Numerical Simulation of Contact Explosion on Reinforced Concrete Slabs Based on Different Algorithms

Abstract

The explosion dynamics process is complex and difficult to carry out accurate analytical analysis. Numerical simulation with controllable boundary conditions and initial conditions can continuously and repeatedly display the explosion development process over time. In this paper, we take a single-layer reinforced concrete slab contact explosion as an example and use Lagrange method, Arbitrary Lagrangian–Eulerian (ALE) method, and smoothed particle hydrodynamics (SPH) method in LS-DYNA software to simulate the explosion of explosives. The results show that Lagrange method can display a clear material interface, but the explosives elements in the simulation of explosion process could cause serious distortion easily leading to the termination of the calculation; the explosives and air in the ALE method are Eulerian materials, there is no element distortion, but the number of the explosives and air elements that need to be established is relatively large; SPH algorithm can simulate the large deformations caused by explosion, making it more realistic in simulating the splashing of explosive materials. In the comparison of three algorithms, the ALE algorithm matches the experimental results the best.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Due to the complexity of the explosion dynamics process, the accurate analytical analysis is difficult to carry out, so the model test and numerical simulation have become the most commonly used methods at present. But the explosion test is not only expensive but also highly dangerous. More importantly, the duration of the explosion is short, and some physical phenomena during the process cannot be observed during the test. On the contrary, numerical simulation is an effective and economical test, with controllable boundary and initial conditions, and can continuously and repeatedly display its development process over time, facilitating in-depth analysis and understanding of the test results. Some scholars have carried out relevant research [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Numerical simulation of explosion effects is widely used and has solved many practical engineering problems with obvious effect. There are four main methods for explosion analysis in LS-DYNA: Lagrange method, Euler method, Arbitrary Lagrange–Euler method, and smooth particle hydrodynamics method.

In finite element calculation, the Lagrangian method is commonly used in solid mechanics, while the Eulerian method is used in fluid mechanics, the Lagrangian grid is fixed in the object and moves with it, meaning that the grid points and material points always coincide during the deformation process of the object. Therefore, there is no relative motion between the material points and the grid points, which greatly simplifies the solving process of controlling the equations, it can accurately describe the object’s moving interface and can track the trajectory of the material points; while the Eulerian grid is fixed in space and the computational grid remains unchanged during the deformation of the object, so the distortion of the matter can be easily handled. Both pure Lagrangian and pure Eulerian descriptions have not only serious drawbacks but also have their own advantages. If the two can be organically combined to give full play to their respective advantages and overcome their respective shortcomings, many problems that cannot be solved by using only pure Lagrangian and pure Eulerian descriptions can be solved [11, 12]. The ALE method was first proposed in terms of coupled Eulerian–Lagrangian for this purpose, and the ALE method is further divided into two levels, i.e., the single-material arbitrary Lagrangian–Eulerian method and the multimaterial arbitrary Lagrangian–Eulerian method. The SPH method is a meshless numerical simulation method proposed internationally in recent years, which was first proposed by Lucy [13], and was initially applied to astrophysical and cosmological problems, and more recently, has been widely used in the fields of hydrodynamics, explosions, penetration, and high-speed collisions, and so on [13, 14, 15, 16].

This article takes a single-layer reinforced concrete slab with contact explosion as an example and uses the Lagrange, ALE, and SPH algorithms in the LS-DYNA program to simulate explosive explosions. The differences, advantages, and disadvantages of the three different algorithms are compared and analyzed from three aspects: model establishment, calculation results, and calculation time, providing some reference and guidance for numerical simulation methods of explosive effects.

2. Contact Explosion Test on Single-Layer Reinforced Concrete Slabs

2.1. Test Model

The reinforced concrete slab is 300 mm thick, with a length and width of 2,000 mm × 1,500 mm. The slab has the upper and lower two layers of steel bars, the spacing of layers is 250 mm. The grade of concrete is C60, the diameter and grade of the steel bar are 12 mm and HPB300, respectively, and the spacing of two-way reinforcement is 200 mm. The two ends of the reinforced concrete slab are placed flat on concrete piers with a width of 200 mm, as shown in Figure 1. A 3.0 kg TNT charge is located at the center point of the upper surface of the slab.

