Mechanics Based Design of Structures and Machines, 35: 113 125, 27 Shape Optimization of Impactors Against a Finite Width Shield Using a Modified Method of Local Variations # G. Ben-Dor, A. Dubinsky, and T. Elperin Department of Mechanical Engineering, Pearlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, Beer-Sheva, Israel Abstract: We suggest a modification of the method of local variations of Banichuk and Chernous ko that expands further the scope of applications of this method beyond the applications considered in our previous study. This modification allows us to solve numerically the variational proem of determining the shape of the impactor having a minimum ballistic limit velocity (BLV) when the criterion of optimization is a doue integral with an integrand including a doue integral over a variae area. The integrands in both integrals depend on the unknown function determining the generatrix of the impactor. We found that impactors with flat or conical nose have a minimum BLV and derived an analytical criterion that determines the choice between these two shapes depending on the parameters describing the size of the impactor s nose and properties of the material of the shield. The proposed approach can be useful in solving other optimization proems. Keywords: Ballistic limit velocity; Impact; Method of local variations penetration; Optimization; Perforation. Received December 9, 26; Accepted January 2, 27 # Communicated by N. Banichuk. Correspondence: T. Elperin, Department of Mechanical Engineering, Pearlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 8415, Israel; Fax: +972-8-6472813; E-mail: elperin@bgu.ac.il
114 Ben-Dor et al. 1. INTRODUCTION Overview of the studies on shape optimization of penetrating impactors using engineering models of impactor-shield interaction can be found in recent review and monograph of Ben-Dor et al. (25, 26). It must be emphasized these studies considered variational proems arising for semi-infinite shields whereas shape optimization of impactors against shields with a finite thickness was not considered at all (except for Ben-Dor et al., 22). The latter study is an exception in this respect although it employed very simple model, which allowed application of the classical calculus of variations. The absence of studies on shape optimization of impactors against shields with a finite thickness can be explained by the following reasons. Analysis of perforation of thick shields with finite thickness must take into account plug formation and the incomplete immersion of the impactor at the initial and final stages of penetration. However, incorporation of these factors even in the simple two-term model of impactor-shield interaction (Ben-Dor et al., 25, 26) yields a formula for the ballistic limit velocity (BLV) of the impactor (body of revolution) in the form of a nonintegral and cumbersome functional depending on the generatrix of the impactor. Variational proem for the functionals of this type cannot be solved analytically, and to the best of our knowledge, commonly used numerical methods cannot be directly used for the solution of the proem. The method of local variations (Banichuk et al., 1969; Chernous ko and Banichuk, 1973) was successfully employed in its original version for shape optimization of bodies of revolution penetrating into semiinfinite shields when the special two-term impactor-shield interaction model is used (Ben-Dor et al., 23). Modification of the method of local variations (Ben-Dor et al., 27) allowed solving similar proem for a wide class of impactor-shield interaction models. In this study, we suggest a further modification of the method of local variations (MLV) for more complicated functionals and apply the suggested modification for numerical optimization of the shape of the impactor against a finite width shield. 2. FORMULATION OF THE PROBLEM AND MATHEMATICAL MODEL Consider a high-speed normal penetration of a rigid body of revolution into a ductile shield with thickness b. The notations are shown in Figs. 1(a) (b). The coordinate h, the depth of the penetration, is defined as the distance between the nose of the impactor and the front surface of the shield. The coordinates x and are associated with the impactor.
