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https://www.eduzhai.net International Journal of Composite M aterials 2013, 3(6B): 17-22 DOI: 10.5923/s.cmaterials.201310.03 A Modified Analytical Model for Analysis of Perforation of Projectile into Ceramic Composite Targets GH. Liaghat1,2,*, H. Shanazari1, M. Tahmasebi1, A. Aboutorabi2, H. Hadavinia2 1Department of M echanical Engineering, Tarbiat M odares University, Tehran, Iran 2M aterial Research Centre, SEC Faculty, Kingston University, London, SW15 3DW, UK Abstract In this paper, based on Woodward model[1], an analytical model has been developed for perforation of projectile into ceramic co mposites targets. In the new model, contribution of different phases of projectile during perforation (erosion, mushrooming and rigid phase), modificat ion of semi-angle of ceramic cone during perforation process, modification of the shape of the nose of projectile and changes in yield strength of ceramic during perforation are considered. The ballistic limit and residual velocity of projectile by presented model have a good agreement with e xperimental and other theoretical results of other researchers. Keywords Penetration, Ceramic Armour, Analytical Model, Pro jectile, Layered Materials 1. Introduction Ceramic materials are widely used in armour systems as well as aircraft structures and military vehicles fo r the advantages of low density, high compressive strength, hardness and heat resistance. Response of ceramics to projectile impact and other types of high-speed loading conditions is an important issue for these applications. Ballistic performance of many types of ceramics was investigated in many experimental, theoretical and nu merical studies. A review of penetration/perforation process of ceramic targets can be found in[2,3,4]. A great amount of these studies regarding ceramic targets subjected to high velocity impact investigate the behaviour of materials under impact load. The ceramic destroys the projectile t ip, slows it down, and distributes the load over a large area of the back-up plate. The back-up plate supports the ceramic and brings the comminuted ceramic and projectile to rest. The back-up plate material is selected on the basis of structural, ballistic, and weight considerations. Kevlar, fib reglass, spectra, and alu miniu m are most co mmonly used as the backing material. The mechanical properties of a ceramic determine its ballistic efficiency. The hardness of the ceramic causes the erosion and disintegration of the projectile, thus, preventing further penetration. The armour plate is exposed to very high bending stresses; hence, the ceramic must have high flexural and tensile strength. If the fracture toughness of the ceramic is too low, the crack propagation might be too severe after the impact wh ich could * Corresponding author: ghlia530@modares.ac.ir (GH. Liaghat) Published online at https://www.eduzhai.net Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved damage the ceramic significantly. This can reduce the degree of multih it protection offered by the ceramic armour system. Ceramics with lower density is preferred as it reduces the overall weight of the armour. An ideal armour ceramic material must have a combination of these desired properties and should also be easy to manufacture. Tate[5] has presented a model that is considered a basis for research in long rod penetration on thick targets. In Florence’s model[6], a global energy balance is proposed leading to the derivation of the ballistic speed limit. The Woodward model[1] investigates penetration mechanism considering the lu mped mass approach. This model presents analytical solutions for calculat ion of velocity and residual mass of a projectile at any instant of time after impact. In 1991, Den Reijer[7] developed an analytical model based on Woodward work[1]. He proposed a set of governing equations to model the main physical mechanism during the penetration process. In 1998, Chocron and Galvez et al.[8] presented a model where the back plate of the armour is made of poly mer composite material such as Kevlar/Epo xy. The model allows the calculation of residual velocity, residual mass, the projectile velocity and the deflection and strain histories of the back-up plate. In 1998, Zaera and Sanchez-Galvez[9] developed a new analytical model to simulate the ballistic impact of a projectile into cera mic/ meta l armour. This model is based on Tate[5] equation for the projectile penetration into ceramic tile. In this model, the values of residual velocity, residual mass and ballistic limit velocity, are consistent with the e xperimental results. In 1999 Fellows[10] developed Woodward's model for ceramic-faced semi-infinite armour. In 2010 Feli et al.