Introduction

Nowadays, rubber-like materials are widely used both in civil and military area due to several reasons, costs and mechanical behavior being among the most important ones.

Among military activities, indoor shooting range practice sessions are widely spread. Since during this kind of practice sessions the interaction between high or low velocity projectiles and the building columns, beams, ceilings, walls or floors are highly probable, special measures must be emplaced. To achieve that, the building plans for the indoor shooting range include special structures and materials designed to stop the bullets and prevent ricochets.

The mechanism involved in bullet stop can vary depending on the nature of the used materials. When hard materials like armor plates, concrete walls or ceramic tiles are used, the goal is to sharply decelerate or stop the bullet on impact or to force it due to successive ricochets in a spinning movement inside a special helical chamber called bullet trap chamber. However, for this material class the risk of ricochets is high. As opposed to the first category, the relatively soft materials like wood, elastomers or soil slowly decelerate the bullets mainly by friction, mitigating also the risk of ricochet1,2. Among this second material class, the rubber-like materials are widely used. In fact, the uses of rubber like materials not only for military application but also for civilian application became so extensive that standards were developed to standardize the way in which their mechanical properties are investigated3,4,5,6. Mainly, this trend is due to several advantages that this type of material possessed. Thus, if we consider the manufactures cost of the material, the refurbishing of already existing shooting facilities can be managed with low expenditure. The low specific weight can also play an important role in the refurbishing process since no important supplementary stress is induced to the existing building resistance structure. The easy replacement of the damage rubber like material plates also advocate for the use of such material instead of classic metallic materials.

Starting with the 1940’s the conducted studies regarding mechanical properties of rubber like materials pointed out that basically those materials have a hyperelastic with viscoelasticity behavior7,8,9,10,11,12. Along with the visco-elasto-plastic behavior, the fracture of rubber like materials was also of interest since many applications intentionally or not implies large deformation and material fracture. Thereby, over time many studies were targeted to reveal the mechanism that stands behind failure and to establish failure criteria. The developed fracture criteria unfortunately cannot be applicable to the entire spectrum of applications hence their high numbers. Some of them consider the strain energy density at breaking, stresses at breaking or elongations at breaking13.

The struggle to correctly define the constitutive material model is understandable since the model is essential for numerical studies based on FEM (Finite Element Method) which are quite common nowadays and can prove to be valuable tools in the designing process. However, this type of studies is conditioned by the correct identification of constitutive material model parameters14. Therefore, the need to perform a test program that allows the calculus/measurement of material parameters is mandatory and common to all materials, hyperelastic ones included. Nevertheless, this type of approach for hyperelastic materials can prove itself more or less difficult depending on the selected constitutive models (Neo-Hookean, Ogden or Mooney-Rivlin being the most often used). In a general context the mechanical tests that must be performed refer to static or/and dynamic tests, uniaxial or biaxial tension and compression tests, shear tests and so on performed at different temperatures and strain rates. In addition to previously mentioned mechanical tests, ballistic limit and ricochet tests must be performed since are of most relevance for military and forensics applications. Hence, the numerous performed studies over the years regarding the ricochet inflicted by hard targets (steel, concrete, ice and so on15,16,17,18,19,20) and more recently by soft materials (water, soil, gelatin, polycarbonate and so on21,22,23,24,25,26), also.

As mentioned before, in the last decades the use of elastomers or more specific the rubber-like materials (REGUPOL, Dura-Bloc, Coru, Flip-Lok, Sportec Fragsafe and Linatex being some of the most known ballistic rubbers commercially available) for both bullet trap devices and ricochet mitigation surfaces became notorious. Thus, in many shooting facilities the use of layered ballistic rubber panels (ballistic rubber made of recycle rubber and metal backing plate) or simple ballistic rubber panels both for ballistic purpose and acoustic mitigation is something common. Despite this growing trend data regarding ballistic limit and critical ricochet angle for that type of materials are not easily available.

