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Analytical solution for penetration of deformable blunt projectiles into the metallic targets/Analitinis ardanciojo buko sviedinio isiskverbimo i metalini taikini skaiciavimas.

1. Introduction

Impact is one of the complicated problems in solid mechanics and in recent years, it has changed into an interesting subject for many researchers [1-6]. For example, designers and production engineers are interested in this area due to its applications at high speed blanking and hole-flanging processes. Aerospace scientists also need to understand the penetration process to design structures that are more efficient against the projectile impact. Vehicle manufacturers also use this knowledge to improve the performance and safety of their products.

Due to the complicated nature of the subject, most of the research projects were done using experimental and numerical techniques that are more expensive and time-consuming than the analytical solutions. Therefore, development of the analytical models can help the researchers to verify their experimental or numerical results as well as the designers to avoid doing expensive experiments at the start of the design process.

Penetration of a blunt projectile into metallic targets is one of the impact problems that can be complicated due to plugging and cutting phenomena. Plugging occurs when a cylindrical part is separated from a target due to the cutting at the impact edges [1].

In addition to plugging, material of the target starts to stretch out from the cutting edges during the penetration process and therefore, the transverse displacement increases at the target surface; this phenomenon is named dishing [7].

The first step to establish a good analytical model is defining correctly the penetration stages. In 1970, Awerbuch [8] defined the penetration process of a flat projectile at two stages. Later on in 1974, he modified his definition by adding the third stage to the penetration process [9]. These three stages are.

First stage: Primary compression and elongation of the target in front of the projectile without cutting at the vicinity of impact edges.

Second stage: Penetration and plug elongation.

Third stage: Plug separation from the target.

In 1983, Shadbolt et. al [10] modified Awerbuch & Boodner [9] model by considering the absorbed energy to calculate the target area deformation. The results of this model also have no good agreement with the experimental results. Later, the penetration process by Liss & Goldsmith in 1983 was divided to five stages including indentation (erosion), plug formation, plug separation, plug slipping and post-perforation [11, 12]. They used plastic stress wave propagation theory to analyze each of these stages. In 1987, Woodward developed a new solution for thin layers under impact loading using the plastic shear deformation theory [13]. In this model, the penetration process includes two stages, plug cutting and stretching of the target.

All of the above mentioned models are based on the rigid projectile assumption. For the first time, the projectile deformation was investigated using stress wave propagation theory at the normal impact of the projectile to the rigid target by Taylor at 1948 [14]. Although, the achieved results have no coincidence with the experimental results, but Taylor's work is a basis for other studies. Using Taylor model, Recht [7], developed a simple analytical approach for modeling the problem of on impact between a blunt projectile and one-layer metallic targets. Based on Recht approach, the penetration process takes place at two stages, the first one is erosion, and the second stage is projectile deformation and plug formation.

Taylor model is not valid for the impact velocities greater than the plastic wave propagation speed. The reason is that at these speeds, the projectile weight will be decreased continuously due to the erosion process until the velocity reaches the plastic wave propagation one. This effect is not considered by Taylor and therefore the validity of his model is limited to the impact problems with the projectile speed equal or smaller than the projectile wave propagation velocity.

Recht improved Taylor model by defining material of the target and projectile using the surface hardness. As a result, Recht calculated the projectile deformation, its weight reduction and deformation of the target.

In 1983, by modifying Recht method, Wenxue by considering plugging developed a new analytical model for normal penetration of the deformable blunt projectiles into the one-layer metallic targets considering the plugging [15].

Wenxue assumed that the penetration will occur at six stages. Materials of the projectile and the target are defined to be rigid with linear work hardening and completely plastic, respectively. This model is suitable for speeds greater than the ballistic limit.

At Wenxue model [13], dishing of the target is neglected and it is supposed that when the speed of the un-deformed part of the plug reaches the speed of its deformed part, further penetration of the plug into the target will occur rigidly. Also, for this stage of penetration, there is no description available regarding the cutting process.

