Study on the Similarity Laws for Local Damage Effects in a Concrete Target under the Impact of Projectiles

1State Key Laboratory for Disaster Prevention & Mitigation of Explosion & Impact, PLA University of Science and Technology, Nanjing 210007, China 2School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China 3Engineering Research and Designing Institute of Xinjiang Military Area Command, Wulumuqi 830042, China 4The Fifth Department, 145 Erqi Road, Wuhan 430012, China


Introduction
As the main construction material, concrete is widely used in civil and military engineering.Studies on destruction effects in concrete structures have drawn widespread attentions, especially after World War II.The investigation of damage effects on concrete structures under impact has become a major research field in the weapon manufacturing industry and protective engineering and continues to receive prevalent attention from research groups all over the world.
These formulae provide the basic references for engineering design.However, it should be noted that the majority of them were obtained via dimensional analysis of experimental results.Some dimensionless parameters in these formulae require further physical interpretation.And the dimension mismatch in some formulae also requires further investigation.The analytical calculation models based on impact mechanics are another important aspect.For instance, Luk and Forrestal [15] and Forrestal and Tzou [16] used the cavityexpansion theory to establish the locked, incompressible, and linearly compressible models for concrete targets.Li and Chen [17] proposed a new calculation method for penetration depth by introducing the nondimensional impact factor and warhead shape factor.

Shock and Vibration
Wang et al. adequately considered the destruction and deformation characteristics near the penetration cavity [18] and proposed a rigid-plastic model for thin target impact [19][20][21] and internal friction [22,23] and modified hydrodynamic models [23,24] for thick target penetration.Furthermore, the differential surface stress method [25] and the velocity potential theory [26,27] for concrete penetration are advanced by other researchers.
The above models provide a framework for the impact process study on concrete targets, but the destruction mechanisms hiding in the empirical and analytical models still require reasonable physical explanations, and the inherent regularities also need to be discovered.In this study, based on the analysis of concrete target local destruction and deformation characteristics, the similarity laws of concrete targets impacted by rigid projectiles are studied with the rigid-plastic, internal friction, and modified hydrodynamic models for a wide range of striking velocities.

The Local Effect Characteristics of Concrete Targets under Impact
The mass, momentum, and energy conservation laws are satisfied in impact process.Within the region of local deformation and destruction under impact, the mechanical energy conservation can be expressed as where   denotes the stress on surface  related to volume  occupied by the material,  is the mass force distribution,  is the displacement vector, and  is the potential energy.
In general, a part of  is transformed into deformation potential energy, while some of the energy is dissipated through the fracture and shearing surfaces induced by irreversible deformation, and the remaining energy is transferred into kinetic energy through the boundaries.
can be described by the principal stresses  1 ,  2 , and  3 along the three perpendicular directions and the corresponding principal strains  1 ,  2 , and  3 .An arbitrary stress state with  1 ≥  2 ≥  3 can be separated into three states: ( For the first stress state, the maximum shearing stress  = ( 1 −  3 )/2 corresponds to a state with compression in the first principal stress direction and tension in the third principal direction.The second stress state corresponds to the hydrostatic pressure   = ( 1 +  3 )/2, with equal pressure in all directions.  = (2 2 −  1 −  3 )/( 1 −  3 ) is the Lode parameter, and the third stress state shows the influence of  2 .Therefore, any triaxial stress state can be described with ,   , and   .And impact problems can be divided into typical generalized shearing and generalized compression states according to the initial impact and boundary conditions.

