Equivalent loading conditions satisfying pressure time history consistency between close-in blast and mass-block impact

Abstract

Close-in blast tests are plagued by defects such as difficult data measurement, high safety risks, and poor repeatability. A promising method for studying close-in blast is simulating it through impact loading in a laboratory environment. However, current loading equivalence research primarily focuses on reproducing the target plate’s failure mode, with the equivalent criterion adhering to the impulse criterion, which considers only the single contribution of impulse rather than the combined effects of impulse and peak pressure. Few studies have addressed the consistency of pressure time history and the quantitative prediction of equivalent loading conditions between close-in blast and mass-block impact. In this work, the mapping relationship between close-in blast loading conditions and pressure time history parameters is established, based on the fluid-solid coupling model and strong shock wave theory. An incomplete elastic collision model for the mass-block impact is also developed. Peak pressure and impulse transmission are regulated by adjusting impact velocity and mass, while the restitution coefficient and impulse transmission upper limit are determined. Finally, a quantitative prediction method for equivalent loading conditions is proposed. The results show that the pressure time history curves of close-in blast and mass-block impact under equivalent loading conditions exhibit good agreement.

Keywords

Close-in blast mass-block impact equivalent loading conditions impulse transmission

Introduction

Close-in blast is the most common loading form in terrorist attacks and industrial accidents (Adhikary and Dutta, 2019). Unlike the far-field blast, the close-in blast has high pressure peak, short time duration, and non-uniform distribution, which cause severe local damage and destruction to building structures, industrial facilities, and protective armor (Zhang et al., 2020). Consequently, the damaging effect of close-in blast loading on target structures has always been a research hotspot in protective design (Adhikary and Dutta, 2019).

However, explosives are inherently dangerous and have high requirements for operational safety (Gan et al., 2020). Detonation products, fireball flashes, dust, and debris clouds will interfere with the recording and tracking by high-speed cameras. The high temperature and high pressure environment in the near field may cause severe damage to the test equipment and sensors, and even the test signal cannot be measured (Durant, 2013). More advanced test measurement techniques are under development and not yet widely available (Barr et al., 2023; Rigby et al., 2019). Therefore, the non-explosive impact test based on the laboratory environment is a promising simulation method for close-in blast research.

The University of California, San Diego has designed and developed the Blast Simulator that uses mass impact loading instead of close-in blast loading, which is the first facility to utilize ultra-fast, hydraulically driven, computer-controlled actuators to generate impulsive loadings on full-scale structures (Stewart et al., 2014). Tonatiuh (Rodriguez-Nikl, 2006) used the Blast Simulator to conduct impact tests on composite armor, simulating the damage mechanism of the structure by the impulse load of 6.8 MPa·ms ∼ 15.7 MPa·ms. This impulse level is equivalent to the explosion load on the armored vehicle under the charge weight of 560 kg, the height of 0.9 m, and the detonation distance of 3.5 m ∼ 6.1 m. Oesterle (Oesterle, 2009) realized the explosion simulation loading with impulse load in the 1 MPa·ms ∼ 2 MPa·ms. Chen (Chen, 2010) studied the dynamic response of composite sandwich structures to blast loads using impact tests and numerical simulations. Stewart (Stewart, 2012) carried out impact tests and numerical simulations on the I-beam steel structure. Compared with the real blast test results, there is good consistency in the failure mode. Freidenberg (Freidenberg, 2013) further demonstrates the Blast Simulator’s capability in generating blast-like loading on wall systems. The above studies all simulated the plane shock wave load, rather than the close-in shock wave with short duration and high curvature. Whisler (Whisler, 2014) carried out impact tests on traditional steel and flexible armor plates using a discrete module impactor based on the impulse equivalent criterion. The results show that the discrete module design has a good reproduction effect on the close-in blast load distribution. Still, it is also challenging to balance the accuracy of pressure peak and time duration. While it was demonstrated that short-duration “impulsive” blast-like loading could be achieved on structures with planar geometry, the Blast Simulator has some limitations relating to pulse shape and duration (Freidenberg et al., 2014). The impulses between the Blast Simulator and the field test display good agreement, but the overall time duration of loading is shorter than the time duration of loading in the field test.

To sum up, the current equivalent research based on the impact test machine to simulate the close-in blast mainly focuses on reproducing the target plate’s dynamic response and failure mode. The equivalent criterion mostly follows the impulse equivalent criterion, which means the impulse value is used to evaluate the loading’s destructive capability to target structures. The consistency of the pressure time history is reflected in the consistency of the pressure peak and specific impulse (or time duration) of the triangle-shaped loads, which together determine the loading’s destructive ability. The effect of pressure peaks will be more significant for the materials and structures with shorter natural vibration periods (Jones, 2011). The close-in blast load is determined by the charge weight, the detonation distance, and the incidence angle, which together serve as the loading conditions for the close-in blast. In comparison, the mass-block impact load is determined by the recovery coefficient, velocity, and mass, which together serve as the loading conditions of the mass-block impact. In order to establish the pressure time history equivalent relation and quantitatively solve the equivalent loading conditions, it is necessary to predict the close-in blast loads and regulate the mass-block impact loads, respectively.

At present, there are many definitions of the explosion range in the close-in field. As some studies define explosion with scaled distance z < 1.2 m/kg^1/3 (Enstock and Smith, 2007; Longinow, 2013) as a close-in blast. Others define scaled distance z < 1.05 m/kg^1/3 as a close-in blast (Gel’fand et al., 2004). An empirical limit value scaled distance z = 0.5 m/kg^1/3 is used to demarcate close-in and far-field blast effects in UFC 3-340-02 manual (The unified facilities criteria of the structures to resist the effects of accidental explosions) (Department of Defense, 2008). The most widely recognized criterion is whether there is direct action of explosive products. In the case of an air explosion resulting from a spherical charge, the limit of volume expansion for the explosion product is typically 10–12 times (Baum et al., 1961) the radius of the charge. However, this limit applies only once the fluctuating explosion products have stabilized. During the initial stages of the explosion, the maximum distance the explosion product can reach may increase by 30–40% (Baum et al., 1961) due to the inertial effect. Consequently, the maximum range of the explosion product from the spherical charge is typically 13 to 17 times the charge radius, with an average of 15 times the charge radius. When converted from this range to the spherical charge, the scaled distance is 0.8 m/kg^1/3. That means, if the scaled distance exceeds 0.8 m/kg^1/3, the target is less susceptible to direct action from the explosion product. Compared with the far-field blast, the time duration of the close-in shock wave is very short, only a few milliseconds or tens of milliseconds, but the peak pressure will reach tens or even hundreds of standard atmospheres (Moszynski, 1983). In such a strong shock environment, the air medium will be dissociated and ionized, and the expansion exponent will change, resulting in a substantial increase in the load on the surface of the target structure, which shows a different law from the far-field blast (Kinney and Graham, 2013). It means the close-in blast phenomenon is more complicated, resulting in significant differences in the prediction of the close-in blast loads in the existing literature, manuals, and codes (Association, 2012; Brode, 1959; Department of Defense, 2008; Mills, 1988; Wu and Hao, 2005; Yang et al., 2008). Based on the theory of regular reflection and Mach reflection, Zhou et al. (Zhou et al., 2004, 2018) established a strong shock wave reflection theory applicable to air medium and analyzed the shock wave propagation characteristics of spherical TNT charges that conform to the real air state equation. On this basis, combined with numerical simulation and test data, this paper attempts to propose a more reliable close-in blast loads prediction method to provide the necessary load parameters for establishing the blast-impact equivalent loading relation.

