Damage of a large-scale reinforced concrete wall caused by an explosively formed projectile (EFP)

Li-ki Ho , Wen-bin Gu , Y-ong Zhng , Qi Yun , Xing-bo Xie , Sho-xin Zou ,Zhen Wng , Ming Lu

a College of Field Engineering, Army Engineering University of PLA, Nanjing, 210000, China

b Unit 31539 of PLA, Beijing,100000, China

c Engineering Research Center of Safety and Protection of Explosion and Impact of Ministry of Education, Southeast University, Nanjing, 210000, China

d Southwest University of Science and Technology, Mianyang, 621000, China

Keywords:Reinforced concrete Explosively formed projectile (EFP)Penetration Explosion shock wave Numerical simulation

ABSTRACT To quickly break through a reinforced concrete wall and meet the damage range requirements of rescuers entering the building,the combined damage characteristics of the reinforced concrete wall caused by EFP penetration and explosion shock wave were studied.Based on LS-DYNA finite element software and RHT model with modified parameters, a 3D large-scale numerical model was established for simulation analysis,and the rationality of the material model parameters and numerical simulation algorithm were verified.On this basis, the combined damage effect of EFP penetration and explosion shock wave on reinforced concrete wall was studied, the effect of steel bars on the penetration of EFP was highlighted,and the effect of impact positions on the damage of the reinforced concrete wall was also examined.The results reveal that the designed shaped charge can form a crater with a large diameter and high depth on the reinforced concrete wall.The average crater diameter is greater than 67 cm (5.58 times of charge diameter), and crater depth is greater than 22 cm (1.83 times of charge diameter).The failure of the reinforced concrete wall is mainly caused by EFP penetration.When only EFP penetration is considered,the average diameter and depth of the crater are 54.0 cm (4.50 times of charge diameter) and 23.7 cm(1.98 times of charge diameter), respectively.The effect of explosion shock wave on crater depth is not significant, resulting in a slight increase in crater depth.The average crater depth is 24.5 cm (2.04 times of charge diameter)when the explosion shock wave is considered.The effect of explosion shock wave on the crater diameter is obvious, which can aggravate the damage range of the crater, and the effect gradually decreases with the increase of standoff distance.Compared with the results for a plain concrete wall,the crater diameter and crater depth of the reinforced concrete wall are reduced by 5.94%and 9.96%,respectively.Compared to the case in which the steel bar is not hit,when the EFP hit one steel bar and the intersection of two steel bars, the crater diameter decreases by 1.36% and 5.45% respectively, the crater depth decreases by 4.92% and 14.02% respectively.The EFP will be split by steel bar during the penetration process, resulting in an irregular trajectory.

1.Introduction

Over the recent years, accidents such as fires, explosions, and building collapses have occurred occasionally,which have caused a significant number of fatalities and injuries[1].Rescuers often need to break through walls to establish quick rescue channels and enter the buildings.The tandem penetration warhead is composed of a front-stage shaped charge and a rear-stage follow-up warhead[2].The front-stage shaped charge opens up a favorable channel for the follow-up of the rear-stage warhead.The rear-stage warhead continues to penetrate along this channel by relying on its kinetic energy,and then the charge of the rear-stage warhead is detonated by a delay fuse or an intelligent fuse [3].Buildings are usually reinforced concrete structures with limited thickness, which is lower than that of important military protective facilities.Therefore, for reinforced concrete walls with limited thickness, the penetration depth and the crater diameter must be considered in the design of front-stage shaped charge to meet the destruction range requirements of rescuers entering the building,which is the key to the design of such warheads.Therefore, for developing an effective tandem penetration warhead, it is necessary to study the damage mechanism of reinforced concrete subjected to shaped charge loading.

Concrete materials are widely used in military facilities and civilian buildings, so several experimental studies have been conducted on the penetration of shaped charges into concrete targets in the past, and a basic understanding of the penetration mechanism has been obtained.Among them,some studies focused on the influence of shaped charge structure on the penetration efficiency,mainly including the influence factors of liner material [4,5], liner structure[6,7],charge diameter[8],and length-to-diameter ratio of the charge[9].At the same time,the influence of concrete materials and structures on the penetration efficiency has also been investigated,and experiments have been carried out on the penetration of shaped charge into concrete targets with different strength grades such as ultra-high performance concrete (UHPC) [10-12]and reactive powder concrete (RPC) [13,14] as well as layered and spaced concrete targets [15-18].Although a large number of experimental studies have been reported,they have mainly focused on improving the penetration ability of shaped charge and obtaining penetration holes with higher depth and larger diameter,and only a few studies have attempted to improve the crater area.In addition, plain concrete has been mostly utilized as the target material of the experiment, and the damage characteristics of reinforced concrete has been rarely examined.Therefore, a clear understanding of the experimental phenomenon and damage mechanism is lacking in the existing literature.Further, due to economic reasons, the current research conclusions are usually based on small-scale experiments, and most of the targets have a cylindrical shape with finite size.However, concrete has a strong size effect[19,20],and erroneous conclusions may be drawn when small-scale experiments are extrapolated to full-size structures.Therefore, it is necessary to conduct experiments on a full-scale reinforced concrete wall for facilitating theoretical analysis and engineering application, which is also of great significance to further improve the design and safety evaluation of protective structures based on reinforced concrete.