Generalized Shearing State.
The concrete deformation characteristics are shown in Figure 1 [11].Figure 1 can be used to study the material's response for the generalized shearing state.In Figure 1,  max is the maximum shear stress,  =  1 − 3 is the principal shear strain, and   =  1 + 2 + 3 is the volume strain.
Furthermore, in Figure 1, the segment  corresponds to the elastic deformation zone under impact.The material deformation does not violate the deformation harmony condition, and in the reversible deformation state the material deformation is mainly related to elastic, viscous, and frictional properties.If the load on material increases, despite the decrease in cohesion, the strength rises due to the internal friction increase.The shearing strength limit increases with the hydrostatic pressure acting on the shearing surface.This reflects the internal friction increasing. determines the elastic energy stored in the material which is separated by the surface, and the energy storage shows a volume characteristic.
In segment , after the maximum shearing stress, the material enters the irreversible deformation state, and the surfaces of local shearing deformation appear and expand.The material is divided into blocks or pieces with certain dimensions, and the boundary conditions are satisfied through the abrupt discontinuous slipping of single surface.The residual strength of material has close relationship with the block dimension which influences the later stress and strain states.Thus, this state corresponds to the material's comminuted zone under impact.The shearing slips result in an asymmetry of the stress tensor, but the deformation harmony condition is still approximately satisfied because enough forces are acting on the local deformation surface.Therefore, the distributed moment appears on the discontinued lines and the material is in a quasi-continuous internal friction state.
After the peak point, with the rotation of blocks, mutual sliding occurs within the blocks; both the total strain and the shearing strain increase.Thus, the volume change and dilatation are related to the size of the destroyed blocks.The response to the change in volume is the key information.A part of  is dissipated through the cracks and slip lines, which are irreversible deformations, and the energy dissipation shows superficial characteristics.The residual energy is transformed into kinetic energy through the boundary and the transferred kinetic energy shows a volume characteristic.

Generalized Compression State.
Concrete and rock penetration experiments [11,18] showed that the stress wave near the penetration cavity has the shape shown in Figure 2. The duration time of the material's positive movement is approximately one order longer than that taken to reach the maximum velocity; that is,  + / − ≈ 0.1 ∼ 0.2.The higher stress and the rapid velocity change occur only in a certain narrow zone; the stress wave is almost identical to a shock wave and exhibits "short wave" properties.The difference between short wave and shock wave is the lateral deformation.For a short wave, the material moves with tiny lateral displacement, and the shearing and volume deformation are of the same order of magnitude.Therefore, the boundary of the maximum particle velocity at a certain time can be regarded as the wave front, and the following equation is satisfied: where   is the radial stress,  0 is the density,  0 is the longitudinal wave speed,   is the particle velocity, and   is the radial strain.And, as Figure 2 shows, although the radial stress decreases after its maximum  max , the radial displacement increases until it is equal to the maximum  max .This means that the work is dissipated in order to overcome the internal friction.Figure 3 shows the responses of columniform test specimen under uniaxial compression.The wall  1  1 of cylinder in Figure 3(a) is rigid and those in Figures 3(b) and 3(c) are confined.Figure 3(d) shows the unconfined uniaxial compression.In Figure 3,  is the bulk modulus,  is the shearing modulus,   is the radial stress,   =   are the circumferential stresses,   ,   , and   are the radial and circumferential strains, ] denotes Poisson's ratio, and   is the shearing strength.Controlled by the impact velocity and the boundary conditions, the material's response has four states shown as Figure 3.
Figure 3(a) corresponds to segment  in Figure 1, and the material is in one-dimensional strain state, which can be regarded as uniaxial compression with rigid lateral constraint.The friction between particles is high enough and the cohesion has not been destroyed.Therefore, under compression, the material deformation is elastic and The stress wave and deformation of material near penetration cavity [11,18,23].
the longitudinal wave speed is the propagation speed of small perturbations.With Hook's law, the stress and strain can be yielded as Figure 3(a).
Figure 3(b) corresponds to segment  in Figure 1, and the cohesion between the material particles is destroyed.However, the friction cannot be ignored, and the material is in a quasi-continuous internal friction state.Mohr-Coulomb criterion can be used and the longitudinal wave speed decreases from the small perturbation propagation speed.The material can deform laterally because of the finite constraint by the wall  1  1 .But   and   are very small compared with   .Therefore,   ,   , and  max are almost the same.
If the load enhances in Figure 3(b), the material response changes into Figure 3(c) which corresponds to segment  in Figure 1.In Figure 3(c), the friction between materials is almost constant and the three stresses   ,   , and   are almost the same.The friction between material particles is so small that it can be ignored.The longitudinal wave speed decreases to the bulk sound speed.This condition can be regarded as a fluid-like state.
Therefore, the stress state of the material changes from an elastic state to an internal friction state and eventually reaches a fluid-like state, and the deformation is the result of the gradual disappearance of the local shearing stability on multiple stages and levels.The energy near the penetration cavity is mainly dissipated because of internal friction with lateral constraint and the energy for material destruction is only a small fraction.Therefore, the stress state changes and the internal friction should be taken into account in penetration analysis.Because of the internal friction, when the material changes from elastic to plastic, it does not change to an ideal plastic state as a metal and changes to an obviously enhanced plastic state instead [11].During the material conversion process from a state with internal friction to a fluid-like state, the longitudinal wave speed decreases from the small perturbation propagation speed to the sound speed.Figure 3(d) represents the critical state under uniaxial compression without constraint.The material has enough time to deform laterally, and severe volume compression is impossible, which is similar to the deformation state of a concrete target with finite thickness impacted at low velocity [11].The target's free surface provides the precondition for using the incompressibility condition and rigid-plastic model.

