With the rapid development of high-energy explosive materials and charging technology, the large equivalent high-energy warhead explosion will produce strong shock vibration on the ground, which will change the instruments and equipment working performance on the test site. When the shock vibration reaches a certain strength, it will lead to vibration damage or even buildings collapse [1–3]. At present, there is no relatively mature technical data to analyze the intensity of ground impact vibration caused by warhead explosion, and the research content in this field is relatively lacking. Therefore, it is necessary to study the ground impact vibration intensity caused by high-energy warhead explosion, and clarify the impact vibration propagation and distribution law in different ground media, so as to evaluate the ammunition explosion damage power, as well as instruments and equipment design of buildings anti shock and vibration performance provides scientific and accurate theory and data support.

At present, researchers in the explosion field damage testing technology and evaluation have carried out less research on the ground shock vibration strength during the ammunition explosion, the existing research data mainly focus on the analysis of seismic waves generated by ammunition explosion. For example: Vorobiev, O [4] carried out numerical simulation of underground chemical explosion of hard rock under different constraint stresses, compared the various generation mechanisms and seismic characteristics of shear wave with the simple fault rupture process observed in the earthquake, and made clear the change law of shear wave during ammunition explosion. Ma K et al. [5] found that buildings will affect the explosive seismic waves propagation. Therefore, they established a reinforced concrete plant model, conducted explosion tests outside and inside the model, and measured the corresponding seismic wave signals. The analysis results show that the building structure existence changes the seismic wave propagation environment, makes the peak velocity have a greater attenuation, and reduces the energy contained in the high-frequency components of the frequency. Nepeina, KS. et al. [6] analyzed the acquired seismic wave curve, constructed the travel time curve of seismic wave, obtained the linear function of body wave propagation within a certain range of epicenter distance, and estimated the overall velocity of the Earth’s upper mantle and outer box. Chen Y H et al. [7] tested the seismic wave induced by ground explosion, and analyzed its time-space evolution and frequency characteristics according to the test results. It is clear that when the surface is covered with loose soil, P, S and R waves will be dispersive, and the explosive seismic wave dominant frequency band is 5∼30 Hz, higher than that of natural seismic wave. Wang L Q et al. [8] collected and analyzed the seismic wave signal generated during the cloud explosion by using the seismic wave testing system, clarified the seismic wave propagation attenuation law during the cloud explosion, and fitted the particle vibration velocity theoretical formula with the measured data. The fitting function relationship can well explain the cloud bomb explosion seismic wave propagation and distribution law. Kholodilov, AN et al. [9] analyzed the ground vibration velocity caused by large equivalent explosion, gave the effectiveness of the elliptic filter on the second-order low-frequency characteristics in the process of modeling using the velocity waveform, proved the efficiency of the model in detecting the delay error, and was able to predict the load caused by the explosion. By analyzing the site effects of blast vibration velocity and blast vibration frequency, the relationship between blast vibration velocity and blast equivalent and seismic wave propagation distance, Yang L et al. [10] obtained the characteristics of blast seismic waves in terms of wave main frequency, duration, near-field initial motion and the blast seismic waves propagation law in rock and soil and the difference between these characteristic laws and natural seismic waves. Pytel, W. et al. [11] applied the numerical simulation method to analyze the elastic wave generated by the explosion of explosives in the mining face, and verified it through the field test of the ground particle velocity and acceleration parameters, clarifying the relationship between the numerical simulation of the seismic particle velocity value and the in-situ measurement using the seismic three-component geophone. Liu H Q et al. [12], to better study the wave process in the rock and soil mass under the shallow buried underground explosion action, established a calculation model for the seismic waves movement on the free surface of semi infinite medium under the shallow buried explosion action, and clearly obtained the source displacement change laws in the horizontal and vertical directions under the same burial depth and different burial depths. Wang X D et al. [13] used ABAQUS finite element software to conduct numerical simulation on the grid structure dynamic response under the action of explosion seismic wave alone, air shock wave alone and the combination of both, analyzed three load cases impact on the grid structure dynamic response, and clarified three loads impact on the grid structure internal force. To improve the seismic waves excited quality by explosive sources in seismic exploration, Mou J et al. [14] selected air, water, fine sand and rock soil as coupling media, and conducted explosion tests in soil to compare and analyze the coupling media influence on the seismic waves characteristics. The test results show that the seismic wave energy generated by explosive source with water as coupling medium is relatively high, especially in the range of 25∼100 Hz, the energy contrast difference is more obvious. Fan P X et al. [15] adopted the common deformation theory and matrix force method to consider the interaction between lining and surrounding rock, established a fast calculation method for the underground protective structures dynamic response under the explosive seismic waves action, and analyzed the impact of surrounding rock grade, cushion parameters and load action forms on the typical straight wall circular arch structures dynamic response. The results show that the surrounding rock grade has a significant impact on the structure dynamic response. In the same year [16], they discretized the circular tunnel into a finite freedom system degree by using the concentrated mass method, considered the structure self deformation under various internal forces and the elastic half space deformation by using the matrix force method, and established the structural dynamic equation matrix. By solving the structural dynamic equation, the structural dynamic concentration factor, internal force, bending moment, displacement and other parameters are obtained. Zhao S et al. [17] took the 15 story frame structure as an example, established the space plate beam structure finite element model by using the finite element analysis software, calculated the structure natural frequency and natural vibration mode, and analyzed the high-rise frame structure dynamic response under the simulated explosion seismic waves action. According to the structural dynamic response results, the structure weak story is judged, which provides a reference for the project seismic design. Zhang X J et al. [18] used LS-DYNA to conduct explosion process numerical simulation in the closed rock mass, and obtained the law of the ground vibration acceleration changing with time, which provides a certain reference for the research and calculation method of the explosion seismic wave response effect on the ground structure.