[11] based on Zaera and 18 GH. Liaghat et al.: A M odified Analytical M odel for Analysis of Perforation of Projectile into Ceramic Composite Targets Sanchez-Galvez[9] work presented an analytical model which employed the mo mentum equation to describe frag mented ceramic conoid. In this model, the ballistic limit and residual velocity of pro jectile have a good agreement with the experimental results. In the analytical models mentioned above, the mushrooming of project ile ’s tip has not been included and its erosion during penetration has only been taken into account. Also the authors assumed that a conoid of comminuted ceramic with a semi-angle of about 650 is developed which pushes forward a circu lar area of metallic plate with dimensions equal to the base of the ceramic conoid. 2. Analytical Model Using the lumped mass approach and according to Woodward[1] theory, the following equation can be formed. The equations of motion of the system shown in Fig.1 are: .. FP = −M P U P (1a) . FI − FP = −∆M P UP ∆t (1b ) . FC − FI = −∆M C UC ∆t (1c) .. FT − FC = −M T UT (1d) Where F is force, M is mass, ∆t is the time incrementand . .. ∆M is the change in mass. U and U are velocity and acceleration respectively, and subscripts P, I, C and T refer to penetrator, interface, ceramic and target respectively as shown in Fig.1. column of material to occur, a value of flow stress, Y, must be exceeded, whether th is is governed by uniaxial yield stress, hardness or some other strength measure. ???????????????? = ???????????????? ????????0 ( 3a ) ???????????????? = ????????????????????????0 (3b) To solve the equations, elimination of ????????̇???????? and ∆???????????????? fro m the equations yields a quadratic which can be solved at each time step for the only unknown parameter, the penetrator mass loss ∆???????????????? . At each time step all other parameters can be updated and the solution then repeated. At first step, we have just velocity of pro jectile. The sign in equation (1b) is such that if ???????????????? > ???????????????? , ∆???????????????? is negative, thus there is a mass loss from the projectile. The sign in equation (1c) is such that if ???????????????? > ???????????????? , ∆???????????????? ????????̈???????? is positive, thus there is an increase in momentu m of the ceramic which is moved out of the way. Woodward et al.[15] derived an equation for the work W, to dishing a plate o f thickness b and flo w stress YT to a displacement h, ???????? = ????????????????ℎ???????????????? �2 3 ???????? + 1 2 ℎ� (4) And it is shown that this equation can be used with reasonable accuracy to calculate the work done on dished back up plate fro m actual impacted ceramic co mposite targ ets . The resisting dishing force is obtained by differentiat ing equation (4) to give; FT = πbYT �2 3 b + h� (5) Failure of back-up plate is considered for two cases. If ceramic erosion occurs during the accelerat ion phase, the effective kinetic energy Ek is equated to the work done by equation (4) i.e.: ???????????????? = 1 2 ???????????????? ????????̇ ????2???? + ???????? 8 � ???????? ???????????????? 5 + ???????????????? ???????? � 3 ????????????4???? ????????????2???? /(???????????????? − ???????????????? )2 (6) Where c is the reduced ceramic thickness after erosion, Dp is the projectile diameter. If the ceramic eroded to zero thickness during acceleration phase, the equation to determine whether the projectile would perfo rate is 1 2 ???????????????? (????????̇???????? − ????????̇???????? )2 = ???????? ????????????2???? ???????? ???????????????? 2 (7) The equations for the accelerat ion and failure phase will be solved and all penetration parameters such as residual velocity and mass of projectile will be determined. Figure 1. Lumped mass model Penetrator is considered as a flat ended cylinder with cross sectional area A0. We can assume a colu mn of penetrator and ceramic being squeezed out giving continuity equations of the form: ∆???????????????? ???????? ???????? ????????0 = −�????????̇???????? − ????????̇???????? �∆???????? ∆???????????????? ???????? ???????? ????????0 = −�????????̇ ???????? − ????????̇???????? �∆???????? ( 2a ) (2b) where ρ is the material density. Constitutive equations for the failure or penetrator and ceramic can be established by requiring that for erosion of a 3. Modifications on Woodward Model Woodward model[1] has been further developed by considering different phases of projectile during perforat ion, i.e. erosion, mushrooming, rigid ity, and applying interface force FI, modify ing the shape of project ile nose, changing the semi-angle ceramic conoid and considering reduced compressive strength during penetration into target. 3.1. Different Phases of Projectile during Perforation The projectile behaviour during impact, based on work by International Journal of Composite M aterials 2013, 3(6B): 17-22 19 Reijer[7], is divided into three phases; a mass erosion phase, a mushrooming phase and a rigid phase (Fig. 2) dependent on the projectile veloc ity and the materia l strengths. Initially during high-velocity impact, the pressure at the interface between the projectile and ceramic will exceed the erosion strength of the projectile, and the projectile will erode. If the impact velocity is h igh enough, the erosion strength of the ceramic will be exceeded, and the ceramic will also erode. At high impact velocities the projectiles are eroded at ceramic/project ile interface. Therefo re, ???????????????? = �???????????????? ????????0 �????????????