Numerical investigation on ballistic performance of coarse-aggregated layered UHPFRC

The impact resistance of coarse-aggregated layered Ultra-High Performance Fiber Reinforced Concrete (UHPFRC) is investigated numerically with LS-DYNA. The Holmquist Johnson Concrete (HJC) model is employed to describe the dynamic behavior of the UHPFRC, and the effect of the coarse aggregates is reflected in the pressure-compaction relation. Mechanical tests are conducted to obtain the materialrelated inputs for the numerical model, and ballistic experiments are applied to calibrate the model parameter as well as to validate the simulation results. After valuation, the ballistic histories of the projectile and the penetration processes in the UHPFRC targets are analyzed. Furthermore, the study discusses the effects of the target thickness on the depth of penetration, showing the possibility to replace a thicker single-layered target by a thinner triple-layered one to achieve the same level of protection. Finally, perforation limits of the singleand tripled-layered UHPFRC at different impact velocities are estimated, based on which the ACE formulae are modified to accurately predict the perforation limit of the coarse-aggregated layered UHPFRC. The numerical simulations in this study reveal that the triplelayered target requires fewer dosages of cement and steel fibers in comparison to its single-layered counterpart with the same level of ballistic protection. 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
The increasing risks of terrorist attacks in the current world clearly show the necessity of improving the impact resistance of important buildings and infrastructures. The advanced mechanical properties [1][2][3] make Ultra-High Performance Fiber Reinforced Concrete (UHPFRC) a promising candidate for these infrastructures. The superior resistance of UHPFRC has been revealed in literature. For instance, the impact responses of UHPFRC against deformable and non-deformable projectiles were evaluated by Máca et al. [4]; the significantly improved resistance of UHPFRC was confirmed in their study and an optimal fiber volume fraction of 2% was obtained. Liu et al. [5] analyzed experimentally and numerically the ballistic resistance of UHPFRC, the results of which presented that UHPFRC has advantages regarding the depth of penetration (DOP), crater diameter and volume loss. Yu et al. [6] investigated the ballistic performances of UHPFRC against 7.62 mm projectile at 830 m/s, and claimed that the UHPFRC containing hybrid fibers is more efficient than that with only the hooked-end fiber.
Generally, conventional UHPFRC, e.g. those mentioned above, only utilizes fine particles [7,8]. On the one hand, using only fine https://doi.org/10.1016/j.conbuildmat.2020.118867 0950-0618/Ó 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). aggregates in UHPFRC can promote its strength enhancement, thus benefiting the reduction of DOP [9]. On the other hand, the advantages of coarse aggregates on enhancing its impact resistance, e.g. eroding the projectile and deviating the ballistic trajectory, are consequently sacrificed [10,11]. Only a few studies are available concerning the penetration resistance of coarse-aggregated UHPFRC, and positive results are obtained in these studies, showing that the coarse aggregates are beneficial to reduce both the DOP and the area of damage [12][13][14]. Nevertheless, the compressive strength r c,s of these coarse-aggregated UHPFRC are relatively low, e.g. the 28-day r c,s of the UHPFRC with coarse basalt aggregates are around 106 MPa [12] and 119 MPa [14], and the 160day r c,s in [13] is about 128 MPa. That being the case, the coarse-aggregated UHPFRC in these studies actually fail to meet the strength requirement of UHPFRC, i.e. having a compressive strength of at least 150 MPa at 28 days [15].
In addition to taking the advantages of coarse aggregates, another innovative approach to improve the impact resistance of UHPFRC may be applying the concept of layered-structure. Our previous study concerning static bending of UHPFRC [16] exhibited that the double-layered UHPFRC has an enhanced energy absorption capacity over its single-layered counterpart. It was revealed that the layer interface in the double-layered UHPFRC facilitated the initiation of multiple micro-cracks, promoting the energy dissipation of the specimen [16]. With regard to the ballistic impact, Quek et al. [17] claimed that the four-layered concrete target (r c,s -110 MPa) developed in their study achieved a superior ballistic resistance, and that the layer delamination benefited the consumption of the projectile's kinetic energy. Nonetheless, the subject of their study is high-strength concrete instead of UHPFRC, and the effects of the layer interface on the penetration process are not addressed.
The above findings inspired us to design coarse-aggregated layered UHPFRC to achieve an excellent ballistic resistance. Understanding the dynamic performances of this coarse-aggregated layered UHPFRC is therefore a significant issue. Ballistic experiments can provide direct results; however, the expensive test facilities and the time-consuming test process put limits on conducting massive experiments in this field [18,19]. Empirical and theoretical formulae [20] can predict some important ballistic responses, e.g. the DOP, perforation limit and residual velocity; nevertheless, these formulae cannot reproduce the dynamic stress and damage distributions in the impacted concrete. On the other hand, with the advancement of computer technology, numerical simulations of concrete against ballistic impacts become feasible. A number of commercial software, such as LS-DYNA, ABAQUS and AUTODYN, can be utilized to model the response of concrete subjected to ballistic impacts, providing valuable information on the real-time interaction between the projectile and the concrete.
Modelling the dynamic response of UHPFRC under penetration is challenging [19]. As a result of the rate effect, the behavior of UHPFRC subjected to dynamic loads is more complicated in comparison to that under static load [1]. Moreover, UHPFRC is practically idealized as a homogeneous material in macroscale simulations. However, concrete is actually inhomogeneous and has multiple phases (e.g. matrix, aggregates and steel fibers), which makes modelling the constitutive relation of the ''homogenized" UHPFRC rather difficult [21]. As a consequence, choosing the appropriate material models and determining the suitable parameters to reflect the influences of important UHPFRC components, e.g. the coarse aggregate, under dynamic loads are of great significance for the success of these macroscale simulations. Peng et al. [22] modelled the ballistic behaviors of concrete target at both macroscale and mesoscale (with coarse aggregates explicitly presented). Comparing the relative DOP in their simulations exhibits that the macroscale model can give results close to that from the mesoscale model as long as appropriate material models were applied, especially for the cases where the ratio between the aggregate size and the projectile diameter is smaller than 2. This confirms the possibility of applying homogenous models to simulate the ballistic performance of concrete at macroscale.
As with UHPFRC, several studies have been conducted to simulate its ballistic response at macroscale. For example, Liu et al. [19] applied the K&C model in LS-DYNA to investigate the effects of UHPFRC compressive strength, projectile velocity and nose shape on the DOP and the cratering damage size. Yu [23] used the HJC model to describe the ballistic response of UHPFRC by LS-DYNA, and it was claimed that the obtained numerical simulations agreed well with the experimental results. Sovják et al. [24] numerically evaluated the resistance of slim UHPFRC panels with the RHT model in AUTODYN, and the effect of fibers was incorporated in the model by increasing its fracture energy. It is noteworthy that the above numerical studies only focus on conventional UHPFRC with a single-layer and fine particles, i.e. the influences of the layered structure and the coarse aggregates are not addressed.
This study investigates numerically the ballistic performances of coarse-aggregated layered UHPFRC at macroscale. LS-DYNA is employed to execute the simulations, and the effect of the coarse aggregates is reflected by the crushing pressure in the HJC model. After validating the model against experiments, the dynamic penetration processes inside the coarse-aggregated layered UHPFRC are analyzed. Moreover, influences of the target thickness on the DOP are discussed, and the perforation limits of the single-and tripled-layered UHPFRC at different penetration velocities are evaluated. The study confirms the advantages of the coarse-aggregated layered UHPFRC and advances the understanding of its ballistic performances.