Shape Optimization of Impactors 115 Figure 1(a) (b). Coordinates and notations. Two-stage peneration model. The part of the lateral surface of the impactor between the cross-sections x = and x = interacts with the shield where (see Figs. 2(a) (b)) { if h b h b L = h b if b h b + L { h if h L h b L = L if L h b + L (1) It is assumed that the impactor generally consists of a noncylindrical nose with the length L and of a cylindrical part. Only the nose interacts with the shield, and interaction between the impactor and the
116 Ben-Dor et al. Figure 2(a) (b). Description of the area of impactor-shield interaction. shield occurs when h b + L. In coordinates x associated with the impactor the surface of the nose is described by the following equation: = x = r L = R (2) where the increasing function x determines the generatrix of the impactor s nose. We assume that the impactor-shield interaction is described by the following widely used model (for details see, e.g., Ben-Dor et al., 25, 26; Recht, 199): d F = [ a 1 v n 2 v 2 + a n ds (3) where d F is the force acting at the lateral surface element ds of the projectile that is in contact with the shield, n is the inner normal unit vectors at a given location on the projectile s surface, v is a unit vector of the surface element velocity of the projectile, v (translational
Shape Optimization of Impactors 117 velocity of the impactor), the parameters a and a 1 characterize the properties of the material of the shield. It is usually assumed that a 1 equals to the density of the material of the shield, sh. The drag force acting at the lateral surface of the impactor, D imp, is found by integrating v d F over this surface. Using formulae of differential geometry we obtain: where D imp h v = 2 ( a I + a 1 I 1 v 2) (4) I h = h b L h b L G dx = L = d /dx x h b L G dx (5) G = 2 / 2 + 1 = 1 (6) if x h b L x h b L = 1 if h b L x h b L if h b L x L (7) For a unt impactor, we use a simplified model of plugging (see, e.g., Goldsmith and Finnegan, 1971; Sagomonyan, 1988) whereby it is assumed that a cylindrical plug with the radius r, height b, and mass m plug = sh r 2 b (8) is formed at the beginning of penetration, and this plug moves together with the impactor while h b (the first stage of penetration, see Fig. 1(a)). Formula for the resistance force associated with the plug during this stage of the penetration reads: D plug h = 2 sh r b h (9) where sh is the shear strength of the material of the shield. If h = b, the impactor and the plug break apart, and the impactor with mass m moves under the action of the force D imp (the second stage of penetration, see Fig. 1(b)). Consequently, equations describing the motion of the impactor read: m + m plug ḧ + D imp h ḣ + D plug h = h<b (1) mḧ + D imp h ḣ = b h b + L (11)
118 Ben-Dor et al. Equations (1) and (11) can be replaced by one equation: m + m plug h b ḧ + D imp h ḣ + h b D plug h = (12) where { 1 if h b h b = if h>b (13) Let us introduce a new function w = v 2 = ḣ 2 of the independent variae h and take into account that 2ḧ = dw/dh. Then Eq. (12) can be written as 1[ m + mplug h b ] dw 2 dh + D imp h w + h b D plug h = (14) Substituting D imp, m plug, and D plug from Eqs. (4), (8), and (9), respectively, into Eq. (14) we obtain: [ m + sh r 2 b h b ] dw dh + 4 a 1I 1 h w + a I h + sh r b h h b = (15) The ballistic limit velocity (BLV) is defined as the initial velocity of the impactor that is required in order to emerge from the shield with zero velocity. Let w = w h be the solution of Eq. (15) with the initial condition w b + L =. Then the BLV, v, is determined by the formula v 2 = w = w. Using the dimensionless variaes Equation (15) can be rewritten as follows: h = h L w = w w w = a sh (16) where d w d h + f h w + g h = (17) f h = k I 1 h h g h = r b h h b + I h h h = 25 m + b r 2 h b r = J I h = 1 x h b 1 G d x J = 1 d x (18)
Shape Optimization of Impactors 119 = L k = a 1 sh = d d x x = x L b = b L = R L = sh a m = m sh L 3 Equation (17) yields the following general expression for the dimensionless BLV: ( b+1 h ) v 2 = w = g h exp f H d H d h v = v w (19) Equation (19) yields formulas for the particular cases of the sharp cone nosed impactor [ ( ) ] w cone = 2 + 1 2k exp 4 b 1 k 2 2 + 1 m x = x (2) and a cylindrical impactor with a flat untness w cyl = 2 b 2 m + b x = (21) 2 Let us consider the proem of minimizing the BLV. It is assumed that the model describing impactor-shield interaction, parameters determining the mechanical properties of the shield, the mass of the impactor, and the length and the shank radius of the nose of the projectile, is given. In dimensionless variaes, the proem is reduced to the determining the non-decreasing function x that satisfies the constraints 1 = x x x x x<1 (22) 1 x and minimizes the functional in Eq. (19). The parameters b k, and m are assumed to be known. The second constraint in Eq. (22), stipulates that the optimum shape is sought between the sharp cone and the cylinder. The third inequality represents the necessary (but not sufficient) condition of convexity of the generatrix: the tangent in every point of the generatrix must be located not lower than the straight line segment between this point and the final point of the generatrix. Hereafter we assume that the parameter a 2 has the meaning of the density of the shield material, i.e., k = 1. 3. ADAPTATION OF THE METHOD OF LOCAL VARIATIONS The functional w x in Eq. (19) is quite involved and the known methods of minimization cannot be applied directly for its minimization.