In addition to experimental research, numerical simulation is indispensable for the in-depth understanding of the penetration mechanism of shaped charge, and researchers generally adopt the combination of simulation and experiment for relevant analysis.Although penetration depth and crater diameter are both important evaluation indexes of concrete damage, most numerical simulation studies seek to match the simulated penetration depth results with the experimental results, and the crater diameter has not been accurately simulated yet.Under near-field explosion, the damage to concrete caused by shaped charge includes the penetration of shaped charge, damage of explosion shock wave, and joint damage enhancement effect [21], which indicates the characteristic of combined damage under multiple loads[22].However,most of the reported two-dimensional(2D)and three-dimensional(3D) numerical models mainly focused on the penetration effect and a small air range was considered.Therefore, the damaging effect of explosion shock wave caused by the initiation of the shaped charge was ignored or simplified, so these models were not effective for engineering design and calculation.In addition,the existing numerical models are rarely based on reinforced concrete,and the large-scale,3D simulation of shaped charge penetrating reinforced concrete is still in its infancy.In particular,the influence of steel bar on the penetration of shaped charge has been rarely investigated,and the corresponding interaction mechanism has not been revealed yet.It should be noted that the 3D numerical modeling of shaped charge penetrating reinforced concrete considering the damage of explosion shock wave requires huge amount of computations, which leads to practical problems that cannot be supported by conventional computers.Fortunately, with the rapid development of computational technologies, large-scale finite element simulation becomes possible.

Appropriate concrete constitutive model is the key to obtain reasonable simulation results.Concrete models commonly used in numerical analysis of impact and explosion problems include K&C model [23], HJC model [24], RHT model [25], etc.However,although the above concrete models can well describe the damage behavior of concrete under compression, it is difficult to truly simulate its tensile softening behavior, and there is a great difference between the simulation results and the test results.In view of this, many scholars [26-29] have studied the reliability and applicability of the concrete models mentioned above,pointed out their limitations and made corresponding improvements.In addition, scholars have also established a variety of new concrete constitutive models based on the dynamic mechanical behavior of concrete material.For example, Xu [30] developed a dynamically enhanced uniaxial unconfined compression/tension concrete model considering strain hardening and softening, excluding inertia effects.Kong [31-33] developed a fluid elastic-plastic damage concrete model (Kong-Fang model), which systematically considers important factors such as tensile and compressive different strain rate effects, third stress invariants, and shear expansion effects.Huang [34] developed a cap elastic-plastic damage concrete model (Huang-Kong model), which systematically considered dynamic tensile, shear and volumetric compression damage.The RHT model can reflect the properties of materials such as strain rate sensitivity, strain softening, and damage evolution and is widely used to simulate the dynamic mechanical behavior of brittle materials such as concrete at high strain rates[35].However,several studies have shown that the RHT model with default parameters has some problems in describing the strainsoftening behavior of concrete under strong impact and explosion loads, but it can be properly corrected by modifying some default parameters of the model.Tu et al.[36],Abdel et al.[37,38],Hu et al.[39], and Nystrom et al.[40] have modified the RHT model from several angles.The modified parameters mainly include uniaxial compressive strength (FC), normalized tensile strength (FT*), residual strength surface parameters (AF and NF), damage parameters(D1), and effective plastic strain (EPSF).

The conventional shaped charges mainly include shaped charge jet (JET), jetting projectile charge (JPC), and explosively formed projectile(EFP).Relevant studies[41]have indicated that compared with JET and JPC, EFP has the advantages of good stability, small velocity gradient, large penetration aperture, and large damage aftereffects,and it can realize better damage to concrete walls and brick walls.Therefore, in this study, an EFP is designed, and its damage effect on a reinforced concrete wall is experimentally analyzed.Further, a large-scale 3D numerical simulation model is established based on the LS-DYNA finite element software.By simulating the combined damage of EFP penetration and explosion shock wave, the destructive effect of shaped charge on the reinforced concrete wall can be analyzed more comprehensively and realistically.In addition, by validating the feasibility of the numerical model,the combined damage effect of EFP penetration and explosion shock wave on the reinforced concrete wall is studied,the influence of steel bars on the penetration of EFP is highlighted,and the effect of impact positions on the damage of the reinforced concrete wall is also examined.Overall, the findings of this study can serve as a useful reference for the optimization design of tandem penetration warheads and protection of reinforced concrete facilities, which has high practical significance and wide application prospects.

2.Charge structure and forming characteristics

2.1.Charge structure

Based on the relevant theory, existing literature, and design experience,to obtain an EFP with a high speed and compact shape,the structure of shaped charge is designed,which is shown in Fig.1.It is mainly composed of a liner and an explosive.The liner is made of red copper and has a hemispherical structure of variable thickness with thick middle and thin edge.The overall thickness of the liner is determined by the top thickness (δ), and the thickness gradient is determined by the internal and external radii of curvature (r1and r2).The main charge is pressed JH-2 explosive(charge diameter is 120 mm), and to reduce the weight of the warhead and improve the energy efficiency of the charge,the sternshaped charge structure is adopted.In addition, to avoid the influence of shell constraint on the EFP formation, improve the experimental safety, and reduce the processing cost, two shaped charges without shells have been processed to conduct the penetration experiment on a large-size reinforced concrete wall.