[figure(s) omitted; refer to PDF]

2.2. Test Results and Analysis

When explosives explode on the surface of a concrete slab, two funnel-shaped craters will form on the front and back sides of the slab. The size of the craters is related to the amount of explosives used, and even the two craters are connected, a penetrating damage will occur. However, the damage mechanisms of the two types of craters are different. The crater on the blast-facing surface is caused by the high-intensity compression wave generated by the explosion of the explosive, which poses the concrete on the surface of the plate to be strongly compressed and damaged (the concrete is crushed into powder), forming a blast crater. The damage on the back blast surface is caused by the tensile wave that is formed by the reflection of the compression wave at the free surface and propagates upwards. Once this tensile wave exceeds the tensile strength of the concrete, it will cause tensile damage to the concrete, forming a spalling crater. The damage of reinforced concrete slabs under different charges of explosives can be divided into four forms: explosion cratering, explosion spalling, explosion penetrating, and explosion punching, as shown in Figure 2. Figure 3 shows test results, it belongs to the type of explosion spalling.

[figure(s) omitted; refer to PDF]

3. Numerical Simulation Methods

3.1. Lagrange Model

In the finite element calculation of single-layer thick plate with contact explosion, considering the symmetry of structure and load and reducing the amount of calculation, a quarter model was taken for calculation. In the solid modeling, TNT charges, concrete and reinforcement are all modeled by solid element solid 164. In the quarter model, the explosive size is 61 mm × 61 mm × 122 mm. There are three types of explosive element sizes, as listed in Table 1. In the solid modeling of reinforced concrete slabs, concrete and reinforcement bars are modeled separately (Figure 4), and the reinforcement bars are connected to the concrete using common nodes connection with no relative slip. In order to facilitate the modeling, round steel bars with a diameter of 12 mm are processed into square steel bars with a side length of 10.63 mm based on equal cross-sectional area. The element size for simulating concrete and steel reinforcement is approximately 9 mm, with 260,580 and 1,500 elements, respectively, and the piers are also concrete elements, divided into 9,000 elements in the quarter computational model, as shown in Figure 5.

[figure(s) omitted; refer to PDF]

Table 1

Division of explosive element for each model.

Model	Element size	Element number
Lagrange	12.20 mm	250
	8.71 mm	637
	4.36 mm	5,488
ALE	12.20 mm	250 (TNT), 18,350 (air)
	8.71 mm	637 (TNT), 23,420 (air)
	4.36 mm	5,488 (TNT), 51,632 (air)
SPH	6.10 mm particle	2,000
	4.07 mm particle	6,750
	3.05 mm particle	16,000

3.2. ALE Model

In the ALE model, the reinforced concrete slab is identical to the Lagrange model, except that the Euler-type element format is used for the explosive and air, that is, the air and TNT explosives elements described by Euler surround the Lagrange concrete slab structure coupled with it, and the two are calculated by the coupling of Euler and Lagrange. When meshing the air element, the calculation time is taken into account while ensuring the calculation accuracy, and the area near the explosive is densely divided, and the mesh is sparse at the distance from the explosive, see Figure 6. Three element sizes were also used for explosives and air in the ALE model, as shown in Table 1.

[figure(s) omitted; refer to PDF]

3.3. SPH Model

The reinforced concrete slab in the SPH model is consistent with the previous two models, the difference is that the TNT explosive is discretized into smooth particles with a size of 61 mm × 61 mm × 122 mm, smooth particles are set with three particle sizes (Table 1). The TNT explosives and the reinforced concrete slab’ model is shown in Figure 7(a). The discretization model of the explosives is shown in Figure 7(b) after enlargement.

[figure(s) omitted; refer to PDF]

3.4. Material Models and Equations of State

The materials involved in the numerical simulation include TNT explosives, air, concrete, and steel bars. For the explosive material, a high explosive material model was used, with a density of ρ = 1.63 × 10³ kg/m³, a detonation velocity of D = 6.93 × 10³ m/s, and a burst pressure of P_cJ = 2.1 × 10¹⁰ Pa [16]. The detonation process is described by the JWL equation of state with the following expression:$$\begin{array}{}\text{(1)}& P=A1-\frac{\omega }{R_{1}V}{e}^{-R_{1}V}+B1-\frac{\omega }{R_{2}V}{e}^{-R_{2}V}+\frac{\omega }{V}E,\end{array}$$where A = 3.712 × 10¹¹ Pa, B = 3.231 × 10⁹ Pa, R₁ = 4.15, R₂ = 0.95, ω = 0.30, and E₀ = 7.0 × 10⁹ J/m³ [16].

The air element in the coupling is modeled using the null material (Null) model and linear polynomial equation of state [17, 18]. The concrete utilizes the Holmquist–Johnson–Cook concrete model in LS-DYNA [19, 20], which is more suitable for calculating the reinforced concrete penetration and explosion problem. Reinforcing steel was simulated using an elastic–plastic model (MAT_PLASTIC_KINEMATIC) with a modulus of elasticity E = 2.1 × 10¹¹ Pa, Poisson’s ratio ν = 0.3, a density of ρ = 7.86 × 10³ kg/m³, a yield strength σ_y = 2.1 × 10¹⁰ Pa, and a tangent modulus E_tan = 4.6 × 10⁸ Pa [21].