Considering the aforementioned research studies and in order to develop a more accurate model, in the current work effects of the plugging and dishing are considered simultaneously in the analytical model of normal penetration of the deformable blunt projectiles into one-layer metallic targets. The supposed stages of penetration are shown in Fig. 1.

Based on this figure, by increasing the penetration stages and relating them to the target deformation, the previous models are modified and the results are improved. To verify the results obtained by solving the presented analytical model here, the analytical and experimental results of Wenxue [15], are used. Also, to verify the predicted amount of dishing of the target by the current model, a comparison is performed between these results and the available experimental and numerical results [16,17].

By doing these comparisons, it is shown that the analytical model developed here is valid and more accurate than the previous models.

[FIGURE 1 OMITTED]

2. Modeling

Fig. 2 shows a typical impact problem between a projectile with initial diameter Do and a target with thickness of To. The geometrical parameters and velocities of different areas are shown in the figure. Stress state at the projectile and strain state at the target are considered having one dimension.

It can be seen from Fig. 2 that the projectile flattening will occur at the impact area. The amount of flattening, [mu], is equal to A/[A.sub.o] and can be calculated from the following equation

[mu] = [(1-[V.sub.s]/[C.sub.p]).sup.-1] (1)

where [V.sub.S] = [V.sub.P] - [V.sub.C] and [C.sub.P] are defined as [18]

[C.sub.P] = [square root of 1/[[rho].sub.P] [partial derivative][sigma]/[partial derivative][epsilon]

During impact, based on the initial velocity of the projectile, the erosion and cutting may occur inside the flattened area of the projectile. When the value of [V.sub.S] is greater than the plastic stress wave propagation speed, the erosion will occur and weight of the projectile will be reduced. Based on the experimental results, the amount of flattening is limited. This limitation is measured by [bar.[mu]] and is obtained experimentally for different materials. Therefore, if [V.sub.S] < [C.sub.P] and the calculated value of [mu] from Eq. (1) are greater than [bar.[mu]], then the cutting will occur near the impact edges and the amount of the flattening will remain equal to [bar.[mu]].

Normally, during an impact process, three regions for the projectile and target are defined which consists of projectile, plug and target without plug. Furthermore, each of these three regions includes two areas, nondeformed and deformed. When the plastic stress wave is passing through an area, deformation will occur at that area. As a result, deformations of the three mentioned regions are as below.

Projectile: if [V.sub.S] < [C.sub.P], then a longitudinal stress wave will propagate at the projectile. The deformation will occur at those areas of the projectile which this wave is passing through them and therefore, the diameter of the projectile at the impact location will increase. The velocities of nondeformed and deformed areas are equal to [V.sub.P] and [V.sub.C], respectively; where VP is descending by time.

[FIGURE 2 OMITTED]

Plug based on Fig. 2, thicknesses of the areas deformed by the longitudinal stress wave will be reduced. By assuming one-dimensional strain state at the plug, the speed of axial stress wave propagation can be calculated from the following equation

[C.sub.t] = (K/[rho].sub.t] + 2/3 [partial derivative][sigma]/ [partial derivative][epsilon].sup.1/2]

where K is the bulk modulus of the target and [[rho].sub.t] is its material density. The axial stress at the plug can be obtained from [15]

[[sigma].sub.t] =(K/G + 4/3) [[sigma].sub.Y]/ 2 (4)

where [[sigma].sub.Y] is the dynamic yield strength of the target and G is its shear modulus.

Target without plug: another wave will initiate and propagate from the cutting area around the plug that is called transverse wave [C.sub.S]. Dishing will occur at the area affected by this wave while the other areas will remain rigid. The propagation speed of this wave is [7]

[C.sub.S] = [square root of 1/[rho].sub.t] d[tau]/d[gamma] (5)

where [tau] and [gamma] are shear stress and shear strain, respectively.

