A similarity method for predicting the residual velocity and deceleration of projectiles during impact with dissimilar materials

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1 Research Article A similarity method for predicting the residual velocity and deceleration of projectiles during impact with dissimilar materials Advances in Mechanical Engineering 2017, Vol. 9(7) 1 14 Ó The Author(s) 2017 DOI: / journals.sagepub.com/home/ade Qing Song 1, Yongxiang Dong 1, Miao Cui 2 and Bin Yu 1 Abstract A method for predicting the residual velocity and deceleration of a projectile during normal low-velocity impact on a 2024-O thin aluminium plate is developed based on the similarity theory. Geometric scaling, the dissimilar materials of the projectile and different target thicknesses are considered. By a similitude analysis, the simulation criteria between the modelling and prototype experiments are obtained. The dimensionless velocity and deceleration of a projectile can be predicted by the relationship equations with the dimensionless dynamic pressure, projectile density and target thickness. On the basis of experimental data, the dimensionless residual velocity relationship is obtained and verified. In the range of normalised target thicknesses of 0:5 = 1 (where is target thickness and is projectile diameter), the deceleration time data during penetration is simplified as a triangular wave. Moreover, it can be characterised using the maximum deceleration, the time to the maximum deceleration and the period of the triangular wave. Through a simulation analysis, three dimensionless deceleration characteristics of the projectile are developed to replicate a prototype-like deceleration time process in a scaled model. Keywords Dissimilar material, perforation, residual velocity, deceleration, dimensional analysis Date received: 6 December 2016; accepted: 27 March 2017 Academic Editor: Rahmi Guclu Introduction In dimensional analysis, a replica of a prototype is derived by scaling one or more parameters of the prototype. Scaled replicas enable the economical and easy set-up of the experiments, especially in the engineering field. In terminal ballistics experiments, a scaling law is widely used in researching penetration and perforation, 1 and the dimensionless parameters make the relationship among the parameters intuitive. In terminal ballistics experiments, geometric scaling is a convenient approach to test prototypes. Jones et al. 2 4 conducted geometric-scale (geometric-scale factor = 0.25, 0.5, 0.75) and full-scale experiments to compare the plate perforation and reported that the dimensionless perforation energy and the dimensionless transverse displacement of the ductile target conform to the scaling law. Noam et al. 5 simulated the scaling of dynamic failure using two types of failure criteria, namely, the maximum stress criterion and strain energy density criterion; the obtained results could adequately 1 State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, China 2 Institute of Chemical materials, China Academy of Engineering Physics, Mianyang, China Corresponding author: Yongxiang Dong, State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing , China. dongyongx@bit.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License ( which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages ( open-access-at-sage).