Mechanical and ballistic experiments
Mechanical and ballistic experiments of the coarse-aggregated layered UHPFRC are conducted to provide inputs for the numerical model, as well as to validate the simulations.

Materials and mix design
The recipes of the coarse-aggregated UHPFRC are given in Table 1, including Portland Cement I 52.5R (HeidelbergCement Benelux), limestone powder (CB2M), micro-silica (Elkem Grade 920E D), sand (DIN EN 196-1), basalt aggregates of four size groups, water, and a PCE based superplasticizer (solid content of 35%). The straight steel fiber (Dramix OL 13/0.20) is used, which has a length of 13 mm and diameter of 0.2 mm. The density of the fiber is around 7850 kg/m 3 .
The fractions of the basalt aggregates are calculated by the Brouwers mix design method [25][26][27] to achieve an optimal packing density of the mixture: where q is the distribution modulus, and the value of 0.22 is applied for UHPFRC [23]. D is the particle size; D min and D max are the minimum and maximum values, respectively. P m (D) is the cumulative fraction of the total solid with a size smaller than D, and P t (D) presents the targeted cumulative fraction. An optimization algorithm is applied to achieve an optimal fit between P m (D) and P t (D), i.e. a maximum R 2 (the quality of fit between the designed and the target distributions). Examples of the particle size distribution curves of the designed coarseaggregated UHPFRC mixture are shown in Fig. 1 [28]. More information about the detailed properties of the applied ingredients can be found in [28,29].