2.1. Motion equations

At each time, motion equations can be written for the material with density pi and volume [OMEGA] the area of which S. By using M for the mass momentum at [OMEGA] and [[phi].bar] for the mass flow passed through the area S, the motion equation would be equal to the sum of the momentum rate and area mass flow, as

[F.bar] = [partial derivative]/ [partial derivative]t + [[phi].bar] (6)

where

[M.bar] = [[integral].sub.[OMEGA]] [[rho].sub.i] [V.bar]d[OMEGA] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

At the above equations [n.bar] is the outward normal vector of the area S, and the velocities are measured relative to the initial surface. The external forces [bar.F], which are applied on the area S, are equal to the momentum rate. To extract the motion equations, the penetration process is assumed based on Fig. 1.

2.2. Stage of projectile erosion ([V.sub.S] > [C.sub.P]), plug formation (T > 0) and target dishing ([V.sub.d] > 0)

When the relative speed between the impact area and nondeformed part of the projectile is greater than the propagation speed of plastic stress wave, the erosion will occur at the projectile. Assuming that the target width is infinite, the motion equations will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[phi] = [[rho].sub.P] [A.sub.o] [V.sub.S] [.V.sub.C] (10)

F = 0 (11)

Now, by applying the above equations to the motion equation and simplifying the result, it can be obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [[sigma].sub.P] and [[sigma].sub.t] are defined as below: [[sigma].sub.P] is stress at the projectile that will reduce the speed of its nondeformed part; [[sigma].sub.t] is stress at the plug that will increase the speed of its nondeformed part as well as the speed of the target at dishing area.

The other equations for the penetration process and deformation are the following

d[V.sub.P]/dt = - [[sigma].sub.P]/ [[rho].sub.P] L (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

dL/dt = -[V.sub.s] (15)

dT/dt = -[C.sub.t] (16)

d[R.sub.S] = [C.sub.t] (17)

d[Z.sub.C]/dt = [V.sub.C] (18)

d[Z.sub.t]/dt = [V.sub.t] (19)

The value of [V.sub.C] at t = 0 can be found from Eq. (12), considering the fact that the multiplied parameter to the time derivative of [V.sub.C] is equal to zero at this time

[V.sub.C] (0) = [psi] - [square roort of [[psi].sup.2]-[phi]] (20)

where

[psi] = A [[rho].sub.t][C.sub.t]/2[A.sub.o][[rho].sub.P] + [V.sub.o] (21)

and

[phi] = [V.sup.2.sub.o] + [A.sub.o] [[sigma].sub.P] - A [[sigma].sub.t]/[A.sub.o][[rho].sub.P] (22)

The other required boundary conditions to solve the above differential equation set, are as below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

This stage of the penetration process will end when one of the below conditions becomes true:

1 -[V.sub.S] < [C.sub.P] that means having no erosion (erosion

stop);

2 -[V.sub.t] = 0 (the speed of the plug rigid part);

3 -T = 0 that means the longitudinal wave has

passed through the plug thickness.

2.3. Stage of projectile deformation ([V.sub.S] < [C.sub.P]), plug formation (T > 0) and target dishing ([V.sub.d] > 0)

Again, by defining the nondeformed and deformed areas for the projectile, the Eq. (9) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Note that the value of F and [phi] are zero. At the above equation, [L.sub.1] is the length of projectile after finishing the erosion stage but if [V.sub.S] < [C.sub.P], at initial stage [L.sub.1] = [L.sub.o]. Therefore, by substituting at the Eq. (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

The other required equations for finding the parameters of this penetration stage are the following ones in addition to the Eqs. (13), (14) and (15) to (19)

dL/dt = -[C.sub.P] (26)

dH/dt = [C.sub.P] - [V.sub.S] (27)

But, if this stage would be the first penetration stage, the following initial conditions are required for solving the obtained differential equation set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

If this penetration stage starts after the projectile erosion stage, the final values of the erosion stage will be considered as the boundary conditions. Also, the height of the primary mushroom type area of the projectile, H(0), will be zero.