2 2 Advances in Mechanical Engineering describe the separation and adiabatic shear failure, respectively. In hypervelocity impact experiments, impact velocities higher than 7 km/s are difficult to achieve using two-stage light-gas guns; thus, velocity scaling is used in order to reduce the impact velocity. Mullin et al. 6 used a dissimilar-material model and lower velocities through velocity scaling; and this approach yielded high correlations for the debris cloud structure, materials, and velocities. In a structure impact, the approach VSG-D 7 (initial Velocity, V 0 ; dynamic Stress, s d and impact mass, G; with density factor) makes the target response (including the scaled force, displacement and final plate profile) in scaled replicas with different materials highly consistent with a prototype. Rosenberg and Dekel 8 determined the relationship between the effective resisting stress s r and the thickness of the target in perforation for a wide range of plate thicknesses; they found that s r and are normalised by the target flow stress Y tf and the diameter of the projectile D, respectively, and presented a dimensionless piecewise equation for three =D values, from which the perforation energy can be calculated. For terminal ballistics research, the geometric scaling, velocity scaling and normalised target thickness have been used, as mentioned above. The present work extends previous work by coupling these scaling methods for perforation research. In perforation experiments, the deceleration time data and the residual velocity of the projectile are valuable output response parameters. The residual velocity of the prototype can be estimated through geometric-scaling experiments. owever, in geometric scaling, a smaller model results in a higher projectile deceleration, which means that a downscaled experiment cannot adequately reflect the real-world (prototype) environmental and deceleration characteristics. Thus, in engineering experiments, the anti-overload capabilities of the projectile components (e.g. the fuse and warhead) of a downscaled geometricscaling model may be difficult to assess. Therefore, in this study, the geometry (i.e. the geometric scaling and target thickness), dissimilar projectile materials and impact velocity are considered for the prediction of the residual velocity and deceleration (specifically, for rigid projectile perforation of a thin aluminium plate at low velocities). The target thickness range in this study ranges from half of the projectile diameter to the projectile diameter. The rest of this article is organised as follows: first, the experiments and results are introduced; then, an equation for predicting the projectile residual velocity through a dimensional analysis is obtained; after describing and validating the simulation process, the dimensionless prediction equations for deceleration of the projectile are obtained by simulation; finally, using the obtained equations, the deceleration time data of projectiles in the prototype model can be appropriately represented using a scaled model. Experiments Experimental setup A gas-gun apparatus (Figure 1) with an inner diameter of 16 mm is used to launch the projectiles. The projectile impact velocity is less than 500 m/s. A steel chamber with a metal target is fixed in front of the gun barrel; this chamber has a circular hole in line with the barrel, through which the projectile could pass into and Figure 1. Gas-gun experimental system.

3 Song et al. 3 Figure 2. Metal target, C-shaped support and clamping rig. penetrate the target. The target is sandwiched between a C-shaped clamping rig and a C-shaped support plate using 10 bolts, and the support plate is welded with steel plates to ensure its rigidity (Figure 2). Rubber plates are placed behind the target as a buffer to stop and recover the projectile after perforation. On one side of the target chamber is a transparent acrylic plate window through which a high-speed camera equipped with Similitude analysis and experimental design For a projectile perforating a plate, the three characteristic parameters are the residual velocity v r, the deceleration of the projectile a and the penetration time T pene, which is the time taken to completely perforate the plate. These parameters can be determined using the main parameters as follows v r = f r r p, L eff,, N, r t,, v i, E py, E ty, Y p, Y t, E pt, E tt, Y ps, Y ts, e pf, e tf, m a = f a r p, L eff,, N, r t,, v i, E py, E ty, Y p, Y t, E pt, E tt, Y ps, Y ts, e pf, e tf, m T pene = f pene r p, L eff,, N, r t,, v i, E py, E ty, Y p, Y t, E pt, E tt, Y ps, Y ts, e pf, e tf, m ð1þ ð2þ ð3þ a flash unit can record the projectile action. The camera is placed in such a way that it can photograph both the front and back of the target. The flash unit is placed between the window and the camera and can provide an exposure time of approximately 21 ms. A copper wire connected to a trigger mechanism is connected to the gas-gun muzzle. The wire is cut when the projectile passes through the muzzle, which in turn triggers the flash; simultaneously, photographs are manually captured using the camera at a frame rate of 44,000 fps. In this study, all the targets were composed of O aluminium alloy. Due to it is a ductile metal with unchanged elastic modulus E ty, tangent modulus E tt, yield strength Y t, ultimate strength Y ts and failure strain e tf during the experiments. ence, the only measurable constant, the yield strength Y t, is used to express the behaviour of the target material. The ultimate tensile strength of the 2024-O aluminium alloy is 220 MPa. 9 We used projectiles fabricated from dissimilar highstrength materials, AISI 1045 steel and 7055-t77 aluminium, the ultimate tensile strength of which are 625

