The penetration of limestone targets by rigid projectiles: Revisited

Abstract

We reanalyze the results from a set of terminal ballistics tests in which large limestone targets were impacted by similarly shaped rigid projectiles having different sizes. Our first goal is to show that the data in terms of penetration depths as a function of impact velocity can be accounted for by a model in which the limestone’s resistance to penetration is constant throughout the penetration process. It turns out that the actual values of this penetration resistance, as derived from the data, decrease with the projectile’s size. This is a clear manifestation of a scaling issue in the terminal ballistics of the limestone rock. In order to account for this issue we assume that it is closely related to the size effect in the compressive strengths of rock specimens. Finally, we offer a simple procedure by which one can define a surrogate material model for the targets in numerical simulations, in order to predict the penetration depths in limestone, and possibly other rock targets, under both normal and oblique impacts.

Keywords

Terminal ballistics limestone rocks scaling issue size effect Hoek–Brown relation

Introduction

The penetration of rigid projectiles into concrete targets has been the subject of numerous works for many decades, due to its importance for both military and industrial applications. These works include experimental data, empirical and semi-analytical formulations, and numerical simulations with various constitutive relations for concrete. The review articles of Dancygier (2000), Li et al. (2005), Ben-Dor et al. (2014), and Yankelevsky (2017) summarize these issues in a detailed fashion, covering hundreds of relevant papers on concrete penetration. It is clearly evident that the unconfined compressive strength (f_c) of the concrete is the main constitutive parameter which governs the resistance to penetration of these targets, in all of the empirical relations and the analytical models which deal with their penetrations by rigid projectiles.

In sharp contrast with the data for concrete targets, there are very few published studies devoted to the penetration of rigid projectiles into rock targets. Layers of rock are usually used as protective means for buried bunkers and other structures, but one can find only a few works in the open literature on projectiles impacting rock targets, as in Rohani (1975) for example. However, these data do not span a large range of impact velocities, and they often lack detailed specifications for the projectiles and the rock targets. An exception for this statement is the work of Frew et al. (1999) who impacted similarly shaped projectiles, with different masses and diameters (D), at large Salem (Indiana) limestone targets. Impact velocities of these CRH3 ogive-nosed steel projectiles were in the range of 0.5–1.5 km/s, and their penetration depths (P) were accurately measured. It turned out that these penetration depths do not scale linearly with the size of the projectile, as the larger projectiles resulted in higher normalized penetrations (P/D). Thus, a clear scaling issue is evidenced in the data of Frew et al. (1999) for these projectiles, with diameters of: 7.1, 12.7, and 25.4 mm. The main purpose of our work is to reanalyze the results of these tests, and to offer an explanation for the scaling issue in the terminal ballistics of these limestone targets. We also present a simple scheme to predict penetration depths into limestone targets, at normal and oblique impacts, by using numerical simulations with equivalent-strength surrogate targets. This algorithm circumvents the difficulties with the constitutive relations and the equations of state for rocks, which have to be carefully calibrated in order to be used in numerical simulations.

Analysis of the penetration data in limestone targets

One of the most important observations by Frew et al. (1999) is the fact that the penetration profiles in all of their tests consisted of two distinct parts: a shallow crater at the impacted face of the target, having a depth of about two diameters (2D), followed by a long penetration channel with a diameter which is equal to that of the projectile. This experimental fact implies that the resisting forces on these projectiles, beyond the short entrance phase, were constant throughout the penetration process, independent on their velocity. This conclusion is the result of a simple consideration based on energy conservation, as was shown by Rosenberg and Dekel (2016) for both metallic and concrete targets. Their argument starts with the realization that the kinetic energy lost by the projectile (dE_k) as it advances for a distance of dX is equal to the work done (dW) in order to displace the volume A_tdX of target material, where A_t is the cross-section of the penetration channel. Thus, one can write dE_k = dW = KA_tdX, where K is a constant. This relation is simply an energy-conservation statement which holds for every element dX, wherever it is located. In principle, A_t can change along the projectile’s trajectory so that one should have A_t(X) in the above equation The next step is to realize that if A_t(X) is constant throughout the whole process, one can integrate this equation, from E₀ at X = 0 to E = 0 at X = P, and obtain E₀ = KA_tP where P is the final penetration depth. This relation states that the initial kinetic energy of the projectile is proportional to the volume of the penetration channel. The initial kinetic energy is: E₀ = MV₀²/2, and the projectile’s mass is given by M = A_pρ_pL_eff, where A_p, ρ_p, and L_eff are the cross-sectional area of the projectile, its density, and its effective length, respectively. Thus, one obtains the following relation: A_pρ_pL_effV₀²/2 = KA_tP. The crucial point now is to realize that if A_p = A_t these terms cancel out and we get P = kV₀², where k is another constant. So far we have shown that if the penetration channel has the same diameter as the projectile, then energy conservation leads to a quadratic dependence of P on V₀. This dependence is the well-known Robins–Euler relation for a penetration process with a constant resisting stress on the projectile, independent on its velocity. The Euler–Robins relation is obtained by integrating the equation of motion (Newton) F = Ma, assuming that the deceleration (a) is constant. Integration of this equation results in P = kV₀², as shown in Rosenberg and Dekel (2016). Stated differently, the observation of a quadratic relation between P and V₀ means that the resisting stress is independent on the projectile’s instantaneous velocity. We have two indications that the resisting stresses on the projectiles in the work of Frew et al. (1999) were constant. The first is their observation that the penetration channels had constant diameters, which equal that of the projectile. Another indication can be obtained by checking the dependence of the penetration depths on the impact velocity, namely P = P(V₀). If this relation turns out to be quadratic (P = kV₀²), as derived above, one can safely assume that the resisting stress on the projectile does not depend on its velocity. We should note that analytical models which include the target’s inertia (ρ_tV²), in their resistance to penetration, follow the so-called Poncelet model. The resisting stresses on the projectiles, according to these models, depend on the target’s density and on the projectile’s velocity. The different penetration models for both metallic and concrete targets are reviewed in Rosenberg and Dekel (2016). They show that quadratic penetration relations, as well as constant channel diameters, enhance the constant deceleration (resisting stress) conjecture.