In order to simulate the spalling failure of reinforced concrete slabs, in the HJC concrete model the erosion criterion (MAT-ADD-EROSION) is added here to control the failure of the element. This criterion can define seven failure modes, namely pressure, principal stress, equivalent stress, shear strain, threshold stress, stress pulse, and failure time. Through analysis and calculation, it has been found that using the pressure failure criterion can achieve good results in the test of reinforced concrete slabs subjected to contact explosion. When the pressure reaches the failure pressure, the element is deleted:$$\begin{array}{}\text{(2)}& P\le P_{\min },\end{array}$$where P is the pressure (positive in compression), and P_min is the minimum pressure at failure. Here in the calculation, P_min is taken as −4 MPa.

3.5. Boundary Conditions and Contact Settings

Symmetry constraints are set at the nodes on the symmetry plane in all three computational models, and fixed support constraints are applied to all nodes on the bottom of the piers. In the ALE model, the air domain is set as a transmitting boundary on the outer boundary surfaces except for the symmetry plane, which is used to simulate the infinite domain of the air. Surface-to-surface contact is used between the explosives and the reinforced concrete slabs in the Lagrange model (CONTACT_SURFACE_TO_SURFACE), and point-to-surface contact is used between the explosives and the reinforced concrete slabs in the SPH model (CONTACT_AUTOMATIC_NODES_TO_SURFACE), and the interaction between explosives, air, and reinforced concrete slabs in the ALE model is defined by setting the keyword CONSTRAINED_LAGRANGE_IN_SOLID (CTYPE = 4, DIREC = 2), and surface-to-surface contact is also used between the concrete slabs and piers in the three models.

The penalty method is used in the surface-to-surface contact of Lagrange model as well as in the point-to-surface contact of SPH model. The basic principle of the penalty method is to first check whether each slave node penetrates the master surface at each time step, and if not, no further processing is performed. If penetration occurs, a large interface contact force is introduced between the slave node and the penetrated master surface, which is proportional to the penetration depth and the stiffness of the master surface. This is physically equivalent to placing a normal spring between the two to limit penetration from the node to the master surface. The contact force is called the penalty function value.

The penalty coupling is selected in the fluid–structure interaction of ALE model. Like the contact algorithm in the Lagrange and SPH model, the penalty coupling tracks the relative displacement d between the Lagrangian node (structure, i.e., slave material) and the fluid material location (master material) in the Eulerian material, as shown in Figure 8. Check the penetration of each slave node into the surface of the master material. If there is no penetration of the slave node, no operation is performed. If there is penetration of the slave node into the surface of the master material, the interface force F will be distributed to the nodes of the Eulerian fluid and structure, and the magnitude of the interface force is proportional to the degree of penetration that occurs:$$\begin{array}{}\text{(3)}& F=K\cdot d,\end{array}$$where K is computed based on the constitutive material model properties.

[figure(s) omitted; refer to PDF]

4. Results

4.1. Explosive Blast Processes

In Figure 9, the explosive uses the Lagrange algorithm. The element grid of Lagrange algorithm is attached to the material, and the deformation of the element grid occurs as the material flows. It can be found that the degree of grid distortion of the explosive also increases during the continuous explosion and expansion process, especially for the grid located on the periphery of the explosive. However, this mesh distortion often results in a decrease in the calculation accuracy, the calculation of the step size becomes small and other problems, and in severe cases, negative mass can even result in termination of the calculation. The advantage of this method is that it can obtain a clear material interface.

[figure(s) omitted; refer to PDF]

In Figure 10, the concrete slab structure is surrounded by explosives and air composed of Eulerian grid, the red area in the figure shows the explosives, you can see that with the passage of time the red area continues to become larger and outward expansion. In the ALE algorithm, due to the explosives and air using the Eulerian grid, the interface of the material cannot be seen directly. The interaction interface between explosives and the structure can only be seen in the post-processing after a special treatment. The advantage of this algorithm is that the explosives and airflow in the Eulerian mesh, without grid distortion problems. The maintenance of calculation is very good and rarely appears in the phenomenon of termination of calculation. For this reason, the ALE algorithm is mostly used in the numerical simulation of explosion impact problems.