4 4 Advances in Mechanical Engineering Figure 3. Dimensions of the two types of projectiles: (a) l = and (b) l =1. and 645 MPa, respectively. 9,10 These projectiles can be considered rigid, as their ultimate tensile strengths are higher than that of the target materials. Because of this assumption, the effects of the elastic modulus E py, tangent modulus E pt, yield strength Y p, ultimate strength Y ps and failure strain e pf, which are constant for the projectile materials, can be ignored. Because the densities of AISI 1045 steel and 7055-t77 aluminium differ greatly, inertial effects should be considered r p. To simplify the analysis, the coefficient of dynamic friction between the projectile and the target is also assumed to be a constant irrespective of the projectile material. In addition, because the size ratio of the projectiles remains unchanged, the projectile s nose-shaped factor N can be ignored. Regarding the effective length L eff and the diameter of the projectile, a constant projectile size ratio is maintained in the dimensional analysis; this means that L eff = is a constant. Therefore, is investigated in this study. According to the foregoing assumptions, equations (1) (3) can be simplified to v r = f r r p,, r t,, v i, Y t ð4þ a = f a r p,, r t,, v i, Y t ð5þ T pene = f pene r p,, r t,, v i, Y t ð6þ Equations (4) (6) are the dimensionless forms of equations (1) (3) v r v i = f r a v 2 i, r p, r pv 2 i r t Y t = f a, r p, r pv 2 i r t Y t T pene v i = f pene, r p, r pv 2 i r t Y t ð7þ ð8þ ð9þ Thus, three dimensionless pi terms, namely, =, r p =r t and r p v 2 i =Y t, are obtained. On the basis of equations (7) (9), the following conditions should be satisfied to ensure that the phenomena or responses in the replicas are similar to the prototype 8 D = p rep >< prot r p r = r p t rep r t ð10þ prot rep = r pv 2 >: r p v 2 i Y t where = represents the dimensionless target thickness, r p =r t is the ratio of the inertial effect and r p v 2 i =Y t is the dynamic pressure relative to the target strength. The perforation experiments are designed considering these three pi terms and the scaling factor l. In the geometric-scaling experiments, r p =r t is maintained constant and AISI 1045 projectiles are used. Two types of projectiles are used in the experiments (Figure 3): for the 16- and 10-mm-diameter projectiles, the dimensions of the targets are 140 mm mm 3 8 mm and 140 mm mm 3 5 mm, respectively. The 16-mm projectile perforating the 8-mm-thick plate is considered the prototype, and the latter projectile target combination is considered its geometric-scale model (hereafter, replica-g). In the dissimilar-materials experiments (hereafter, replica-d), 7055-t77 aluminium projectiles perforate 2024-O aluminium plates; the projectile and target dimensions are the same as those of the prototype in the geometric-scaling experiment. The normalised projectile density in the prototype is r p =r t = 2:84 (7:85=2:78) and that in replica-d is r p =r t = 1:02 (2:84=2:78). In experiments where the target thickness is varied (hereafter, replica-t), we use 9.5- and 12-mm-thick targets. owever, the projectile material and dimensions are the same as those of the prototype. The normalised target thicknesses in the prototype and in replica-t are = = 0:5 (8=16) and = =0:59 (9:5=16), 0:75 (12=16), respectively. i Y t Experimental results and analysis The experimentally obtained projectile residual velocities are presented in Table 1 and Figure 4, and prot