As mentioned above, the penetration channels in the limestone targets in Frew et al. (1999) had the same diameter as the projectile and, as we show below, the empirical penetration depths of the three types of projectiles can, indeed, be represented by simple quadratic relations: P = kV₀². We elaborated on this constant penetration resistance issue because it is not self-evident that this is also the case for the limestone targets which are quite porous, as compared with metallic and concrete targets. We should also note that the deceleration of a rigid projectile can be velocity-dependent, if its impact velocity is higher than a certain threshold which marks the cavitation phenomenon, as was shown by Rosenberg and Dekel (2016). In these cases, the diameter of the penetration channel is larger than that of the projectile, and cavitation (or separation) takes place. At these higher velocities a target inertia term (ρ_tV²) has to be added to the strength-related resisting stress, in order to account for the target’s resistance to penetration.

We can conclude that apart from a short entrance phase, which takes place at the spalled front layer of the limestone target, the resisting stresses on the projectiles in the work of Frew et al. (1999) were constant throughout their penetration. The penetration process through the spalled-off layers can be accounted for by the method suggested in Warren et al. (2014) for concrete targets. According to this scheme, the resisting stress during the entrance phase increases linearly up to a depth of about 2D, which is the extent of this phase for projectiles with CRH3 ogive nose shape. This procedure was implemented by Rosenberg and Kositski (2016), who proposed a semi-analytical model that accounts for many sets of penetration data in concrete targets. They derived the following relation between the penetration depth (P) in a thick concrete target and the impact velocity (V₀) of a rigid projectile

P = \frac{ρ_{p} L_{e f f} V_{0}^{2}}{2 R_{t}} + D

(1)

where R_t is the concrete’s resistance to penetration, ρ_p and L_eff are the projectile’s density and effective length, respectively, and their product is given by ρ_pL_eff = 4M/πD². We should note that the constant resisting stress (R_t) depends on the nose shape of the projectile and on the concrete’s strength, but not on its density (ρ _t ). As shown by Rosenberg and Kositski (2016), the addition of the term D in equation (1) is due to the entrance phase (the spalled-off layer) in the concrete target, which has a thickness of about 2D when impacted by a CRH3 ogive-nosed projectile. The concrete is characterized by its unconfined compressive strength (f_c) as measured statically with cylindrical or cubic test specimens. By using the data for R_t and f_c of various projectile/concrete pairs, Rosenberg and Kositski (2016) obtained the following empirical relation for the penetration resistance of concrete targets

R_{t} = 0.25 l n (\frac{f_{c}}{2.7})_0.046 (\frac{D}{d})

(2)

where R_t is given in GPa and f_c in MPa, and d is the maximal diameter of the aggregates in the concrete. We should note that the second term in this equation, with the projectile/aggregate diameter ratio (D/d), accounts (empirically) for the scaling issue in concrete penetration. Relation (2) includes the unconfined compressive strength of the concrete (f_c) as its main constitutive parameter, in determining its resistance to penetration. In this sense it is not different than the all the models for concrete targets, both empirical and semi-analytical, as reviewed by Rosenberg et al. (2020). One may argue that this entity cannot be considered as a true material property for the resistance to penetration, because the target material around the projectile is experiencing a complex, three-dimensional (3D) state of stress. However, one should note that f_c itself is not identified as the resistance to penetration, and it is only a measure of this resistance. The true penetration resistance is found to be in the range of (10–20)f_c as can be seen in equation (2), as well as in other models for concrete penetration. The fact that f_c has to be multiplied by such factors means that the pressure dependence of the concrete’s strength is, effectively, taken into account in these semi-empirical models. Moreover, these factors are different for concretes with various f_c values, pointing to their different sensitivities to pressure. This issue will be further discussed when we deal with the pressure dependence of the limestone’s strength, as well as the role of its equation of state on the penetration process.