̇ ???? − ????????̇???????? 2 � � + ???????????????? (8) As the projectile veloc ity falls, the relative impact velocity (????̇ ???????????? − ????????̇???????? ) will fall below the plastic wave velocity, UPLAS. The relative d isplacement between the end of the projectile and the projectile ceramic interface can then start to be accommodated by plastic deformat ion of the projectile. Thus the projectile mushrooms as illustrated in Fig.2[10]. ???????????????? = ???????????????? ????????0 �???????????????????????????????????????? �????????̇???????? − ????????̇????????� − ????????̈????????(???????????????????????????????? − ???????????????????????????????? )� + ???????????????? (9) Where LERO and LELA are the length of projectile at start of mushrooming phase and the length of projectile unaffected by the plastic wave, respectively. At some point in time the velocity of project ile will become equal to the velocity o f the mushrooming stage which is penetrating the ceramic. At this point it is assumed that the projectile becomes a rigid body. Where LP is the actual length of the projectile and D(z) the diameter for position z. Equivalent length Leq is determined fro m mass of the projectile: ???????????????????????? = 4 ???????????????? ????????????????2 ???????????????? ???????? ???????? (12) Figure 3. Ogival projectile 3.3. Modification to the Conoi d Semi-angle at Initial Impac t Changing in semi-angle ceramic conoid formed after impact is next modification on Woodward model[1]. The ceramic conoid semi-angle is an important parameter. There are different ideas about this angle[9,11,12]. Woodward[1] considered this angle about 68º (the angle formed in quasi-static state). According to different approaches and regarding the fact that in high impact velocit ies, because of high energy of projectile the projectile-ceramic interface force is more than ceramic erosion stress. Therefore the erosion of ceramic and then penetration of projectile into ceramic will occur. Original conoid Reduced d imen s io n s Figure 2. Projectile behaviour during impact as projectile velocity falls[10] Projectile rigid stage: ???????????????? = −????????̈???????? ???????????????? ???????????????????????????????? ????????0 (10) While in Woodward's model FI is only determined by equation (1). 3.2. Modification to the Projectile Nose Shape Next step is about shape of projectile tip. Woodward model considers flat ended projectile with perfect cylindrical shape, while actual pro jectile usually have ogival nose (Fig. 3). Fo r non-cylindrical pro jectiles, equivalent diameter and length will be defined in Woodward model for such projectiles. Equivalent diameter Deq can be found by weighting each differential element of the projectile with its dia meter[ 9] , ???????????????????????? = ∫0???????? ???????? ∫0???????? ???????? ????????3 ????????2 (???????? )???????????????? (???????? )???????????????? (11) Cera mic ero s io n Figure 4. The assumed reduction in effective dimensions of the conoid and backing plate as a result of ceramic erosion Both the erosion of ceramic and pro jectile are considered in Woodward model which makes the results closer to e xperimental data. When erosion of ceramic occurs, the effect ive dimensions of ceramic conoid reduced (see Fig.4). In the fact, when ceramic eroded, new ceramic conoids with smaller dimensions are formed. The higher impact velocity, results in smaller ceramic conoid. The semi-angle of ceramic can be approximated by equation (13). In this equation, this angle changes linearly between 68º in Woodward[1] model and 63º in Florence model[6]: 20 GH. Liaghat et al.: A M odified Analytical M odel for Analysis of Perforation of Projectile into Ceramic Composite Targets 68 ????????̇???????? < 600 ???????? = � ???????? 180 �3500 � −????????̇ ???????? + 900� + 63� 600 < ????????̇ ???????? < 900 (13) 63 ????????̇???????? > 900 i.e. for impact velocities less than 600 m/s the semi-angle φ is equal 68º and for velocities more than 900 m/s is 63º and between them, changes linearly according to equation (13). Zaera[9] showed that by increasing the impact velocity the failure part increase and the dimensions of the cone decrease. Other researches verified the accuracy of equation (13). 3.4. Modification to the Conoi d Semi-angle during Perforati on at a S pecified Impact As mentioned before, in the Woodward model[1] when the ceramic erosion occurs, the effective dimensions of the ceramic cone decrease (Fig. 5). Sides of newly formed cones are parallel to the original cone. In words, we assume the angle remains constant during perforation process. The semi-angle φ is now changed in the newly formed cone and considered less than the init ial value. Results of this model are closer to the experimental values. In other words, sides of newly formed cone are not parallel to the original cone. This change of the angle approximately can be found form: ???????? = ???????? 180 �???????? 