At present, the commonly used formula for calculating the explosive seismic wave ground particle vibration velocity is Sadovsky’s particle vibration velocity formula [19]. This formula does not consider the ground medium effect on the vibration velocity, and the calculation formula is only applicable to the small equivalent calculation, conventional warhead explosive seismic wave vibration velocity. When the type of ground medium, charge type and charge mass of warhead change greatly, the theoretical calculation formula accuracy will be greatly reduced, and the calculation results cannot reflect the ground impact vibration strength. In view of these problems in the current research process, it is very necessary to carry out research on the ground impact vibration intensity model during the high energy warhead explosion.

In this study, the finite element numerical simulation was carried out for the explosion shock wave and ground shock vibration velocity of the high-energy warhead with the equivalent TNT explosive mass in the range of 100 kg∼1,000 kg in the ground medium of sandy soil, C35 and C140 concrete by using the display explosion dynamics simulation software, and the propagation and distribution laws of the shock wave pressure and ground shock vibration velocity at different measuring points were analyzed. Based on the explosion similarity law and the dimensional analysis method, the impact vibration strength model with clear physical significance is established by introducing the ground propagation medium properties, and the strength model calculation accuracy is verified by using the measured data.

When the warhead explodes, the explosive products sharp collision results in the surrounding air intense compression, resulting in the local high-pressure wave front at the interface between the explosive products and the air. This wave front is the initial shock wave formed. As the explosion time goes by, the wave front gradually spreads to the distance. If the wave front does not encounter the ground or other obstacles in the propagation process, the wave front will continue to move towards the distance in the form of spherical waves. When the wave front contacts the ground, the shock wave will collide with the ground, reflecting, and form a reflected shock wave. During the collision between the shock wave and the ground, part of the incident shock wave energy will be transferred to the ground to form explosive seismic wave, which will make the ground vibrate; Because the shock wave front propagates at hypersonic and supersonic speeds in the initial propagation process, the shock wave front has a strong impact effect when contacting the ground, and the impact effect will also cause the ground to vibrate. It can be seen that, as the explosion time goes by, the shock wave front will continuously act on the ground to generate shock vibration during the propagation process. Therefore, the shock vibration source in the whole ammunition explosion process can be shown as Figure 1.

Through the above analysis and in combination with the explosion impact dynamics relevant theory, the main factors that affect the ground impact vibration velocity in the ammunition explosion process are: shock wave pressure peak value P , explosive quality ω , distance between measuring point and explosion center r , ground medium density ρ , sound velocity C , ground medium elasticity modulus E . According to the explosion similar law, the ground medium shock vibration velocity V when the ammunition explosion can be expressed as a relationship function shown in Eq. 4. V = f P , ω , r , ρ , C , E . (4)

The three basic physical quantities L (length)—M (mass)—T (time) are selected to analyze the expressions of the above physical quantities each scale, and the each physical quantity scale expressions are shown in Table 3.