We now use equation (1) in order to account for the limestone data in Frew et al. (1999) for their three different-sized CRH3 ogive-nosed, L/D = 10, projectiles with masses of 20, 117, and 931 g and diameters of 7.1, 12.7, and 25.4 mm, respectively. Most of these projectiles were made of 4340 steel (hardness of 43–45 Rc), and several were made of AerMet100 steel (hardness of 53Rc). The projectiles remained essentially rigid throughout their penetration process, with relatively small mass losses at the highest impact velocities. Using a least-square procedure we found the following values of R_t that best fit the experimental data of Frew et al. (1999), through equation (1): R_t = 1.1, 0.99, and 0.825 GPa, for the D = 7.1, 12.7, and 25.4 mm projectiles, respectively. Figure 1 shows the good agreement between the data for the three projectiles and the corresponding predictions through equation (1) with these values for R_t. Particularly, the dependence of P on V₀² is very important here since it enhances our constant-resisting-stress conjecture. In fact, the agreement between the model and the data is even better if one considers some experimental details in Frew et al. (1999). For example, the highest velocity point in the data for the 7.1 mm projectile resulted in a projectile mass loss of about 11%, which can account for the downward offset of this point from the predicted curve in Figure 1. Since the experimental data span the whole ordnance velocity range from 0.5 to 1.6 km/s, one can safely state that the constant-resisting-stress conjecture has been proven for the whole range of relevant velocities, and it is not limited to low impact velocities. The limestone targets used by Frew et al. (1999) were taken from three batches having densities of 2.30–2.32 g/cm³ and porosities in the range of 14.4%–15.1%. Their nominal unconfined compressive strengths ranged between 58 and 63 MPa, with an average value of f_c = 60 MPa.

Figure 1.

Fitting equation (1) with the limestone data for projectiles with D = 7.1 mm (squares), D = 12.7 mm (circles) and D = 25.4 mm (triangles).

The three different values of R_t, which are needed to fit the model with the data, highlight the scaling issue in the limestone’s penetration process. Obviously, larger projectiles result in higher normalized penetrations (P/D or P/L_eff) since they encounter lower resisting stresses (R_t). Note that unlike concrete, the limestone targets did not include hard aggregates which may have caused this scaling issue. We should also note that our R_t values are somewhat higher than the corresponding values in Frew et al. (1999), since their analysis is based on the assumption that the resisting stress on the projectile includes a term related to the target’s inertia (ρ_tV²), for all impact velocities. In addition, both our model and that of Frew et al. (1999) ignore the possible effect of friction between the projectile and the limestone. This is also the case in the models of Forreatal and his colleagues for concrete targets. The friction issue was discussed in Rosenberg and Dekel (2016) for penetrations in metallic targets, and it was found to have a small effect (at most 3%) on the penetration depth. On the contrary, Chai et al. (2019) claim that friction can play a role of up to 10% on the penetration into metallic and concrete targets. We do not deal with this issue here, mainly because we do not have data to assess the effect of friction on the limestone targets.

It is clear that in order to derive an empirical penetration relation for rock targets, which is equivalent to equation (2) for concretes, one needs penetration data on other rocks with various compressive strengths. Moreover, even if such an empirical relation can be constructed, one would still have to explain the physical reason for the scaling issue in the limestone’s penetration data. In the next section we offer such an explanation by relating the scaling issue to the well-known size effect in rock specimens, namely the size dependence of their unconfined compressive strengths. This connection was made recently by Rosenberg et al. (2020) in order to account for the scaling issue in the terminal ballistics of concrete targets. Unlike the situation with rock targets, there is a lot of data on the penetration of concrete targets by rigid projectiles, and the analysis in Rosenberg et al. (2020) makes use of this vast amount of data. Both the scaling issue and the size effect are due to the brittle behavior of rocks and concretes, and they have to be analyzed through fracture-mechanics considerations. Such analyses are outside the scope of the present work, and readers are referred to the review article by Bazant (2000), for various size effects in brittle materials like rocks and concretes. We should also note that the scaling issue in the terminal ballistics of concrete targets has been recently approached by several groups using meso-scale numerical simulations, as in: Peng et al. (2018), Feng et al. (2018), and Wu et al. (2019). These simulations used targets with different aggregates, both in size and strength, looking for differences in the penetration efficiencies of different-sized projectiles, in terms of their normalized penetrations (P/L or P/D). These simulations showed that the scaling issue is related to the D/d ratio, as in the empirical model of Rosenberg and Kositski (2016), but they did not account for the full extent of the scaling effect. Thus, one has to look for other reasons for the scaling issue, as we do next by following the approach in Rosenberg et al. (2020) for concrete penetration.