5 Song et al. 5 Table 1. Experimental data for the perforation of the 2024-O target. Experiment Test ID t (mm) (mm) M p (g) v i (m/s) v r (m/s) Prototype S S S Replica-G S S S S S S S Replica-D A A A A Replica-T S S plug was very small, as shown in Figure 5, and its mass was therefore negligible; hence, equation (11a) becomes v bl = v 2 i v 2 0:5 r ð11bþ Accordingly, the perforation energy 12 can be estimated as the loss of projectile s kinetic energy Figure 4. Residual velocity as a function of the impact velocity. illustrative high-speed camera images are depicted in Figure 5. After the perforation experiments, the steel projectiles did not exhibit any substantial deformation (Figure 6). The deformation of the nose tip of the steel projectile used in experiment S16-20 was not caused by penetration but by impact with the steel box after perforation; the aluminium projectiles exhibited only slight nose-tip deformation. Figure 4 illustrates the impact velocity as a function of the residual velocity using equation (11) M p v bl = v 2 i v 2 0:5 r ð11aþ M p + M s This equation was proposed by Recht and Ipson. 11 ere, M s is the mass of the plug formed on the perforation, v i is the impact velocity of the projectile and v r is the residual velocity of the projectile and the plug. The W p = M pv 2 i 2 M pv 2 r 2 ð12þ As shown in Figure 4, the results of replica-g experiments have a good agreement with results of prototype experiments (S16-1, S16-3 and S16-5). According to the fitted curves (red line and green line), the ballistic limit velocities are and m/s for prototype and replica-g, respectively. But the results of replica-t (S16-15 and S16-20) are offset from the fitted curve (red line) of prototype experiments, which means that the perforation of thicker target costs higher kinetic energy. In replica-d experiments (A16-1, A16-4, A16-5 and A16-8), the ballistic limit velocity is m/s and the average perforation energy is J. The average perforation energy for prototype (S16-1, S16-3 and S16-5) is J t77 aluminium projectiles cost more kinetic energy than AISI 1045 steel projectiles to perforate the 2024-O aluminium plate of the same thickness. Different dynamic frictional coefficients of different projectile materials may cause the deviation, but the deviation is lower than 5%. Figure 7(a) shows the normalised impact and residual velocities obtained through the experiments. The v r =v i curve overlaps that of r p v 2 i =Y t for the prototype, replica-g and replica-d experiments for = = 0:5. ence, v r =v i of the rigid projectiles with dissimilar materials and different geometric-scaling models can be predicted using

6 6 Advances in Mechanical Engineering Figure 5. Illustrative images captured using the high-speed camera. Figure 6. Dimensions of example projectiles after the perforation experiments, as well as the standard projectile. 0 0:987 10:458 rp v2i vr A 1:750 vi Yt ð13þ Figure 7(a) shows that when rp v2i =Yt for the 16-mm steel and aluminium projectiles are the same, the normalised projectile density (rp =rt ) does not substantially

7 Song et al. 7 Figure 7. Normalised residual velocity as a function of (a) r p v 2 i =Y t and (b) (r p v 2 i =Y t )=(= ). affect v r =v i. For replica-, where the values of = of the projectiles are different (0.59 and 0.75), the actual curve is offset from the fitted curve. If the abscissa in Figure 7(a) is changed from r p v 2 i =Y t to (r p v 2 i =Y t)=(= ), the = curve overlaps the fitted curve (Figure 7(b)). Thus, the new prediction equation can be expressed as 0 0 v r r p v 2 i =Y 1 B t 1 2:753@ A v i = 0:775 1 C A 0:341 ð14þ From Figure 7(b), the residual velocity prediction for rigid projectiles for different target thicknesses, dissimilar materials and different geometric-scale factors can be achieved using a single dimensionless relationship equation. As explained in section Introduction, the effective resisting stress (s r ) 8 is used to estimate perforation energy W p = pd2 p s r ð15þ 4 According to Rosenberg and Dekel, 8 the relationship between the normalised effective resisting stress (s r =Y tf ) and the normalised target thickness (=D) for the perforation of ductile plates by sharp-nosed rigid projectiles is s + 4 D for : >< D 1 3 r = 2:0 for : 1 Y 3 D 1:0 ð16þ tf >: 2:0 + 0:8 ln D for : D 1:0 Combining equations (12) and (16), the fit curve of equation (17) is also listed in Figure 7(b). This method of residual velocity prediction 8 has demonstrated that equation (15) is valid M p v 2 i 2 M pv 2 r 2 = pd2 p s r 4 ð17þ On the basis of the research of residual velocity prediction, for a rigid projectile perforating a 2024-O target, replicas with different geometries (geometric scaling and target thickness), dissimilar projectile materials and impact velocities can be similar to the prototype. The effect of the normalised projectile density r p =r t on a short projectile (spheres and L=D = 1 cylinders, where L is the height of the cylinder and D is diameter of the cylinder.) should be considered. 12 In our experiments, the projectiles are not short, and the nose shape of the projectiles is conical. This is why it does not substantially influence the normalised residual velocity v r =v i. Numerical simulation Numerical model and validation Numerical model. A two-dimensional axisymmetric model is used to simulate normal penetration using the finite element code LS-DYNA (Figure 8). The projectiles can be assumed to be rigid; the constants used in the rigid material model for the steel and aluminium projectiles obtained from the literature are listed in Table 2. 9,10 Similar to the target dimensions used in the experiments, the radius of the targets in the simulation models is set as 50 mm. The target is fully clamped at the boundary. *Contact_2D_Automatic_Single_Surface is used for the friction setting, and the targets are assigned the *ourglass setting with the Flanagan Belytschko