[figure(s) omitted; refer to PDF]

Among the three algorithms, the SPH algorithm provides the most realistic detonation effect. Figure 11 graphically shows the material dispersion process during the explosion of explosives. The SPH algorithm was initially developed to simulate astrophysical phenomena, such as the collision of two stars, the explosion of supernovas, and the formation of the moon, etc. Therefore, the SPH algorithm is ideal for simulating hypervelocity collisions and explosive impact phenomena and for solving high-speed impact dynamics problems involving large material deformation.

[figure(s) omitted; refer to PDF]

4.2. Damage Evaluation of Cratering and Spalling

Figure 12 shows the schematic diagram of a damage evaluation including the diameter and depth properties of cratering and spalling. The damage results of reinforced concrete slabs under different algorithms are listed in Table 2. It should be pointed out that the reinforced concrete slab models used in different algorithms are the same, with the difference being that explosives use different algorithms. In the Lagrange algorithm, the area around the explosion cratering on the blast-facing surface is severely damaged, especially in the 4.36 mm grid. This is inconsistent with the experimental results, and the reason for this result is that when using the Lagrange algorithm for explosives, there is a tendency for element distortion in the calculation, resulting in abnormal contact between the explosive element and the reinforced concrete slab element (where contact should not occur), leading to damage that differs from the experimental phenomenon. In the ALE algorithm, the failure simulation of concrete slabs is closest to the experimental phenomenon. In the SPH algorithm, there is also a significant amount of damage to the upper surface of the concrete slab, which is the result of the contact and collision between the SPH particles of the explosive and the surface of the concrete slab. In terms of crater diameter, the results of the three algorithms differ significantly from the experiment, with Lagrange algorithm being the largest, SPH algorithm being the middle, and ALE algorithm being the smallest; in terms of crater depth, the ALE algorithm is the largest, and there is not much difference between the Lagrange algorithm and the SPH algorithm; in terms of the spalling diameter, the ALE algorithm is the largest, followed by the Lagrange algorithm, and the SPH algorithm is the smallest; in terms of the spalling depth, the three algorithms are equivalent.

[figure(s) omitted; refer to PDF]

Table 2

Results of experiment and simulation.

Method	Damage evaluation of cratering and spalling (mm)					Plastic strain1	Contact force2 (N)
	a	b	c	d	Results of simulation
Experiment	520	85	650	70	—	—	—
Lagrange
12.20 mm mesh	262	96	344	67		0.0839	4.931 × 107
8.71 mm mesh	276	80	400	76		0.0803	5.430 × 107
4.36 mm mesh	274	78	494	58		0.0678	6.019 × 107
ALE
12.20 mm mesh	244	92	632	67		0.0858	4.517 × 107
8.71 mm mesh	244	100	612	58		0.0785	5.384 × 107
4.36 mm mesh	254	99	570	67		0.0766	6.316 × 107
SPH
6.10 mm particle	254	79	306	67		0.0687	4.250 × 107
4.07 mm particle	260	88	344	58		0.0839	5.740 × 107
3.05 mm particle	260	95	326	57		0.0880	5.550 × 107

¹Maximum effective plastic strain of steel bars. ²Maximum vertical resultant force between TNT and reinforced concrete slab.

4.3. Vertical Displacement of Explosion Crater

Eight points on a line were taken at the center position of the surface of the concrete slab (Figure 13), and the vertical displacement of each point at 0.15 ms under different algorithms was calculated. The calculation results are shown in Figure 14. From the graph, it can be seen that the consistency of the calculation results under different algorithms is good, and the overall difference is not significant except for a few points, which verifies the effectiveness of the algorithm. In the Lagrange algorithm, the nodes’ displacement generated by the coarse explosive element is the largest, and as the element becomes smaller, the displacement by the nodes decreases; in the ALE algorithm, the influence of explosive mesh size is similar to that of the Lagrange algorithm; in the SPH algorithm, the influence of smooth particle size shows opposite results to the Lagrange and ALE algorithms, with the displacement of nodes gradually increasing as the number of particles increases. Comparing the three algorithms, the ALE algorithm has the highest node displacement, and the results of the Lagrange and ALE algorithms are not significantly different.

[figure(s) omitted; refer to PDF]

4.4. Damage of Steel Bars and Contact Force

Figure 15 shows that only the upper longitudinal and transverse steel bars closest to TNT exhibit significant plastic deformation at the middle part under contact explosion, with the maximum plastic strain occurring at the middle (element #229,402) of the short-direction steel bars. The effective plastic strain response time curve of the steel bars at this location is shown in Figure 16(a), which shows that the time from deformation to reaching the maximum value is approximately 60–70 μs. The maximum plastic strain values of the steel bars in different algorithms and models are shown in Table 2. The influence of different mesh densities and particle sizes on the maximum effective plastic strain of steel bars is consistent with the analysis in the vertical displacement of explosion crater.