The size effect in the unconfined compressive strength of rocks

The unconfined compressive strengths of rock specimens depend significantly on their size, resulting in lower strengths for larger specimens. This empirical observation has been reported by many researchers, and it was thoroughly analyzed by Hoek and Brown (1980), who used results from various sets of data for different rocks (including limestone). They suggested the following empirical relation for the unconfined compressive strength f_c(D_s) of a rock specimen with a diameter of D_s

f_{c} (D_{s}) = f_{c} (50) {(\frac{50}{D_{s}})}^{0.18}

(3)

where f_c(50) is the corresponding strength of a reference specimen with a diameter of 50 mm. As pointed out by Darlington et al. (2011), this type of relation results in zero strength value for very large specimens, and one has to add some limiting value to f_c at large values of D_s. A close examination of the various data sets in Darlington et al. (2011) shows that, indeed, the compressive strengths reach an asymptotic value for specimen diameters of about 300 mm. Still, the Hoek–Brown relation in equation (3) accounts very well for specimens with diameters in the range of 10–100 mm and we shall use it below. It is also clear by the nature of equation (3) that one can use any other reference specimen, with a diameter of D₀ instead of 50 mm, as long as the compressive strength of this specimen, f_c(D₀), is known. Thus, one can rewrite equation (3) for the compressive strength of a given rock specimen with a diameter D

f_{c} (D) = f_{c} (D_{0}) {(\frac{D_{0}}{D})}^{0.18}

(4)

We should note that similar relations were found for concrete specimens by Bazant (1984) and by other workers, who used fracture-mechanics arguments to account for these size effects. These arguments are usually related to the stress concentrations around in-homogeneities in the brittle solid, such as the aggregates in concrete. In fact, fracture theories attach a characteristic length to the brittle solid, which is responsible for these size effects, as discussed by Hoek and Martin (2014). More data on the size effect in concrete specimens, and their modeling, can be found in Del-Viso et al. (2008), Sim et al. (2013), and the references therein.

We now make the heuristic assumption that during penetration the projectile has to overcome a rock cylinder which has (roughly) the projectile’s diameter. Thus, the larger projectile has to push aside a larger cylinder which has a lower strength due to the inherent size effect. This simplistic view can be justified, at least qualitatively, by noting that during penetration the target elements which are severely strained are those inside the penetration channel or very close to it. This observation was noted by Heuze (1990) who reviewed empirical, analytical, and numerical models for the penetration process in rocks. He stated that “an order of magnitude decay in the stresses induced around the projectile’s body, takes place over a radial distance of about 2.5 times the projectile diameter; this gives the scale of the volume of target material involved in controlling the penetration.”

We turn now to the limestone targets in the work of Frew et al. (1999), with an average unconfined compressive strength of f_c = 60 MPa. The cylindrical test specimens which they used in order to determine this strength were 107 mm long with diameters of 50 mm, as reported in Frew (2001). Thus, we can use D₀ = 50 mm as the reference diameter, together with the value of f_c(50) = 60 MPa, for their test specimens. Inserting these values in equation (4) results in the values of: f_c = 85.2, 76.8, and 67.8 MPa, for limestone specimens with diameters of 7.1, 12.7, and 25.4 mm, respectively. Following the nomenclature of Rosenberg et al. (2020) we designate these f_c values, for cylinders with the projectiles’ diameters, as: f_c* = f_c(D_p). This entity is, in fact, a property of the given projectile/rock pair which is more physically based than f_c, since it includes the size effect in the compressive strength of the rock, through the projectile’s diameter. By defining this new entity we circumvent a major problem in the terminal ballistics of these materials (concretes and rocks), which is due to the fact that their compressive strengths (f_c) are not determined with a universally accepted size for the test specimens. Different workers use various specimen sizes ranging from D₀ = 50 mm to D₀ = 200 mm, which may result in different compressive strengths even for the same material.

We now look for an empirical relation between the R_t and f_c* values for the three projectiles impacting the limestone targets. We are aware of the scarcity in the data points which we have here, as compared with the data for concrete targets in Rosenberg et al. (2020). Still, following their logarithmic relation for concrete targets, we find the following relation between R_t and f_c* for the limestone targets

R_{t} = 1.2 l n (\frac{f_{c}^{*}}{34})

(5)

where R_t is given in GPa and f_c* in MPa. Obviously, this relation cannot be applied for rock targets having very low compressive strength, and it should be checked against other data for rock penetrations, when these become available. In section “Granite targets,” we compare the predictions from this relation with experimental results from two impact tests on granite targets which were reported by Li et al. (2018).