The current study is focused on REGUPOL E43 mechanical properties evaluation and the investigation of both ballistic limit and ricochet risk posed by a 9 × 19 mm ball impacting a REGUPOL E43 sheet. The ballistic limit and ricochet studies are approached both experimentally and numerically.

Experimental section

Mechanical behavior investigation

As pointed out in Section "Introduction" the complete mechanical characterization of a material implies a wide range of tests. Nevertheless, given the nature of the application being studied, one can reduce the number of tests depending on the relevance of the investigated property to the application. For common civilian applications related with the use of elastomers type materials (REGUPOL included) the investigation of quasi-static engineering stress versus engineering strain dependency and storage modulus are the main priorities. Instead, for military applications as it was mentioned ballistic limit and critical ricochet angle are meaningful.

Quasi-static uniaxial tests

Usually, the mechanical properties investigation of a given material starts with the quasi-static uniaxial testing followed by dynamic tests relevant to the intended application. The geometrical configuration of the REGUPOL E43 specimens for the quasi-static uniaxial investigation is presented in Fig. 1. As depicted in Fig. 1, two types of specimens were used for the compression investigation: a 44 mm side cubical specimen for Poisson’s ratio evaluation (Fig. 1a) and a cylindrical specimen (ϕ28 mm × 24 mm height) for the identification of engineering stress versus engineering strain characteristic curve (Fig. 1b). For the tensile tests, dog bone specimens were used with an active length of 45 mm, a 6.5 mm width and 43 mm thickness (Fig. 1c). The strain rate imposed for both tensile and compression test was 1.6 × 10−3 s−1.

Fig. 1
figure 1

REGUPOL E43 test specimens.

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Dynamic mechanical analysis

REGUPOL E43 stands for rubber granulates bound with polyurethane (PU). Since PU is basically a class of polymer, a comprehensive mechanical investigation must include tests specific to this type of materials. The most common mechanical quantities aimed to be measured for this type of materials refers to complex modulus as a function of temperature or applied frequency in a dynamic mechanical analysis (DMA). Basically, knowing those parameters one can fully understand the viscoelastic behavior of such materials which can prove handy for different types of applications. In fact, quantifying the ability of the material to store or to dissipate mechanical energy, identifying the phase lag between loads and deformations and the phase transition temperature one can predict the response of the material to different kind of stimulus in real life applications.

To evaluate the change in stiffness of the REGUPOL E43 material, its damping ability and phase transition, certain dynamic mechanical analysis (DMA) tests were performed using a Discovery 850 DMA TA Instruments analyzer. Specimens cut from a REGUPOL E43 sheets were analyzed in single cantilever setup (for determination of glass transition temperature). For testing, rectangular specimens with the following dimensions: 37 mm × 10 mm × 2 mm (length × width × thickness) were investigated. The imposed test conditions assumed a sinusoidal load (20 µm amplitude and 1 Hz frequency) over temperature interval -90 °C to 100 °C and 5 °C/min heating rate (maintained by cooling with liquid nitrogen).

Morphological and structural characterization

To analyze the structural bonding between the rubber granules and the binder, before and after mechanical testing, the REGUPOL E43 specimens were structurally analyzed using a Dino-Lite Digital microscope (Weller, Germany). Furthermore, to investigate the quality of the interaction of the rubber granules with the organic binder, samples were cut from the tested specimens and the bonded area was investigated using a Scanning Electron Microscopy (SEM) analyzer. Electronic images were acquired with a Nova NanoSEM 630 scanning electron microscope (FEI Company, Hillsboro, OR, USA) at an accelerating voltage of 10 kV, under high vacuum conditions. The samples were coated with a thin layer of conductive metal (Au) to prevent charging during the investigation.