2.2.Forming characteristics

The stable forming shape of EFP is the basis of the design of shaped charge.Firstly,the numerical simulation technology is used to verify the effectiveness of the structural design of shaped charge.The forming shape and change law of EFP are analyzed, and the standoff distance is determined based on this reference.

The numerical model is composed of three parts: explosive,liner,and air,and the arbitrary lagrangian-eulerian(ALE)algorithm is used in all units.Considering the symmetry of the model and to reduce the amount of calculation, the LS-DYNA finite element software is used to establish a quarter-symmetric 3D model, as shown in Fig.2.The detonation point is located at the center of the top surface of the charge, the symmetry constraint is set on the symmetry axis and plane of the model, and the transmission boundary condition is imposed on the air.The unit system of cm-gμs is used for calculation.The explosive is described by MAT_HIGH_EXPLOSIVE_BURN and EOS_JWL;the liner is described by MAT_JOHNSON-COOK and EOS_Gruneisen; the air is described by MAT_NULL and EOS_LINEAR_POLYNOMIAL.The values of the material parameters are shown in Tables 1-3.

Fig.2.Numerical model of EFP formation.

The formation process of EFP is shown in Fig.3.The top of the liner is first affected by detonation products; it starts to move forward along the axis before the edge of the liner,and the top of the liner gradually forms the head of EFP.The axial speed of the microelement on the top of the liner is higher than that on the edge,resulting in a continuous stretching at the top of the liner, relative lag in the movement of the edge, and gradual compression from both sides of the edge to the axial center to form the tail of EFP.Due to the large and persistent velocity gradient, the liner gradually turns backward during the formation process.Finally, under the action of axial tension and radial compression,the designed shaped charge forms a long rod-shaped EFP with a dense center and tail fin,which has a large length-diameter ratio and good flight stability.

Fig.1.Schematic diagram of the EFP.

Table 1Material parameters of air [42].

Table 3Material parameters of red copper [44].

Fig.3.Formation process of EFP (t = 16, 20, 38, 54,152 μs).

Fig.4.Formation form of EFP.

It can be seen in Fig.4 that EFP presents continuous,stable and highly dense form within a large range of standoff distance, indicating that the shaped charge structure designed in this paper is effective.Here, H represents the standoff distance; V1, V2, and V3represent the head velocity, tail velocity, and the difference between them,respectively;L,D,and L/D represent the length of EFP,middle diameter,and aspect ratio,respectively.As shown in Fig.5,with the increase of standoff distance, V1, V3and D gradually decrease, while V2, L and L/D gradually increase, and the change trend is gradually decreasing with the increase of blast height.The necking phenomenon becomes increasingly obvious in the process of axial elongation, and the head and tail show a fracture trend,which can lead to instability in the penetration of EFP.Considering the forming shape and velocity distribution of EFP, the standoff distances are determined to be 30 cm (2.50 times of charge diameter) and 45 cm (3.75 times of charge diameter).At this time, the aspect ratio of EFP is moderate,the shape is dense and the head has no obvious necking phenomenon.

3.Experimental conditions and numerical simulation

3.1.Experimental conditions

Fig.5.Formation data of EFP.

The reinforced concrete wall used in the experiment is shown in Fig.6,which consists of upper and lower parts.It is built with C35 concrete and equipped with steel bars,and the grade of steel bar is HRB335.The lower part of the wall, which is the object of this experiment,has a length of 400 cm,height of 130 cm,and width of 80 cm.Five layers of steel bars are arranged along the direction of penetration, and each layer is composed of steel bars with 1.2 cm diameter interwoven horizontally into a square with side length of 24 cm.The five layers of steel bars are 5,21,40,59,and 75 cm away from the front of the wall,respectively.Due to the large size of the wall and the uniform layout of the steel bars, only the partial structural diagram is shown in Fig.7.The shaped charge was placed on a wooden support frame, and the mouth of the liner was vertically facing the reinforced concrete wall to ensure that the EFP penetrated the wall vertically.The layout of the experimental site is shown in Fig.8.The standoff distances were set to 30 cm and 45 cm,and the shaped charge was detonated by a standard electric detonator after calibrating the experimental settings.Through the damage effect experiment, the macroscopic failure mode of reinforced concrete wall was obtained.

3.2.Numerical simulation of experimental conditions

3.2.1.Numerical model

Fig.7.Partial structure of the reinforced concrete wall.

Fig.8.Schematic layout of the experimental site.

During the penetration process, steel bars endure a large deformation and fracture, which consume a part of the kinetic energy of EFP and improve the anti-penetration performance of concrete.However,in most of the existing studies on the damage of reinforced concrete caused by shaped charge,the influence of steel bars on the penetration of shaped charge has not been considered.Steel bars are often ignored in numerical simulation,and reinforced concrete is considered equivalent to plain concrete [45], which leads to a certain deviation between the simulation results and experimental results.Considering the symmetry of the concrete wall structure with steel bars, to balance computational efficiency and computational accuracy, a quarter-symmetric 3D numerical model is established(Fig.9)to quantitatively analyze the influence of steel bars on the penetration process and damage results.The reinforced concrete wall is modeled by a separate method,and the concrete and steel models are established separately.The length,width and height of the geometric model of the reinforced concrete wall are 55, 55 and 80 cm respectively.A large-scale air domain with a radius of 80 cm covering the entire reinforced concrete wall is constructed for the transfer of explosive pressure and EFP, and the coupled damage to the reinforced concrete wall caused by the EFP penetration and explosion shock wave is considered.Accordingly,the entire damage process of reinforced concrete wall under the combined load of EFP penetration and explosion shock wave is simulated using the LS-DYNA finite element software.The simulation and experimental results are compared and analyzed from the aspects of failure pattern, crater depth, and crater diameter.