Engineering Calculation Models for
Different Impact Conditions

Rigid-Plastic Model for Thin Target Impacted at Low
Velocities.For the thin target under impact, overall and local damage effects will occur.This study focused on the local ones, and the velocity varying from 30 m/s to 150 m/s was regarded as the low velocity considering both the engineering status quo and striking threats.For structures with finite thickness in a state as shown in Figure 3(d), impact occurs with free surfaces, and severe material volume compression cannot occur.For large deformation, the elastic component occupies a much smaller part than the plastic one, and the material near penetration cavity is in a local plastic state.Therefore, the material can be supposed as rigid-plastic and incompressible.

The Ultimate Resistance on Projectile.
Wang et al. [19,20] and Chen et al. [21] considered the characteristics of thin target impact and regarded the material as incompressible and rigid-plastic in order to obtain the permissive dynamic velocity field in the dynamic process by utilizing the virtual work principle and ultimate load theorem.The permissive dynamic velocity fields of flat-nose projectile impact for penetration, scabbing, and perforation are as Figure 4 shows.With the permissive dynamic velocity fields, the ultimate resistances on the projectile for penetration, scabbing, and perforation are obtained as follows: where   is the normal component of the ultimate resistance on unit area of the projectile nose,  is the target thickness, ℎ is the penetration depth, and  is the projectile shank diameter.Equation (4a) shows the ultimate resistance   for penetration depth ℎ in semi-infinite target impact.Equations (4b) and (4c) are the ultimate resistances for scabbing and perforation in impact on the targets with penetration depth ℎ, respectively.With (4a), (4b), and (4c), the change trend of   /  is calculated by setting / as 4.5 and 1.5, as shown in Figure 5.Each ultimate resistance curve satisfies the corresponding permissive dynamic velocity field.
As Figure 4(a) shows, if the impact velocity is low, the target material in region IV does not move and the target material in regions I, II, and III move.However, the material in region I moves along the projectile velocity direction and that in region II begins changing the moving direction.At last the material in region III moves along the opposite direction of projectile velocity.With the impact velocity increasing, the target rear surface starts to influence the impact process and the target material in region IV of Figure 4(a) begins moving, and the target material does not move along negative direction of -axis any more.The material in region II moves along positive directions of -axis and -axis.Therefore, Figure 4 shows.This means on the left side of intersection point  perforation does not happen and on the right side of it perforation appears.In this condition, the critical perforation thickness ℎ  can be obtained.In Figure 5(b), the curves 1 and 2 do not intersect, which means that penetration will not occur because the target thickness is too small.Therefore, for this condition, only scabbing and perforation appear at the same time.

The Critical Scabbing and Perforation
Thicknesses.The normal resistance on projectile nose  can be calculated with the following equation: The differential equation for the motion of projectile is where  is the projectile mass and  is the time.The initial boundary condition is where  0 is the impact velocity.With (4a), (4b), (4c) ( 5), ( 6), (7a), and (7b), the critical scabbing and perforation thickness can be yielded.For (4a), (4b), and (4c), it is found that if / ≤ 1.82 the curves 1 and 2 have no intersection point and scabbing and perforation appear together.Therefore, the intersection point  is used to calculate the critical perforation thickness.With (4b), (5), and (6), the following equation is yielded: According to (7a) and (7b) and the intersection point 's condition, it can be obtained that If / ≤ 1.82, scabbing and perforation happen together; therefore the critical perforation thickness is concerned.
The extremum of / can be calculated to get the critical perforation thickness as If / ≥ 1.82, the critical scabbing and perforation thicknesses can be obtained with (4a), (4b), (4c), ( 5), ( 6), (7a), and (7b) by the same method above.Consider  1, where   = 0.5√   st .It is found that the results obtained using (10a), (10b), and (10c) are in good agreement with the experimental results (except for test NC-F0-2), suggesting that, in general, the rigid-plastic model can be used to predict the damage caused by the impact of a projectile on a thin concrete target at low velocities with a high degree of accuracy.