For the research on the regulation principle of impact loads, Radford (Radford et al., 2005) first performed a one-dimensional plastic shock wave analysis for a foam projectile impacting a free rigid mass block. It is shown that the pressure versus time pulse exerted on the mass depends upon the ratio of foam mass to impact mass. The pulse’s magnitude and duration can be regulated by adjusting the length, density, and velocity of the foam projectile. Then Yang et al. (Yang et al., 2023) proposed the nonlinear dynamic model of polyurethane waveform generators. If used as a load regulation method, (Radford et al., 2005; Yang et al., 2023) are fundamental and rigorous. Still, it relies too heavily on one-dimensional assumptions and precise material model parameters, which is unsuitable for practical application. It is a more concise method to analyze the regulation principle of impact loads from the macroscopic energy and momentum perspective. It’s worthy to focus on that despite the same impact energy level, the maximum values of impact behavior (The maximum response of the structure under impact loading, such as rotation angle, displacement, etc.) depends on the impacting momentum (Yu et al., 2021). The force time history depends on the combination of the impact velocity and mass. If the mass is heavy and the impact velocity is low, the maximum impact force is small, but the time duration is large. If the mass is small and the impact velocity is high, the maximum impact force is large, but the time duration is small (Yu et al., 2017). It was found that the total contact time is more affected by the impact momentum than the impact energy. At present, few research has addressed the influences of velocity and mass under the equivalent level of impact energy. Various combinations of impact mass and velocity can lead to different structural responses for the same level of energy (Aryal et al., 2019; Papa et al., 2022).

This paper combines a fluid-solid coupling numerical model, strong shock wave theory, and an incomplete elastic collision model to quantitatively analyze the equivalent loading conditions between close-in blast and mass-block impact. The section on Equivalence criterion of triangle-shaped loads highlights the limited applicability of the impulse criterion, which is only valid when the loading time duration approaches zero, making it insufficient for modeling close-in blast loading. Therefore, the consistency of pressure time histories between close-in blast and mass-block impact is proposed as a more suitable criterion. The sections on Close-in blast loads prediction and Mass-block impact loads regulation focus on predicting and regulating the load parameters of close-in blast and mass-block impact, respectively, with the consistency reflected in pressure peak, specific impulse, and time duration. We establish a mapping relationship between close-in blast loading conditions and pressure time history parameters, using the fluid-solid coupling model and shock wave theory. An incomplete elastic collision model is developed for mass-block impact, where peak pressure and impulse transmission are adjusted through impact velocity and mass. The section on Solving for equivalent loading conditions proposes a quantitative prediction method for the equivalent loading conditions between the two forms of loading.

Equivalence criterion of triangle-shaped loads

It is known that triangle-shaped loads are generally considered to obey the peak pressure criterion when the loading time duration is long, and it is considered to obey the impulse criterion when the loading time duration is short. But it’s only a rough conclusion of engineering experience. According to the analysis results in Appendix A, it can be found that the applicable range of the impulse criterion is too tiny, so it is applicable only when loading time duration t_m approaches zero, as shown in Figure 1. The result also shows that even under looser conditions with an allowable engineering error of 5%, for the target plate with a conventional size or smaller (with a large natural frequency), the t_m obeying impulse criterion t_max is insufficient to cover the close-in blast loading completely, as shown in Table 1.

Figure 1.

The dynamic amplification factor curve of triangle-shaped loads: where k_d is the dynamic amplification factor, p_e is the equivalent static load, t_m is the loading time duration, T is the structural free vibration period. p_r is peak pressure of triangle-shaped loads, I_r is the specific impulse, determined by the area under the triangle-shaped load curve.

Table 1.

The maximum loading time duration obeying the impulse criterion for the reinforced concrete plates: t_max is the maximum value of loading time duration t_m obeying the impulse criterion, L is the short span length of the plate, and the thick-span ratio is the ratio of the plate thickness to the short span length.

Duration-span ratio t_max/L (ms/m)
Boundary conditions	Thick-span ratio	Length-width ratio
Boundary conditions	Thick-span ratio	$\frac{1}{3}$	$\frac{1}{2}$	$\frac{1}{1.5}$	$\frac{1}{1.2}$	1.0
Simple support	0.1	38.7	34.7	30.1	25.7	21.7
	0.2	20.5	18.7	16.5	14.3	12.4
	0.3	14.7	13.4	11.9	10.5	9.1
	0.4	11.8	11.0	9.9	8.7	7.7
Clamped support	0.1	18.2	17.3	15.8	13.9	11.9
	0.2	10.2	9.9	9.3	8.3	7.3
	0.3	8.1	7.9	7.4	6.7	5.8
	0.4	7.2	7.1	6.7	6.1	5.4

The above means that, within a reasonably wide range of t_m, establishing the equivalent relationship should consider the combined effects of impulse and peak pressure rather than the single contribution of impulse. That is, the pressure time history consistency of close-in blast and mass-block impact is a more reasonable equivalent criterion, especially under the long time duration of loading or large natural frequency of target. However, there are few studies on the consistency of pressure time history and quantitative calculation of equivalent loading conditions at present, which is the ultimate goal of this paper: Considering the combined effects of impulse and peak pressure, propose a quantitative prediction method for the equivalent loading conditions satisfying pressure time history consistency between the two loading forms: close-in blast and mass-block impact.

Close-in blast loads prediction

The close-in blast phenomenon is complicated, resulting in significant differences in existing predictions. In this Section, the combined method based on the fluid-solid coupling numerical analysis and the strong shock wave theory shows a good agreement with the test data in the UFC manual and various references.

Treating the UFC manual proposals at the close-in field as reference values would introduce big uncertainties and should be avoided. But, so far, due to the defect of measurement technology and the large discreteness in close-in blast tests, there are few credible test data. Therefore, the UFC manual is still the most widely used in many close-in blast studies (Cheng et al., 2013; Nagata et al., 2018; Zhao et al., 2019). Although we cannot use the UFC manual as an absolutely correct reference standard, it is still the most valuable reference basis for this work’s data comparison besides a large number of close-in blast tests data obtained through the new measurement technology (Barr et al., 2023; Rigby et al., 2019). In short, the UFC manual is only used as a comparison reference for the reliability of the close-in blast prediction method in the section on Close-in blast loads prediction. The final equivalent loading condition prediction process (in the section on Solving for equivalent loading conditions) will not use the UFC manual.

Fluid-solid coupling simulation

Because the research focus of this Section is not on the dynamic behavior of materials, the materials selected are all the most widely used models, in order to avoid the test work on material constants, and also make it easy for readers to reproduce the numerical simulation work of this paper. The free-field one-dimensional wedge models of 1 kg and 8 kg spherical charges are established, as shown in Figure 2(a). The farthest measurement distance is 4m from the detonation position, and fifteen gauge points are arranged. The scaled distance of each gauge position is arranged in sequence from 0.1 m/kg^1/3 to 2.0 m/kg^1/3. A reinforced concrete plate with an area of 1 m² is used as the target plate. The steel bars have a diameter of 6 mm and a spacing of 80 mm. The reinforcement ratio is not an important parameter, and the pressure time history is not affected noticeably by increasing the reinforcement ratio (Othman and Marzouk, 2016; Said and Mabrook Mouwainea, 2022). The gauge points are arranged at the interface between the target plate and the air region at an incident angle interval of 10°, as shown in Figure 2(b). Using the 3D remapping method (Li et al., 2014; Taha et al., 2018), the output file of the 1D model before the spherical blast wave reaches the target plate’s surface is imported as the initial energy into the 3D model. The complete fluid-solid coupling model includes the air Euler domain, explosive solid filling, and RC (Reinforced Concrete) plate as shown in Figure 2(c) and Figure 2(d).

Figure 2.