Reasonable material models and calibration parameters are the keys to obtaining high-fidelity simulation results.Moreover, the establishment of 3D model and simulation algorithm cannot be ignored.Based on the mesh sensitivity analysis, the local mesh refinement of the reinforced concrete wall is carried out.The element size on the penetration path is 2 mm.After the distance exceeds 6 times the radius of the EFP,a proportional gradient mesh is used,and the maximum element size at the edge is 13 mm Solid 164 element is used to establish the model of concrete, and the element algorithm is Lagrange.The element type of steel bar is Beam 161, and the interaction between steel bar and concrete is defined by CONSTRAINED_BEAM_IN_SOLID [46].The intersection of horizontal and vertical steel bars is modeled by the common node, and the load can be transferred between the steel bars through the common node, which is consistent with the experiment and theory.The interaction between the explosive, liner, air,and reinforced concrete wall is modeled using the fluid-solid coupling algorithm, and constrains the displacement and rotational degrees of freedom in the directions perpendicular to the two symmetry planes.All the surfaces of air except the two symmetry planes are defined as non-reflection boundaries, simulating the infinite air domain and avoiding pressure reflections on the boundaries.Since the size of the reinforced concrete wall used in the experiment is much larger than the diameter of the shaped charge,the boundary effect is small and can be ignored.Therefore,the two sides of the reinforced concrete wall, except the two symmetric surfaces, are defined as non-reflective boundaries, and the infinite size of the reinforced concrete wall is simulated without considering the influence of reflected waves on the boundary.

3.2.2.Model parameters for concrete and steel bar

Fig.9.3D numerical model of reinforced concrete wall damaged by the shaped charge.

As mentioned above,the strain softening behavior of RHT model can be improved by modifying the parameters of the RHT model,such as uniaxial compressive strength (FC), normalized tensile strength (FT*), residual strength surface parameters (AF and NF),damage parameters (D1), and effective plastic strain (EPSF).FT*refers to the ratio of tensile strength to compressive strength, the default value is 0.1.AF is the initial slope of the residual surface,and NF is the residual strength index,whose default values are 1.6 and 0.61, respectively.D1and D2are damage parameters that control the cumulative damage rate, and their default values are 0.04 and 1.0, respectively.EPSF represents the minimum equivalent plastic strain of failure, beyond which the concrete element will fail, and the default value is 2.In this study, based on the conclusion of Hu et al.[39], D1is taken as 0.02, D2is 1.0 by default.Here, the simulation results are compared with the experimental results under different values of EPSF, and it is found that the simulation results under the EPSF value of 1.5 are relatively consistent with the experimental results.Since the residual surface cannot be measured,the values of AF and NF cannot be determined according to the experimental data [35], so there is no widely recognized conclusion at present.Therefore,based on the existing studies and combining the experimental results,several simulation trials were conducted to find that the modified results of Abdel et al.[37,38]were more suitable for the research conditions used in this study.According to Abdel et al.[37], cubic compressive strength (fcu)rather than cylindrical compressive strength(fc’)is adopted for FC,and NF is 0.3.Furthermore, according to another study of Abdel et al.[38],FT*and AF need to be modified in combination with FC.The fitting relationship between FT*, AF, and FC is expressed as follows:

After modifying the above parameters,the modified RHT model is obtained, as shown in Table 4.

The steel bar is described by the MAT_PLASTIC_KINEMATIC model, and the failure criteria included in the model are used to define the failure of the steel bar.When the failure strain of the steel bar is greater than 0.15,the element of the steel bar is deleted,and the material parameters are shown in Table 5.

3.2.3.Simulation results

The RHT model with modified and default parameters is used for the numerical simulation of experimental conditions, and a comparison between the corresponding results is presented in Figs 10-13.The damage distribution of the reinforced concrete wall is displayed in the form of a cloud map.The damage degree ranges from 0 to 1,and the larger the value,the higher the damage degree.Different colors represent different damage degrees.Specifically,red color represents the most serious damage and the corresponding damage degree is 1.Blue color represents the least damage or no damage, and the value of damage degree is 0.As shown in Fig.14,the overall damage area of the reinforced concrete wall obtained through numerical simulation is regarded as the crater shape.The damage mode is consistent with the experimental results, indicating the overall characteristics of the damage.

The experimental results show an irregular shape, which is because the concrete is a heterogeneous material, the internal aggregate size is inconsistent and randomly distributed, and the penetration resistance is also different.The numerical simulation is based on the continuity assumption,and the material model is also an idealized description of the real material.In the experimental results, the crater is not a regular symmetrical circle, so the crater diameter is measured from four directions:horizontal,vertical,45°,and 135°,which are defined as D1,D2,D3,and D4,respectively,and the average value is defined as D.The measurement definition is shown in Fig.15, and the diameter and depth of the crater are shown in Table 6.