This stage of the penetration process will finish when one of the below cases occurs:

1-the relative speed between the rigid and the de formed areas of the projectile reaches zero ([V.sub.S] = 0) what means that the projectile will penetrate rigidly;

2-the longitudinal stress wave crosses over the plug thickness (T = 0) and the projectile deformation goes on.

In addition to above situation, it is possible that the speed of the plug rigid area becomes equal to zero ([V.sub.t] = 0) and the penetration process stops.

2.4. Stage of projectile erosion ([V.sub.S] > [C.sub.P]), plug ejection (T = 0) and target dishing ([V.sub.d] > 0)

This stage of the penetration process will start when during the projectile erosion, the longitudinal stress wave crosses over the whole plug thickness and as a result, the plug formation stops. At this condition, deformation of the areas near the plug will continue by transverse stress wave propagation through them. To extract the governing equations, two control volumes are considered; the first control volume is the projectile-plug and the second one is the target dishing area. The force F exists as action and reaction at the contact area of these two control volumes. Equations for the M, [phi] and F at the projectile-plug control volume are as below

M = [[rho].sub.P] [A.sub.o]L[V.sub.P] + [[rho].sub.t] A[T.sub.o] [V.sub.C] (30)

[phi] = [[rho].sub.P] [A.sub.o] [V.sub.S].[V.sub.C] (31)

F = -[pi]DY([Z.sub.d] + [T.sub.o] -[Z.sub.C]) (32)

where [Z.sub.d] + [T.sub.0] - [Z.sub.C] is thickness of the plug as well as contact area of the target and Y is yield strength of the target material. By substitution of these equations at the motion equations and by simplifying them

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

Relationships of M, [phi] and F for the target dishing area are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

[phi] = 0 (35)

F = [pi] DY ([Z.sub.d] + [T.sub.o] - [Z.sub.C]) (36)

With substituting these relations at the motion equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

Also, dishing growth rate can be found from the below equation

[dZ.sub.d]/dt = [V.sub.d] (38)

This stage of the penetration process will finish when [V.sub.S] < [C.sub.p]. When [Z.sub.d] + [T.sub.o] - [Z.sub.C] = 0, the penetration process will complete and the plug will be totally separated from the target.

If the dishing speed becomes equal to zero and target deformation stops, the penetration will occur based on this stage equations by ignoring the target dishing control volume. Also, it is possible that the plug exit speed VC becomes zero and the plug completely does not separate from the target.

2.5. Stage of projectile deformation ([V.sub.S] < [C.sub.P]), plug separation (T = 0) and target dishing ([V.sub.d] > 0)

When the longitudinal stress wave is passing through the target thickness, the plug will exit from the target, rigidly. In this case, if [V.sub.S] be smaller than the propagation speed of the plastic stress wave, using the same control volumes as the previous penetration stage, the

governing equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

[phi] = 0 (40)

F = -[pi]DY ([Z.sub.d] + T - [Z.sub.C]) (41)

By substituting these equations into the motion equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

Dishing of the target will be described with the equations similar to the Eqs. (34) to (37). This stage will finish when [V.sub.S] = 0 or [Z.sub.d] + [T.sub.o] - [Z.sub.C] = 0. In the first case, the plug and the projectile will move rigidly and dishing will continue. In the second one, the penetration process will be completed and the plug will exit totally from the target. If the dishing speed becomes zero, the governing equations will be rewritten by ignoring the target dishing controls volume. It is also possible that before the complete ejection of the plug from the target, [V.sub.C] becomes zero and the penetration process stops.