Ballistic properties investigation

Ballistic limit assessment

Along with the development of military equipment and facilities, the study of materials in regard to their ability to stop or deflect ballistic threats (as bullets or projectiles) became of huge interest. The comparative analysis of two or more materials for ballistic application can be carried out mainly but not exclusively by assessing the so called “ballistic limit” (the minimum impact velocity required for a given type of projectile for a 50% chance to penetrate a given target material) and the critical angle for ricochet.

To assess the ballistic limit of a 1000 × 500x43 mm REGUPOL E43 sheet, six ballistic normal temperature (≈15 °C) impact tests were carried out using a test configuration as the one depicted in Fig. 2.

Fig. 2
figure 2

Ballistic impact test configuration. (1—Photron Fastcam SA-Z; 2—embedded REGUPOL E43 sheet; 3—mounted gun; 4—Doppler radar).

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To point out the ballistic limit of REGUPOL E43 sheet against FMJ 9 × 19 mm ball (STANAG 4090 standard ammunition) some of the rounds were altered in terms of propellant mass. By varying the propellant mass and keeping the distance up to the target constant, the instrumentation of several impact velocities was enabled. The impact velocities were recorded with the use of both Doppler radars and ultra high-speed camera.

Ricochet risk assessment

Over time extensive studies were carried out regarding ricochet phenomenon. The main focus of those studies was to assess the ricochet risk (a critical angle of impact) and other parameters like angle, velocity and rotational velocity of projectile (ball) after impact. Due to the complexity of impact phenomenon, a unified model is not yet possible. This is why usually the papers aim to study particular cases instead of a general one. In line with this concept, several studies regarding 9 × 19 mm ricochet risk were developed considering different target material like water27,28, concrete28, ceramic plates29, steel plates29, soil30, glass31 and so on.

The present study aims to assess the critical angle (α) above which a FMJ 9 × 19 mm ball impacting a REGUPOL E43 sheet will no longer pose the risk of ricochets for meaningful ball velocities. To enable the critical angle identification, tests were performed at several impact angles using a test configuration as the one presented in Fig. 3. The tests for each impact angle were carried out at two different average impact velocities (three tests for each average impact velocity) at normal temperature. The ricochet occurrence and ball velocity were assessed with the help of an ultra-high-speed camera and a Doppler radar.

Fig. 3
figure 3

Ricochet test configuration.

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Experimental results and discussions

Quasi-static uniaxial tests

As indicated in section "Experimental section", the evaluation of REGUPOL E43 sheet mechanical behavior requires both quasi-static (compression and tension) and dynamic (dynamic mechanical analysis) mechanical tests.

Figure 4 depicts the schematic of the quasi-static compression test design to provide information regarding the material volume compressibility (cubic specimens) and compression curve (cylindrical specimens).

Fig. 4
figure 4

Schematic representation regarding the determination of compression behavior for cubic and cylindrical specimens.

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The material volume compressibility is characterized by Poisson’s ratio. In the case of hyperelastic materials, due to their non-linear elastic behavior, the Poisson’s ratio exhibits a strain dependency. This is why, in the case of hyperelastic materials, instead of Poisson’s ratio it is more appropriate to talk about Poisson’s function32,33. To state if the REGUPOL E43 exhibits a similar behavior as hyperelastic materials, using a camera, the compression of the cubic specimens was recorded and instrumented in terms of cube longitudinal and transverse dimension for several engineering strain values. The Poisson’s ratio variation for 5% to 20% engineering strain interval is depicted in Fig. 5. Even though for ballistic impacts, due to high strains involved in the process, the relevance of the Poisson’s function is low, for other applications which involve only elastic deformations it becomes relevant.

Fig. 5
figure 5

Poisson ratio versus Engineering strain.

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The recorded compression behavior of REGUPOL E43 in terms of engineering stress versus engineering strain is detailed in Fig. 6 for both loading and unloading stage. As indicated by the engineering stress versus engineering strain curve, the behavior is typical for a hyperelastic material. The performed tests refer only to one load-unload cycle and were not designed to capture insights regarding Mullins effect since the specific studied application (ballistic application) does not involve such approaches.