Internal Friction Model for Thick Target Impacted at
Intermediate Velocities.For a concrete target, if the condition / 0 < 0.2 is satisfied, the material near penetration cavity is in an internal friction state [11,22] indicated in Section 2.2.Therefore, 150 m/s <  0 < 0.2 0 can be regarded as the intermediate velocity range.For thick target under impact, the material's destruction and deformation states can generally be divided into the comminuted, cracked, and elastic ones [11,15,16,22] as shown in Figure 6.

The Resistance on Projectile
Nose.The short wave in the comminuted zone represents the fastest part of the stress wave and reflects the compressibility and irreversible deformation of the material [11,22] according to the following formula: where (  ) is the polynomial of   with |(  )| ≪ 1 corresponding to the weak linear characteristic of (11).With (3) and the mass conservation law, we obtain where  is the shear dilatation parameter, which is in the range from 1.6 to 1.8 for concrete and hard rock [11,22], and  is the radial Euler coordinate.With (11) and (12), the general solution of radial stress in the comminuted zone is where    is the radial stress in comminuted zone and  is the function to be ascertained.
In the cracked zone, the circumferential stress is zero.And because the dynamic properties of material in the cracked zone have potty influence on the solution, the inertia effect can be ignored and the equation of momentum conservation can be written as On the boundary of the cracked and comminuted zones ( = ), Tresca criterion is satisfied.Consider With ( 14) and ( 15), the radial stress    in the cracked zone is The radial stress on the boundary of the cracked and comminuted zones is continuous which means ( 13) and ( 16) are equal if  = .Therefore, function  is obtained as Combining ( 13) and ( 17) and setting  = , the cavity pressure    can be obtained as If / is known, the resistance on projectile nose can be calculated.

Calculation of 𝑎/𝑏.
In the elastic zone, the following equations are satisfied: The material can be regarded as incompressible in the elastic zone: where  is the radial displacement.Combining (19a), (19b), and (20), the radial stress in the elastic zone    is obtained as The radial stress on the boundary of elastic and cracked zone is continuous which means ( 16) and ( 21) are equal if  = .With this condition,  is obtained: With ( 20) and ( 22), the displacement of elastic zone boundary can be calculated: With the incompressible condition ( 3 −  3 = ( − ) 3 ), the displacements in comminuted and cracked zones can be obtained as On the boundary of the elastic and cracked zones ( = ), the displacement is continuous, and, with ( 23) and ( 24), the following equation is yielded: In the cracked zone, because the energy for plastic deformation can be ignored, the energy for crack growth  1 is mainly the elastic potential energy which is half the work from outside force.Consider The whole surface energy used for new surface forming in crack growth  2 is where   is the surface energy for unit area and  is the number of cracks grown.With ( 25), ( 26), (27a), and (27b), the energy conservation condition is With ( 24) and ( 25), we obtained Combining ( 25), (28), and (29), the following equation is yielded: If the derivative of  on the left side of ( 30) is equal to zero, the cracked zone has the extremum.This can be regarded as the condition of cracked zone boundary.Consider According to (31), it is found that The fracture toughness   and surface energy for unit area   have the relationship in condition of plane strain state as With ( 25), (32), and (33), the following equations can be yielded: Equation (34b) reveals the scaling relationship in penetration of projectiles with different calibers.In scaling experiments, the higher the scaling parameter is, the larger the difference between the real and experimental results is.Therefore, the scaling parameter should be in a certain range in order to ensure that the scaling experiment reflects reality.