2.6. Stage of projectile rigidity ([V.sub.S] = 0), plug formation (T > 0) and target dishing ([V.sub.d] > 0)

This stage of the penetration process will start when the relative speed between rigid and deformed areas of the projectile [V.sub.S] becomes zero and any further penetration continues with the projectile rigid motion and the plug formation. The equation of momentum for this stage is as below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

This stage will finish when the longitudinal stress wave passes through the completely deformed thickness of the plug and [Z.sub.d] + [T.sub.o] - [Z.sub.C] = 0; and will result in the plug separation. Here, again incomplete plug ejection may occur when [V.sub.t] = 0.

2.7. Stage of no dishing ([V.sub.d] = 0), projectile erosion ([V.sub.S] > [C.sub.P]) and plug rigid motion (T = 0)

When speed of the target area excluding the plug reaches zero and the target dishing stops, this stage of the penetration process will start and continue with the plug rigid motion and the target erosion. Assuming the control volume including the projectile and the plug, the related motion equations can be obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

d[Z.sub.t]/dt = d[Z.sub.C]/dt = [V.sub.C] (46)

dL/dt = -[V.sub.S] (47)

d[V.sub.P]/dt = - [[sigma].sub.P]/[[rho].sub.P]L (48)

where [Z.sub.d1] is the remained amount of dishing at the end of the target deformation stage. This penetration stage will finish when the plug is separated completely from the target ([Z.sub.d1] + [T.sub.o] - [Z.sub.C] = 0) or the projectile erosion stops and its deformation starts.

2.8. Stage of no dishing ([V.sub.d] = 0), projectile deformation ([V.sub.S] < [C.sub.P]) and plug rigid motion (T = 0)

This stage of the penetration process starts with the projectile erosion, the plug rigid motion, the projectile deformation, and stop of the target dishing. Therefore, the changes of [V.sub.C] with time would be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

This stage will finish when the plug separates from the target completely ([z.sub.d1] + [T.sub.o] - [Z.sub.C] = 0) or the relative speed between the rigid and deformed areas of the projectile reaches zero.

2.9. Stage of projectile rigid motion ([V.sub.S] = 0), plug rigid motion ([V.sub.t] = 0) and target dishing ([V.sub.d] > 0)

This stage starts when the relative speed between the rigid and the deformed areas of the projectile become equal to zero. Thus, the momentum equation is

M = ([[pi].sub.P][A.sub.o][L.sub.1] + [[rho].sub.t]A[T.sub.o])[V.sub.C] (50)

and

[phi] = 0 (51)

F = -[pi]DY ([Z.sub.d] + [T.sub.o] - [Z.sub.C]) (52)

By substituting these equations into the motion equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

The target dishing equations would be similar to Eqs. (34) to (37). This stage will finish when the target dishing stops or the plug separation process is completed.

2.10. Stage of no dishing ([V.sub.d] = 0), projectile rigid motion ([V.sub.S] = 0) and plug rigid motion ([V.sub.t] = 0)

This is the last stage of the penetration process and by completing the plug separation, it will end. The governing equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54)

d[Z.sub.c]/dt = [V.sub.c] (55)

The above differential equation set can be solved using the Rung-Kutta method based on the algorithm presented in the Fig. 1. The inputs to solve the penetration equation are geometrical and physical parameters of the projectile and the target added to the initial velocity.

The outputs of the solution are remained velocity of the plug, the projectile and the plug mass, the projectile deformation, the target dishing amount, the penetration duration and the plug thickness.

3. Results and discussions

Awerbuch et. al. [9] did a series of experiments for the impact problem between lead projectiles and steel and aluminum targets that the obtained results are used by Wenxue to verify his model results. Wenxue also assumed a rigid-linear hardening behavior for the projectile and rigid-completely plastic ones for the plug [15]. Therefore, the Eqs. (2) and (3) can be used as below

[C.sub.P] = [square root of [E.sub.P]/[[rho].sub.t]] (56)

[C.sub.t] = [square root of K/[[rho].sub.t]] (57)

Tables 1 and 2, give physical and geometrical properties of the projectile and target during these experiments.

