plates can be obtained from the high-speed images shown in Figure 5. A close-view of the typical high-speed side-view image is shown in Figure 6. Curve fitting methods, such as cubic spline curve fitting, can be used to pick up the position of the front face of the specimen. An example of the 7-point cubic spline curve fitting is shown in Figure 6. Since the specimen did not show any compression during the shock wave loading process, this curve can be used to represent the position of the specimen. Then the displacement profile of each point on the specimen can be obtained by correlating the position of the specimen in each image to that in the image at time t = 0. The differential of this displacement profile with respect to the time gives the velocity profile. The momentum of the plate can be evaluated from the velocity profile and the areal density of the plate. It can be expressed as:

tmpA76_thumb

where, ux is the x direction (horizontal) velocity and ds is the areal element of the plate.

Specimen under blast load with cubic spline

Fig. 6 Specimen under blast load with cubic spline

MOMENTUM OF PLATE FROM MESURED PRESSURE PROFILES

The momentum of the plates can be also obtained from the measured pressure profiles. Figure 7 shows the measured reflected pressure profiles of the aluminum plates for different shock loadings. These pressure profiles are considered to be the same as the pressure applied upon the plates. Since the cross-sectional area of the muzzle is known. The pressure impulse applied on the plates can be calculated as,

tmpA78_thumb

where, p0 is the atmospheric pressure.

Consider the momentum conservation during the one dimensional shock wave loading, the momentum of the plate should be identical to the pressure impulse applied to it. Figure 8 shows the momentum of the plates measured from the high-speed images and the pressure impulse of aluminum plates for different shock wave loadings. It can be seen that the momentums measured from the high-speed images agree with the pressure impulses very well. This verifies that the momentum is conserved during the loading process.

Reflected pressure profiles of the aluminum plates for different shock loading

Fig. 7 Reflected pressure profiles of the aluminum plates for different shock loading

Comparison between the measured momentums and the pressure impulses

Fig. 8 Comparison between the measured momentums and the pressure impulses

COMPARISON WITH THEORETICAL PREDICTION

The reflected pressure profile has been theoretical predicted by Taylor [2]. Though Kambouchev, et al [3-4] have claimed to have extended Taylor’s model to consider the compressibility of the gas, their results are compatible with Taylor’s model under the acoustic limit. Therefore, only Taylor’s model is given here. Assume the expression of an ideal incident pressure profile as following,

tmpA81_thumb

where, ppeak is the peak pressure and Q is the time constant.

Through the analysis in Taylor’s classic paper [2], the reflected pressure profile can be calculated as,

tmpA82_thumb

where,

tmpA83_thumb

is a non-dimensional parameter. c0 is the wave speed in the gas. p0 And ps are the densities of the gas and the specimen, respectively. Thus, when t=0, the reflected pressure reach its peak value,

tmpA84_thumb

Then, the impulse imparted on the specimen can be calculated as,

tmpA85_thumb

The relation between the peak reflected pressure and the peak incident pressure is shown in Figure 9. It can be seen that the prediction is close to the experimental result when the peak incident pressure is very low. However, when the incident shock wave is very intense, there are large differences between the predicted and experimental results. This indicates that the compressibility of the gas will play a crucial role in a highly intensive shock wave loading process.

The relation between the peak reflected pressure and peak incident pressure

Fig. 9 The relation between the peak reflected pressure and peak incident pressure

Comparison between the experimental and predicted reflected pressure profile

Fig. 10 Comparison between the experimental and predicted reflected pressure profile

The relation between the peak reflected pressure and peak incident pressure

Fig. 11 The relation between the peak reflected pressure and peak incident pressure

The experimental and predicted pressure profiles of an aluminum plate under a shock loading with a 1.03 MPa incident peak pressure are plotted in Figure 10. In the first 750 ^s, which is the most important region for the fluid structure interaction, the predicted pressure profile is completely different than the experimental one no matter the amplitude or shape. It shows that it will induce a large error if the compressibility of the gas in a shock wave loading process is ignored. It should be noted that after 750 ^s, the prediction agrees with experimental results very well. This is due to the experimental setup. From Figure 5, it can be seen that the gap between the specimen and muzzle was very large, compared to the specimen’s dimensions. The gas can escape from this gap and the pressure transducer, which is located on the muzzle, is far from the specimen. Therefore, the gas at the pressure transducer is not disturbed too much by the specimen. The compressibility of the gas can thus be ignored for these times/distances. Figure 11 shows the impulses calculated from the pressure profiles in Figure 10 and Eq. (6). It can be clearly seen that Taylor’s model under predicted the impulse imparted to the plates.

CONCLUSIONS

In this paper, a series of shock wave loading experiments on free-standing monolithic plates was conducted using a shock tube apparatus. The impulses imparted onto the plates were calculated from the measured reflected pressure profile and verified by the momentum of the plates measured from the high speed images. These experimental pressure and impulse results were compared with the results predicted by existing theoretical models [2-8]. The results show that in the early time of the shock loading process, which is the most important region for the fluid-structure interaction, the existing theoretical model cannot predicted the experimental results no matter the peak reflected pressure or the pressure and momentum profiles. This indicates that the compressibility of the gas must be considered in future models especially in the very early region of a shock wave loading process.

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