Fig. 6
figure 6

Representative engineering stress versus engineering strain curve in compression.

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Due to the fact that the testing equipment enables force and displacement record, the variation of Young’s modulus as a function of engineering strain was possible to be instrumented as depicted in Fig. 7.

Fig. 7
figure 7

Compressive modulus evaluation.

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Once the variation of Young’s modulus and Poisson’s ratio is known, the bulk modulus, K, and shear modulus, G, can be calculated based on the Eqs. (1) and (2):

$$K=\frac{E}{3(1-2\mu )}$$
(1)
$$G=\frac{E}{2(1+\mu )}$$
(2)

where: E is the Young’s modulus and μ stands for Poisson’s ratio.

Since a higher detailed characterization of REGUPOL E43 mechanical behavior requires assessing both the engineering stress versus engineering strain curve in tension along with tensile fracture stress and strain, several tests were carried out in a test configuration as depicted in Fig. 8. For tension tests INSTRON 68TM-50 equipment was used along with INSTRON 2712–116 Series products for screw side-action grip with serrated face.

Fig. 8
figure 8

Schematic representation regarding the determination of tensile behavior.

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The tests were performed at the same strain rate as in compression case, allowing in this way a complete definition of engineering stress versus engineering strain curve for a given strain rate.

For both tensile and compression tests a value of 3.9 MPa for the initial elastic modulus (the slope of the stress–strain curve in the elastic region) was founded. Furthermore, as Figs. 6 and 9 depict, viscoelasticity behavior is obvious for both tensile and compression loads.

Fig. 9
figure 9

Representative tensile test results.

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As depicted in previous figures, the measurements were performed only for quasi-static regime. Although strain rate sensitivity is expected for the tested material, the mechanical characterization did not include such tests mainly because the stopping mechanism of REGUPOL and other similar materials refers to friction instead of strength.

DMA results

It is notorious that the mechanical properties of composite materials and polymers are strongly related with the mechanical properties of the component materials and the interaction between those materials. Since REGUPOL E43 consists of PU bounded rubber granules, its mechanical behavior will be set by both rubber and PU. A comparative analysis between different rubber like materials recipes in term of the best choice for the binder nature and mechanical properties can be achieved based on DMA tests. In order to highlight the particular behavior of the REGUPOL E43 and the importance of the binder nature in a shock absorber/ballistic rubber material, the REGUPOL E43 DMA results were compared in Fig. 10 against DMA results of a PU sample developed to be a binder in a composite rocket propellant mixture34.

Fig. 10
figure 10

Dynamic mechanical analysis of REGUPOL E43 sample and polyurethane reference.

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As indicated in Fig. 10, the REGUPOL E43 storage modulus (E') curve exhibits lower values in the rubbery plateau region compared against binder curve. E' is a measure of material stiffness and can be used to provide information regarding polymer molecular weight, degree of cure and cross-link density. The difference between the storage modulus (ΔE') in the plateau regions before and after the glass transition is related to the degree of cross-link density. A smaller ΔE' is associated with greater cross-link density. Since in the REGUPOL E43 case ΔE' is higher compared with the PU case, one can conclude that the cross-link density is lower in the REGUPOL E43 analyzed sample. In the same figure it can also be identified that the glass transition temperature (Tg) of REGUPOL E43 is approximately − 57 °C, the same as of the binder (PU). Although the two specimens present a similar Tg, in case of loss modulus (E") the PU exhibits a sharper and higher peak compared against REGUPOL E43. Generally, the value of E" is of great importance since high values of E" suggest greater mobility of the polymer chains associated with dissipation of energy when the polymer is subjected to deformation. Thus, polymers exhibiting a high and broad E" transitions possess the ability to absorb energy associated with impact. In fact, this kind of behavior can also be observed in the tan (delta) versus temperature curves. A sharper tan (delta) transition suggests more uniform cross-links. Due to the solid rubber particles in the REGUPOL E43 structure, the tan (delta) transition suggests, as expected, a high degree of cross-link heterogeneity compared against PU case.