Combine the vibration velocity V of the surface propagation medium and the three basic physical dimensions selected to rank the power exponent of the each physical quantity dimensional expression in Eq. 4, The arrangement results are shown in Table 4.

Comprehensively consider the dimensional form and physical meaning of each physical quantity, the explosive mass, the distance r between the measuring point and the explosion center, and the sound velocity C are selected as the reference physical quantities. Reorder the physical quantities in Table 4, and the sorting results are shown in Table 5.

Based on the selected reference physical quantities, r and C , change the dimension power index of each physical quantity in Table 5, and obtain the dimension power index of each physical quantity as shown in Table 6.

From the each physical quantity power in Table 6, the four dimensionless physical quantities are Π 1 , Π 2 , Π 3 , and Π 4 , respectively, according to Π theorem. Π 1 = ρ ω r − 3 Π 2 = E ω r − 3 C 2 Π 3 = P ω r − 3 C 2 Π 4 = V C . (5)

The above four dimensionless physical quantities combination can be obtained as a function of Eq. 6. V C = f ρ ω r − 3 , E ω r − 3 C 2 , P ω r − 3 C 2 . (6)

The dimensionless functional relationship in Eq. 6 is simplified to obtain the functional relationship shown in Eq. 7. V C = f P × r 3 ω × ρ E . (7)

According to the relevant theory in the materials mechanics, it is known that the product of the material density ρ and the sound speed C is the material acoustic impedance, r 3 / ω is the third power of the explosion proportional distance R . Therefore, the functional relationship of Eq. 7 is processed to obtain the functional relationship shown in Eq. 8. V = f P × r 3 ω × ρ C E . (8)

Equation 8 is the ground shock vibration velocity intensity model during the ammunition explosion. From this intensity model, it can be seen that the ground shock vibration velocity is positively correlated with the peak shock wave pressure P , the blast proportional distance R 3 , the ground medium acoustic impedance ρ C , and negatively correlated with the ground medium elasticity modulus E .

To determine the above functional relationship final expression, the functional relationship in Eq. 8 is expressed in the form shown below. V = a 1 × P × r 3 ω × ρ C E a 2 + a 3 . (9)

The shock wave pressure and ground shock vibration velocity data obtained from the finite element numerical simulations carried out in different ground media with different TNT masses were used to obtain the coefficients values a 1 , a 2 , and a 3 in Eq. 9 by non-linear adaptive fitting. The density ρ , sound speed C and sandy soil elasticity modulus E , C35 and C140 concrete are shown in Table 7. The fitting results are shown in the following Figure 7.

According to the fitting effect diagram of the above function relationship, the data fitting error squares sum is 0.97488, and the adjustment error squares sum is 0.97423. When P × r 3 / ω × ρ C / E is less than 32,500, the fitting data can be evenly scattered on the fitting curve both sides. The fitting data and the fitting curve have very good consistency. The strength model constructed can well reflect the ground impact vibration velocity change law during the ammunition explosion process; When P × r 3 / ω × ρ C / E is greater than 32,500, the strength model calculated result is gradually smaller than the measured value, mainly due to the TNT explosive large mass and the high pressure of shock wave generated by explosive explosion. When the shock wave collides with the ground, it will directly lead to the ground propagation medium density change, thus affecting the elastic modulus and sound velocity of the propagation medium. Specifically, the high intensity shock wave pressure will compact the soil, increase the density and sound velocity, and reduce the ground impact vibration velocity attenuation speed in the transmission process, and spread farther. Therefore, in order to ensure the calculation accuracy, P × r 3 / ω × ρ C / E should be less than 32,500 for the ground impact vibration intensity model established above in the actual use of ammunition explosion.

In Eq. 9, the undetermined coefficients a 1 = 0.0013 , a 2 = 0.95256 , a 3 = 29.35633 , so Eq. 9 can be expressed as the functional relationship shown below. V = 0.0013 × P × r 3 ω × ρ C E 0.095256 + 29.35633 . (10)

To verify the above functional relationship model calculation accuracy, we carried out the field test under the condition that the surface propagation medium is sandy soil and the high-energy warhead equivalent TNT explosive mass is 300 kg and 500 kg. The surface reflection pressure measurement system and the ground vibration velocity measurement system are shown in Figures 8, 9.