The quarter model of the fluid-solid coupling simulation for close-in blast loading: (a) The free-field one-dimensional wedge model of spherical charges; (b) steel reinforcement and gauge points layout; (c) RC plate’s Lagrange model; (d) Euler-Lagrange coupled model by 3D remapping method.

The explosive charge is TNT, and the state equation of its detonation product adopts the JWL (Jones-Wilkins-Lee) state equation (Menikoff, 2015), which is developed based on the C-J detonation theory. It is a form that describes the mechanical power exportation ability of explosive detonation products without obvious chemical reactions expression. The parameter settings are shown in Table 2.

Table 2.

JWL state equation parameters of TNT charge (Menikoff, 2015).

JWL parameters	Values
Reference density, ρ (g/cm³)	1.63
C-J detonation velocity, v_CJ (m/s)	6.9 × 10³
C-J pressure, P_CJ (kPa)	2.1 × 10⁷
C-J energy/unit volume, E (kJ/m³)	6.0 × 10⁶
Parameter A (MPa)	3.74 × 10⁵
Parameter B (MPa)	3.75 × 10⁵
Parameter R₁	4.15
Parameter R₂	0.9
Parameter W	0.35
Maximum expansion	0.1
Minimum density factor	1.0 × 10⁶
Maximum temperature (K)	1.01 × 10²⁰

The RHT model (Abdel-Kader, 2019) is used for concrete, a dynamic constitutive model proposed by Riedel, Hiermaier, and Thoma, including the description of strength and damage, and is generally used together with the p-a state equation. The strength model of RHT has divided into three stages: the elastic stage, linear strengthening stage, and damage softening stage. The Johnson-Cook model (Wan et al., 2021) is used for steel bars, which can simulate the strength performance of materials with large strains and high strain rates. The RHT and J-C parameter settings (Wang et al., 2013) are shown in Tables 3 and 4.

Table 3.

RHT model parameters (Abdel-Kader, 2019).

RHT parameters	Values
Reference density, ρ (g/cm³)	2.55
Porous density (g/cm³)	2.314
Young’s modulus (GPa)	28.3
Intact failure surface exponent N	0.76
Intact failure surface constant A	1.6
Compressive strength, f_c (MPa)	40
Tensile strength, f_t (MPa)	4
Elastic strength/fc	0.53
Elastic strength/ft	0.7
Shear strength (MPa)	7.2
Shear modulus (MPa)	1.67 × 10⁴
Failure strain, ε′	0.001
Fractured strength constant B	0.7
Fractured strength exponent M	0.8
Brittle to ductile transition	0.0105
Tens./Comp. Meridian ratio	0.69
Compressive strain rate exp. Alpha	0.02
Tensile strain rate exp. Delta	0.028
Damage constant, D₁	0.04
Damage constant, D₂	1
Minimum strain to failure	0.001
Principal tensile failure stress (MPa)	4

Table 4.

J-C model parameters (Wang et al., 2013).

J-C parameters	Values
Reference density, ρ (g/cm³)	7.83
Gruneisen coefficient	2.17
Young’s modulus (GPa)	200
Parameter C1 (m/s)	4569
Parameter S1	1.49
Reference temperature (K)	300
Yield stress, A (MPa)	600
Hardening exponent, n	0.26
Strain rate constant, C	0.014
Reference strain rate, έ (s⁻¹)	1.0
Failure strain, ε′	0.1
Shear modulus (MPa)	81800
Hardening constant (MPa)	384
Thermal softening exponent	1.0
Ref. Strain rate (/s)	1.0
Maximum expansion	0.1
Minimum density factor	0.0001

The close-in shock wave velocity is high, and the time duration of the positive pressure in the whole occurrence process is only a few milliseconds or even a few tenths of a millisecond. The calculation accuracy depends on the grid quality sensitively. The mesh size is set to 0.5 mm, considering calculation accuracy and efficiency (Draganić and Varevac, 2018). Figure 3 shows the transmission process of the stress wave of the target plate, and the vertical scaled distance is equal to 0.5 m/kg^1/3.

Figure 3.

The transmission process of the stress wave of the target plate.

Figure 4 shows the comparison between the experimental test results and numerical simulation results. Figure 4(a) is the damage effect of the RC plate under the close-in blast testing carried out by Wang (Wang et al., 2012, 2013). There are mainly circumferential cracks on the front blast surface, and radial cracks on the back blast surface. Figure 4(b) and (c) show the numerical simulation results of this work, and the damage mode is in good agreement with the above test results, which further verifies the reliability of the numerical model in this work.

Figure 4.

Comparison of numerical results with reference test results: (a) damage mode of reference tests (Wang et al., 2012); (b) simulation crack distribution of this work; (c) simulation damage state of this work.

Incident load fitting

In the section on Fluid-solid coupling simulation, the fluid-solid coupling model is established, and in this Section, the model is used to compute the incident loads at different scaled distances, as represented by the blue line in Figure 5. For a spherical charge, the control parameters that determine the characteristics of the shock wave strength come from the following three aspects, including explosive parameters (charge weight Q, charge density $ρ_{e}$ , chemical energy released per unit mass of explosive E_e, expansion exponent of explosion products $γ_{e}$ ), air state (initial pressure P_a, initial density $ρ_{a}$ , adiabatic exponent $γ_{a}$ ), and detonation distance r. The (p-p_a)_m is the peak overpressure, and the t⁺ is the positive pressure time duration. Therefore, the peak overpressure and positive pressure duration should be a function of the above parameters, and the dimensionless expression is:

{\begin{cases} {(p - p_{a})}_{m} / (ρ_{e} E_{e}) = f ({r / (Q / ρ_{e})}^{1 / 3}) \\ t^{+} / [{(Q / ρ_{e})}^{1 / 3} / {E_{e}}^{1 / 2}] = g ({r / (Q / ρ_{e})}^{1 / 3}) \end{cases}

(1)

Figure 5.

Numerical simulation data and UFC manual’s test data of close-in incident peak pressure: (a) the scaled distance from 0 to 1 m/kg^1/3; (b) the scaled distance from 1 m/kg^1/3 to 2 m/kg^1/3.

Equation (1) is the explosion scaling law of spherical charges (Langhaar, 1962) in the air medium, which expresses the relation among free-field peak pressure, loading time duration, and scaled distance. In this section, based on the relation, combined with the numerical simulation results, the following load prediction formulas suitable for the free-field close-in blast are obtained through nonlinear fitting:

P_{m} = 0.0817 - \frac{0.5637}{z} + \frac{1.721}{z^{2}} - \frac{0.273}{z^{3}} + \frac{0.014}{z^{4}} M P a (0.1 \leq z \leq 2)

(2)

\frac{I_{m}}{\sqrt[3]{w}} = \frac{0.171}{z} - \frac{0.084}{z^{2}} + \frac{0.017}{z^{3}} - \frac{8.756 e^{- 4}}{z^{4}} M P a \cdot m s \cdot k g^{- 1 / 3} (0.1 \leq z \leq 2)

(3)

z = \frac{r}{\sqrt[3]{w}} m \cdot k g^{- 1 / 3}

(4)

where r is detonation distance, w is TNT charge weight, P_m is incident peak pressure, I_m is incident specific impulse, and z is scaled distance.

It can be seen from equations (2) and (3), the peak pressure decreases with increasing scaled distance. The specific impulse is jointly determined by the binary function of the scaled distance and the charge weight.

Figure 5 shows the numerical simulation data and the UFC manual’s test data (Department of Defense, 2008) of close-in incident peak pressure. The comparison shows that the numerical simulation data and its fitting results agree with the test data given by the UFC manual.