UHPFRC targets
The coarse-aggregated UHPFRC with single and triple layers were tested, as shown in Table 2. The thickness of the UHPFRC target is 90 mm and the diameter is about 275 mm. U in the identifications represents UHPFRC. The identification of the third is divided by dashes, showing that it is a layered target. The layer thickness is given in the brackets (in mm). The numbers before a and s are the maximum aggregate size (in mm) and the fiber volume fraction, respectively. For instance, U16a1s(30)-8a1s(30)-16a1s(30) is a triple-layered UHPFRC target with an equal layer thickness of 30 mm; the maximum aggregate sizes of the three layers are 16 mm, 8 mm and 16 mm, respectively; and 1% (volume fraction) of the steel fibers are incorporated in each of the three layers (see Table 2). As can be noticed, the larger aggregates are arranged in the two outer layers of the triple-layered UHPFRC target. This aggregate size arrangement is determined based on: (1) the more severe damage of the front and rear surfaces of the target compared to its middle region [23,30], and (2) the contribution of the larger aggregates to decreasing the damage size [12][13][14]. A reverse order of the layers, e.g. U8a1s(30)-16a1s(30)-8a1s (30), is expected to result in an inferior impact resistance due to the less efficient effect of the 8 mm aggregates on reducing the damage of the two outer surfaces.

Sample preparation and testing
The mixing of the UHPFRC mixture was conducted at room temperature, and the procedure is illustrated in Fig. 2.
Cylindrical moulds were utilized to cast the UHPFRC targets. For the triple-layered UHPFRC, the layers were cast subsequently after a short time interval of 45 min [16]. The UHPFRC samples were demolded 24 h after casting and they were cured in water at room temperature of 20 ± 1 ℃. The ballistic tests were conducted at 56 days with in-service 7.62 mm Â 51 AP projectile. The impact velocity V 0 was around 840 m/s. The test setup is shown in Fig. 3. A specially designed fixing frame was used to hold the cylindrical UHPFRC target, and a white board was put behind the frame to check the trace of the projectile in the case of ballistic perforation. The targets were cut to measure the DOP after the ballistic tests [31].
Additionally, 100 mm UHPFRC cubes were prepared to test the compressive, splitting tensile and bond strengths of the designed

Penetration test results
The DOP and the crater size D eq of the single-and triple-layered UHPFRC targets are given in Table 3. D eq was calculated as , where D 1 and D 2 are the crater size measured at two perpendicular directions. With regard to the single-layered UHPFRC, increasing the maximum aggregate size from 8 mm to 16 mm increases the ballistic resistance. This enhanced resistance is because that the higher strength and hardness of the coarse aggregates contribute to consuming more kinematic energy of the projectile, which reduces its penetration velocity [10,12,22]. Moreover, comparing U16a1s(30)-8a1s(30)-16a1s (30) and U8a1s (90) in Table 3 presents that applying a layered structure can further reduce the DOP. As expected, discontinuous cracks are observed along the layer interface, as shown in Fig. 4. This interfacial cracking can further facilitate the energy dissipation of the projectile [17], leading to the reduced DOP in the triple-layered target.

Overview of the model
Three dimensional models of the UHPFRC targets and the projectile are built in LS-DYNA, as shown in Fig. 5. In order to improve the computational efficiency, only one quarter of the test arrangement is modelled. Symmetric boundary conditions are imposed on models and fixed boundary conditions are imposed on the back edge of the targets. The penetration process is simulated based on the explicit time integration. To achieve the stability of the solution, a scale factor of 0.67 for the computed time step is applied [36].
3D solid 164 element is used in the simulation. As shown in Fig. 5, the target is divided into 45 indices in the direction of penetration and 30 indices along its radius direction. The length of the element gradually increases along the target radius, and it is finer in the region where the projectile contacts the concrete. The minimum element sizes of the target and the projectile are comparable, i.e. about 1.2 mm and 1 mm, respectively. The results of mesh convergence study are presented in Fig. 6, in which the minimum mesh size of the target changes from 1.2 to 2.0 mm. The figure shows that 1.2 mm is sufficiently fine and the mesh is therefore adopted in the simulations. For the triple-layered target U16a1s (30)-8a1s(30)-16a1s(30), a thin layer of interfacial elements (with a thickness of 0.5 mm) is added in between the U16a1s layers and the U8a1s layer, and the interface elements have shared nodes with the elements of the U8a1s and U16a1s layers. In other words, there is no direct contact between the U8a1s and U16a1s layers, and their connection is considered by the thin interfacial layer fixed to the target layers.