8 8 Advances in Mechanical Engineering Table 2. Material constants for rigid projectiles. Materials r (g/cm 3 ) E (GPa) y AISI T Table 3. Material constants for the 2024-O target. r 9 (g/cm 3 ) E 9 (GPa) y 9 s 0 (MPa) E t (MPa) b FS Figure 9. Stress as a function of the strain in 2024-O aluminium, obtained through an SPB test. Figure 8. Finite element model showing a conical projectile and Zones I and II of the target plate. stiffness. For reducing the computing time, refined shell elements are used as shown in Figure 8. The dynamic frictional force influences the perforation behaviour of the conical-nosed projectiles but is difficult to determine, as it varies with factors such as the stress state, temperature, material(s) between the contact surfaces, contact direction, projectile velocity and melting point of the material. This coefficient has been assumed to be in simulations and theoretical analyses of aluminium-plate perforation in the literature To simplify the model, ignoring the difference in the dynamic frictional coefficients of dissimilar materials, this coefficient is assumed to be 0.05 in all simulations in this study. Using a bilinear elastic plastic model (*MAT_PLASTIC_KINEMATIC), the yield stress and tangent modulus of the target material are obtained from its dynamic stress stain curves (Figure 9), which were measured using a split- opkinson pressure bar (SPB) test. owever, the Young s modulus from the fitted curve substantially deviates from that of aluminium; hence, in Table 3, the Young s modulus is assumed to be E = 73:1 GPa according to MatWeb data. 9 The 2024-O aluminium alloy has no obvious strain-rate effect; 21,22 hence, this effect is not considered in the simulation. In practice, the element size influences the effective plastic strain for erosion; therefore, the effective plastic strain for erosion in the simulation was adjusted on the basis of earlier ballistic experiments. 23,24 According to the experimental data, element erosion occurs at Zone I during perforation; thus, the effective plastic strain for erosion is set as e f = 1:68. Validation of the simulation method. The simulation can be validated by comparing the experimental and simulated impact velocity v i and penetration time T pene, as shown in Figure 10(a) and (b). The experimental and simulated perforation processes are depicted in Figure 10(c); as is evident, the penetration position and time of the projectile in the simulation are similar to those in the experiment. Moreover, these validations of the velocity and time validate the projectile deceleration data indirectly. Deceleration filtering In the simulations, the deceleration time history often contains high-frequency components, one of which is an elastic wave oscillating along the projectile axis. 25 For the A16-1 simulation, high-frequency components can be substantially reduced using a fast Fourier transform and low-pass filtering (57,000 z) of the deceleration time history (Figure 11). Using the authentication method of Fasanella and Jackson, 26 we then integrated the filtered deceleration curve (red solid curve) to obtain the velocity curve (green dashed dotted line), which fits well with the velocity time history curve (black solid line) derived from the simulation (Figure 11).