Reflected load calculation

As mentioned in the section on Introduction, the close-in blast phenomenon is more complicated, resulting in significant differences in the prediction of the close-in blast loads in the existing literature, manuals, and codes. It is necessary to describe the reflection law in a strong shock environment in more detail and propose a more reliable close-in blast loads prediction method to provide the required load parameters for establishing the blast-impact equivalent loading relation. In this section, based on the incident loads obtained in the section on Incident load Fitting, the pressure impulse reflection coefficient at any incident angle is calculated using the close-in strong shock wave theory, (Zhou et al., 2004, 2018). The solving process is as follows, from equations (5) to (10):

Define the pressure reflection coefficient, which is the ratio of the reflected pressure to the incident pressure:

N_{P} = \frac{P_{r}}{P_{m}}

(5)

where N_p is the reflection coefficient, P_m and P_r are incident pressure and reflected pressure, respectively.

When the shock wave is incident at a zero-degree angle, a normal reflection pressure will be formed on the target plate surface and the peak pressure is as follow,

P_{r} = 2 P_{m} + \frac{\frac{γ + 1}{γ - 1} {P_{m}}^{2}}{P_{m} + \frac{2 γ}{γ - 1} P_{0}}

(6)

where γ is the air adiabatic exponent related to the incident pressure intensity. P₀ is the environmental pressure.

When the shock wave is incident at a non-zero angle, oblique reflections will occur on the target plate surface. The oblique reflection is regular if the incident angle is smaller than the Mach limit angle. The oblique reflection is irregular if the incident angle exceeds the Mach limit angle.

The regular reflection pressure in the air medium is as follow:

P_{r} = [\frac{γ (B - \sqrt{B^{2} - 4 A C})}{(γ + 1) A}] (P_{m} + P_{0}) - P_{0}

(7)

The parameters A, B, C, and their built-in parameters σ, ξ in the equation are as follows:

{\begin{cases} A = {(γ + 1)}^{2} σ^{4} \\ B = 2 {(γ + 1)}^{2} σ^{4} + 4 γ σ^{2} {(σ^{2} - 1)}^{2} \cos^{2} α + σ^{2} {[(γ - 1) σ^{2} + 2]}^{2} \cot^{2} α \\ C = {(γ + 1)}^{2} σ^{4} - 4 {(σ^{2} - 1)}^{2} \cos^{2} α + {[(γ - 1) σ^{2} + 2]}^{2} \cot^{2} α \\ σ^{2} = [(γ + 1) ξ + (γ - 1)] / 2 γ \\ ξ = p_{0} / (P_{m} + P_{0}) \end{cases}

(8)

where the incident angle α should not be greater than the Mach limit angle α_c, which can be determined according to the following equations:

{\begin{cases} D \sin^{6} α_{c} - E \sin^{4} α_{c} - F \sin^{2} α_{c} - G = 0 \\ D = 16 γ^{2} {(σ^{2} - 1)}^{4} \\ E = 16 {(γ + 1)}^{2} (γ σ^{2} + 1) {(σ^{2} - 1)}^{2} + 16 γ^{2} {(σ^{2} - 1)}^{4} - 8 γ {(σ^{2} - 1)}^{2} {[(γ - 1) σ^{2} + 2]}^{2} \\ F = 4 {(γ + 1)}^{2} (σ^{2} - 1) {[(γ - 1) σ^{2} + 2]}^{2} - {[(γ - 1) σ^{2} + 2]}^{4} + 8 γ {(σ^{2} - 1)}^{2} {[(γ - 1) σ^{2} + 2]}^{2} \\ G = {[(γ - 1) σ^{2} + 2]}^{4} \end{cases}

(9)

The Mach reflection pressure conforms to the exponential function, which is the power function of the incident angle α.

P_{r} = ψ e^{- n α}

(10)

where the coefficients ψ and n can be determined by the boundary conditions of the above equations.

This section will propose the reflected load prediction formula for close-in blast shock based on the above. The reflected load applied on the target plate surface at any incident angle can be expressed by the incident load and the reflection coefficient:

{\begin{cases} P_{r} = N_{p} P_{m} \\ I_{r} = N_{i} I_{m} \end{cases}

(11)

where P_r is the reflected overpressure at any incident angle. I_r is the reflected specific impulse at any incident angle. N_p is the pressure reflection coefficient, and N_i is the impulse reflection coefficient.

Based on the triangular load assumption, the increase in impulse and peak pressure should be synchronous, that is, the values of N_i and N_p are close. However, in order to obtain a more accurate impulse value, there is a N_id-fold relation for the effect of the dynamic pressure between N_i and N_p can be assumed. Then according to the reflection relationship in (Baum et al., 1961), when the incident angle of the shock wave is smaller than the Mach limit angle, the impulse reflection coefficient of the regular reflection is:

N_{i} = 2 + (N_{p} - 2) N_{i d}

(12)

where N_p

- 2

is the effect of the dynamic pressure, and the effect of the static pressure is always equal to 2-fold. Similarly, when the incident angle of the shock wave is greater than the Mach limit angle, the impulse reflection coefficient of the Mach reflection is:

N_{i} = 1 + \frac{\cos α}{\cos α_{c}} + (N_{p} - 1 - \frac{\cos α}{\cos α_{c}}) N_{i d}

(13)

where α is the incident angle. α_c is the Mach limit angle. N_id is the enhancement coefficient which is related to adiabatic exponent of real air, scaled distance, and incident angle, so the following fitting formula in variable separation form can be assumed and obtained:

{\begin{cases} N_{i d} = f (k) \cdot g (z) \cdot h (α) \\ N_{i d} = (2 - 0.65 k) (e^{- \frac{{(z - 0.35)}^{2}}{0.04}} + 1) \sqrt{\cos α} \end{cases}

(14)

where k is the nonlinear fitting relation of real air adiabatic exponent γ under high pressure environment (Brode, 1959; Chock and Kapania, 2001), as follow:

k = 1.2 + 0.2 \times {(0.94)}^{P_{m}}

(15)

Since the solution of the reflection coefficient is complicated, nonlinear fitting is used here to separate the two variables of incident angle α and scaled distance z to form a quartic polynomial, which is convenient for engineering calculation and parameter control. The fitted data is calculated by equations (5) to (10).

N_{p} = A_{1} α^{4} + A_{2} α^{3} + A_{3} α^{2} + A_{4} α + A_{5}

(16)

where the coefficients A_i are functions of the scaled distance z. The fitting results are as follows:

{\begin{cases} A_{1} = (- 0.4775 z^{4} + 3.297 z^{3} - 8.426 z^{2} + 8.942 z - 0.09669) \times 10^{- 7} \\ A_{2} = (0.8374 z^{4} - 6.25 z^{3} + 16.41 z^{2} - 18.28 z + 2.004) \times 10^{- 5} \\ A_{3} = (- 0.6753 z^{4} + 4.436 z^{3} - 10.69 z^{2} + 11.88 z - 2.835) \times 10^{- 3} \\ A_{4} = (1.116 z^{4} - 5.558 z^{3} + 10.03 z^{2} - 8.024 z - 1.407) \times 10^{- 2} \\ A_{5} = 1.324 z^{4} - 6.366 z^{3} + 12.4 z^{2} - 14.14 z + 12.18 \end{cases}

(17)

Figures 6 and 7 show the predicted and simulated values compared with test values given in the UFC 3-340-02 manual. By comparison, the predicted value of reflected pressure is low at the scaled distance close to 0.1 m/kg^1/3 as shown in Figure 6(a). This is because the prediction process does not consider the direct effect of the detonation products, but the numerical simulation establishes a real solid model of the TNT charge, and the energy generated by the entire state change of the explosive is wholly released, resulting in the high peak pressure. However, the UFC manual’s test data are obtained from experimental measurements. Although detonation products have a direct effect, it is not as ideal as the simulation process. The testing process has phenomena such as incomplete chemical reactions of explosive and charge fragments’ energy consumption, which will significantly influence the load intensity and reduce the close-in peak pressure. The predicted value of reflected specific impulse is slightly higher in the range of scaled distances greater than 1 m/kg^1/3 as shown in Figure 7(b) and is in good agreement with the UFC manual in close-in range as shown in Figure 7(a).