The values we obtained above for the expected compressive strength (f_c*) of the three limestone cylinders correlate with the empirical values for the penetration resistance (R_t) which the limestone targets exerted on the corresponding projectiles. In fact, the ratios of R_t/f_c* for the three pairs range between 12.2 and 12.9, suggesting that, indeed, there may be a relation between the scaling issue, as manifested by the different R_t values, and the size effect in the expected compressive strength (f_c*) of the three limestone cylinders with the different dimensions. This correlation can account for the scaling issue in the limestone’s penetration data observed by Frew et al. (1999), as was also found by Rosenberg et al. (2020) for concrete targets penetrated by rigid projectiles of different size. We should note that these R_t/f_c* ratios are much higher than the corresponding ratios (R_t/Y) for metallic targets with strength of Y, which are in the range of 4–6, as shown by Rosenberg and Dekel (2009). As will be discussed below, this difference may be related to the fact that the strength of metals is practically independent on the pressure exerted on them, while the strength of concretes and rocks is very sensitive to the lateral stresses (pressures). These lateral stresses act on the target’s elements along and around the projectile’s nose during the whole penetration process. The same situation holds for concrete targets, as discussed by Rosenberg et al. (2020), and one may assume that their R_t/f_c* values are also dependent on the lateral stresses during penetration.

To summarize, our approach to the scaling issue in the limestone targets is based on the heuristic assumption that the resistance to penetration experienced by a rigid projectile is due to the specific cylinder in the target which has the same diameter as the projectile. This idea was used in our approach to the scaling issue in concrete targets, as shown in Rosenberg et al. (2020), where we had much more data to check its validity. According to fracture-mechanics theories, the fact that cylinders with different diameters have different strengths is due to some characteristic length which controls the fracture process of the brittle solid. In order to support our heuristic claim one should perform numerical simulations in which such a characteristic length is implemented in the constitutive model of the rock. The commonly used constitutive models for rocks (and concretes) include many physical/mechanical features which have to be calibrated by specific tests, but they do not include such a characteristic-length parameter. These lengths are also manifested in the values for the fracture toughness of brittle materials, which is also a non-scaling parameter. Another problematic issue with the properties of brittle materials concerns the rate dependence of their compressive strengths and its influence on the penetration depths of rigid projectiles. This issue is discussed in Rosenberg and Dekel (2016) and in Rosenberg et al. (2020), and we shall deal with it later on.

We see that in order to properly simulate the penetration process in rock targets, such as limestone, one has to determine various constitutive parameters of the rock such as its fracture toughness, the pressure and rate dependencies of its strength, and so on. In addition, the rock’s equation of state which is needed in all numerical codes has to be determined through various high pressures tests. This is a complex procedure and, as we discussed above, most of the data has yet to be determined for each rock. Due to these difficulties we chose a different approach, in order to highlight the scaling issue with the limestone targets. This approach is based on the concept of the equivalent-strength-target which we introduced in Rosenberg et al. (2020) for concrete targets. The basic idea here is to use surrogate targets as replacement for the brittle targets (concretes and rocks), in order to get useful information about their penetration characteristics in a simple way. We do not claim that a surrogate target can replace the rock target, due to the vast differences between the complex behavior of rock and the simple elasto-plastic surrogate material. Rather, we offer an easy way to circumvent the complexities of the mechanical and physical properties of the brittle solid, as described above. The proposed scheme is a suggestion for the defense engineer, either from the armor or from the anti-armor standpoints, to use a simple and quick simulation in order to assess the penetration depths for a given projectile/target pair, particularly for non-normal impacts, as we show next.

Numerical simulations with equivalent surrogate targets

The constant deceleration model described in this work provides an easy and straightforward method to predict the penetration depth of rigid projectiles at normal impact angles. While this method is straightforward for normal impacts it cannot be used to predict the penetration depth at high obliquity angles or other more complex scenarios. In this section, we generate a simple surrogate material model for the limestone that has the proper resisting stress R_t, in order to obtain predictions for penetration depths under non-normal situations. In fact, our scheme follows the ideas of Li and Chen (2008) who showed that one can treat penetration of rigid projectiles into metallic and concrete targets in a similar fashion, concluding that “the penetration results for various targets may be equivalent to each other.”