Figs.10 and 11 show a comparison of the caving area on the front of the wall obtained through simulation and experiment,where thewhite circle indicates the damaged area.When the RHT model with modified parameters is adopted, the simulation and experimental results are reasonably consistent.When the standoff distance is 30 cm, the average crater diameters obtained by experiment and simulation are 71 cm(5.92 times of charge diameter) and 72.4 cm(6.03 times of charge diameter),respectively,and the relative error is 1.97%.When the standoff distance is 45 cm, these values are 67.5 cm(5.63 times of charge diameter)and 68.8 cm(5.73 times of charge diameter), respectively, and the relative error is 1.93%.The experimental and simulated variation trends of diameter are consistent, which indicates that the diameter gradually decreases with the increase in the standoff distance.

Table 4Material parameters of concrete.

Table 5Material parameters of steel bar.

Fig.10.Comparison between the simulation and experimental results of crater diameter when the standoff distance is 30 cm: (a) RHT model with modified parameters; (b) RHT model with default parameters; (c) Experimental result.

Fig.11.Comparison between the simulation and experimental results of crater diameter when the standoff distance is 45 cm: (a) RHT model with modified parameters; (b) RHT model with default parameters; (c) Experimental result.

Fig.12.Comparison between the simulation and experimental results of crater depth when the standoff distance is 30 cm:(a)RHT model with modified parameters;(b)RHT model with default parameters; (c) Experimental result.

Fig.14.Overall simulation results of the reinforced concrete wall: (a) standoff distance is 30 cm; (b) standoff distance is 45 cm.

Fig.15.Measurement definition of the crater diameter.

In the numerical simulation, the damage on the surface of the reinforced concrete wall is obvious under the short-range comprehensive damage of the EFP and the explosion shock wave.Several irregular radial cracks appear on the wall surface, and serious caving occurs,presenting an irregular crater shape,which is consistent with the experimental observation, and the simulated range of caving area is close to the experimental results, which cannot be realized by 2D simulation.There is no obvious crack propagation in the area outside the crater, and the damage is limited to a local region.Further, the damage conditions under different standoff distances are fairly consistent.By analyzing the crater shape of the wall, it is observed that there are concrete fragments at the edge of the crater, which are lifted but not collapsed due to the restraint of steel bars.Furthermore,it is found that the surface damage of the crater is caused by the coupled action of the strong impact compression wave,reflected tensile wave,and circumferential tensile wave.When the RHT model with default parameters is adopted,the simulation results of the average crater diameter at the standoff distance of 30 cm and 45 cm are 59.8 cm(4.98 times of charge diameter)and 54.2 cm(4.52 times of charge diameter), respectively, and the errors with respect to the experimental results are 15.77% and 19.70%, respectively.Compared to the RHT model with modified parameters, the damage mode of the reinforced concrete wall obtained by the RHT model with default parameters does not change, but the caving area is significantly underestimated, and the relative error is considerably larger than that obtained by the RHT model with modified parameters.

The resistance of penetration is determined by steel bar and concrete together,so the influence of steel bar on the penetration of EFP cannot be ignored and it becomes the next research focus.

In the experiment,one steel bar in the first layer was hit by EFP,the middle section of the single steel bar was broken,and the steelbars near the fracture were bent and deformed with an obvious diameter shrinkage phenomenon.This is because the steel bar had a certain ductility,and it was bent toward the inner wall along the penetration direction by the extrusion action of the EFP and concrete,which yielded but did not break.The area outside the fracture of the single steel bar that was hit and multiple steel bars that were not hit expanded and deformed outward in the opposite direction of penetration.This is because the steel bars were embedded in the concrete and were pulled, stretched, and restrained by the concrete.The concrete fragments flew outward under the action of reflected tensile stress,which caused the steel bar to deform in the opposite direction of penetration.Fig.16 shows the fracture and deformation of the steel bar when the standoff distance is 30 cm,and dotted lines represent the initial state of the steel bar.It can be observed that the simulation model successfully describes the fracture characteristics as well as the outward bulge and expansion of the steel bar in the opposite direction of penetration, which is consistent with the experimentally obtained deformation mode of the steel bar.

Table 6Experimental and simulation results of damage to the reinforced concrete wall.

Fig.16.Fracture and bending deformation of steel bars.

When the standoff distance is 30 cm and 45 cm, there is little difference in the crater depth,which is basically concentrated in the position of the second layer of steel bars.As shown in Figs.12 and 13, the interaction between the EFP and the second layer of steel bars can be intuitively observed in the experiment.The second layer of steel bars is locally bent under the strong impact,and this phenomenon is also well described by the simulation.When the RHT model with modified parameters is used and the standoff distance is 30 cm,the experimental and simulated crater depths are 24.6 cm(2.05 times of charge diameter)and 25.1 cm(2.09 times of charge diameter), respectively, and the relative error is 2.03%;When the standoff distance is 45 cm,these values are 22.4 cm(1.87 times of charge diameter) and 23.0 cm (1.92 times of charge diameter), respectively, and the relative error is 2.68%.When the RHT model with default parameters is used, the simulated crater depths at the standoff distance of 30 cm and 45 cm are 22.2 cm(1.85 times of charge diameter) and 21.3 cm (1.78 times of charge diameter), respectively, and the errors with respect to the experimental results are 9.76% and 4.91%, respectively.Compared to the RHT model with modified parameters,although the crater depth is slightly underestimated and the relative error is slightly larger,this model can also effectively reflect the fracture and deformation of the steel bar.