Figure 6.

Numerical simulation data and UFC manual’s test data of close-in reflection peak pressure: (a) the scaled distance from 0 to 1 m/kg^1/3; (b) the scaled distance from 1 m/kg^1/3 to 2 m/kg^1/3.

Figure 7.

Numerical simulation data and UFC manual’s test data of close-in reflection specific impulse: (a) the scaled distance from 0 to 1 m/kg^1/3; (b) the scaled distance from 1 m/kg^1/3 to 2 m/kg^1/3.

Load prediction program

According to the calculation method in the section on Reflected load calculation, the reflection coefficients of pressure and impulse are determined by the close-in blast parameters: charge weight, detonation distance, and incident angle. Figure 8 shows the close-in reflected load distribution diagram and pressure time history curves of a finite target plane (1 m²) output by the prediction program under the TNT charge weight of 10 kg and vertical detonation distance of 0.6 m.

Figure 8.

The output diagram of prediction program for close-in blast loads.

The results show that the close-in blast loads are distributed in a curved surface on a finite plane. The reflected peak pressure in the loading center reaches 90 MPa. The incident angle at the plane’s edge is the largest, the reflected pressure drops to 40 MPa, the specific impulse drops from 6.5 MPa·ms to 3 MPa·ms, and the positive pressure time duration extends from 0.54 ms to 0.67 ms.

To compare the predicted values of peak pressure and specific impulse at various representative positions (with different incident angles), we select the test data provided by the UFC manual. Figure 9 shows that the maximum and minimum errors observed are 11.4% and 0.7%, respectively.

Figure 9.

Comparison of prediction program results with the test data in UFC manual.

Figure 10 shows the close-in blast test data given by Hokanson (Hokanson et al., 1978), Baker (Dezhi et al., 2009), and Huffington (Huffington and Ewing, 1985), all of which are measured within scaled distances less than 1 m/kg^1/3. The maximum and minimum errors between the predicted values by the program in this section and the test data in the above references are 20.9% and 3.48%.

Figure 10.

Comparison of prediction program results with the test data in references.

Mass-block impact loads regulation

The previous impact loading regulation method relies too heavily on one-dimensional assumptions and precise material model parameters (Radford et al., 2005; Yang et al., 2023), which is unsuitable for practical application. This Section regulates impulse transmission by the macroscopic momentum variation. Unlike blast shock wave loading, the mass-block impact load is generated by the interaction between the mass block and the target plate. Whether the impact loading can be equivalent to the blast loading is closely related to the impact parameters: mass and velocity. The section on Impact velocity regulates pressure peak will discuss the influence of impact velocity on the pressure peak firstly, and on this basis, the section on Impact mass regulates impulse transmission will continue to examine the effect of impact mass on impulse transmission.

Impact velocity regulates pressure peak

Build an impact system consisting of the steel impact module and reinforced concrete target plate. Figure 11 shows the quarter model fixing both ends. The side length L = 1m, the target plate thickness h₂ varies from 0.05 m to 0.3 m, and the impact module thickness h₁ is determined by the impact module mass. The material parameters are the same as in the section on Close-in blast loads prediction.

Figure 11.

The quarter numerical impact model consisting of the impact module and RC target plate.

The pressure time history curves under different impact velocities are shown in Figure 12. The peak pressure increases as the velocity increases. When the impact velocity reaches 60 m/s, the peak pressure reaches more than 200 MPa, which has reached the peak pressure under the close-in blast loading with a scaled distance of 0.2 m/kg^1/3. The pressure peak can be reduced if the Young’s modulus of the impact module is reduced or an elastic/plastic cushion (Li et al., 2019) is added between the impact module and the target plate.

Figure 12.

The effect of impact velocity on pressure time history curves: (a) velocity from 3 m/s to 30 m/s; (b) velocity from 40 m/s to 100 m/s.

It is worth noting that although the peak pressure increases with the velocity increases, the time duration hardly changes. This is because the time duration and the impulse transmission are mainly affected by the impact mass (The surface area is constant) and the target plate mass. The relation between them will be discussed in the section on Impact mass regulates impulse transmission.

Because the impact model of this section does not meet the one-dimensional assumption, it is unsuitable for calculation with a one-dimensional stress wave theory. Instead, it uses non-linear fitting to obtain the relation between velocity and peak pressure. The peak pressure can be expressed as a partition function of the impact velocity:

P_{r} = {\begin{cases} 0.2 {v_{0}}^{2} + 3 v_{0}, 1 \leq v_{0} \leq 10 \\ 4 v_{0} + 10, 10 \leq v_{0} \leq 100 \end{cases}

(18)

Then, the impact velocity corresponding to the peak pressure can be predicted as:

v_{0} = {\begin{cases} \frac{\sqrt{0.8 P_{r} + 9} - 3}{0.4}, 3.2 \leq P_{r} \leq 50 \\ \frac{P_{r} - 10}{4}, 50 \leq P_{r} \leq 410 \end{cases}

(19)

Figure 13(a) shows the relation between impact velocity and peak pressure under different impact masses. The three dashed lines are simulations of three different masses, and it can be found that the differences between them are very small, and they all agree well with equations (18) and (19), indicating that the pressure peak is mainly affected by the impact velocity and is not sensitive to changes in impact mass. This conclusion is similar to that under the one-dimensional stress wave hypothesis. The study of reference (Jin et al., 2023) also verified this conclusion. The section on Impact mass regulates impulse transmission will show that the impact mass mainly affects impulse transmission. The three dashed lines in Figure 13(a) are simulations of three different masses.

Figure 13.

Fitting relation between impact velocity and load parameters: (a) peak pressure; (b) coefficient of restitution.

The coefficient of restitution c_r (Ahmad et al., 2016) is a crucial impact parameter ranging from 0 to 1, which is used to describe the relation between the approach velocity and separation velocity of the impact module and the target plate in the non-perfect elastic collision model. When the impact mass and velocity are uniformly distributed on the target plate surface, the coefficient of restitution is only determined by the material properties and impact velocity:

c_{r} (v_{1 i} (x)) = \frac{v_{2 f} (x) - v_{1 f} (x)}{v_{1 i} (x)} = \frac{v_{2 f} - v_{1 f}}{v_{0}}

(20)

where v_2f and v_1f is the velocities of the target plate and the impact module after they are separated, and the difference value between them is the separation velocity. c_r can be determined by the single value of the approach velocity v_1i when the material is known. It should be noted that the approach velocity v_1i is equal to the difference value between the impact velocity v₀ and the target plate’s initial motion velocity. In this problem, the initial state of the target plate is stationary, so the approach velocity v_1i is equal to the impact velocity v₀.

According to the numerical simulation results in this section, the following non-linear fitting formula for the coefficient of restitution applicable to the steel impact module and the reinforced concrete target plate is obtained:

c_{r} = 0.3213 e^{- {(\frac{v_{0} - 0.5189}{25.92})}^{2}} + 0.05033 e^{- {(\frac{v_{0} - 44.49}{22.12})}^{2}}

(21)

Figure 13(b) shows the coefficient of restitution’s numerical data and its fitting curve under different impact velocities. It can be seen that as the v₀ increases, the c_r gradually decreases. When the v₀ increases from 1 m/s to 60 m/s, the c_r drops from 0.32 to 0.025.