Morpho-structural analysis results

Since in the case of REGUPOL E43 samples subjected to mechanical compression the optical instrumentation of failure is more difficult, the analysis was firstly performed on tensile specimens. In Fig. 11 are depicted the images recorded with the use of an optical microscope on the fracture surfaces of the material. The purpose of the investigation was to clarify if when subjected to quasi-static mechanical load, the rubber granules are damaged (break) or if the binder is affected by detaching from the surface of the rubber granules. To highlight the difference between a granule without PU and one coated with PU, in Fig. 11.C two granules are compared. Thus, as depicted in Fig. 11 (from detail A.1 to detail B.8), one can observe areas on the surface of the rubber granules, detailed in yellow, where the binder is detached, and the rubber granules remains structurally unaffected, which suggest that most likely the failure is controlled by PU detach.

Fig. 11
figure 11

Structural optical microscope analysis of REGUPOL specimens.

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To fully elucidate the nature of the failure of the specimens, the optical analysis was supplemented with a micron level analysis using scanning electron microscope (SEM) technology. The SEM analysis was performed on three types of specimens: rubber granules (without PU), REGUPOL E43 sample part of a cubic specimen prior compression test and a REGUPOL E43 sample part of a cubic specimen after performing the compression test. For a detailed representation, the three configurations were investigated at different scales (500 µm, 100 µm and 50 µm), as depicted in Fig. 12. The images recorded following the analysis pointed out that the surface of the rubber granules is rough (see Fig. 12, details a.2 and a.3), while for the REGUPOL E43 samples both untested and tested the surface of the rubber granules is covered with a thin layer of PU (see Fig. 12, images b.2, b.3 and c.1). Furthermore, in the image b.1 (yellow color contours) one can see how the rubber granules bond. As indicated in details c.2 and c.3 one can observe that after testing the rubber granules did not suffer any fracture and only the PU is affected as highlighted with a blue outline.

Fig. 12
figure 12

SEM images.

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The morphological results allowed us to conclude that at low strain rate the failure of the REGUPOL E43 is due to PU failure. Instead, when high strain rate and high strains are induced the failure mechanism is changing both PU and rubber granules exhibiting failure (Fig. 13).

Fig. 13
figure 13

Dynamic test (a. Impact (specimen initial strain rate 4 × 103 s−1), b. SEM image of the recovered debris – fracture propagation through rubber granule circle in yellow).

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Ballistic results

Ballistic limit test results are listed in Table 1. According to the listed values with the decrease of impact velocity an increase in velocity variation (variation between impact and residual velocity) occurs.

Table 1 Impact versus residual velocity.
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Usually, the identification of ballistic limit by experimental means is ammunitions expensive especially due to the mixed results one obtains around the ballistic limit value (rounds with impact velocity below ballistic limit will completely penetrate the target while rounds with a higher impact velocity than the ballistic limit will not completely penetrate the target). This is why over the time several models were developed to approximate with some degree of confidence the ballistic limit based on a reduced number of tests. Among those models Recht–Ipson35 and Lambert–Jonas36 are widely used. In fact, as proved in37 the results provided by the two models are similar. For the present study the Lambert–Jonas equation was considered. It allows a raw initial evaluation of the ballistic limit based on just three tests which result in a complete penetration. Thus, according to Lambert and Jonas the residual velocity in normal impact can be express as

$${v}_{r}=\alpha {({v}_{i}^{p}-{v}_{bl}^{p})}^{1/p}$$
(3)

where vi is the impact velocity, vbl stands for ballistic limit velocity and α and p are model constants.

By solving Lambert–Jonas equation based on the data listed in Table 1 one obtains α = 0.977, p = 1.921, and a ballistic limit of 113 m/s (Fig. 14). The correlation coefficient between the Eq. 3 results and the measured results is 0.999589 with an average squared difference of 12.626 m/s.