We used the 2D Lagrangian version of the LS-DYNA code to explore the penetration characteristics of the CRH3 ogive-nosed projectiles in Frew et al. (1999), with diameters of 7.1, 12.7, and 25.4 mm. The steel projectiles, with a Young’s modulus of 200 GPa and Poisson’s ratio of 0.3, were given a yield strength of 100 GPa in order to keep them rigid throughout the penetration process. The surrogate material is based on aluminum that was chosen as the target in these simulations since its density is close to the density of limestone. We should point out that any other surrogate material other than aluminum could also be chosen, as long as its yield stress maintains the correct value of R_t. The constitutive equation for the surrogate target was the simple elasto-perfectly-plastic von-Mises criterion, without rate dependence, and the erosion parameter was set at a value of 1.0. We checked other values for this parameter and found small changes in the corresponding penetration depths. For example, a value of 1.5 for the erosion parameter resulted in a decrease of about 2% in the final penetration depths. The size of the elements in the central part of the target (at a radius of 65 mm) was 0.3 mm, and it increased gradually away from this region. The dimensions of the targets were large enough to be considered as semi-infinite.

Our first step was to determine the proper strength values for the surrogate targets which will represent the penetration resistance values of the limestone rock. In order to achieve this goal, we use the analytical model of Rosenberg and Dekel (2009) for the penetration depths of rigid rods in metallic targets, as given by the following relation

P = \frac{ρ_{p} L_{e f f} V_{0}^{2}}{2 R_{t}}

(6)

where the target’s resistance to penetration (R_t) depends on its strength (Y), its Young’s modulus (E), as well as on the projectile’s nose shape. For CRH3 ogive-nosed rigid rods Rosenberg and Dekel (2009) derived the following numerically based empirical relation for R_t

R_{t} = Y [1.1 l n (\frac{E}{Y}) - 1.15]

(7)

We now use the three values derived above for the resistance to penetration of the limestone targets impacted by the three projectiles in Frew et al. (1999), namely: R_t = 1.1, 0.99 and 0.825 GPa. Inserting these values in equation (7), together with E = 69 GPa for the 6061 aluminum alloy, results in the values of Y = 210, 185 and 147 MPa, for the strength of the surrogate targets impacted by the three projectiles with D = 7.1, 12.7, and 25.4 mm, respectively. We then performed numerical simulations for the three projectiles at the corresponding impact velocities, as given by Frew et al. (1999) for the limestone targets. The three sets of simulations resulted in penetration depths in the surrogate targets which are very close to the penetration data for the limestone targets. In order to demonstrate this similarity we list in Table 1 the penetration depths in both the surrogate and the limestone targets for the D = 12.7 mm projectiles. The differences between the two sets of penetration depths are less than 10%, except for the V₀ = 1134 m/s shot. The projectile in this “problematic” test had relatively large inclination angles (yaw = 1.9°, pitch = 1.0°), and we shall discuss this test later on.

Table 1.

Penetration in limestone and surrogate (E = 69 GPa Y = 185 MPa) for the 12.7 mm projectile.

Impact velocity (m/s)	P in limestone (mm)	P in surrogate (mm)
459	141	132
608	232	215
853	362	383
956	523	472
1134^a	562	643
1269	812	790
1404	924	945
1502	1017	1070

The projectile in this shot had relatively large yaw and pitch angles, see text.

The corresponding results for the other projectiles, with diameters of 7.1 and 25.4 mm, showed similar agreements for the penetration depths in the surrogate and the limestone targets. This is the expected result since the corresponding penetration formulas, equations (1) and (6), are very similar, and we used the same R_t value for the surrogate material and limestone targets in each pair.

We notice that the values for the ratio Y/f_c* between the two sets of compressive strengths, are in the range of 2.2–2.4 with an average value of 2.35. Thus, in order to achieve a given R_t value, the equivalent aluminum target that was chosen as the base of the surrogate material model has to be stronger than the corresponding limestone target by this factor. This means that the effective strength of the rock during penetration is higher than its unconfined compressive strength. The enhanced effective strength is responsible for the limestone’s large values of R_t/f_c*= 12.5 which we found above, as compared with the corresponding values of R_t/Y = 4–6 for metals. This strength enhancement can be explained by the fact that the compressive strengths of rocks (and concretes) are strongly dependent on the confining pressures (lateral stresses) which act during penetration. These pressure sensitivities were determined by various triaxial tests for many rocks, as reported by Mogi (1974), Hoek (1983), Haimson and Chang (2000), and Singh et al. (2011), for example. In particular, the pressure dependence of Indiana limestone specimens with an unconfined compressive strength of f_c = 62.5 MPa is clearly seen in Heard et al. (1974) and in Singh et al. (2011) for f_c = 44.5 MPa limestone specimens.