The residual plug velocity, the penetration duration, the plug thickness for different combinations of the projectile and target and different velocities are supplied at Table 3 and a comparison is made between the experimental results, Wenxue model results, and the results of the new developed model. In addition, dishing amount is calculated for both models. As it is shown, a good coincidence exists between the results.

Here, the results of the new analytical model developed for the penetration time of steel targets, in comparison to Wenxue model, are closer to the experimental results. However, the penetration times of this model are longer than the times obtained from Wenxue model due to the dishing effect. As a result, for the equal plug separation speeds, the existence of dishing at target will prolong the separation process. Based on the same reason, it can be predicted that the plug thickness in the case of dishing presence is smaller than the case of no target deformation. This is verified by calculating the plug thickness using the new analytical model and comparing them with Wenxue's results. Also, the calculated plug thickness obtained using the new analytical model is more accurate than the Wenxue's result. The offset between the results of the new analytical model and the experimental results can be related to the plug bulging.

Considering the dishing values calculated using the new analytical model; it can be observed that for a specified projectile and target, the ratio of the dishing amount to the target thickness decreases by increasing the initial impact velocity. The comparison between the results of the A-R projectile impact to the AL-6 and the AL-1 targets is presented in Table 4. From this table, it can be considered that the targets with greater shear strength have more deformation than the weaker targets for the same projectile impact. Therefore, the targets with less shear strength against the projectile penetration will have less dishing during the plug formation.

Borvik et. al. [17] did some experimental tests on 12mm thick Weldox-460E targets and compared the obtained results with the numerically calculated results using Ls-Dyna. They used a steel projectile with dynamic strength, [[sigma].sub.P], equal to 1100MPa; 20mm diameter and 80mm length to generate software models. The target behavior supposed to be based on Johnson-Cook model as below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

where [A.sub.JC], [B.sub.JC], [C.sub.JC], n and m are constant values described in Table 5 for this material and

T* = T - [T.sub.room]/ [T.sub.melt] - [T.sub.room] (59)

Assuming this behavior for the target material, Eqs. (3) and (5) can be written as below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (61)

As shown in Fig. 3, there is a good agreement between the results of these methods at high impact speeds. However, at low impact speeds, the plug formation will not occur simply due to the low impact energy and therefore, there are some discrepancies between the analytical and experimental results. At these experiments, the ballistic limit velocity was near 190m/s.

Fig. 4 shows a comparison for the target dishing between the results of above mentioned methods. By increasing the impact velocity, the amount of dishing will be reduced. Referring to the Fig. 4, it can be considered that the agreement of experimental and analytical results is good for the impact speeds near the ballistic limit. But results of the numerical analysis using the traditional software shows some differences between the analytical and experimental results.

Fig. 5 shows the penetration times for three different situations. Considering Fig. 5, the results of the new analytical model and experimental investigations have a good agreement at the available experimental speeds.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

4. Conclusion

It was shown that the results obtained using the plastic stress wave propagation method for the penetration problem of the deformable projectiles into the metallic targets was in a good agreement with the experimental results.

The model developed here, can be used accurately to solve the problem of normal penetration of the deformable projectiles into thin and medium metallic targets and predict the amount of penetration.

Considering the speed of the code developed based on this model, it can be used easily for designing projectiles and armors. In addition, the proposed model is able to predict the amount of target dishing and its effects on other parameters of the penetration process. Therefore, the introduced model is effective for thin targets when the speed of the projectile is near or above the ballistic limit.

Received December 02, 2009 Accepted March 15, 2010

References

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[2.] Backman, M.E., Goldsmith, W. The mechanics of penetration of projectiles into targets.--Int. J. Eng. Sci., 1978, 6, p.1-99.

[3.] Liaghat, G.H., Malekzadeh, A. A comment on the penetration theory by Dikshit and Sundararajan.--Int. J. Impact. Eng., 1995, 16, p.691.

[4.] Liaghat, GH, Malekzadeh, A. A modification to the mathematical model of perforation by Dikshit and Sundararajan. -Int. J. Impact Eng., 1999, 22, p.543-550.