Fig.17.Stress cloud diagram of failure of reinforced concrete wall (standoff distance is 30 cm).

Fig.18.Stress cloud diagram of failure of reinforced concrete wall (standoff distance is 45 cm).

Due to the limitations of the RHT model,only the damage range and crater cracks can be predicted by numerical simulation,but the collapse and spraying of concrete cannot be described.Concrete is a brittle material,and there is a certain deviation in the macroscopic failure phenomenon.Therefore, for this study, the error between the simulation and experimental results can be ignored.This suggests that the proposed numerical modeling method, material parameters, and selected algorithm are reasonable, and the RHT model with modified parameters is effective for simulating the damage effect of shaped charge on the reinforced concrete wall.

4.Analysis of influencing factors

4.1.Influence of explosion shock wave

There is a combined effect of explosion and penetration on the target when the shaped charge explodes near the target, so the single damage load cannot fully reflect the damage effect of the shaped charge.Taking the standoff distance of 30 cm as an example,the combined action mechanism of EFP penetration and explosion shock wave is analyzed.As shown in Fig.17, when t = 1 μs, the shaped charge detonates at the center of the top of the charge.When t = 12 μs, the detonation products begin to act on the liner.From 22 to 36 μs, the liner produces obvious overall forward movement and gradually turns over and deforms.From 36 to 104 μs, the propagation speed of explosion shock wave is greater than the flight speed of EFP,and the front of explosion shock wave is always in front of the head of EFP.When t=104 μs,the explosion shock wave reaches the wall surface before the EFP, and the EFP head is still 4.8 cm away from the wall surface.The intensity of the incident explosion shock wave is 6.49 MPa, and the shock wave front will cause reflection after contacting the wall surface.Compared with the incident shock wave, the intensity of the reflected wave is significantly increased to 32.31 MPa, which is 4.98 times of the incident shock wave.When t=112 μs,the stress wave generated by the explosion shock wave propagates inside the wall,causing damage to the local area of the wall.When t=118 μs,EFP reaches the surface of the wall, and the pressure at the collision point is as high as 2.6 GPa, which is much higher than the compressive strength of concrete, causing local crushing damage,and stress waves continue to propagate to the wall during penetration.After t = 124 μs, the external explosion shock wave gradually propagates to the wall surface, and the cambium cracks and collapses on the front of the wall under the action of the reflected tensile wave generated by the EFP penetration and the explosion shock wave.

Fig.19.3D numerical model of reinforced concrete wall damaged by the shaped charge after reducing air domain.

Fig.20.Simulation results of large air domain: Standoff distance is (a) 30 cm, (b)35 cm, (c) 40 cm, (d) 45 cm, (e) 50 cm, (f) 55 cm, (g) 60 cm.

After the initiation of shaped charge, the explosion shock wave propagates outward in an elliptical shape.As the propagation process progresses, the front curvature radius of the shock wave front gradually increases, and the strength and propagation speed of the shock wave gradually decrease.As shown in Fig.18,when the standoff distance is 45 cm, the EFP gradually catches up with the front of the explosion shock wave from 136 to 170 μs, and acts on the reinforced concrete wall before the explosion shock wave.When t=180 μs,the explosion shock wave reaches the wall surface and begins to destroy.At this time, the intensity of incident shock wave is 4.96 MPa,and the intensity of reflected wave is 20.47 MPa,which is 4.13 times of the incident shock wave.Compared with the standoff distance of 30 cm,when the standoff distance is 45 cm,the pressure of the shock wave front is obviously attenuated,the wave front shape is more gentl, the energy density decreases, and the intensity of the incident shock wave decreases by 23.57%.

After the initiation of shaped charge,it is difficult to separate the explosion shock wave,and its destructive effect cannot be studied directly.Therefore, to quantitatively describe the damage effect of EFP penetration and explosion shock wave on reinforced concrete wall as much as possible,this paper establishes a numerical model of small air domain(Fig.19)based on the numerical model of large air domain(Fig.9).The top radius of the small air domain is 6.6 cm and the bottom radius is 2.5 cm.Through the numerical model of small air domain,only the single damage effect of EFP penetration is considered, and the damage effect of explosive shock wave is investigated from the side.To further study the effect of explosion shock wave on the failure of reinforced concrete wall, more numerical simulations are carried out, and the standoff distances are set as 35, 40, 50, 55 and 60 cm, respectively.Simulation results of large air domain are shown in Fig.20 and Table 7, and simulation results of small air domain are shown in Fig.21 and Table 8.