In a detailed simulation study, Fossum (2004) examined the dependence of the penetration depth for a test projectile impacting a Salem limestone target, on the physical parameters in the rock’s constitutive relations. These parameters were calibrated for the same limestone which was used by Frew et al. (1999) in their ballistics tests. Fossum (2004) checked the role of these parameters by systematically turning off each one of them in the simulations, and he followed the corresponding change in the penetration depth of the test projectile. These parameters included the pressure and strain-rate dependencies of the rock’s strength, tensile cracking, nonlinear elasticity, kinematic hardening, compaction, and the Lode angle dependence. He found that turning off the pressure dependence of the rock’s strength resulted in the most significant effect (68%) on the penetration depth. For all the other features of the rock he found much smaller changes in the penetration depth (from 0.1% to 6%). It is interesting to note that turning off the rate sensitivity of the limestone resulted in an increase of only 6% in the penetration depth of the test projectile. Thus, the rate dependence of the limestone’s strength in Fossum’s study is an order of magnitude less important than its pressure dependence. Noting that the strain rates in the limestone targets of Frew et al. (1999) differed by factors of only 2–3, we may assume that the rate was not an important factor in the observed scaling issue. Fossum (2004) concluded that “the most important physical feature of a rock, with respect to penetration resistance, is its ability to get stronger with increasing mean stress..” This conclusion was also reached by Heuze (1990) who reviewed the empirical data for rock penetration and their numerical simulations. He stated that “Regarding constitutive laws and material properties, the most desirable strength formulation is that which describes the complete variation of shear strength with mean stress.” These assertions mean that one should not be too concerned with all the other properties of these materials as far as penetration depths are concerned. Thus, the use of equivalent-strength surrogate targets to represent the limestone targets seems reasonable, in spite of the many differences between the real rock and the surrogate model. This statement holds as long as one correctly represents the pressure-hardening properties of the rock by assigning the proper (higher) strength to the surrogate target.

Our next step was to check the validity of the surrogate target concept with experimental data and numerical simulations for inclined impacts on limestone targets, at 15° and 30° obliquities, as given by Warren et al. (2004). The limestone targets were taken from the same site as those of Frew et al. (1999), and the L/D = 10 ogive-nosed steel rods were the same as the D = 7.1 mm VAR 4340 projectiles in that work. The impacts at an obliquity of 15° were very similar to the normal impacts, in terms of penetration depths as a function of impact velocities. The penetration channels were straight and the projectiles behaved in a rigid-like manner, as most of them did not bend during penetration. However, the impacts at 30° obliquity resulted in considerable bending of the projectiles in these tests. Thus, for the oblique impact simulations we did not use the high-strength projectiles which were used in the simulations described above. Instead, we used a constitutive relation which represents the real properties of the VAR 4340 steel rods in Warren et al. (2004). This is the well-known Johnson-Cook strength model, which includes both strain and strain-rate hardenings

Y = (A + B ε^{n}) (1 + C l n (\frac{\dot{ε}}{{\dot{ε}}_{0}}))

(8)

with the following values for the material parameters: A = 1462 MPa, B = 510 MPa, n = 0.19, and C = 0.01. These values resulted in stress–strain curves for the VAR 4340 steel which are similar to those of Warren et al. (2004), through their power-law constitutive model. Note that the value of A is the same in both our model and that of Warren et al. (2004). We performed several 3D simulations with the LS-DYNA code, for the D = 7.1 mm projectiles impacting the surrogate targets with a strength of Y = 210 MPa. For the impacts at an obliquity of 15° we obtained penetration depths which are very close to the data for the limestone targets, as shown in Table 2 for several impact velocities. The good agreement between these two sets of values is due to the fact that the impacts at an obliquity of 15° are very similar to normal impacts.

Table 2.

Penetration depths at 15° obliquity in limestone and surrogate targets.

Impact velocity (m/s)	P in limestone (mm)	P in surrogate target (mm)
411	59	55
773	153.7	153
1030	241	251

The next step was to check our simulation results with those of Warren et al. (2004) for an impact at 30° obliquity. They implemented the spherical cavity expansion model in their 3D code, in order to define the force on the surface of the projectile as it penetrates the target. We chose to simulate the V₀ = 610 m/s test at 30° obliquity, because of its low inclination angles (pitch = 0° and yaw = 1.25°). We obtained a penetration depth of 101 mm in the surrogate target, which is very close to the value of 103 mm obtained by Warren et al. (2004) in their simulation for the limestone target. Figure 2 compares the shape and position of the projectile during its penetration in the surrogate target from our simulation, with those of Warren et al. (2004) in their simulations for the limestone target. The difference between the projectiles’ orientations, at the later times, can be due to the fact that in order to simplify our simulation, we did not include the yaw angle in this shot. The overall agreement between the two sets of simulations, as far as the projectile’s deformation and position are concerned, enhances the usefulness of the surrogate target concept presented here.

Figure 2.

Comparing our simulation results (black) for the 30° oblique impact, with those of Warren et al. (2004) (red).