9 Song et al. 9 Figure 10. Validation of the simulated (a) residual velocity and (b) penetration time; (c) images of the perforation process in the S16-1 experiment (captured using the high-speed camera) and the corresponding simulation. After filtering, the profile of the deceleration time curve can be simplified as a triangular wave, which can be characterised by the maximum deceleration during penetration (a max ), the time to a max (T max ) and the time to complete perforation by the projectile nose (T all ). After complete perforation by the projectile nose, only the frictional force acts on the cylindrical surface of the projectile, and this frictional force is very low. Therefore, T pene is not influential in filtering the deceleration time curve. a and T pene in equations (8) and (9) can be substituted with a max, T max and T all a max v 2 i = f a, r p, r pv 2 i r t Y ð18þ T max v i = f Tm, r p, r pv 2 i r t Y t T all v i = f Ta, r p, r pv 2 i r t Y t ð19þ ð20þ In the next section, the deceleration time curve is filtered by the same manner. For all simulations, before analyses, the (integrated) velocity time curve is matched with the (simulated) velocity time curve. And for simplicity, the effects of the Young s modulus and Poisson s ratio of the rigid projectile have been ignored. In all subsequent simulations, the Young s modulus and Poisson s ratio of the projectiles are assumed to be 203 GPa and

10 10 Advances in Mechanical Engineering Figure 11. Projectile velocity and deceleration as a function of time. Black solid line: velocity time curve from the simulation; blue solid line: deceleration time curve from the simulation; red solid line: filtered deceleration time curve; green dashed dotted line: velocity time curve integrated filtered deceleration time curve. Table 4. Simulation results. No. r p (g/cm 3 ) v i (m/s) (mm) r p =r t r p v 2 i =Y t = a max (m/s 2 ) T max (ms) T all (ms) , ,884, ,234, , , , , , , , , , , , , ,155, ,274, , respectively (the same as those of AISI 1045 steel in Table 2). In the following simulations of dissimilarmaterial projectiles, only the densities are assumed to be different. In addition, in the next section, the target diameter is determined using the equation =D t = 0:16 (16/100), where D t is the target diameter. Deceleration Similarity analysis of the projectile deceleration. Considering three deceleration characteristics (a max, T max and T all ), dimensional analysis is used to obtain the deceleration time data for downscaled models and to make it similar to those in the prototype. Let simulation 1 in Table 4 be the prototype simulation. The 17 listed simulations were performed as follows: when changing one pi term, the other two pi terms were unchanged. When varying r p =r t, the projectile impact velocity was adjusted so that r p v 2 i =Y t remained unchanged. According to the results of simulations, the dimensionless relationship for a max, T max and T all is as follows a max v 2 i r p = 63:882 f 1 f 2 r t r p v 2 i Y t f 3 ð21þ