Projectile
The penetration velocities in the experiments are around 840 m/s, which fall within the range of non-deformable (rigid) penetration (V 0 1000 m/s) [19]. Furthermore, the hard core of   the projectile after the ballistic test is compared to that before the test. Despite the slight scratches on the surface, the main body of the hard core after the ballistic test has very limited deformation thanks to the stiff material of the hard core and its small size. Therefore, the deformation of the projectile is neglected in this simulation, and it is modeled as a rigid body using the material model *MAT_RIGID (MAT_020) in LS-DYNA. The input parameters for the projectile are listed in Table 4. The contact between the projectile and UHPFRC is modelled with ''CONTACT_ERODING_SUR FACE_TO_SURFACE".

UHPFRC target
The HJC constitutive model *MAT_JOHNSON_HOLMQUIST_CON CRETE (MAT_111) is adopted for the UHPFRC and the layer inter-face. This model is suitable to describe the response of cementitious materials subjected to high strain rates, large deformations, and high hydrostatic pressures [37,38], and it has been widely employed to model the ballistic performance of concrete [22,23].
In the HJC model, the yield surface is [37]: where r* is the normalized equivalent stress, P* is the normalized pressure._ e Ã ¼ _ e=_ e 0 is the dimensionless strain rate ( _ e and _ e 0 = 1.0 s À1 are the actual and reference stain rates, respectively).
A c , B c , C c and N c are input parameters. The damage D is accumulated as a function of the equivalent plastic strain e p , plastic volumetric strain l p , and the total plastic strain until fracture e p f + l p f [39]: where D 1 and D 2 are the damage parameters. T* = T/r c,s is the normalized maximum tensile hydrostatic pressure and T is the maximum tensile pressure. EFMIN is the amount of plastic strain before fracture [38,39]. The pressure-compaction relation of the material can be separated into three phases:  The first phase is linear elastic [39]: where l v is the volumetric strain, P crush and l crush are the crushing pressure and the corresponding volumetric strain.
The second phase corresponds to the transitional region [39]: where P lock and l lock are the compaction limit and the locking volumetric strain, respectively.
The third phase describes the response of the fully densified material [39]: where K 1 , K 2 and K 3 are material parameters, l À is the modified volumetric strain.
The input parameters for the HJC model are listed in Table 5.
The density q, uniaxial compressive strength f c and the maximum tensile pressure T of the U8a1s and U16a1s mixtures are obtained from the experimental results in Section 2. The density and compressive strength of the plain UHPC U8a are used for the layer interfacial element. The tensile strength of the interface is assumed to be equal to the bond strength between the U8a1s and U16a1s layers, as measured by experiments in Section 2. The input shear modulus in Table 5 is determined by G = E/2(1 + v) with E the Young's modulus and v the Poisson's ratio.
The strength parameters A c , B c , N c , SFMAX can be determined by triaxial compressive tests [39]. The test data in the literature [40][41][42][43] concerning concrete from 92 MPa to 171 MPa is given in Fig. 7, in which the normalized equivalent stress is r* = (r 1 -r 3 )/f c and the normalized pressure is P* = (r 1 + 2r 3 )/3f c [39] (r 1 and r 3 are the major and minor principal stresses, respectively). As presented in the figure, the different concrete strength does not cause signif-icant deviations on the relation between r* and P*. Therefore, it is assumed that the developed UHPFRC in this study also possesses a similar trend as that shown in Fig. 7. At a static strain rate _ e = 1.0 s À1 and without considering the damage, Eq. (1) with A c = 0.79, B c = 1.6, N c = 0.61 and SFMAX = 7 is presented by the solid curve in Fig. 7. These are the default values in the original HJC model [39]. A fair agreement is observed between the experimental data [40][41][42][43] and Eq. (1) with the default parameters. That being the case, these values are further used in the HJC model in the present study (see Table 5). Similarly, the strain rate parameter C c , EPS0, and the damage parameters D 1 , D 2 , EFMIN also remain the same as the default values in the original HJC model [39]. These values are widely used for concrete with strengths from 12 MPa to 200 MPa [22,39,44,45], and the strength of the UHPFRC developed in this study also falls within this range.
As regards to the equation of state (EOS) parameters, for the layer interface and the U8a1s mixture, the crushing pressure P crush and the corresponding volumetric strain l crush are determined by P crush = r c,s /3 and l crush = P crush /K, in which the bulk modulus K = E/3(1-2v) [39]. On the other side, for the U16a1s mixture, the crushing limit and the stiffness of the material are manually enhanced to represent the influences of the larger aggregates. The contributions of the coarse aggregates on improving the ballistic resistance of concrete can be mainly attributed to the following reasons [46]: (1) The stronger coarse aggregates can act as barriers to cracking, as the cracks are driven to cut through the aggregates due to the high stress rate in the case of penetration. (2) The higher hardness of the coarse aggregate increases the kinetic energy loss of the projectile. (3) The direction of the projectile may be changed after it hits the coarse aggregate, which further promotes the energy consumption of the projectile. In the macroscale simulation with symmetric boundary conditions applied to the projectile, the trajectory of the projectile has no deviation, i.e. the direction of the projectile always remains perpendicular to the impact surface of the target. Although the third effect of the coarse aggregates, i.e. due to material heterogeneousness, cannot be reflected in the macroscale model, it is still possible to represent the first two effects of the coarse aggregates, i.e. the improved strength and the enhanced hardness provided by the coarse aggregates. These two effects can be reflected in the macro model by increasing the crushing limit and the stiffness of the material. As given in the HJC model, P crush and l crush are the thresholds beyond which the voids in the concrete material are gradually compressed [39]. An enhancing factor of 2.5 is applied to the P crush of U16a1s, which is calibrated by the DOP of the U16a1s target from the ballistic tests in Section 2 (see Fig. 8). Since the crushing volumetric strain l crush is not changed in the simulation, this enhancing factor improves both the crushing pressure and the stiffness, corresponding to the improved strength and the enhanced hardness of U16a1s. Furthermore, the erosion algorithm *MAT_ADD_EROSION is applied to remove the elements with large distortions during the penetration process. The maximum principal strain e max of 0.5 and the failure volumetric strain e vol of 0.01 are adopted as the erosion criteria, i.e. the element will be removed if e max or e vol reaches the corresponding critical values. These erosion criteria of e max and e vol are determined by preliminary study to fit the experimental DOP and D eq of U8a1s. And they are further used for the other targets, which also show good agreements. Additionally, it is noteworthy that the steel fibers are mainly to increase the tensile strength of the UHPFRC and to decrease the damage scale of the target. Therefore, the influences of fiber addition can be reflected in the model by the maximum tensile pressure T and the above erosion criteria, viz. e max and e vol (the crater size D eq is related with these parameters).