The Penetration Depth and Comparison.
For a conicalnosed projectile, the penetration resistance   can be expressed as where  is the penetration velocity,  is the friction coefficient, and  is half cone angle of projectile nose.For a concrete target, the calculation results show that the third summand on the right side of (35a) can be neglected if / 0 < 0.3.The penetration depth ℎ  can be calculated with the following equation [11,22]: In (36), the linear term of  0 reflects the influence of the material's motion in the comminuted zone on penetration, and the logarithmic term shows the effect of material strength.
In order to validate the correctness of internal friction model, the results calculated with internal friction model will be compared with results obtained from empirical formulae, for example, the Bernard I and modified NDRC equations.The concrete target and projectile parameters are listed in  Tables 2 and 3, where   is the projectile nose length.The friction coefficient influences the predicted result.Forrestal [32] suggested that the friction coefficient between steel and porous rock can be ascertained as With (37), it is found that  is no more than 0.08 for the velocity considered here.Therefore, 0 and 0.08 were chosen as the limiting values of , and the predicted results for different projectile and target groups obtained with the internal friction model, the Bernard I, and modified NDRC equations are shown in Figures 7-12, respectively.As shown in Figures 7-12, the majority of results obtained with the two empirical formulae are located between the results predicted with the proposed model for  = 0 and  = 0.08, which means that the internal friction model describes the penetration process well.Furthermore, the slopes of the curves obtained from the internal friction model and the Bernard I formula are almost identical, especially for  = 0, which strengthens the conclusion that the internal friction model can predict the penetration depth well for thick target impacted at intermediate velocities by rigid projectiles.

Modified Hydrodynamic Model for Thick Target
Impacted at High Velocities.The calculation revealed that, for a concrete target, if / 0 ≥ 0.2, the comminuted material near the penetration cavity is in fluid-like state [11,23], as illustrated by Figure 13.Therefore, 0.2 0 ≤  0 < 2500 m/s was selected as the high velocity range, with both current and future weapon velocities considered.For a high velocity penetration, the comminuted material can be considered as an ideal incompressible fluid with density  0 .If the pressure on the fluid-like boundary Γ is equal to the target material dynamic hardness  [33][34][35][36], the impulse equation and the Bernoulli equation of fluid with the continuous condition can be written as [37] where   is the contraction coefficient of the fluid-like material,  ∞ is the fluid-like material velocity limit, and  is the penetration resistance.Using (38a), (38b), and (38c), the following equations can be obtained: Therefore, the penetration depth can be calculated according to From (38a), (38b), (38c), (39a), (39b), and (39c), it is clear that   is the key parameter required to solve the equations.According to Slepyan [37], for this axisymmetric problem,   can be directly calculated by solving the plane problem of a wedge by replacing  2 and  2 with 2 and 2, respectively.Then The penetration depth is determined with (39a), (39b), (39c), ( 40), (41a), (41b), and (41c).However, it is impossible to obtain an analytical solution for the penetration depth, and then the numerical method is used.Forrestal et al. [29] performed penetration experiments on concrete targets with 0.90 kg 4340 steel rods whose CRH and diameter are 2 and 26.9 mm, respectively.The projectiles were heat-treated in order to obtain a hardness of Rc 43-45, so that they remained rigid during the experiments.The target parameters are as follows:  was calculated with Wang's formula [38]: where  is the dynamic shear strength.
In foregoing conditions, the predicted results were compared with the experimental ones and are shown in Figure 14.The predicted results are in good agreement with the experimental values for  0 > 500 m/s, except for case 4.However, the experimental result obtained for case 4 is aberrant when compared with the whole range of experimental results.If  0 ≤ 500 m/s, the predicted penetration depths are lower than the experimental values obtained for cases 1-3 in Figure 14, which means that the model cannot accurately predict the penetration depths at lower velocities.Furthermore, for this concrete target, 500 m/s is close to 0.2 0 = 494 m/s, which is the critical value for high velocity penetration.Therefore, it can be concluded that the modified hydrodynamic model can accurately predict the penetration depth for high velocity impact by rigid projectile. 0 (m/s) Figure 14: The experimental results [29] and predicted penetration depth.