Fig. 14
figure 14

Ballistic limit velocity.

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As proved in the case of cylinders rebounding from water, ricochet risk at angle below critical angle is correlated with the impact velocity38. In order to elucidate if the same issue can be related to the case of REGUPOL E43 and 9 × 19 mm ball impact phenomenon, tests for ricochet risk assessment were performed for two different average impact velocity: 362 m/s (value around initial velocity of 9 × 19 mm ball involved in trials) and 181 m/s. The 181 m/s velocity is not arbitrary but instead was chosen having in mind that due to ball velocity drop during ricochet the ball energy will most likely drop around critical lethal energy which is about 80 J (ball mass is 7 g). The results recorded during tests are listed in Table 2.

Table 2 Ricochet assessment.
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As indicated in Table 2 the ricochet often results in ball flight stability alteration when the 9 × 19 mm ball sinks and then reemerges from the REGUPOL E43 sheet. Most likely this result is related with the fact that, during movement through target, the ball interacts with rubber granules resulting in a flight path alteration that varies from case to case. This phenomenon has been revealed only for 8, 9, 10 and 12 degrees impact angle tests (Fig. 15).

Fig. 15
figure 15

Flight instability of the 9 × 19 mm ball following the impact with the inclined REGUPOL sheet.

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The tests results point to a critical angle of 12 degrees for the investigated average velocities and to a dependency between impact velocity and impact angle in terms of ricochet risk, also. As the results listed in Table 4 underline the critical angle increases with the decrease of impact velocity. It is worth mentioning that for the instrumentation of ball trajectory stability a high-speed camera was used and for impact velocity a radar equipment (unidirectional system).

Numerical approach

Nowadays, FEM based codes are often used being considered useful engineering tools both in the pre-design and design stage of a product. The main FEM approach advantage is directly related with the fact that once the numerical model is set and calibrated based on some specific mechanical tests one can numerically investigate different configurations or/and set different initial data with minimum effort and costs. Thus, the costs and time associated with new products development or testing are significantly decreased.

When it comes to numerical simulation, using a 3D model with a finer mesh and plasticity and failure model associated to materials is more likely to ensure a high precision of the calculus. Nevertheless, sometimes a fair correlation with the experimental tests can be achieved with more simple approaches (2D models, coarse mesh, Eulerian methods, or ALE solver, and so on) which tend to be more time efficient. In fact, this is what is aimed for the studied application, namely the ballistic impact between 9 × 19 mm ball and REGUPOL E43 sheet.

In order to reduce the time needed to complete the numerical calculus as much as possible, a 2D plane-symmetric model was adopted (Fig. 16). The imposed boundary conditions were similar to the ones in the experiment (Figs. 2 and 3). The numerical model was set using LS-DYNA software.

Fig. 16
figure 16

FEM model.

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A hyperelastic material type, namely *MAT183-SIMPLIFIED_RUBBER_WITH_DAMAGE was chosen for the target. The huge advantage of MAT183 is that no additional calculus for material coefficients is needed since the implementation in numerical calculus requires a single uniaxial load curve for loading and a single uniaxial load curve for unloading. Along with the mentioned curve transverse and bulk modulus and material density are also required. Thus, a non-homogenous material structure like REGUPOL E43 can be converted numerically into an isotropic and homogenous material. In order to accommodate this approach, several measurements were carried out to estimate the average density of REGUPOL E43, the results pointing to a value of 0.84 gm/cc. In regard to single uniaxial load curve for loading and unloading average values of the experimental recorded curves (both tensile and compression) were used.

Worth mentioning that *MAT183 allows strain rate effect consideration also if the single uniaxial load curve is replaced by a table that accounts for different strain rates. However, for the current study no strain rate effect was considered since the main goal is to investigate if a simple robust time saving model can ensure fair correlation with experimental tests and consequently enable a starting point in the test program. Moreover, the stopping mechanism of REGUPOL E43 and other similar materials are strongly related with the friction between ball and target plate.