0 −34 ???????????????? (???????????????? − ????????) + 34� (14) Where x is ceramic eroded length, tc is the thickness of ceramic and φ0 is the semi-angle o f in itial cone determined by equation 13. In the other wo rds: ● In case there is no erosion of ceramic the angle φ is as its maximu m value. ● In case there is highest possible erosion (total cera mic thickness) the angle φ is 34º (min imu m value in Fellows[10] model). In summary the angle of a newly formed ceramic cone with semi-angle φ0 is determined by equation (13) and the angle will change during perforation accord ing to equation (14). Origin al conical ϕ′ The last modification on Woodward model[1] is the determination of an expression for the co mpressive frag mented ceramic strength. Ceramic penetration strength is intensely lowered after frag mentation[13]. Furthermore, the back-up plate is deformed and some particles being expelled fro m the crater. More space for the frag ments motion is available and thus the penetration resistance decreases[13]. If YCO represent the compressive strength of the intact ceramic p late and YC denotes a lower value after frag mentation, the expression for frag mented ceramic strength can be found: ???????????????? = ????????????????0 �???????? −???????? 2 � ???????? 0 (15) where u is the penetration velocity of pro jectile, w is the velocity of back-up plate and u0 is the initial penetration velocity of projectile. 4. Results and Discussion The results of modified analytical model have been compared with other published analytical and experimental results. In order to show the effect of ceramic and back-up plate thickness, a target-projectile system has been considered[14] where the steel projectile is a small calib re 7.62AP with 8.3gr mass and its diameter is 7.62 mm. A t wo layer target has been made of A l2O3 alu mina with variab le thickness and Al6061-T6 as a back-up p late. Fig.6 shows, the ballistic limit velocity versus thickness of back-up plate for ceramic thicknesses of tc=4.05 mm. Present model shows good results in comparison with Woodward model. According to the figure in a specified ceramic thickness when the thickness of back-up plate increases, the ballistic limit velocity will increase. Th is is due to the fact that when the back-up plate thickness increases, mo re energy will be spent for bending and stretching the back-up plate. In a low thickness, stretching and bending forces are effective in the resistance to penetration of back-up plate but when thickness increases, the behaviour of back-up plate changes from petaling to plugging and shear forces increase. ϕ Reduced dimension X Ceramic erosion Figure 5. Reducing the semi-angle ceramic cone in an impact 3.5. Modification to Ceramic Strength after Impact Figure 6. Ballistic limit velocities for a surrogate steel 7.62AP projectile impacting AD85+Al6061-T6 with thickness of ceramic 4.05 mm International Journal of Composite M aterials 2013, 3(6B): 17-22 21 Fig.7 shows, the ballistic limit velocity of modified analytical model versus ceramic thickness for two different backing thicknesses. According to this figure, the ballistic limit velocity of project ile will increase when the thickness of ceramic increases. In fact, when the thickness of ceramic increases, the projectile will remain longer behind ceramic tile due to more volu me of frag mentation of ceramic and its velocity will be reduced. the ceramic is not eroded there will be little d ifference between the ballistic performance of the two ceramics (AD85 and B4C). Fig. 9 shows that when the ballistic limit velocity increases due to increase in backing thickness, the difference between the behaviour of two types of ceramic will be mo re pronounce. The effect of decreasing comp ressive strength of the fractured ceramic conoid in the process of penetration is shown in Fig.10. In this figure, ballistic limit velocity in a target made of Alu mina-Alu miniu m is co mpared in t wo states: first Yc is constant and second Yc is reducing during the penetration process according to equation (15). According to the present model shown in Fig.10, as result of decreasing compressive strength of ceramic during penetration process, the ballistic limit velocity will be decrease. Moreover for the higher impact velocity (h igher ballistic limit velocity) it decreases more rapid ly because of more erosion in ceramic. Figure 7. Ballistic limit velocity versus ceramic thickness when a projectile impacting the two-layer target AD85+Al6061-T6 Fig.8 shows, ballistic limit of the analytical model in comparison with experimental data by Wilkins[14], Woodward model[1] and Zaera and Sanchez-Galvez model [9]. The modified analytica l mode l is in good agree ment with measured ballistic limit velocity than Woodward[1] and Zaera and Sanchez-Galvez[9] models. Figure 9. Comparison of ballistic limit velocity of two type of ceramic with different hardness for the same impact scenario on the target with Al6061-T6 as a back-up plat e Figure 8. Analytical and experimental results of ballistic limit of 7.62AP projectile impacting the target AD85+Al6061-T6 One of the most important parameters to determine the ballistic performance of ceramic co mposites is the ceramic hardness. To compare