Similarity Laws for the Impact on a Concrete Target
4.1.Dimensional Analysis.Dimensional analysis is the effective method in physics research and needs cogent analysis on the dimensions of main variables in the problem [39].At present, majority of the widely used empirical formulae for concrete impact were obtained via dimensional analysis with experimental results.In different impact conditions, the main factors which influence the impact results are different.In this paper, the research object is the rigid projectile impact on a concrete target, and the relative parameters of projectile deformation are not considered.The projectile mass , the diameter , and the nose shape parameter  * are taken into account.For a concrete target, thickness , material density  0 , shearing strength   , Young's modulus , longitudinal wave speed  0 , fracture toughness   , and dynamic hardness  are the main parameters that are considered.Furthermore, the impact velocity  0 and the friction coefficient  are also included in the analysis.Then, the penetration depth ℎ  , the critical scabbing thickness ℎ  , and the perforation thickness ℎ  can be written as ℎ  =   (, ,  * , ,  0 ,   , ,  0 ,   , ,  0 , ) , ℎ  =   (, ,  * , ,  0 ,   , ,  0 ,   , ,  0 , ) , ℎ  =   (, ,  * , ,  0 ,   , ,  0 ,   , ,  0 , ) .
Li and Chen [17] deemed that Young's modulus is not the main influence factor and found that Young's modulus is not included in the majority of the empirical formulae and ignored this parameter in their analysis.Young's modulus describes the target material's property in elastic state, while the material's property in plastic state is the main factor for penetration resistance, so it is reasonable to ignore Young's modulus.
Li and Chen [17] suggested that the friction coefficient between projectile and target can be ignored in rigid projectile penetration, and as the case stands,  is difficult to measure [40].According to (37),  is very small, so the influence of  can be ignored.  and  as strength representations are included in (44a), (44b), and (44c); however, they have different effects on impact.  reveals the area character in energy transferring and has local plastic property.However,  is the energy consumption per unit volume for cavity and reflects the volume character of energy consumption and is the embodiment of the material's overall plastic state and dynamical cratering.It contains much important information, such as the strain rate effect, the inertia, and strength effects [41].Therefore,   and  should be preserved in (44a), (44b), and (44c).In thick target penetration,  is not the main influence factor which controls the process, but it plays a very important role in scabbing and perforation of thin target, and the critical target thicknesses for scabbing and perforation in thin target impact are equal to the target's thickness .Therefore,  can be eliminated in (44a), (44b), and (44c), and it can then be rewritten as ℎ  =   (, ,  * ,  0 ,   ,  0 ,   , ,  0 ) , ℎ  =   (, ,  * ,  0 ,   ,  0 ,   , ,  0 ) , ℎ  =   (, ,  * ,  0 ,   ,  0 ,   , ,  0 ) .
Equations ( 45a), (45b), and (45c) can be transformed to In (46a), (46b), and (46c),  =  2 0 /( 3   ), and it shows the same form as the impact factors suggested by Chang [9], Reid and Wen [13], Hughes [42], Haldar and Hamieh [43], Kojima [44], and Li and Chen [17,45].The only difference is the fact that the uniaxial compressive strength is replaced by the shearing strength.Therefore,  =  2 0 /( 3   ) is also defined as the impact factor and indicates the ability of projectile to destroy the target. * represents the influence of the projectile's nose shape on the impact.Ψ =   /( 0  0  0 ) shows the relationship between the shearing strength and the dynamic stress and essentially represents the energy ratio of destruction and particle motion and shows the distribution of impact energy.Therefore, it can be defined as dynamic factor.Λ =    0.5 /  , defined as toughness factor, shows the relationship between the thickness of the comminuted and cracked zones. =   / represents the energy distribution between the local destruction and overall plastic destruction and can be defined as dynamic hardness factor.For targets consisting of the same material, impact factor is the main parameter that determines the destruction effect and can be used to guide the design of scaling experiments.

Similarity Laws in the Calculation Models for Concrete
Target Impact.At present, many empirical formulae for concrete target impact were obtained by dimensional analysis and more or less show the effect of impact factor, with only slight differences in form or power [3].To reveal the influence extent of , Ψ, and  in (46a), (46b), and (46c), the similarity laws for impact on a concrete target will be studied based on the models in Section 3.

Similarity Laws of the Impact on a Thin Target at Low
Velocities.According to the general forms given by (46b) and (46c), by curve fitting, (10a), (10b), and (10c) can be rewritten as and the parameters in (47a), (47b), and (47c) are listed in Table 4.
According to (47a), (47b), and (47c), the impact factor  is shown to be the unique factor that determines the critical scabbing and perforation thicknesses, and Ψ, Λ, and  have no influence on these critical thicknesses.The local plastic deformation state of the target material caused by the free surface brings the results above, and the rear surface of target seriously influences the impact process.The exponents of the impact factor are close to 0.5 for  ≥ 6, suggesting an approximately linear relationship between the target critical thicknesses (scabbing and perforation) and impact velocity.