Model validation
The DOP obtained by the numerical simulations and the experiments are compared in Table 6 to validate the model. As shown in the table, the simulation results coincide very well with the experimental data, i.e. the deviations are less than 0.5% for both the single-and triple-layered UHPFRC targets. Furthermore, the penetration reductions caused by the 16 mm coarse aggregates and the layered-structure are also successfully captured by the model, showing the reliability of the HJC material model and the corresponding inputs.
The effective strain distribution in the impact surface of the UHPFRC is utilized to denote the crater damage, and the compar-isons between the experimental and the numerical D eq are shown in Fig. 9. In general, the numerical simulations are able to reproduce the major damage distribution in the targets, i.e. the difference between the numerical and experimental D eq is less than 5%. Additionally, the comparison between Fig. 9a and 9b indicates that the reduced D eq in the UHPFRC containing larger aggregates, as observed in the experiments, is also successfully shown in the simulations. As demonstrated in Fig. 9, D eq of U16a1s(90) and its destruction level (represented by the color fringe) are more moderate compared to those of U8a1s(90), which are in line with the experimental results. In summary, considering both DOP and D eq , fair agreements are achieved by the numerical model and it is hence utilized for the further analysis. Fig. 10 shows the displacement and velocity histories of the projectiles during the penetration process. The projectiles in U8a1s(90), U16a1s(90) and U16a1s(30)-8a1s(30)-16a1s (30) have comparable displacements at the early penetration stage (0-40 ls after impact) because of their similar initial velocities around 840 m/s. However, with the development of the penetration process, the effects of the three targets on the projectile become increasingly obvious. Among the three targets, U16a1s(30)-8a1s(30)-16a1s(30) most effectively reduces the projectile velocity, leading to the smallest projectile displacement in the target, i.e. the simulated DOP. Moreover, the declining rate of the projectile velocity is influenced by the layer interface in U16a1s(30)-8a1s(30)-16a1s (30). Differing with the straight velocity curves of the single-layered targets, the velocity curve of the triple-layered UHPFRC starts to bend when the projectile reaches the layer interface at around 42 ls after impact. This may be attributed to the fact that the layer interface influences the wave propagation and reflection between the layers, affecting the force on the projectile and therefore its velocity. As with the single-layered UHPFRC, the projectile in U16a1s(90) yields a smaller velocity compared to that in U8a1s(90) at the same penetrating moment. The DOP in the U16a1s(90) target is, consequently, smaller.