The input deck used for the target is exemplified in Table 3.

Table 3 *MAT183-SIMPLIFIED_RUBBER_WITH_DAMAGE.
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The uniaxial curve load used for MAT183 is the one depicted in Fig. 17 and consist in 2818 × 2 values.

Fig. 17
figure 17

FEM model.

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For the ball parts, an elasto-plastic material type with strain rate and temperature dependency was adopted, namely *MAT015-JOHNSON_COOK, the material coefficients (Table 4) being adopted from39,40,41. Since MAT015 requires the use of an equation of state, Mie-Grüneisen equations of state were added (Table 5) for the lead and gliding metal (jacket materials). Failure or fracture models were not considered since as experimental tests proved the ball did not experience fracture.

Table 4 *MAT015-JOHNSON_COOK material data39,40,41.
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Table 5 Mie-Grüneisen equation of state coefficients39.
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The interaction between a projectile and a target can be numerically modeled using different approaches, each one with its own advantages and disadvantages. A core characteristic of using Lagrangian approach is that in order to allow the target penetration one must consider the erosion of the target’s failed elements. In fact, this kind of approach can be used even for calibrating the model for the target material as pointed out in40. A second option is to use Eulerian approach in which the erosion criteria can be avoided, both the projectile and target being considered fluids. Unfortunately, in this case some difficulties regarding the accurate representation of projectile/target interface occurs. A way to overcome in some degree the shortcomings of the first two options is to consider either SPH (Smooth Particle Hydrodynamics) or ALE (Arbitrary Lagrangian Eulerian) approaches. Even though the SPH option on first sight seems appealing, the computational resources demanded by this particular solver could prove prohibitive for large models. Having in mind that REGUPOL material is a classic example for friction bullet-stopping mechanism1 and based on previously mentioned reasons for current study, the ALE method was adopted. Thus, the 9 × 19 mm ball was modeled as Lagrangian part while the REGUPOL E43 sheet was considered a fluid being modeled as Eulerian part. Since in ALE approach the fluid must be able to flow, above and beneath the REGUPOL E43 sheet a VOID part with the same dimension as the target was defined. For the VOID part a MAT_VACUUM characterized by a density of 1.2 × 10−12 tons/cm3 was assigned. For the Lagrangian part a plane strain shell element was consider and for the target and void part a multi-material ALE element was adopted.

Considering FEM models mesh sensitivity, simulations were carried out for three different mesh dimensions of the target (1 × 1 mm, 0.5 × 0.5 mm and 0.1 × 0.1 mm) regarding ballistic limit test case. Moreover, since no erosion criteria for target material was assigned, five different values for the friction coefficient (0, 0.05, 0.1, 0.2 and 0.3) between 9 × 19 mm ball and REGUPOL were used as second parameter to calibrate the model regarding ballistic limit tests. The calculated results are listed in Table 6.

Table 6 Mesh and friction coefficient sensitivity analysis.
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For 359 m/s impact velocity case the results listed in Table 6 points out that the relative error between measured and calculated values decreases as mesh dimensions decreases. However, for the other two cases (172 m/s and 117 m/s) mesh refinement results in an increase of results mismatch. In fact, the only case in which the results for all three cases are around the measured one (differences of up to roughly 10%) corresponds to a 1 × 1 mm mesh case and a 0.3 coefficient of friction. As a result, the 1 × 1 mm mesh and 0.3 coefficient of friction FEM model was adopted for the numerical investigation of the ricochet risk.

In order to furthermore evaluate the suitability of the selected model the ballistic limit was investigated in terms of numerical results which pointed to a 108 m/s impact velocity. Comparing the numerical versus theoretical predicted ballistic limit (102 m/s vs. 113 m/s) is obvious that numerical results fall short with less than 10%.