[5.] Borvik, T., Dey, S., Clausen, A.H. Perforation resistance of five different high-strength steel plates subjected to small-arms projectiles.--Int. J. Impact Eng., 2009, 36, p.948-964.

[6.] Chen, X.W., Zhou, X.Q., Li, X.L. On perforation of ductile metallic plates by blunt rigid projectile.--Eur. J. Mech.--A/Solids, 2009, 28, p.273-283.

[7.] Recht, R.F. Taylor ballistic impact modeling applied to deformation and mass loss determinations.--Int. J. Eng. Sci., 1978, 16, p.809-827.

[8.] Awerbuch, J. A mechanics approach to projectile penetration.--Israel J. Tech., 1970, 8, p.375-383.

[9.] Awerbuch, J., Bodner, S.R. Analysis of the mechanics of penetration of projectiles in metallic plates.--Int. J. Solid. Struct., 1974, 10, p.685-699.

[10.] Shadbolt, P.J., Corran, R.S.J., Ruiz, C. A comparison of plate perforation models in the subordnance range.--Int. J. Impact Eng., 1984, 2, p.283.

[11.] Liss, J., Goldsmith, W., Kelly, J.M. A phenomenological penetration model of plates.--Int. J. Impact Eng., 1983, 1, p.321-341.

[12.] Liss, J., Goldsmith, W. Plate perforation phenomena due to normal impact of blunt cylinders.--Int. J. Impact Eng., 1984, 2, p.37-64.

[13.] Woodward, R.L. A structural model for thin plate perforation by normal impact of blunt projectiles.--Int. J. Impact Eng., 1987, 6, p.129-140.

[14.] Taylor, G.I. The used of flatted-end projectile for determining dynamic yield stress.--Proc. Roy Soc. Lond. A. 1984, 194, p.289-299.

[15.] Wenxue, Y., Lanting, Z., Xiaoqing, M., Stronge, W.J. Plate perforation by deformable projectiles--a plastic wave theory.--Int. J. Impact Eng., 1983, 1, p.393-412.

[16.] Borvik, T., Langseth, M., Hopperstad, O.S., Malo, K.A. Ballistic penetration of steel plates.--Int. J. Impact Eng., 1999, 22, p.855-886.

[17.] Dey, S., Borvik, T., Hopperstad, O.S., Langseth, M. On the influence of constitutive relation in projectile impact of steel plates.--Int. J. Impact Eng., 2007, 34, p.464-486.

[18.] Johnson, W. Impact strength of materials.--Edward Arnold. 1972.

D. Radmehr *, G.H. Liaghat **, S. Felli ***, D. Naderi ****

* Department of Mechanical Engineering, Tarbiat Modarres University, Tehran, Iran, E-mail: da_radmehr@yahoo.com

** Department of Mechanical Engineering, Tarbiat Modarres University, Tehran, Iran, E-mail: Ghlia530@modares.ac.ir

*** Department of Mechanical Engineering, Razi University of Kermanshah, Kermanshah, Iran

**** Department Aerospace and Space Engineering, Sharif University of Technology, Tehran, Iran
Table 1

Geometrical and mechanical properties of the projectile [15]

                       [rho],      [[sigma].sub.P],
Symbol   Material   kg/[m.sup.3]         MPa

TT-R       Lead        11340              20
S-R        Lead        11340              20
A-R        Lead        11340              20

         [C.sub.p],   [D.sub.[??]],   [L.sub.[??]],
Symbol     m/s            mm              mm

TT-R       119            5.6             9.41
S-R        119            5.6             9.41
A-R        119            5.6             9.41

Table 2

Mechanical properties of the target [15]

                          [rho],      [[sigma].sub.Y],
Symbol   Material      kg/[m.sup.3]          MPa

SA-A     Steel alloy       7890              980
SA-B     Steel alloy       7890             1200
SA-C     Steel alloy       7890             1050
AL-1     AL 1100-H14       2780              120
AL-6     AL 6061-T6        2780              260