11 Song et al. 11 Table 5. Dimensions and parameters used in the simulations of the prototypes and replicas. Group Parameters Prototype Replica 1 Replica 2 Replica 3 A r p (g=cm 3 ) (mm) (mm) v i (m=s) B r p (g=cm 3 ) (mm) (mm) v i (m=s) where f 2 r p v 2 i Y t where f 5 f 1 r p r t = 0:136 0:025e 3:178 r t rp = 0:632e 0:981 rpv2 i Yt + 0:259e 0:229 rpv2 i Yt + 0:038 = 0: :029 f 3 T max v i r p = 0:425 f 4 f 5 r t r p v 2 i Y t where f 8 f 4 r p r t r p v 2 i Y t f 6 = 1:556 0:264e 0:903 r p r t = 4:060e 1:870 rpv2 i Yt ð22þ + 0:420e 0:240 rpv2 i Yt + 1:420 f 6 = 1:553 0:044 T all v i r p r p v 2 i = 0:126 f 7 f 8 f 9 r t Y t r p v 2 i Y t f 7 r p r t = 1:035e 2:627 rpv2 i 1:835 Yt f 9 = 2:896 0:208e 0:343 r p r t ð23þ + 0:794e 0:327 rpv2 i 1:835 Yt + 2:554 = 0:482e 1:483 Dp + 1:803 Discussion. To verify the validity of equations (21) (23), we modelled two simulations as prototypes: a 64-mm steel projectile penetrating a 32-mm-thick plate and a 64-mm steel projectile penetrating a 64-mm-thick plate. The projectile is assumed to be rigid, the initial impact velocity is 500 m/s, and the plate material is 2024-O aluminium. a max, T max and T all of the prototypes were obtained through simulations and the dimensionless relationship equations (Groups A and B in Table 5). Substituting the prototype results (a max, T max and T all ) obtained using equations (21) (23) into equations (21) (23) again but with different projectile diameters, r p, v i and were calculated for the projectile diameters = 32, 16 and 8 mm. According to the obtained values, model perforation simulations were performed for the downscaled models. The theoretical and simulation results are listed in Tables 6 and 7 for Groups A and B, respectively. The theoretical and simulated results deviate slightly from each other, as shown in Tables 6 and 7; nevertheless, the deceleration time data in most of the downscaled models are the same as those in the prototype models, as shown in Figures 12 and 13. owever, when the projectile diameter = 8 mm, the deceleration time data of the downscaled models do not agree well with those of the prototype. In Group B, the 8-mm-diameter projectile could not perforate the target plate. For deceleration time curve similarity research, a dimensional analysis with the geometry (i.e. geometric scaling and target thickness), dissimilar projectile materials and impact velocity is still a valid way to find a prototype-like deceleration time process in a downscaled model. Conclusion This study focused on the similarity of the residual velocity and deceleration time data in dissimilar projectile materials and geometric-scaling models with = = 0:5 1 under low-velocity impact loading. The conclusions are as follows:

12 12 Advances in Mechanical Engineering Table 6. Theoretical and simulated results for Group A. Group A Group A Prototype Replica 1 Replica 2 Replica 3 Equation a max (m=s 2 ) 204, , , ,736.3 T max (ms) T all (ms) Simulation a max (m=s 2 ) 185, , , ,800.6 T max (ms) T all (ms) Table 7. Theoretical and simulated results for Group B. Group B Group B Prototype Replica 1 Replica 2 Replica 3 Equation a max (m=s 2 ) 362, , , ,657.3 T max (ms) T all (ms) Simulation a max (m=s 2 ) 337, , , ,696.6 T max (ms) T all (ms) Figure 12. Deceleration time curves of Group A: (a) before and (b) after filtering. Figure 13. Deceleration time curves of Group B: (a) before and (b) after filtering.

13 Song et al. 13 The similarity of rigid projectiles normally impacting 2024-O aluminium plates at low velocity was evaluated in this study. On the basis of the similarity criteria of the residual velocity and deceleration time data, replicas with geometric scaling, different target thickness and dissimilar projectile materials can be similar to the prototype. Through experiments and simulations, the residual velocity can be predicted by the single relationship equation with the main dimensionless parameters. Only the dimensionless pi term (r p v 2 i =Y t)=(= ) has an effect on the dimensionless residual velocity v r =v i in the residual velocity equation. According to the simulation results of a rigid projectile perforating a 2024-O aluminium plate in the range of 0:5 = 1, the deceleration time data are simplified as a triangular wave and characterised using a max, T max and T all. The dimensionless equations for the three characteristic parameters a max, T max and T all were developed by numerical and dimensional analyses. A similar deceleration time process can be replicated by these three dimensionless equations. On the basis of the experimental and numerical results, the dimensionless term (r p =r t ) has a negligible effect on the dimensionless residual velocity v r =v i for predicting the residual velocity. owever, the term r p =r t should be considered for predicting the deceleration. 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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the National Natural Science Foundation of China (no ). References 1. Corbett GG, Reid SR and Johnson W. 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