It should be pointed out that since the coefficient of restitution is jointly determined by material properties and impact velocity, the above conclusions and their fitting formula are only applicable to the impact behavior between steel modules and reinforced concrete selected in this section. If other materials and boundary conditions are chosen, the analysis process given by this section needs to be re-executed. In addition, the span and boundary conditions of the target plate are no longer important influencing parameters when the impact module’s surface area is much smaller than the target plate’s surface area. Research shows (Li et al., 2021; Pham and Hao, 2018) that the response during the impact process could be limited to a localized area around the impact point. Even though the structures have different boundary conditions, the pressure peaks are similar. It indicates that the pressure peaks are independent of the boundary conditions and the global stiffness. Table 5 presents the impact load parameters determined by the materials of steel and concrete at different impact velocities.

Table 5.

Impact load parameters and the corresponding blast scaled distances determined by the materials of steel and concrete.

Impact velocity v₀ (m/s)	Coefficient of restitution, c_r	Peak pressure P_r (MPa)
3	0.32	10
5	0.31	20
10	0.28	50
15	0.24	70
20	0.2	90
30	0.12	130
40	0.08	170
50	0.05	210
60	0.03	250
80	0.004	330

Impact mass regulates impulse transmission

Rodriguez (Rodriguez-Nikl, 2006) have done valuable study on impact load regulation. Based on the momentum conservation theorem, the impact process is regarded as a momentum transfer process. The mapping relation among impact velocity, total mass, and total impulse is established, and the conclusion obtained is consistent with the classic incomplete elastic collision model.

On this basis, this section takes the velocity condition determined in the section on Impact velocity regulates pressure peak as a constant, replaces total mass of impact system with the areal density, and takes it as the independent variable of the incomplete elastic collision model. Then, the specific impulse is taken as the dependent variable, and the mapping relation between areal density and specific impulse is established. The purpose of converting total mass into areal density is to avoid the influence of geometric area on module mass and target plate mass, and reduce two dependent variables so that the equation group can be solved.

As shown in Figure 14(a), the spatial coordinates x, b, and h are the dimension directions of the rectangular impact module and target plate. Set three impact variables, namely impact mass, impact velocity, and target plate mass, changing along the positive direction of the x-axis. By the momentum conservation relation can be obtained:

\int_{L} ρ_{1} b h_{1} (x) v_{1 i} (x) d x = \int_{L} ρ_{1} b h_{1} (x) v_{1 f} (x) d x + \int_{L} ρ_{2} b h_{2} (x) v_{2 f} (x) d x

(22)

where ρ₁h₁ and ρ₂h₂ represent the areal density of the impact module and the target plate, respectively, change along the x direction.

Figure 14.

Incomplete elastic collision model with impact module and target plate: (a) general form with variable impact mass, variable impact velocity, and variable target mass; (b) co-velocity form with variable impact mass, constant impact velocity, and variable target mass.

Use the shape function to express the distribution of the approach velocity and the target plate’s post-separation velocity along the x direction:

v_{1 i} (x) = v_{0} ψ_{1} (x)

(23)

v_{2 f} (x) = {\dot{z}}_{0} ψ (x)

(24)

where v₀ is the maximum approach velocity, and Ψ₁ is the velocity shape function. z₀ is the displacement at the position with maximum displacement on the target plate, and the position is with the maximum velocity at the moment when the impact module is separated from the target plate.

{\dot{z}}_{0}

is the velocity at the position. Ψ is the velocity shape function of the target plate, which can be expressed as a static load displacement shape function according to the boundary constraints. If the boundary condition is clamped support on opposite sides, the static load shape function is:

ψ (x) = \frac{16 x^{2} {(L - x)}^{2}}{L^{4}}

(25)

Obviously, the impulse transmitted to the target plate during the impact process is:

I = \int_{L} ρ_{2} b h_{2} (x) v_{2 f} (x) d x

(26)

By combining the above equations, the general form of the impulse transmission can be obtained:

(v_{0} / I) \int_{L} ρ_{1} b h_{1} (1 + c_{r}) ψ_{1} d x = \frac{\int_{L} (ρ_{1} b h_{1} + ρ_{2} b h_{2}) ψ d x}{\int_{L} ρ_{2} b h_{2} ψ d x}

(27)

Then the specific impulse expression required in this section can be obtained:

I_{r} = \frac{v_{0} ρ_{1} \int_{L} h_{1} (1 + c_{r}) ψ_{1} d x \cdot ρ_{2} \int_{L} h_{2} ψ d x}{\int_{L} (h_{1} ρ_{1} + h_{2} ρ_{2}) ψ d x \cdot L}

(28)

As shown in Figure 14(b), if the impact module has the same velocity along the x-direction (v_1i is a constant), the equation (28) can be simplified to the following form:

I_{r} = \frac{v_{0} (1 + c_{r}) ρ_{1} \int_{L} h_{1} d x \cdot ρ_{2} \int_{L} h_{2} ψ d x}{\int_{L} (h_{1} ρ_{1} + h_{2} ρ_{2}) ψ d x \cdot L}

(29)

It should be noted that in this co-velocity form, the areal density of the impact module still changes along the x direction. If the areal density of the impact module and the impact target are also uniformly distributed (ρ₁h₁ and ρ₂h₂ are constants), the impulse transmission can be further simplified as:

I_{r} = \frac{v_{0} (1 + c_{r}) h_{1} ρ_{1} \cdot h_{2} ρ_{2}}{h_{1} ρ_{1} + h_{2} ρ_{2}}

(30)

Figure 15 shows the relation between the impact mass and impulse transmission. It is observed that, at a certain impact velocity, the impulse transmission has an upper limit, which means although the greater the impact mass, the more impulse is transmitted, when the impact mass reaches a certain level, the increase in impulse will become slower, resulting in the little effect of increasing mass on impulse transmission. Therefore, it can be inferred that since v₀ is mainly determined by P_r, the larger the charge weight, the more difficult it is to simulate the blast loading pressure time history by mass-block impact. It can be seen from the expression that increasing the target plate’s areal density can increase the upper limit value of impulse transmission.

Figure 15.

Impulse transmission curve of uniform mass co-velocity impact represented by areal density.

Take 60% of the upper limit value as the impact mass condition to ensure a high impulse transmission efficiency. Then the areal density relation between the impact mass and the target plate’s mass can be obtained:

\frac{v_{0} (1 + c_{r}) h_{1} ρ_{1} \cdot h_{2} ρ_{2}}{h_{1} ρ_{1} + h_{2} ρ_{2}} = 0.6 \cdot v_{0} (1 + c_{r}) h_{2} ρ_{2}

(31)

\frac{h_{1} ρ_{1}}{h_{2} ρ_{2}} = 1.5

(32)

This means that when the impact mass exceeds 1.5 times the target plate mass, the efficiency of impulse transmission becomes extremely low. In the section on Solving for equivalent loading conditions, Figures 17 and 18 will indicate this low-efficiency range.

Solving for equivalent loading conditions

Based on the load prediction process of close-in blast in the section on Load prediction program, combined with the load regulation process of mass-block impact in the sections on Impact velocity regulates pressure peak and Impact mass regulates impulse transmission, the equivalent loading conditions between close-in blast and mass-block impact can be obtained. Taking charge weight, detonation distance, impact module density, target plate density, and target plate thickness as input variables. The impact velocity and mass as output variables. The solving process is shown in Figure 16, and the symbol meanings are shown in Table 6. A necessary conditional judgment can be found in the flowchart, which restricts the simulated impulse cannot exceed the upper limit because the target plate mass is insufficient to absorb the required impulse at the current velocity. The applicable range of blast scaled distance z (m/kg^1/3) is from 0.1 to 0.8. However, the applicable range of TNT charge weight is determined by the impulse transmission limit, which is determined by the target plate’s areal density (h₂ρ₂) and restitution coefficient (c_r).