Penetration process
The penetration processes in the UHPFRC targets are presented by the pressure development in Figs Furthermore, the simulation of the triple-layered UHPFRC shows that the pressure distribution is affected by the layer interface when the stress wave reaches it, as can be observed by comparing the light blue fringe level in Fig. 11b and Fig. 12b. Furthermore, when the projectile leaves the first layer and enters the second one, e.g. at 63 ls (Fig. 12d), the pressure distribution  varies. The distinct pressure distribution in the figure also explains the bending velocity curve of the triple-layered UHPFRC target in Fig. 10b. As exhibited in Fig. 12d, the influenced region is divided into two spheres by the layer interface. This may be attributed to the complicated propagation and reflection of the stress wave near the interface, the distinct pressure bearing capacities of the interface and the layer matrix, as well as the different confinement levels of the two layers. Moreover, the existence of the layer interface in the triple-layered UHPFRC dampen the strength of the impact waves [47], which contributes to the smaller DOP and D eq of the target. In addition, the affected area in the triple-layered target is larger than that in the singe-layered counterpart (see the light blue area in Figs. 11 and 12), indicating a higher energy dissipation in U16a1s(30)-8a1s(30)-16a1s (30). This, therefore, further promotes the enhanced impact resistance of the triple-layered UHPFRC.

Influences of layer thickness on DOP
Using the validated models, the influences of the target thickness on DOP are discussed. As suggested by the experimental and the simulation results, DOP of U8a1s(90), U16a1s(90) and U16a1s(30)-8a1s(30)-16a1s (30) are less than 60 mm, indicating that the 90 mm targets are sufficient to resist the 7.62 mm projectile at around 840 m/s. To achieve a more efficient utilization of the targets, their thicknesses can be reduced. Fig. 13 demonstrates the DOP of the targets with their thickness decreasing from 90 mm to 60 mm. U8a1s and U16a1s correspond to the series of the singlelayered UHPFRC targets containing the maximum aggregates of 8 mm and 16 mm, respectively (note that their thicknesses are changing). U16a1s-8a1s-16a1s represents the series of the triple- layered UHPFRC with an equal layer thickness, e.g. when the total thickness of an U16a1s-8a1s-16a1s target is 60 mm, the individual layer thickness is 20 mm. It is noteworthy that the penetration depth in the U16a1s target with a thickness of 60 mm is around 60.2 mm. In other words, the tip of the projectile slightly crosses the rear surface of the target, as shown in Fig. 14. Fig. 13 illustrates that the DOP decreases with the increase of the target thickness, nonetheless, the decrease becomes insignificant after the target thickness reaches a critical value. To be more specific, an U8a1s target with a thickness of 60 mm is totally perforated by the projectile, hence no point corresponding to this case is shown in Fig. 13. However, increasing its thickness from 60 mm to 67.5 mm improves the ballistic resistance remarkably, i.e. the projectile is stopped inside the 67.5 mm target with a DOP of 63.1 mm. The failure pattern of this target is shown in Fig. 15a, in which a major crack appears on the rear surface of the target because of the reflected tensile wave. Further increasing its thickness to 75 mm results in a 2.6 mm reduction in DOP, whereas the reduction becomes insignificant when the target thickness further increases to 82.5 mm and 90 mm. Nevertheless, the rear surface cracking is effectively prevented in these targets thanks to their sufficient thicknesses (see the failure pattern of the 82.5 mm target in Fig. 15b). Similar tendencies can be obtained in the U16a1s and U16a1s-8a1s-16a1s targets. Moreover, the critical target thickness differs for the three target series: it is around 75 mm for the U8a1s series, while for U16a1s the value is around 67.5 mm; as with the triple-layered target, the critical target thickness is smaller than 60 mm. The reduction of DOP by increasing the target thickness can be attributed to the higher confinement on the projectile provided by the thicker concrete [9]. The larger thickness can generate a more prominent confinement on the inner part of the target, enhancing the energy absorption capacity of the inner concrete [48] and reducing the damage level of the target [9]. Furthermore, the amplitude of the impact wave decreases more significantly when it travels a longer distance inside the thicker target. The impact wave transforms into a tensile one when it reaches the rear surface of the target and reflects at this free boundary. Since the impact wave has a smaller amplitude in the thicker target, the reflected tensile wave is also weaker. Consequently, the caused damage, e.g. cracking, on the rear surface of the thicker target is reduced (see Fig. 15). Nevertheless, the contribution of the target thickness become less significant when the critical target thickness is reached. This critical thickness may be related to the necessary travel length to attenuate the impact wave and to reduce its amplitude below a certain value, so that the reflected tensile stress will be too weak to cause cracking on the rear surface of the UHPFRC. Further increasing the target thickness beyond this critical value will have very limited effects since the strength of the tensile wave is already sufficiently small.
Additionally, comparing the DOP of U8a1s and U16a1s series in Fig. 13 reveals that the larger coarse aggregates play an important role on improving the ballistic resistance of the UHPFRC, and this influence is more obvious with a smaller target thickness. For instance, the DOP in the 67.5 mm target has an approximate 12% reduction when the maximum aggregate size increases from 8 mm to 16 mm, while the reduction is around 9% when the target thickness is 90 mm. Moreover, Fig. 13 shows that the U16a1s-8a1s-16a1s series has the smallest DOP among the three simulated target series with the same thickness. For example, the DOP in the U8a1s, U16a1s and U16a1s-8a1s-16a1s targets with a thickness of 75 mm are about 60.5 mm, 54.8 mm and 52.6 mm, respectively. The lower DOP of the U16a1s-8a1s-16a1s series in the whole thickness range reveals that the triple-layered UHPFRC possesses an enhanced resistance compared to its single-layered counterparts with the same target thickness. This also indicates the possibility of using a thinner triple-layered target instead of a thicker single-layered one to achieve the same level of protection, e.g. the 60 mm U16a1s-8a1s-16a1s target may be used to replace the 90 mm U16a1s target considering their comparable DOP.

