Keywords

1 Introduction

Kinetic energy projectile is a weapon mostly used in defence and military applications which contains an explosive charge. It works on a principle to attain a high velocity, generally up to hypervelocity and strike the target material, converting kinetic energy into shock waves and heat. Hence, they are designed in such a way to defeat military bunkers, high aircraft shelters, deep buried targets, etc. Penetration of projectile into a concrete target is an interesting topic for many scholars. Many researchers were investigating various parameters of penetration by calculating projectile depth of penetration, the effect of concrete target diameter on the depth of penetration, etc. Forrestal [1] derived an empirical formula to examine the depth of penetration into the concrete target. From the last three decades, three major techniques were castoff to envisage the interaction of projectile with concrete material. Concrete is remembered because of its low cost and wide availability. It was well defined by its unconfined compressive strength and its ubiquity, but has certain demerits like brittleness, weak in tension, inelastic deformation before failure, etc. Due to the formation of cracks, spalling and scabbing on the target arise erratically. A continuous analysis on the physical and mechanical behaviour of concrete and its composition at the macroscopic level is obedient as concrete is a complex material widely used in structural applications such as bunkers, buildings, dams. The reinforced concrete, i.e. a mixture of steel and concrete, is used to enhance the bending stress of concrete and improve its tensile strength.

Rajput et al. [5] studied the effect of plain and reinforced concrete by impacting hard ogive nose steel projectile at a tremendous velocity. Here, crater diameter was estimated and residual versus impact velocity was observed and related amongst these two targets. Wu et al. [6] prophesied the velocities which are remained as a residue of a projectile afterwards normally perforating the thin UHP-SFRC barriers. Wu et al. [7] analysed high-velocity penetration of the concrete target material with different shapes of a projectile using two constituents. Over here, penetration depth and trajectory stability of these projectiles were analysed. Outcomes disclosed that double-nose projectile had specific drag reduction, high penetration ability. In the interim, projectile with groove-tapered owned good trajectory stability as its curved part had tapering shank which helped in restoring moment for stabilization. Feng et al. [8] replicated sets of experiments to study the effect of thick PCT to high-velocity penetration of an alloy steel projectiles by varying its striking velocities. The crater diameter increased linearly with the striking velocity. Oucif and Mauludin [9] evaluated the geometric modelling of high-speed impact of ogive nose-shaped projectile against reinforced concrete panel using Johnson–Holmquist damage model.

In the facts mentioned above, the present work is conducted by impacting conventional ogive-nose-shaped projectiles, as shown in Fig. 1 with two shapes of the projectile of different CRH and penetrating into the concrete target. Moreover, the penetration depth and the crater damage on the impact face of the targets are analysed.

Fig. 1
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Design of ogive-nose [10]

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2 Numerical Simulation Tool

ANSYS AUTODYN user interface is an analytical tool required to model nonlinear dynamics of solids, fluids, gas, and its interaction. It is easy to use, fully integrated, and enable us to set up, run, and post-process given problems. AUTODYN has been functionalized in a vast range such as enhancing and designing of anti-armor systems, building protection in contradiction of blast marvels, risk identification in contrast to aircraft impact, material characterization lay open to impact and dynamic loading, drop test, etc.

3 Design

3.1 Projectile

A 100-mm-diameter, 1-m-long ogive-nosed projectiles are designed for impacting the concrete target. They are machined from steel 4340 rods. Here, two projectiles of 1.5 and 3.0 CRH are tested against the concrete target. Ogive-nose CRH outlines the ratio of ogive radius to the projectile or shank diameter. Figure 2 simply shows the difference projectile nose shape. It is accelerated at an impact velocity of 350 m/s towards the target. Here, it is assumed that the projectile is kept rigid, i.e. deformation or damage free. Therefore, no failure criteria were applied to this material.

Fig. 2
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Ogive-nose shape with 1.5 CRH

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3.2 Concrete Target

In this study, plain concrete target (PCT) and reinforced concrete target (RCT) have taken as target materials. The material applied for the target is M35 grade of concrete, namely CONC-35 MPa; see Table 1.

Table 1 Mixture ratio of M35 grade of concrete
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Concrete is good in compression rather than tension. The reinforcement is suitable to provide strength against tension. Figures 3 and 4 display the front and isometric view of the reinforced concrete target. The thick PCT size of 500 × 500 × 1500 mm is modelled in ANSYS AUTODYN, and reinforcement is filled in case of RCT. The reinforcement material is preferred as cast iron.

Fig. 3
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Front view of RCT

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Fig. 4
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Isometric view of RCT

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3.3 FEM for Meshing

In the analysis, meshing is necessary to achieve accurate results. A fine mesh is required to compute complex problems. The finite element method (FEM) is the best numerical technique which generates fairly accurate solutions by discretizing a complicated problem into a small number of elements. The point connecting between them is defined by nodes and the complex system of these points, which produce a grid named as mesh. The element size for projectile mesh is 7 mm, as shown in Fig. 5 while that of the concrete target is 25 × 25 × 25 mm.

Fig. 5
figure 5

Four-node tetrahedron projectile meshing

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3.4 Material Model

Material model is distributed based on three major criteria:

  1. (a)

    Equation of state

  2. (b)

    Material strength model

  3. (c)

    Material failure model

  1. (a)

    Equation of State (EOS)

    For projectile and reinforcement material, linear EOS criteria are used which express P = f(\(\rho , {e}\)). Here, \(\rho\) = density and e = internal energy.