Figure 16.

Solving process of equivalent loading conditions.

Table 6.

Symbols and meanings of the solving process.

Parameters	Meaning
w	TNT charge weight
P _r	Peak pressure
v ₀	Impact velocity
ρ ₁	Density of impact module
h ₁	Thickness of impact module
r	Detonation distance
I _r	Specific impulse
c _r	Coefficient of restitution
ρ ₂	Density of target plate
h ₂	Thickness of target plate

Figure 17, 18, 19, 20, 21 and 22 show the solving results for different target plate masses. In each figure, (a) shows “the equivalent relation between blast charge weight and impact parameter”, and (b) shows “the equivalent relation between blast impulse and impact parameter”, which together determine the equivalent loading conditions.

Figure 17.

Equivalent loading condition curves with target plate thickness h₂ = 0.05 m: (a) determine the blast charge weight; (b) determine the blast specific impulse.

Figure 18.

Equivalent loading condition curves with target plate thickness h₂ = 0.1 m: (a) determine the blast charge weight; (b) determine the blast specific impulse.

Figure 19.

Equivalent loading condition curves with target plate thickness h₂ = 0.15 m: (a) determine the blast charge weight; (b) determine the blast specific impulse.

Figure 20.

Equivalent loading condition curves with target plate thickness h₂ = 0.2 m: (a) determine the blast charge weight; (b) determine the blast specific impulse.

Figure 21.

Equivalent loading condition curves with target plate thickness h₂ = 0.25 m: (a) determine the blast charge weight; (b) determine the blast specific impulse.

Figure 22.

Equivalent loading condition curves with target plate thickness h₂ = 0.3 m: (a) determine the blast charge weight; (b) determine the blast specific impulse.

It can be seen from (a) and (b) of each figure, as the impact mass (or module thickness) increases, the charge weight and specific impulse increase more and more slowly, forming a “low effective” region. This is because the impact mass is getting closer to the upper limit (Refer to Figure 15 in the section on Impact mass regulates impulse transmission). A heavier target plate mass will result in a higher upper limit. It is worth noting that the increase in impact velocity does not necessarily lead to a continuous increase in the TNT charge weight, and it has an inflection point near v₀ = 30 m/s, which is caused by the combined effect of the blast load’s dimensional relation in the section on Close-in blast loads prediction and the impulse transmission relation in the section on Mass-block impact loads regulation. The essential reason is that the increase in impact velocity has a greater effect on the pressure compared with the impulse, so the increase in impulse cannot keep up with the increase in pressure. In order to match the insufficiently powerful impulse, reducing the TNT charge weight is the only way. The occurrence position of the inflection position is determined by the material densities (ρ₁, ρ₂) of the impact module and the target plate, as well as the restitution coefficient (c_r), which can be solved using equations (11), (16), (19), (21), and (30).

The accuracy of the equivalent loading conditions is verified by numerical simulation. Firstly, ten groups of close-in blast conditions are designed. Then, the mass-block impact conditions are calculated, as shown in Table 7. Finally, Numerical simulations are carried for two loading forms :close-in blast and mass-block impact, and the pressure time history curves are compared.

Table 7.

Equivalent loading conditions for close-in blast and mass-block impact.

Conditions	Close-in blast loading		Mass-block impact loading
Conditions	w (kg)	r (m)	v₀ (m/s)	h₁ (m)
E01	8	0.8	9.6	0.07
E02	8	1.0	6.2	0.068
E03	8	0.4	43	0.054
E04	8	0.6	19	0.054
E05	1	0.6	4.68	0.033
E06	8	1.2	4.68	0.078
E07	1	0.4	9.6	0.056
E08	1	0.5	6.2	0.052
E09	1	0.2	43	0.033
E10	1	0.3	19	0.034

Figure 23(a) shows the pressure time history curves of loading condition E01, where the charge weight is 8 kg, the detonation distance is 0.8 m, and the target plate thickness is 0.25 m. The blast curve’s rising front is steep, peak pressure reaches 40 MPa, and the time duration is about 0.45 ms. The mass-block impact velocity calculated is 9.6 m/s, and the impact module thickness is 0.07 m. It can be seen that the steepness of the rising front is similar to the blast curve, and the pressure peak also reaches 40 MPa. The time duration is highly consistent. For other loading conditions, the comparison results also exhibit a good consistency as shown in Figure 23(b)∼(j). This work regulates impulse transmission by the macroscopic momentum variation, which is good at the regulation of peak pressure, specific impulse, and time duration. However, more detailed features, such as the descending stage of the pressure time curves, shows room for further fine-tuning.

Figure 23.

Pressure time history curves of close-in blast and mass-block impact under various loading conditions: (a) loading condition E01; (b) loading condition E02; (c) loading condition E03; (d) loading condition E04; (e) loading condition E05; (f) loading condition E06; (g) loading condition E07; (h) loading condition E08; (i) loading condition E09; (j) loading condition E10.

It is worth noting that there are often two peaks in the close-in blast loading curves. The first peak is the effect of the shock wave, and the second peak is the direct action of the explosion products. However, due to the influence of boundary constraints in the mass-block impact process, an instantaneous secondary impact and a secondary pressure peak will also occur. This phenomenon is more obvious when the impact velocity and mass are low. As the velocity and mass increase, the secondary peak will gradually approach or even overlap with the primary peak. However, according to the triangle-shaped loads assumption, under the consistency of the primary peak and the impulse transmission, the secondary peak will not interfere the equivalent relation in the time domain (the pressure time history) and space domain (the spatial distribution of loading) but will affect in the frequency domain (the distribution of energy on the frequency spectrum). The consistency in the frequency domain will be studied in the follow-up work.

Conclusions

This work abandons the impulse criterion that the applicable range of the loading time duration strictly approaches zero, but adopts the consistency of the pressure time history, which considers the combined effects of impulse and peak pressure rather than the single contribution of impulse. The pressure time history consistency provides a more reasonable loading equivalent criterion under the long time duration of loads or large natural frequency of targets.

The previous impact loading regulation method relies too heavily on one-dimensional assumptions and precise material model parameters, which is unsuitable for practical application. It is a more straightforward method to analyze the regulation principle of impact loads from the macroscopic energy and momentum perspective. This work regulates impulse transmission by the macroscopic momentum variation, which is good at regulating peak pressure, specific impulse, and time duration. Different from previous expressions of impact momentum, this work replaces the total mass of the impact system with the areal density and takes it as the independent variable of the incomplete elastic collision model, which effectively avoids the influence of geometric area on impact module mass and target plate mass, and reduce two dependent variables so that the equation group can be solved. However, more detailed features, such as the descending stage of the pressure time curves, shows room for further fine-tuning.

The results indicate that whether the impact loading can be equivalent to the blast loading is closely related to the impact parameters: mass and velocity. The peak pressure is mainly affected by the impact velocity. The time duration is more sensitive to the impact mass. The impulse transmission has an upper limit under a certain impact velocity, which can be inferred that the larger the charge weight, the more difficult it is to simulate the blast loading by mass-block impact. A heavier target plate mass can result in a higher upper limit. The increase in impact velocity does not necessarily lead to a continuous increase in the TNT charge weight, and it has an inflection point. This is caused by the combined effect of the blast load’s dimensional relation and the impulse transmission relation. The essential reason is that the increase in impact velocity has a greater effect on the pressure compared with the impulse, so the increase in impulse cannot keep up with the increase in pressure. In order to match the insufficiently powerful impulse, reducing the TNT charge weight is the only way.