Influences of penetration velocity on perforation limit
Perforation limit h e is defined as the minimum target thickness required to prevent perforation [21], and it is an important characterization for ballistic engineers to design safety protective structures. A number of empirical formulae have been developed to predict the perforation limit of concrete material, among which the BRL formula, ACE formula, modified NDRC formula and CEA-EDF formula are extensively utilized. The expressions of these formulae are given as follows [21].   The h e of the U8a1s, U16a1s and U16a1s-8a1s-16a1s target series under impact velocities from 640 m/s to 940 m/s are approximately estimated by the numerical models. The results are illustrated in Table 7 together with the calculated h e by the above empirical formulae. Note that for the triple-layered UHPFRC, equivalent concrete properties obtained according to the composite theory [49] are used as the input parameters for the empirical formulae. For instance, the equivalent compressive strength of the triple-layered UHPFRC r c,eq used in the empirical formulae is the arithmetic mean of the layers' compressive strength: r c;eq ¼ r c;s;1 H 1 þ r c;s;2 H 2 þ r c;s;3 H 3 where r c,s,i and H i are the compressive strength and thickness of the i th layer, i = 1, 2, 3. As shown in Table 7, the ACE formula gives results that are closest to the numerical h e of the U8a1s series at the whole velocity range, and predictions for the U16a1s series at high velocities namely 840 m/s and 940 m/s are also acceptable. The modified NDRC formula yields results comparable to the numerical h e of U16a1s at 640 m/s and 740 m/s, as well as those of U16a1s-8a1s-16a1s at 640 m/s, 740 m/s and 840 m/s. In contrast, the numerical h e of U16a1s-8a1s-16a1s at 940 m/s falls between the estimations of the modified NDRC and ACE formulae. Accordingly, none of these existing empirical formulae is able to predict the h e of the developed UHPFRC from 640 m/s to 940 m/s. This is because these formulae are proposed based on the regression analyses of certain types of concrete under certain impact velocities, which limits their application ranges [50]. Since UHPFRC is a relatively new material, these empirical formulae are not exactly proposed for it and their validity has not been verified against impacts of UHPFRC, not to mention the coarse-aggregated layered UHPFRC. That being the case, the ACE formula is modified to give more accurate predictions for the coarse-aggregated layered UHPFRC developed in this study. The modified ACE formulae are obtained on the basis of the simulation results in Table 7   Projectile tip slightly crosses the rear surface The comparisons between the h e from the numerical simulation and the modified ACE formulae are shown in Fig. 16. Excellent agreements are observed. It is worth mentioning that similar to the existing empirical formulae, the modified ACE formulae have also their applied range. They are proposed for the coarseaggregated layered UHPFRC against the 7.62 mm projectile under impact velocities between 640 m/s and 940 m/s. Using them for larger-caliber projectile may lead to underestimation of the DOP due to the scaling effect, which can be attributed to the inconsistent size variations of the projectile and the aggregates [51].  Furthermore, as depicted by the numerical results in Table 7, h e increases with the increase of V 0 ; and the U8a1s series has the largest h e , while U16a1s-8a1s-16a1s the smallest value. For example, the simulation exhibits that for an U8a1s target, the smallest thicknesses to prevent perforation under 640 m/s and 940 m/s are about 48 mm and 78 mm, respectively. On the contrary, those corresponding to the U16a1s-8a1s-16a1s series are only 42 mm and 67.5 mm, decreasing 14% and 16%, respectively, compared to those of U8a1s.
The smaller h e of the U16a1s-8a1s-16a1s series indicates the feasibility to reduce the target thickness by applying the layeredstructure, which then results in the reductions of the required materials, e.g. cement and steel fibers. Table 8 gives the cement and fiber amounts to cast a UHPFRC target with a diameter of 275 mm and a thickness of h e (obtained from the simulations in Table 7). Thanks to the enhanced impact resistance of the triplelayered UHPFRC, the cement and fiber amounts required to cast the U16a1s-8a1s-16a1s targets are obviously reduced compared with their single-layered counterparts. To be more precise, the cement amounts for the U16a1s-8a1s-16a1s targets under 640 m/s, 740 m/s 840 m/s and 940 m/s are about 19%, 22%, 22% and 20% less than those for the U8a1s targets; and the corresponding fiber amount reductions are 13%, 16%, 16% and 13%, respectively. As a consequence, with the same level of ballistic resistance, the triple-layered UHPFRC consumes less materials than the single-layered one, making the layered target more sustainable and more economic. In addition, the lighter weight of the triple-layered UHPFRC target is also beneficial for its transportation and installation.