    For concrete target, porous-alpha polynomial EOS is designed in AUTODYN due to its compaction and compressibility.

  2. (b)

    Material Strength Model

    Initially, all solid materials react elastically in nature, but over an extreme dynamic loading condition, they can extend to stress where it exceeds yield stress and then deforms plastically.

    Johnson–Cook equation

    $$\sigma = \, [\sigma_{0} + B \epsilon^{{{n}}} ] \times [{1} + c( \epsilon / \epsilon_{0} )] \times [{1} - \, \{ (T - T_{{\text{r}}} ) \, / \, (T_{{\text{m}}} - T_{{\text{r}}} )\}^{{{m}}} ]$$
    (1)

    where σ = yield stress, σ0 = static yield stress, B = hardening constant, n = hardening exponent, c = stress-rate constant, ϵ0 = reference strain rate, Tm = melting point, Tr = reference temperature, m = thermal softening constant.

  3. (c)

    Material Failure Model

    Material may fail under high hydrodynamic loading conditions, resulting in the occurrence of crack. Considering in mind that concrete fails under tensile load rather than in compressive load, the failure model of the M35 grade of concrete is simulated under the hydrodynamic tensile failure criteria. The material model for projectile penetration into concrete target material is presented in Table 2.

    Table 2 Material model
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4 Empirical Formulae

Numerical simulation results are compared with the empirical equation. Forrestal et al. formulated an equation to determine the DOP equation for an ogive-nose-shaped projectile penetrating M35 grade of the concrete target. The below-mentioned penetration equation is obtained from post-test scrutiny of soil and concrete target material [1]. The penetration in concrete is subdivided into two definite processes, viz. formation of crater and tunnelling phenomena. By using this penetration equation, depth of penetration comes out and is revealed in Table 3.

$$P = \frac{m}{{2\pi a^{2} \rho N}}{\text{ln}}\left( {{1 + }\frac{{N\rho V^{2} { }}}{R}} \right) + 4a$$
(2)
$$V^{2} = \frac{{mV_{{\text{s}}}^{2} - 4\pi a^{3} R}}{{m + 4\pi a^{3} N\rho }}$$
(3)
$$N = \frac{8\psi - 1}{{24\psi 2}}$$
(4)
$$\psi = \frac{r}{2a}$$
(5)
Table 3 Calculated depth of penetration (DOP) for both nose shapes of projectiles
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Here, P = depth of penetration, m = mass of projectile, ρ = target density, a = projectile radius, Vs = striking velocity, Ψ = calibre-radius-head (CRH), r = calibre-radius, R = empirical constant (410 MPa), N = nose shape factor.

5 Results and Discussions

The main objective of this study was to observe the behaviour between plain concrete and reinforced concrete and its effect on penetration depth and crater formation on the front as well as rear faces. Numerical simulation is conducting using below mentioned four cases of projectile, viz. shown in Table 4. Lagrange interaction is applied between projectile and concrete target. Figure 6 expresses the preliminary setup of complex problem interaction in AUTODYN-3D.

Table 4 Various cases of ogive-nose-shaped projectile
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Fig. 6
figure 6

Interaction using 1.5 CRH projectile

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Here, it is obtained results of crater diameter and penetration depth through numerical simulations along with their results is compared with empirical results.

5.1 Crater Diameter

Crater diameter is generated due to high-velocity impact of the projectile against the target material. Figures 7 and 8 describe these much value of crater diameter caused during penetration in target for all the cases. The figure illustrates the damage produced at the front face of plain as well as the reinforced concrete target.

Fig. 7
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Crater diameter at t = 9.785 ms (case-I)

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Fig. 8
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Crater diameter at t = 8.002 ms (case-II)

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5.2 Depth of Penetration (DOP)

The penetration depth or depth of penetration (DOP) for all these cases obtained from the AUTODYN is given in Figs. 9 and 10. The results are compared with the empirical approach, viz. known as Forrestal penetration equation. One observation was clarified that there is less variation in penetration depth in both RCT and PCT.

Fig. 9
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Penetration depth at t = 9.785 ms (case-I)

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Fig. 10
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Penetration depth at t = 8.002 ms (case-II)

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Finally, all these results are gathered and displayed as a bar chart shown in Fig. 11 which is appeared from the numerical simulation. Here, maximum and minimum DOP is achieved in PCT with 3.0 CRH projectile and RCT with 1.5 CRH projectile, respectively. PCT with 1.5 CRH projectile has maximum crater damage, while RCT with 3.0 CRH projectile has least crater damage. Due to technical constraints, up to 7% of error is considerable in the value of the results of penetration when compared with empirical equations because of some limiting factors such as mesh size, meshing method, interactions, irrelevant selection of the material model.

Fig. 11
figure 11

Outcomes from numerical simulation

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6 Conclusion

From this paper, the multifaceted interaction of a projectile with target material is an interesting topic of research for ballisticians. From the simulation results, it may be concluded that by adding steel fibres in the concrete, there is an appreciable decrease in spalling and scabbing on the concrete target. Forrestal et al. [1] empirical formulae are used as a benchmark for comparing results with numerical simulation. The results obtained from the penetration equation are validated upright promise with the results of numerical simulation.