The ground reflection pressure measurement system is composed of ground reflection pressure sensor, sensor installation component, self-developed high-precision synchronous trigger controller, TranNet 308S data collector and computer terminal. The measurement system can realize data acquisition at a sampling rate of 1 MS/s for 32 channels at the same time. The ground vibration speed measurement system is composed of unidirectional magnetoelectric speed sensor, speed sensor mounting rod, TC-4850 vibration meter and computer terminal. A TC-4850 vibration meter can be connected to three speed sensors at the same time. The vibration meter has a built-in power supply, which can realize continuous acquisition for a long time.

The above measuring system is used to obtain the ground reflection pressure and the ground impact vibration velocity variation law generated during the ammunition explosion process. When the equivalent TNT explosive mass is 300 kg, the horizontal distance between the measuring point position and the vertical projection point of the blast center is 3.5, 5, 6, 6.5, and 8 m respectively; When the equivalent TNT explosive mass is 500 kg, the horizontal distance between the measuring point position and the vertical projection point of the blast center is 4, 4.5, 7, 7.5, and 9 m respectively. The incident angle of shock wave pressure is equal to the horizontal distance between the measuring point and the vertical projection point of the blast center divided by the arctangent function of the blast height. According to the calculation, the pressure monitoring points at the measuring points of 7, 7.5, 8, and 9 m in the above verification data are located in the Mach reflection zone. The peak value of ground vibration velocity obtained during the test and the peak value of ground vibration velocity calculated by Eq. 10 are shown in Table 8.

Through comparison and analysis of the data in Table 8, it can be found that the peak value of ground vibration velocity obtained by the ground vibration velocity measurement system is very close to the calculation result of the functional relationship model of Eq. 10, and the ground vibration velocity overall change rule is consistent. The individual data calculation results in the above data are somewhat different from the measured results. The main reason is that the test site surface propagation medium is not completely homogeneous. The test site is affected by the mixed stones, rubble in the soil and the elevation difference within the region, resulting in the change of the ground vibration velocity peak value, as a result, the relative error between the calculated and measured results of ground vibration velocity at some measuring points is large. The relative error rate between the ground vibration velocity calculated results at different measuring points and the measured results is shown in Figure 10. The formula for calculating the relative error rate is shown in Eq. 11. φ = C a l c u l a t e d   v a l u e − M e a s u r e d   v a l u e M e a s u r e d   v a l u e × 100 % . (11)

It can be seen from the relative error results between the above measured values and calculated values that the maximum relative error occurs at the TNT explosive mass of 300 kg, the distance between the measuring point and the explosion center is 8 m, which is 1.3%, the minimum relative error occurs at the TNT explosive mass of 500 kg, the distance between the measuring point and the explosion center is 4 m, which is 8.2%, and the relative errors at other measuring points are between them. It can be concluded that the calculation accuracy of the above established ground impact vibration velocity calculation function is better than 91.8%. This function relationship improves the ammunition explosion ground impact vibration velocity calculation accuracy, provides scientific data support for the high-energy warhead explosive damage power test, and has strong engineering application value.

This study carried out a finite element numerical simulation analysis of the shock wave pressure and ground medium shock vibration velocity propagation distribution law during the high-energy warhead explosion. The effects of TNT explosive mass and ground medium type on shock wave pressure and vibration velocity are analyzed. Based on the explosion similarity law and the dimensional analysis method, the high energy warhead explosion ground impact vibration velocity calculation model is established. This study results indicate that:

(1) The ground medium has a significant effect on the propagation and distribution of shock wave pressure and ground shock vibration velocity. When the ground medium is sandy soil, C35 and C140 concrete, under the same explosive environment, the shock wave pressure peak value and the ground vibration velocity peak value are P C 140   c o n c r e t e > P C 35   c o n c r e t e > P S a n d   s o i l , V S a n d y   s o i l > V C 35   c o n c r e t e > V C 140   c o n c r e t e . The main reason for this phenomenon is the large difference in density and porosity of the three ground media, resulting in inconsistent reflection intensity and energy conversion rate when the shock wave collides with the ground.

This research results clarify the distribution law of high-energy combatant explosion shock wave pressure and ground shock vibration velocity propagation, and provide theoretical support for the munition explosion damage power test program design. The established ground shock vibration velocity calculation model provides scientific data support for accurate assessment of the ammunitions damage power and guidance for the instruments design, equipment and buildings against shock vibration.