Similarity Laws of the Impact on a Thick
Target at Intermediate Velocities.Calculations showed that if 0.1 <  0 / 0 < 0.2, the influence of the logarithmic term in ( 36) is below 5%.Therefore, (36) can be rewritten as Equation (48a) means that, for a certain impact velocity range, the penetration depth is approximately directly proportional to the impact velocity.This is consistent with the power of velocity term in the formulae proposed by Young [10] and BLZ [11].Furthermore, for the penetration of a thick target at intermediate velocities, , Λ, and Ψ are the principal factors that determine the penetration process, and the material's internal friction motion in the comminuted zone is the pivotal point for penetration resistance.

Similarity Laws of the Impact on a Thick Target at
High Velocities.As mentioned above, it is difficult to yield an analytical calculation formula for the modified hydrodynamic model.Therefore, the numerical method was applied for calculation.With the parameters given in Section 3.3, the penetration resistance and depth were calculated for different cone angles, and the results are shown in Figures 15 and  16.The resistance shows a linear relationship with  2 / 3  for a certain penetration velocity according to the following formula: where  0 and  1 are dimensionless parameters obtained from the curves in Figure 15.Using ( 40) and (49), the scaling penetration depth can then be calculated as follows: In (50), the scaling penetration depth is a function of the impact factor  and hardness factor . Equation (50) further reveals that the scaling penetration depth has a logarithmic relationship with the impact factor and the hardness factor as shown in Figure 16.However, if 10% can be considered to be an acceptable fitting error, (50) can be fitted with the following function: where  ℎ and  ℎ are the curve fitting parameters.The calculation showed that  ℎ is in the range from 0.3 to 0.35, which means that there exists a 0.6-0.7 power relationship between the scaling penetration depth and impact velocity.This is in agreement with the famous 2/3 power scaling law obtained for spherical projectile experiments on metal [46] and rock [47] targets, which indicates that, for the impact of a rigid projectile on a concrete target, the 2/3 power scaling law also exists.The hardness factor is determined by target material only.In contrast, the impact factor is determined by both the projectile and target.Therefore, the impact factor represents the similarity law in penetration and can be used as the controlling parameter for the design of scaling experiments.

Conclusion
Controlled by the initial conditions, such as the target dimensions and impact velocities, the concrete target undergoes different destruction and deformation states, resulting in different impact effects and similarity laws.In this paper, the generalized shearing and compression states of the impact on a concrete target were analyzed.Based on the rigidplastic, internal friction, and modified hydrodynamic models corresponding to the destruction and deformation characteristics of the concrete advanced by the authors, the following conclusions were obtained according to dimensional analysis: (1) For the impact on a thin target at low velocities, the impact factor is the only parameter controlling the impact.There is an approximately linear relationship between the critical thicknesses and impact velocity.
(2) For the impact on a thick target at intermediate velocities, the impact, toughness, and dynamic factors together determine the penetration behavior.There is an approximately linear relationship between the penetration depth and impact velocity.
(3) For the impact on a thick target at high velocities, the impact and hardness factors together determine the penetration behavior.There is a 2/3 power relationship between the penetration depth and the impact velocity.
(4) For the impact on a thick target at different impact velocities, different distinct scaling laws exist.For impact at intermediate velocities, there is a linear relationship between the ratio of the projectile and cavity radii, and there is no relationship with the penetration velocity, which reveals the scaling conversion relationship for the impact of projectiles with different calibers on a concrete target.The higher the scaling coefficient in the experiments is, the larger the difference between the simulation and real results is.However, for impact at high velocities, there is only a relationship between the ratio of projectile and cavity radii and the penetration velocity, which means that the scaling coefficient has little influence on the simulation result for the impact of projectiles with different calibers, and therefore the scaling experiments can be flexibly designed.Integration variable in (41c) :

Nomenclature
Dynamic shear strength : Y o u n g ' s m o d u l u s  * : P r o j e c t i l en o s es h a p ep a r a m e t e r : Impact factor Ψ: Dynamic factor Λ: T o u g h n e s s f a c t o r ( V ): P o l y n o m i a lo f V : Cavity radius : Comminuted zone radius : Cracked zone radius : Elastic zone radius : Function to be ascertained in (13) : S h e a r d i l a t a t i o n p a r a m e t e r : Radial

Figure 3 :
Figure 3: The material deformation under compression.

Figure 6 :
Figure 6: The destruction and deformation of target material in penetration.