Conclusions
This study numerically investigates the ballistic performance of the coarse-aggregated layered UHPFRC with LS-DYNA. Mechanical and ballistic experiments are conducted to provide model inputs and to validate the simulations. After model validation, the displacement and velocity histories of the projectile, as well as the dynamic penetration process in the UHPFRC targets are analyzed. Furthermore, the effects of the target thickness on the DOP, and the influences of the impact velocity on the perforation limit h e are discussed. In addition, modified ACE formulae to accurately predict the h e of the coarse-aggregated layered UHPFRC are proposed based on the simulations. The study promotes a better understanding of ballistic responses of the coarse-aggregated layered UHPFRC. The following conclusions can be drawn: (1) The numerical simulations of the U8a1s(90), U16a1s(90) and U16a1s(30)-8a1s(30)-16a1s(30) targets coincide well with the experimental results. The difference between of the DOP and the crater size obtained by the experiments and the simulations are smaller than 0.5% and 5%, respectively. (2) The HJC constitutive model in LS-DYNA can well characterize the dynamic response of the coarse-aggregated UHPFRC subjected to projectile impacts; and the improved ballistic resistance provided by the coarse aggregates can be reflected by increasing the crushing pressure in the model. (3) The layer interface affects the pressure distribution in the triple-layered target and thus the velocity history of the projectile. This can be attributed to the complicated propagation and reflection of the stress wave near the interface, the distinct pressure bearing capacities of the interface and the layer matrix, as well as the different confinement levels of the two layers. (4) The DOP decreases with the increase of the target thickness; however, the reduction becomes very limited when the critical target thickness is reached. The critical thickness is around 75 mm, 67.5 mm and 60 mm for the U8a1s, U16a1s and U16a1s-8a1s-16a1s targets, respectively. (5) The h e increases with the increase of the penetration velocity. Under the same velocity, U8a1s has the largest h e while U16a1s-8a1s-16a1s achieves the smallest value, indicating the feasibility to produce thinner target with the layered structure concept. Consequently, the triple-layered UHPFRC consumes fewer dosages of cement and fibers than the single-layered one with the same level of protection, making the layered target more environmentally and economically attractive.

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