[figure(s) omitted; refer to PDF]

Figure 16(b) shows the time history curves of the resultant vertical contact force between TNT and reinforced concrete slabs under different algorithms. In terms of loading speed, the Lagrange algorithm and SPH algorithm are higher than the ALE algorithm, and the Lagrange algorithm and SPH algorithm are equivalent. The maximum resultant force is 6.316 × 10⁷ N, and the minimum value is 4.250 × 10⁷ N (Table 2). The action time of TNT explosives on reinforced concrete slabs is approximately 25 μs. To contact force, three algorithms exhibit consistent regularity for different grid densities and particle sizes. Overall, as the number of elements and particles increases, the loading rate of the resultant force gradually increases, and the peak value gradually increases.

4.5. Calculation Time

The computer used for the calculation is an Intel (R) Core (TM) i3-7100 with main frequency 3.90, 3.91 GHz, and a memory of 4 GB. The simulation time is set to 300 μs. Table 3 contains comparison of model size and the statistics on the running time for these models. It can be found that the Lagrange algorithm has the longest computation time, mainly due to the serious mesh distortion caused by the expansion of explosives, resulting in the gradual reduction of the computation step, which leads to the prolongation of the computation. Even two of the cases were terminated due to negative volume. The shortest is the SPH algorithm, the reason is that this kind of algorithms is actually a combination of the use of the Lagrange and SPH algorithms, and the use of SPH algorithm avoids the occurrence of distortion in explosive elements, which greatly shortens the calculation time.

Table 3

Statistics of the running time for different models.

Numerical model of the TNT	Total number of elements	Initial timestep (s)	Final timestep (s)	Normal termination or not	Total of CPU time
Lagrange
12.20 mm mesh	271,330	1.28E-07	3.69E-09	Error termination at 0.275 ms	1 hr 1 min
8.71 mm mesh	271,717	8.42E-08	6.48E-09	Normal termination	1 hr 37 min
4.36 mm mesh	276,568	8.66E-08	2.72E-09	Error termination at 0.162 ms	31 min
ALE
12.20 mm mesh	289,680	8.59E-07	4.40E-08	Normal termination	47 min
8.71 mm mesh	295,137	8.59E-07	5.60E-08	Normal termination	44 min
4.36 mm mesh	328,200	5.80E-07	5.35E-08	Normal termination	56 min
SPH
6.10 mm particle	273,080	6.41E-07	1.31E-07	Normal termination	9 min
4.07 mm particle	277,830	6.41E-07	9.51E-08	Normal termination	11 min
3.05 mm particle	287,080	5.82E-07	8.71E-08	Normal termination	13 min

5. Conclusion

Through the simulation of contact explosion on reinforced concrete slabs using the Lagrange, ALE, and SPH algorithms of the LS-DYNA program, the modeling of different algorithms, the results of the calculations, and the calculation time are discussed, and the following conclusions and recommendations can be obtained as follows:

(1) The advantage of the Lagrange algorithm is that the modeling is relatively simple and a clear material interface can be obtained. The disadvantage is that the explosive elements undergo severe distortions during the explosion process, sometimes affecting the solution process and taking the longest time in this calculation.

(2) The advantage of the ALE algorithm is that explosives and airflow in Eulerian mesh without distortion problems in the elements. It also uses fluid–solid coupling to handle interactions, making it easy to establish explosion models (separately established for fluid and solid). The disadvantage is that in the postprocessing process to get a clear material interface needs to do special processing; at the same time, the establishment of the model needs to establish the air elements, compared to other algorithms the number of elements to be a lot larger, so the amount of computation is relatively large.

(3) The advantage of the SPH algorithm is that it does not require grids, has no element grid distortion problems, can simulate large deformations caused by explosions, and is more realistic in simulating the splashing of explosive materials. In this calculation, it takes the least amount of time.

(4) The consistency of the calculation results under the three algorithms is good, which verifies the effectiveness of the algorithms. Considering of three models of the TNT in this paper, the ALE model seems to be the most feasible for the analysis of the event. The element size impact analysis revealed a little influence of mesh size on the results, this indicates that the mesh density is adequate.

Acknowledgments

The research described in this paper was financially supported by the Construction Project of Key Academic Discipline in Henan Province (Zhengzhou University of Technology, civil engineering), the Scientific and Technological Project in Henan Province (No. 222102320383), the Key Research Projects of Higher Education Institutions in Henan Province (No. 24A560023, No. 22A410004), and the Zhengzhou University of Technology High-Level Talent Research Project (No. 24GC02).

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