By comparing Figs.20 and 21, it can be found that the overall failure characteristics of the reinforced concrete wall do not change significantly for the simulation results of reducing the air range,that is,only considering the destruction of the reinforced concrete wall by EFP penetration.The failure phenomenon is still the front crater,but the crater diameter is obviously reduced,the number of cracks is obviously reduced,and the damage degree is reduced.As shown in Fig.22,for the simulation results of large air domain,the crater diameter decreases significantly with the increase of standoff distance, but the change of crater depth is not significant, and theaverage value of crater depth is 24.5 cm (2.04 times of charge diameter).For the simulation results in small air domain, the diameter and depth of the crater do not change significantly under different standoff distances,and the average diameter and depth of the crater are 54.0 cm(4.50 times of charge diameter)and 23.7 cm(1.98 times of charge diameter), respectively.For the crater diameter, the simulation results in large air domain are significantly larger than those in small air domain, and the gap between them gradually decreases with the increase of standoff distance.As shown in , when the standoff distance is 30 cm and 60 cm, the crater diameter in large air domain is 72.4 cm(6.03 times of charge diameter) and 58.6 cm (4.88 times of charge diameter), respectively.The crater diameter in small air domain is 55.6 cm (4.63 times of charge diameter) and 53.1 cm (4.43 times of charge diameter), respectively.The crater diameter decreases by 16.8 cm and 5.5 cm, respectively, when the air domain is reduced.For the crater depth,the simulation results of large air domain are slightly larger than those in small air domain, but the two are relatively close,and the average difference of the crater depth is only 0.8 cm.

Table 7Damage results of reinforced concrete wall in large air domain.

Table 8Damage results of reinforced concrete wall in small air domain.

Fig.22.Failure results of reinforced concrete wall varying with standoff distances.

The above analysis shows that under the working conditions studied in this paper, the failure of the reinforced concrete wall is caused by the coupling effect of EFP penetration and explosion shock wave, and the multi-load damage characteristics of shaped charge determine the failure result of the reinforced concrete wall.The effect of explosion shock wave on the crater depth is not significant,resulting in a slight increase in the crater depth.The effect of explosion shock wave on the crater diameter is obvious, which can aggravate the damage range of crater,and the effect is gradually weakened with the increase of standoff distance.

4.2.Influence of steel bars

To quantitatively analyze the influence of steel bar on the diameter and depth of crater, the damage effect of shaped charge on a plain concrete wall is numerically simulated based on the above verified model.The simulation results are shown in Fig.23 and Table 9.

It can be seen in Fig.19 that when the standoff distance is 30 cm and 45 cm,the average crater diameter of the plain concrete wall is 76.7 cm(6.39 times of charge diameter)and 72.6 cm(6.05 times of charge diameter),respectively,which increases by 5.94%and 5.52%compared with results of the reinforced concrete wall.The crater depth of the plain concrete wall is 27.6 cm (2.30 times of charge diameter) and 24.7 cm (2.06 times of charge diameter), respectively, which is 9.96% and 7.39% higher than that of the reinforced concrete wall.It can be deduced that for the research conditions used in this study,the steel bar has a certain influence on the crater shape.Although the reinforcement ratio of the reinforced concrete wall is not high and the restraining effect of the steel bar is limited,it can still weaken the effect of reflected tensile waves to a certain extent and reduce the caving diameter of the wall.Williams et al.[47] and Li et al.[48] pointed out that under a low reinforcement ratio, the steel bar has a minor influence on the penetration and caving of the reinforced concrete, which is consistent with our simulation results.In addition, the steel bar has a high strength.Therefore,when the EFP hits the steel bar,the kinetic energy of the EFP is consumed during the fracture and bending deformation of the steel bar, so the anti-penetration ability of the reinforced concrete wall is improved to a certain extent.However, it can be observed through numerical simulation that because thepenetration of EFP into the reinforced concrete wall is a high-speed penetration, indicating obvious hydrodynamic characteristics, the effect of steel bars on the penetration ability of EFP is not significant, which is consistent with the study of Esteban et al.[6].

Table 9Simulation results of damage to the plain concrete wall.

4.3.Influence of impact positions

The steel bar causes the anti-penetration ability of the reinforced concrete to be different at different positions, so the influence of impact positions on the penetration ability of EFP is particularly obvious.It is difficult to accurately determine the relative position of the impact position and steel bar before the experiment, so the influence of impact position cannot be reasonably verified through experiments.At the same time, the experimental cost and manual labor involved are high,and it is difficult to conduct a large-scale experiment.To clarify the influence of steel bars on the EFP, based on the verified model and material parameters,numerical simulation is conducted under typical impact positions with a standoff distance of 30 cm, and the influence of impact positions on the penetration process is analyzed.

As shown in Fig.24, three typical impact positions are considered.Case 1 is the center of the reinforcement grid,and Case 2 is the intersection of two steel bars.Case 3 is the midpoint of a single steel bar and corresponds to the experiment condition.The simulation results are shown in Fig.25 and Table 10.

In Case 1,the EFP does not hit the steel bar,and the crater has an average diameter of 73.4 cm(6.12 times of charge diameter)and a depth of 26.4 cm(2.20 times of charge diameter);In Case 2,when the EFP hits the intersection of two steel bars, the crater has an average diameter of 69.4 cm(5.78 times of charge diameter)and a depth of 22.7 cm(1.89 times of charge diameter);In Case 3,the EFP hits one steel bar,and the crater has an average diameter of 72.4 cm(6.03 times of charge diameter)and a depth of 25.1 cm(2.09 times of charge diameter).It can be observed that the average crater diameter when the EFP hits one steel bar (Case 3) and the intersection of two steel bars (Case 2) is relatively smaller than that in the case when the EFP does not hit the steel bar(Case 1),the crater diameter decreases by 1.36%and 5.45%,respectively.This is because the deformation of the steel bar and the bonding between the steel bar and the concrete absorb part of the energy[49].In general,the average crater diameter at different impact positions is not much different.