This study provides a general method for quantitatively determining the equivalent loading conditions of two loading forms—close-in blast and mass-block impact—which holds practical value for the design of protective structures and damage assessment under actual blast conditions: (1) As a general equivalent method for simulating blast loading in a laboratory environment, it can guide the development of large-scale blast simulation systems, indirectly assisting in the rapid damage assessment of typical components under industrial explosions and terrorist attacks. (2) It facilitates the creation of reference manuals, computational programs, and operational platforms for experimental equipment related to equivalent loading conditions. This, in turn, supports the development of reasonable experimental design schemes for laboratory studies on the performance of protective structures against close-in blast, such as those using drop hammers, Hopkinson bars, and gas guns.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The present work is supported by National Science Foundation of China under Grant No. 12472128 and Open Research Fund of State Key Laboratory of Target Vulnerability Assessment, Defense Engineering Institute, AMS, YSX2024KFXY006. The authors also would like to thank Prof. Ruichao Liu and Can Cui from Defense Engineering Institute, AMS for their helpful suggestions and encouragement on the paper.

ORCID iD

Li Ma

Data availability statement

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.*

The applicable range of impulse criterion for triangle-shaped loads

In this Appendix, based on the classical solution of the equivalent single-degree-of-freedom system (Harris and Piersol, 2002; Hodges and Pierce, 2011) under impulse loads, the dynamic amplification factor curve is expanded by the Taylor series to discuss the accurate applicable range of the impulse criterion and the peak pressure criterion. Furthermore, the loading time duration obeying the impulse criterion for typical plates commonly used in engineering tests are provided.

Close-in blast load is nonuniform on target plates. Based on the single-degree-of-freedom assumption, it is necessary to convert the nonuniform load into a uniform load by the condition of equal work, and then convert the uniform load into a concentrated load, as shown in Figure 24. Figure 24.

Conversion between nonuniform load, uniform load and concentrated load on the target plate.

Assuming that the target plate’s elastic stage vibrates obeying a certain vibration mode (Nassr et al., 2012), and the displacement of all points on the plate surface can be expressed by the same displacement function: (A.1)

z (x, y, t) = s (x, y) \cdot z_{1} (t)

where z(x,y,t) is the vertical displacement of any position (x,y) on the plate surface at any time. The displacement determines the stress state and represents the loading strength. The spatial and time variables are separated by the variables separation method and written as the product of the vibration mode function and time function. z₁(t) is the displacement of the maximum deflection position. s(x,y) is the vibration mode function ranging from 0 to 1 as follows: (A.2)

s (x, y) = \frac{z_{s t 1} (x, y)}{z_{s t}}

where z_st is the static displacement at the maximum deflection position. z_st1(x,y) is the static displacement at any position.

The work of nonuniform and uniform loads are expressed as follows: (A.3)

W_{n o n u n i f o r m} = 4 \int_{0}^{L_{x} / 2} \int_{0}^{L_{y} / 2} p (x, y) s (x, y) z_{\max} d x d y

(A.4)

W_{u n i f o r m} = 4 \int_{0}^{L_{x} / 2} \int_{0}^{L_{y} / 2} p_{r} s (x, y) z_{\max} d x d y

where p(x,y) is the nonuniform load on the surface of the target plate, and its value at any position is equal to the peak value of the time history. p_r is a uniform load equivalent to p(x,y). z_max is the dynamic displacement at the maximum deflection position.

The following relation should be obeyed to determine the uniform load: (A.5)

W_{n o n u n i f o r m} = W_{u n i f o r m}

Blast loading has a short time duration, high pressure peak, and steep front, which can be regarded as a triangle-shaped load, and the function form is: (A.6)

p (t) = {\begin{cases} p_{r} f (t) = p_{r} (1 - \frac{t}{t_{m}}), & 0 \leq t \leq t_{m} \\ 0, & t \geq t_{m} \end{cases}

where p(t) represents the change of pressure time history, and t_m is the loading time duration. The pressure-time curve is triangle-shaped with a leaping wavefront when the time is within the duration t_m.

A motion equation consistent with the single-degree-of-freedom vibration equation (Harris and Piersol, 2002; Hodges and Pierce, 2011) can be obtained by combining the energy method: (A.7)

M \frac{d^{2} z}{d t^{2}} + K z = P f (t)

where P, M, and K are the equivalent force, equivalent mass, and equivalent stiffness of the two-dimensional plane plate’s equivalent single-degree-of-freedom system, respectively.

The structural response follows the forced vibration in the first stage and free vibration in the second stage. Define a dimensionless time ratio: (A.8)

μ = \frac{t_{m}}{T}

where T is the structural free vibration period. The dynamic amplification factor can be defined as follow: (A.9)

k_{d} = \frac{z_{\max}}{z_{s t}} = \frac{p_{e}}{p_{r}}

where k_d is the dynamic amplification factor, and p_e is the equivalent static load. The k_d at the first stage and second stage can be expressed as follows: (A.10)

{\begin{cases} k_{d 1} = 2 (1 - \frac{1}{2 π μ} \arctan (2 π μ)) \\ k_{d 2} = \sqrt{1 - \frac{\sin (2 π μ)}{π μ} + \frac{1 - \cos (2 π μ)}{2 π^{2} μ^{2}}} \end{cases}

As shown in Figure 1 in the section on Equivalence criterion of triangle-shaped loads, the Maclaurin series expansion of k_d2 at the origin can be obtained: (A.11)

\lim_{μ \to 0} k_{d 2} = π \cdot μ

Thus, it can be concluded that in the case of t_m << T, the equivalent static load is: (A.12)

p_{e} = k_{d 2} p_{r} = (π \cdot μ) p_{r} = π \cdot \frac{t_{m}}{T} \cdot p_{r} = \frac{2 π}{T} \cdot \frac{1}{2} t_{m} p_{r} = ω \cdot I_{r}

where I_r is the specific impulse, determined by the area under the triangle-shaped load curve.

Similarly, in the case of t_m >> T, the equivalent static load is: (A.13)

p_{e} = k_{d 1} p_{r} = 2 p_{r}

As an inference, the Taylor series can expand the dynamic amplification factor curve at any point after the axis shifting operation: (A.14)

k_{d} = α \cdot μ + β

(A.15)

p_{e} = α \cdot \frac{ω}{π} I_{r} + β \cdot p_{r}

As shown in Figure 1 in the section on Equivalence criterion of triangle-shaped loads, k_d2 and k_d1 intersect at μ = 0.371 to form a complete k_d curve together. The tangent gradually moves from the origin to the positive direction of the μ-axis, the slope α gradually decreases, and the intercept β gradually increases. The slope represents the contribution of the impulse I_r to k_d, and the intercept represents the contribution of pressure p_r to k_d. It shows that as μ increases, the influence of I_r becomes smaller, and the influence of p_r becomes larger.

Taking the engineering error of less than 5% as the standard, calculate the maximum loading time duration obeying the impulse criterion for the reinforced concrete plates with different constraints and spans, as shown in Table 1 in the section on Equivalence criterion of triangle-shaped loads. t_max is the maximum value of loading time duration t_m obeying the impulse criterion, L is the short span length of the plate, and the thick-span ratio is the ratio of the plate thickness to the short span length. For the simple support, the maximum t_max/L is 38.7 ms/m, and is only 18 ms/m for the clamped support. The smaller L is, the smaller t_max is. The larger natural frequency is, the smaller t_max is.

The description of the experimental setup for the referenced studies

In this Appendix, a detailed description of the experimental setup for the referenced studies is provided to enhance the reader’s understanding.

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