The behavior of ceramicmetal protection against a projectile impact is modeled. The model takes into account the mass and velocity for each stage of the phenomenon. A former model was modified considering more realistic parameters such as geometries and deformation profile. To analyze the model, simulations on different parameters have been run. The impact results of different ballistic projectiles were simulated, and the movement was plotted. In addition, a deterministic simulation on the mechanical properties of the back metal plate properties was done.
Gonçalves et al. [
Proposal of a twolayer protection made by Gonçalves et al. [
The model was better understood once the solutions were used to simulate different impacts over the system. Initially, properties of some ballistic projectiles were used to prevent the shock absorption. In addition, the mechanical properties of the metallic plate were modified intending to observe the effect on the final deformation.
A previous work [
The set of data generated in the simulations together with the newly developed equations for impact and shock absorption were, in conclusion, observed as an advance for the understanding of high speed impact phenomena.
The software Maplesoft Maple V12 was used to solve the system. Gonçalves et al. work [
The mathematical model of the penetration process is divided into three stages. The first represents the initial impact and erosion of the head of the projectile. In this situation there is no penetration into the protection. The force against the projectile is
The impact generates a shock wave that travels through the material and reflects back cracking the ceramic plate. This shock wave is extremely fast and runs through the protection in a small fraction of a second. Theoretically, the sum of the incoming with the returning wave generates a region of high stress. This region is easily observed as a cone in the solids, which fits with the cracking region. Figure
First stage of the impact. The collision generates a shock wave that travels through the armor.
The shape and size of the ceramic tile can also affect the protection performance. The crack cone formation allows the necessary scattering to provide enough ceramic particles for the projectile erosion. With smaller geometry, the crack cone would not be properly formed once the constructive interference of the shock waves would not only occur longitudinally but also transversally. Those considerations were not taken into account in this model.
In the second stage the penetration starts pushing an interface projectileceramic with velocity
It is important to note that the erosion ceases when
The modified hydrodynamic theory given by Tate [
The second stage ends when
It is considered a stressstrain curve given by the power law
Assuming that the material will be bulged in an axis symmetric mode, the effective strain can be written in terms of the radial strain as
For a small displacement, the radial strain can be approximated to [
Considering a geometry of a small dimple given by
Considering the physical nature of most constants, it is possible to suppress the negative sign generated by the derivation demonstrated in (
The solution for the expression is given by
After the initial impact, the movement of the interface starts deforming the metallic plate. It can be argued that, initially, the metal layer is compressed due to high pressure generated by the impact. In addition, the plate does not move significantly because of the low interface velocity. In this way, it is possible to affirm that the deflection of stainless steel plate starts at the end of the second stage of penetration.
Considering that the plastic energy absorbed by the metal plate is equal to the kinetic energy of the projectile in the end of the second stage, it is possible to write the final deflection of the plate as
The movement solutions were obtained divided by the stages and by mass and velocity. The equations needed to be manipulated together to find a unique solvable problem for mass and velocity for each stage. First it was necessary to find a differential equation with a unique function. For the first stage it was
Also, the dimple shape after the impact, shown by Figure
Shape of the metallic plate after impact.
The newly deduced metal final deformity equation is now given by
In addition, for a solid twolayered system, it is necessary to consider the time needed for the generated wave to reflect in the back part of the protection and return to the initial point and, furthermore, start to crack the ceramic layer. The time needed for the wave to travel and reflect back is given by
The substitution of (
The first results were obtained using the projectile and system data presented in the former work [
Properties of the steel nucleus of the projectile.
Property  Value 

Initial velocity (m/s)  835 
Mass (g)  9.54 
Vicker’s hardness (HV)  817.5 
Dynamic yield stress (GPa)  2.82 
Density (g/cm³)  8.41 
Diameter (mm)  7.62 
Ceramic compositions.
Composition  Al_{2}O_{3 }A1000SG 
Al_{2}O_{3 }Tubular T60 
TiO_{2} 

B  90  8  2 
C  85  13  2 
Mechanical and physical properties of the ceramic plates.
Property  Composition  

B  C  
Weibull modulus (m)  8.4  8.8 
Mean strength parameter, 
175.0  171.3 
Reference rupture strength, 
182.8  178.5 
Vicker’s hardness (HV)  1551.4  1259.8 
Resistance against penetration on ceramic, 
4.43  3.60 
Density (g/cm^{3})  3.90  3.80 
Average grain size ( 