The proposed numerical model also has the ability to catch REGUPOL E43 hyperelastic behavior in terms of temporary cavity. As indicated in Fig. 18b behind 9 × 19 mm ball trajectory the cavity closes which match the empirical observation regarding no permanent cavity (Fig. 18a).

Fig. 18
figure 18

Permanent and temporary cavity.

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Once calibrated, the FEM model was used to investigate the critical angle for ricochet considering two average impact velocities (362 m/s and 181 m/s) and several impact angles (2°, 8°, 9°, 10° and 12°).

The results of the 2D simulation regarding ricochet risk assessment are depicted in Fig. 19 in terms of ball center of mass trajectory (solid red line indicates ball CM trajectory and doted blue line refers to REGUPOL upper limit).

Fig. 19
figure 19

Ball center of mass trajectory (a impact velocity 181 m/s, b impact velocity 362 m/s).

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As pointed out in Fig. 19, the numerically calculated critical angle is less than the one experimentally observed. Moreover, the difference between the calculated and the measured value increases with the decrease of the impact velocity. Thus, for the 362 m/s average impact velocity the difference is two degrees (10° experimental versus 8° numerical calculated) while for the 181 m/s case the difference increases to three degrees (12° experimental vs. 9° numerical calculated). Even though one can argue that the error is high (up to 25%), the calculated values are of real help since they are able provide with fair accuracy a starting point for the experimental tests reducing thereby the ammunition consumption. The error should also be put in context since as pointed out in42 the yaw angle has influence regarding the impact angle.

Conclusions

This paper investigates the ballistic limit of a REGUPOL E43 sheet impacted by a 9 × 19 mm ball along with the ricochet risk posed by this ammunition impacting the REGUPOL E43 sheet. Using a wide palette of mechanical and ballistic tests it was possible to set a LS-DYNA numerical model to facilitate the assessment of the critical angle above which ricochets are no longer possible despite 9 × 19 mm ball velocity. The numerical data were confirmed by experimental data.

Based on the numerical and experimental analysis performed in the current study the following conclusions can be drawn:

  • Due to its nature (rubber granules and PU mix) REGUPOL E43 has a typical hyperelastic response to mechanical loads.

  • The quasi-static mechanical tests pointed out an initial average Young’s modulus of 3.9 MPa.

  • The Poisson ratio varies with the engineering strain between 0.39 and 0.43.

  • Regupol®E43 has a relatively low ultimate stress in tensile load and a much higher limit in compression.

  • Due to its particular structure REGUPOL E43 has an apparent density of 0.84 g/cm3.

  • As indicated by compression and tension tests REGUPOL E43 has an elasto-visco-plastic behavior and the ability to store energy.

  • For − 45 to 50 °C temperature range DMA tests indicated that REGUPOL E43 response to dynamic load is strongly correlated with the behavior of PU.

  • SEM analysis pointed out for quasi-static load a particular failure mechanism for REGUPOL E43, namely PU peeling. The failure mechanism changes for high strain rate load both PU and rubber granules experimenting failure.

  • The estimated ballistic limit for REGUPOL E43 sheet impacted by a 9 × 19 mm ball is relatively low (113 m/s) as expected since its main destination is ricochet mitigation.

  • The critical angle for REGUPOL E43 sheet impacted by a 9 × 19 mm ball is 12 degrees when impact ball velocities vary between 181 m/s and 362 m/s.

  • A more precise identification of the ricochet critical angle must consider the instrumentation of yaw angle on impact.

  • 2D FEM simulation can be an expeditious way to evaluate critical ricochet angle and reduce ammunition consumption.

  • A 3D FEM simulation (with the consideration of yaw angle) is most likely to ensure a more precise calculus but also a serious increase in computational time.

  • *MAT185 is an appropriate and easy to implement constitutive model for REGUPOL E43 in LS-DYNA numerical simulations.

  • A more accurate results fit most likely could be expected using numerical model that account for strain rate sensitivity.

  • Numerical approach is a quick and costless method to estimate ballistic properties.