Fig.24.Location of the three typical impact positions: (a) Case 1; (b) Case 2; (c) Case 3.

Fig.25.Damage of the reinforced concrete wall under different impact positions: (a) Case 1; (b) Case 2; (c) Case 3.

Table 10Simulation results of the reinforced concrete wall with different impact positions.

The steel bar is primarily cut and broken by the EFP and is bent and deformed by the impact force of the expanded concrete.The deformation and fracture of the steel bar consume the energy and reduce the penetration ability of the EFP.Two steel bars apply the force at the intersection of the steel bars, which consume the maximum energy.In the midpoint of a single steel bar, only one steel bar applies the force, and the energy consumed is relatively reduced.While passing through the center of the reinforcement grid, no steel bar is involved in applying the force, and the energy consumption is minimal.Therefore, compared with the case in which the EFP does not hit the steel bar(Case 1),when the EFP hits one steel bar(Case 3)and the intersection of two steel bars(Case 2),the crater depth is reduced by 4.92% and 14.02%, respectively.

Fig.26.Outline of the EFP trajectory under different impact positions: (a) Case 1, (b)Case 2, (c) Case 3.

Based on numerical simulation,it is found that the steel bar can not only reduce the penetration depth of the EFP and the caving diameter of the wall but also has a significant influence on the penetration trajectory.As shown in Fig.25,under the local action of steel bars,the influence of steel bars on the penetration trajectory is particularly significant.The simulation results can well describe the ballistic deflection due to the asymmetric resistance caused by steel bars.As shown in Fig.26, in Case 1, the penetration trajectory is vertically downward with a relatively regular shape and maximum depth.In Case 2,the penetration trajectory expands in a 4-Pl shape with the least regular shape and minimum depth.In Case 3, the penetration trajectory expands to both sides in a two-lobed shape.This is because the EFP is split by the steel bar during the highspeed penetration process.As shown in Fig.27, the EFP in Case 1 is relatively intact, while the EFP in Case 2 is split into four pieces,and the EFP in Case 3 is split into two pieces, which is consistent with the penetration trajectory in Fig.26.

5.Conclusions

In this study, the damage effect of a shaped charge on a largescale reinforced concrete wall was examined both numerically and experimentally.Further, the failure mode and damage mechanism of the reinforced concrete wall under the combined load of EFP penetration and explosion shock wave were analyzed.Specifically, the influence of steel bars on the EFP penetration was highlighted.The main results are summarized as follows:

(1) The designed shaped charge can form a crater with a large diameter and high depth on the reinforced concrete wall.The average crater diameter is greater than 67 cm(5.58 times of charge diameter), and crater depth is greater than 22 cm(1.83 times of charge diameter),which can meet the damage range requirements of rescuers entering the building.

(2) Numerical simulation based on the RHT model with modified parameters accurately describes the crater diameter and crater depth of the reinforced concrete wall,and it effectively reflects the failure mode of the reinforced concrete wall under the combined action of EFP penetration and explosion shock wave.Compared with the experimental results, the maximum relative error of crater diameter and crater depth is 1.97% and 2.68%, respectively.

(3) The multi-load damage characteristic of shaped charge determines the failure result of the reinforced concrete wall.The failure of the reinforced concrete wall is mainly caused by EFP penetration.When only EFP penetration is considered, the average diameter and depth of the crater are 54.3 cm (4.53 times of charge diameter) and 23.7 cm (1.98 times of charge diameter), respectively.The effect of explosion shock wave on crater depth is not significant, resulting in a slight increase in crater depth.The average crater depth is 24.5 cm (2.04 times of charge diameter) when the explosion shock wave is considered.The effect of explosion shock wave on the crater diameter is obvious,which can aggravate the damage range of the crater, and the effect gradually decreases with the increase of standoff distance.

(4) For the reinforced concrete wall with a low reinforcement ratio used in the experiment,although steel bars can reduce the crater diameter and crater depth to a certain extent,the effect is not significant.Compared with the results for a plain concrete wall, the crater diameter and crater depth of the reinforced concrete wall are reduced by 5.94% and 9.96%,respectively.The average crater diameter of the reinforced concrete wall under different impact positions is not much different,but there is a certain difference in the crater depth.Compared to the case in which the steel bar is not hit,when the EFP hit one steel bar and the intersection of two steel bars, the crater diameter decreases by 1.36% and 5.45%respectively,the crater depth decreases by 4.92%and 14.02%respectively.The steel bar has a significant influence on penetration trajectory, and the EFP will be split by steel bar during the penetration process, resulting in an irregular trajectory.

Fig.27.Deformation of EFP under different impact positions: (a) Case 1; (b) Case 2; (c) Case 3.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was supported by the Scientific and Technological Innovation Project (Grant No.KYGYZB0019003).