Finally, we turn to the “problematic” test in Table 1, for the 12.7 mm projectile impacting the limestone target at normal incidence with V₀ = 1134 m/s. As reported by Frew et al. (1999), the inclination angles in this test were relatively large (yaw = 1.9° and pitch = 1.0°), which, apparently, resulted in the relatively low penetration depth in this test (P = 562 mm). In order to check this assumption we performed a 3D simulation for this projectile, with the J-C constitutive relation, impacting the corresponding surrogate target having strength of Y = 185 MPa. The projectile was given an inclination angle (γ) which is equivalent to the given yaw $(γ_{1})$ and pitch $(γ_{2})$ angles, through the well-known relation between these angles

\sin^{2} (γ) = \sin^{2} (γ_{1}) + \sin^{2} (γ_{2})

(9)

which results in a value of γ = 2.15°. Figure 3 shows several snapshots of the projectile during its penetration in the surrogate target, and one can see its bending during the process. The final penetration depth in this simulation was 576 mm, which is very close to the penetration depth of 562 mm in the limestone target, as given in Table 1. This agreement further enhances the usefulness of the surrogate target concept for non-normal impacts, for which one needs 3D numerical simulations.

Figure 3.

Several snapshots of the impact at 1134 m/s with an inclination of 2.15.

Granite targets

We mentioned above the scarcity of published data on deep penetrations in rock targets by rigid projectiles. Still, the work of Li et al. (2018) provides a few tests which can be used for our present analysis. They impacted granite targets with CRH3 ogive-nosed steel projectiles, with D_p = 10.8 mm, at velocities in the range of 1.2–4.1 km/s. Most of the projectiles were either broken or heavily eroded due to the high impact velocities. However, the projectiles in the low-velocity shots can be considered as nearly rigid since they suffered a relatively low mass loss and only a slight nose deformation. Specifically, we refer to the shots at impact velocities of 1.196 km/s and 1.43 km/s resulting in penetration depths of 119 and 156 mm, respectively. Using their f_c = 150 MPa value, as measured with D₀ = 50 mm test specimens, we find by equation (4) that f_c*= 197.6 MPa for this projectile/target pair. Using the empirically derived relation in equation (5) we get an expected value of R_t = 2.11 GPa for the penetration resistance, which the granite targets exerted on these projectiles. Inserting this value of R_t in equation (1), together with the projectile’s density and effective length, results in the predicted penetration depths of 131 and 172 mm for the two tests. These values are higher by only 10% than the corresponding experimental results which may be due, at least partly, to the mass loss and the change in nose shape of the actual projectiles during penetration. Moreover, the confinement of the granite blocks in relatively thick steel tubes (10 mm) probably added to their resistance to penetration, reducing the experimental penetration depths even further. Overall, we can claim that the relation between R_t and f_c*, which we derived for the limestone targets, accounts fairly well for the penetration depths in the granite targets. Obviously, more data on rock penetrations are needed in order to validate this relation.

Conclusion

We reanalyzed the penetration data of rigid projectiles impacting limestone targets from the work of Frew et al. (1999). Our analysis is based on the observation that, apart from a short entrance phase, the resisting stresses exerted by these targets on rigid projectiles were constant during the penetration process. We also dealt with the scaling issue in the limestone’s penetration data of Frew et al. (1999), for their three similarly-shaped projectiles with different dimensions. We attributed this scaling issue to the different resisting stresses exerted on these projectiles, as they push aside the corresponding cylinders in the limestone targets. Our basic assumption was that these cylinders exert different penetration resistances on the projectiles because they have different compressive strengths. These differences are due to the well-known size effect in rocks, which is summarized by the Hoek–Brown relation for the compressive strengths of rock specimens. We used this relation in order to derive an empirical relation between the penetration resistance of the limestone targets and the compressive strengths of limestone cylinders having the projectiles’ diameters. This relation was found to predict the penetration resistance of granite targets, in another work, as determined by several tests with high-velocity projectiles.

We also proposed a simple scheme by which one can assess the penetration depths in limestone targets (and possibly other rocks), by using numerical simulations with surrogate targets. This scheme is especially beneficial for non-normal impacts where obliquity and inclination angles significantly influence the trajectory of the projectile and its rigidity, as we showed here. We do not claim that our proposed surrogate material model is similar to limestone, or to any other brittle material, and that it captures all complexities of the deformation in such materials. Rather, we offer a simple numerical procedure in order to have a rough assessment for the penetration depths in limestone, without the need to calibrate its equation of state and its constitutive relations. We should point out that the proposed scheme is limited to impacts of non-eroding projectiles with velocities within the ordnance range, and the targets are large enough to be considered as semi-infinite. Finally, we should note that penetration tests on other rock targets, at a large range of impact velocities, are very much needed in order to achieve a comprehensive understanding of the issues raised here.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Roman Kositski

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