In Figures
Velocity against time for (a) this work and (b) previous work [
In both graphs it is possible to observe the different stages of penetration in the graph. During the first stage, there is no interface velocity. In the beginning of the second stage, the interface begins to move tending to reach the same velocity as the projectile. The third and final stage begins with the projectile and interface velocities equalized. During this final fraction of the movement, both projectile and interface decelerate together. With the solutions it is possible to estimate the fraction of mass, velocity, and energy lost in each stage. These values are presented in Table
Fraction of loss of velocity, mass and energy of the projectile in the three stages (adapted from previous work [
Stage  Velocity loss (%)  Erosion (%)  Absorbed energy (%) 

1st  10.62  19.18  35.73 
2nd  30.27  40.77  50.32 
3rd  59.10  —  13.93 
There were more two ceramic compositions presented in the material selection. However, both of these structures were not stable enough to be molded in the system.
The values of the constants were given by the former work [
Values of the constants used in the modified model.
Symbol  Property  Value  


Ceramic density (g/cm³)  Composition B 
Composition C 

Ceramic elastic modulus (GPa)  300  

Ceramic thickness (mm)  11.3  

Resistance against penetration on ceramic (GPa)  Composition B 
Composition C 

Metal density (g/cm³)  7.77  

Metal elastic modulus (GPa)  193  

Metal thickness (mm)  15  

Metal strength (MPa)  935  

Metal hardening exponent  0.29  

Metal modified deflection profile constant (m^{−1})  0.0018  

Projectile dynamic yielding (GPa)  2.82  

Projectile density (g/cm³)  8.41 
These data (
Comparison of the former and new results.
Composition  Thickness (mm)  Duration of the
1st stage ( 
Impact velocity (m/s)  Maximum deflection  Errors  

Experimental data (mm)  Original theory, OT (mm)  Modified theory, MT (mm) 



B  11.3  8.6  792.7  16.5  17.0  17.2  2.9  4.3 
C  11.3  8.7  858.2  20.0  17.6  18.4  13.6  8.8 
B  9.3  8.2  628.9  18.0  17.8  18.8  1.1  4.2 
C  9.3  8.2  651.1  17.5  17.7  18.3  1.1  4.3 
B  7.3  7.7  428.8  15.5  17.3  14.6  10.4  6.0 
C  7.3  7.8  448.4  13.0  16.6  13.0  21.7  0 
However, the results of the maximum deflection showed that the new theories can generate good results not only for the high speed impacts, but also for lower velocities, differently from the former theory. Simulations for different calibers [
Specifications of the different calibers used in the simulation.
Type  Specification  Bullet diameter 
Mass 
Initial velocity 

II  .357 Magnum  9.07  10.2  453 
IIA  9 mm  9  8  373 
IIA  .40 S&W  10.2  12  300 
IIIA  .357 SIG  9.07  8.1  440 
IIIA  .44 Magnum  11.2  16  460 
Results of the simulation for several different calibers.
Type  Specification  Third stage velocity (m/s)  Third stage mass (g)  Final deflection value (mm)  Penetration in the ceramic (%) 

II  .357 Magnum  170  6.66  19.48  87.5 
IIA  9 mm  129  5.80  18.71  41.4 
IIA  .40 S&W  104  9.45  18.63  49.6 
IIIA  .357 SIG  163  5.36  17.40  84.4 
IIIA  .44 Magnum  197  10.38  20.15  90.1 
Movement evolution for the simulation on different calibers.
In Figure
The not highlighted region in Figure
Considering the back metal plate as another important energy absorber and integrity maintainer for the system, a deterministic simulation was run using the new deformation law together with the deflection profile function. Then it gives
The variable
Deformation
Deflection of the metallic plate
Based on Figures
The values obtained from the model demonstrate how important the ceramic plate is in the protection, responsible for absorbing approximately 85% of the total energy. Moreover, the value for the deflection of the metal plate was close to the experimental data, which validates the model. Together with the optimized deflection equation the model could also prevent the effects of different projectile impacts.
In addition, the simulation demonstrated different aspects of the model and some predictable effects in collision phenomena. The control of the properties and its effects were analyzed with the developed computational method. Some of the effects predicted by the program could not be studied experimentally. However, future studies can use the presented results to validate the theory and, then, analyze some internal phenomena in a deeper way.
The present work is a theoretical analysis of the impact phenomenon in a specified type of protection. In the future studies, the current model and its solutions and simulations can be used to perform experimental test to evaluate the reliability of the model. Also, if the model can be considered valid for the impact, the considerations and formulations can be kept or improved to a more advanced modeling as finite element modeling (FEM). In addition, the experimental confirmation of the presented modeling also permits deeper investigation of the impact phenomenon, such as the effect of the interfacial friction between the projectile and sheared surfaces.
The authors wish to thank CAPES and CNPq for supporting this work.