the effectiveness of hardness of ceramics, two types of ceramics, AD85 and B4C, with different hardness are considered for the same impact scenario (Fig. 9). According to the present model, if the energy of projectile is high enough so that the ceramic is eroded, the resistance to penetration is more when B4C is used. But when the energy of projectile is lo w enough so that Figure 10. Comparison of ballistic limit velocity for two compressive strength cases in a similar impact on the target with Al6061-T6 as a back-up plate: a) Constant and b) Reduced 5. Conclusions The modified model presented in this study is based on 22 GH. Liaghat et al.: A M odified Analytical M odel for Analysis of Perforation of Projectile into Ceramic Composite Targets Woodward model[1] and it is a simple way to predict the penetration resistance of ceramic-metal targets and the results of this model are in good agreement with experimental data and other analytical models. The deformation of the projectile is modelled by lumped mass approach and ogival project ile tip and three phases for projectile behaviour, i.e. a mass erosion phase, a mushrooming phase and a rigid phase depending on the projectile velocity based on work by Den Reijer[7] are considered.In additionin this model modification of semi-angle of ceramic cone at in itial impact and during perforation and reduction in the co mpressive strength of the intact ceramic plate after frag mentation of ceramic are considered. According to the modified model, when erosion of ceramic does not occur, there is a slight difference between ballistic performances of ceramics with similar thickness because of density of ceramics which determines the mass of material bounded by the cone crack. If ceramic eroded, with increase in hardness and thickness of ceramic, ballistic limit velocity will increase. In the presented model, change in semi-angle of ceramic cone has maximu m influence on improving the accuracy. Notation ????????0 , DR , DP h M ????????, ????????̇, ????????̈ ????????????????0, ???????????????? P, I, C, T ????????0, ???????? ???????????????????????????????? , ???????????????????????????????? ???????????????????????????????????????? ???????????????? X Diameter of base cone, base of cone after erosion and projectile, respectively Displacement of backing plate Mass Displacement, velocity, and accelerat ion, res p ectiv ely Compressive strength of the intact ceramic plate andcompressive strength after frag mentation, respectively Subscripts: projectile, interface, ceramic and target, respectively Semi-angle of init ial cone, semi-angle of cone after erosion, respectively Length of projectile at start of mushrooming phase and the length of projectile unaffected by the plastic wave Plastic wave velocity Ceramic thickness Ceramic eroded length REFERENCES [1] Woodward, R. L., “A simple one-dimensional approach to modeling ceramic composite armor defeat”. Int J Impact Eng 1990;9(4):455-74. [2] Chen XW, Chen YZ, “Review on the penetration/perforation of ceramic targets”. Adv. M ech 2006;36 (1):1-19. [3] Liaghat GH., Shanazari H., “Analysis of perforation of projectile into ceramic composites”, J. of Amirkabir, 2005; 15(60/2):97-109. [4] LiaghatGH., Tahmasebi AbdarM ., “Experimental and Theoretical Investigation of Perforation Process into Ceramic Targets and Presenting a M odified Theory”, M Sc Thesis, TM U (2013). [5] Tate A. “A theory for the deceleration of long rods after impact”. J. M ech. Phys, Solids, 1967;14:387-99. [6] Felorence AL. “Interaction of projectiles and composite armour”. Internal Report, US Army, August 1969. [7] Den Reijer PC, “Impact on ceramic faced armours”. PhD thesis, Delft University of Technology; 1991. [8] Chocron Benloulo IS, Rodryquez J, Sanchez Galvez V. “A simple analytical model to simulate textile fabric ballistic impact behavior”, Textile Res J Tentative 1997 (July). [9] Zaera R. Sancez-Galvez V. “Analytical modeling of normal and oblique ballistic impact on ceramic/metal light weight armours”, Int J Impact Eng 1998;21(3):133-48. [10] Fellows N.A. “Development of impact model for ceramic-faced semi-infinite armour”. Int. J. Impact Eng; 1999:22;793-811. [11] FeliS., Alami Aaleagha M E. Ahmadi Z., “A new analytical model of normal penetration of projectiles into the light-weight ceramic-metal targets”. Int. J Impact Eng. 2010;37: 561-567. [12] Wilson D. Hetherington JG, “Analysis of ballistic impact on ceramic faced armour using high speed photography”. In: Proceeding of the lightweight armour system symposium. Cranfield: Royal M ilitary College of Sience;1995. [13] Zhang Xiao-qing, Yang Gui-tong, Huang Xiao-qing, “Analytical model of ceramic/metal armour impacted by deformable projectile, Applied M athematics and M echanics”. (English Edition), 2006, 27(3):287-294. [14] Wilkins M L. “M echanics of penetration and perforation”. Int J Impact Eng 1978;16:793-807. [15] Woodward R.L., O'Donnell R.G., Baxter B.J., Nicol B., Pattie S.D. “Energy absorption in the failure of ceramic composite armours”, M aterials Forum, 1989,13(3):174-181.

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