                       [C.sub.t],
Symbol   Material         m/s       K, GPa   G, GPa

SA-A     Steel alloy      4561       164       80
SA-B     Steel alloy      4561       164       80
SA-C     Steel alloy      4561       164       80
AL-1     AL 1100-H14      5204        75       30
AL-6     AL 6061-T6       5300        78       30

Table 3

Comparison between the results of experimental tests, Wenxue model and
the model developed here for the penetration of lead projectiles into
the steel and aluminum targets

                                   Initial impact
Projectile   Target   Thick., mm   velocity, m/s

S-R           SA-A        6             850
S-R           SA-A        8             855
S-R           SA-B       6.35           854
S-R           SA-C        8             855
TT-R          AL-1        2             385
TT-R          AL-1        4             393
TT-R          AL-1        6             387
A-R           AL-6        3             422
A-R           AL-6        5             416
A-R           AL-6       6.35           412
A-R           AL-1        4             416
A-R           AL-1        5             422
A-R           AL-1        6             420

                 Residual plug velocity, m/s

                                    Analytical model
Projectile   Exp. [9]   Ref. [15]   of this present

S-R          500-600       478            522
S-R            460         454            492
S-R          350-550       463            515
S-R          450-470       455            503
TT-R           346         317            307
TT-R           292         297            300
TT-R           186         249            282
A-R            349         330            332
A-R            291         296            309
A-R            234         276            290
A-R            355         346            328
A-R            339         343            329
A-R            330         341            330

             Time of complete perforation, [micro]s

                                    Analytical model
Projectile   Exp. [9]   Ref. [15]   of this present

S-R            26.9       20.4            24.9
S-R            36.2       30.6            34.7
S-R             --        22.9              30
S-R            27.8       30.4            36.2
TT-R            --         8.3            10.6
TT-R            --        18.2            21.3
TT-R            --        34.9            34.1
A-R             --          12            18.4
A-R             24        23.2            32.7
A-R             35        33.2              44
A-R             --          15            19.4
A-R             32          19              24
A-R             19        22.9            28.6

                    Plug thickness, mm

                                    Analytical model
Projectile   Exp. [9]   Ref. [15]   of this present    Dishing, mm

S-R              5         5.7            4.8              1.8
S-R            6.9         7.6            6.6              2.1
S-R            5.3           6            5.6              2.5
S-R              7         7.7            6.8              2.4
TT-R           1.4         1.9            1.7              0.3
TT-R           2.8         3.8            3.5              0.5
TT-R           4.2         5.8            5.3              0.7
A-R            1.6         2.9            2.8              1.3
A-R            3.6         4.8            4.7              1.8
A-R             5          5.8              6              2.2
A-R            2.8         3.8            3.4              0.6
A-R            3.5         4.7            4.2              0.7
A-R            4.2         5.7              5              0.7

Table 4

Comparison of the dishing to thickness ratio for the penetration of
lead projectiles into the aluminum target

                      Thickness,   Initial impact     Dishing to
Projectile   Target       mm       velocity, m/s    thickness ratio

A-R           AL-6        3             422              0.43
A-R           AL-6        5             416              0.36
A-R           AL-6       6.35           412              0.34
A-R           AL-1        4             416              0.15
A-R           AL-1        5             422              0.14
A-R           AL-1        6             420              0.12

Table 5

Values for Johnson-Cook model constants [17]

[A.sub.JC],   [B.sub.JC],                        [T.sub.melt],
MPa               MPa         n     [C.sub.JC]        K            m

499               382       0.458     0.0079         1800        0.893
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Author:Radmehr, D.; Liaghat, G.H.; Felli, S.; Naderi, D.
Publication:Mechanika
Article Type:Report
Geographic Code:7IRAN
Date:Mar 1, 2010
Words:5692
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