12 Ben-Dor et al. We will reduce the proem of minimization of this functional to a variational proem for the functional that is a function of some integral functionals. The latter proem can be solved using an efficient numerical method of local variations (Banichuk et al., 1969; Chernous ko and Banichuk, 1973). The Newton Cotes formulas for approximate calculation of the external integral in Eq. (19) can be written as follows (Korn and Korn, 1968): w h N i= ( h ) N i i g h i exp f H d H (23) where N + 1 is the number of nodes in which the values of the integrand are calculated, h = b + 1 /N h i = i h i = 1 N (24) Coefficients N i depend on the type of the Newton Cotes formula. N 1 = 1 N N = for the rectangle = 1 for the trapezoid rule formula. The integral in Eq. (23) can be approximated in a similar manner and Eq. (23) can be rewritten as follows: In particular, N = N 1 = = N rule formula, and N = N N = 5 N 1 = N 2 = = N N 1 ( ) N i w h N i g h i exp h i j f h j (25) i= where for simplicity we used the same nodes in both sums. The appropriate choice of N allows attaining the required accuracy of the approximation. Inspection of Equations (18) and (25) shows that the functional w x is, generally, a function of 2N + 1 integrals, I h 1, I h 2 I h N ; I 1 h 1 I 1 h 2 I 1 h N and J, taking into account that I = I 1 =. j= 4. NUMERICAL RESULTS, DISCUSSION, AND CONCLUSIONS As most of the existing numerical methods, the method of local variations allows determining only the local extrema. In order to overcome the latter shortcoming, calculations must be performed with different initial functions, and the obtained results must be compared. In our calculations, we use the following initial functions: x = + x (26)
Shape Optimization of Impactors 121 with different and. In the majority of calculations performed using these initial functions, iterations converge to one of the following three functions describing (1) sharp cone, (2) cylinder, and (3) a non-convex generatrix that is very close to the generatrix of a sharp cone and yields practically the same BLV. Therefore, the minimum value of the BLV is attained for the sharp cone if w cone < w cyl or for the cylinder if w cone > w cyl.if wcone = w cyl, then both shapes yield the same BLV. Using Eqs. (2) and (21) one can define the magnitude of the parameter, div is satisfied: w cone = w cyl div = 2 + 1 m + b 2 2 b 2 3 for which the condition [ ( ) ] 2 4 b exp 1 k 2 + 1 m = 1 (27) Clearly, the sharp cone or the cylinder is an optimum shape if > div or < div, respectively. For relatively large m, m b 2 2 4 b 2 + 1 m (28) Equation (27) can be simplified by keeping only two terms in Taylor series expansion for the exponent: div = / b (29) Since the solution is determined by simple analytical relationships, the obtained results are exemplified only in two plots. Figures 3(a) (b) shows the variation of div as a function of b and m for = 25 and = 5. Generally, the shape of the optimum impactor depends on, b, m, and but if / b, then the cylinder is an optimum shape for impactor of any mass. Figures 4(a) (b) compare the BLV of the impactors with sharp conical, flat (cylinder) and ogive-shaped noses. The values of the BLV in these figures, v, are normalized by the BLV of the conical-nosed impactor, v cone. The obtained solution has a clear physical meaning in some limiting cases. If, then the BLV of the cylinder v cyl, whereas the BLV of any non-cylindrical impactor is positive since energy is required for expanding the cavity in the shield during penetration. Clearly, the advantage of the cylinder is retained for small. On the other hand, for large, perforation with plugging becomes not advantageous, and sharp-nosed projectile is an impactor with an optimum shape under these conditions. Therefore, we demonstrated that (1) impactor with the optimum sharp nose is the cone and (2) the minimum BLV is attained if the nose of the impactor is conical or the impactor is a cylinder. The choice of the
122 Ben-Dor et al. Figure 3(a) (b). Behavior of div as a function of m for different b. optimum shape is performed by comparison of the BLVs of these two impactors. Further investigation of the relationships determined in this study requires analysis of the large volumes of data obtained in experiments and numerical simulations. Each cycle of experiments/simulations must be conducted for the same shields varying the shapes of the impactors (including cylindrical and conical impactors) while keeping R L, and m constant. To the best of our knowledge only several series of such calculations and experiments were puished in the literature (Børvik et al., 22a,b; Dey et al., 24; Gupta et al., 27). Analysis of these results allows us to draw a conclusion that they do not contradict our findings.
Shape Optimization of Impactors 123 Figure 4(a) (b). Comparison of the BLVs of the impactors with conical, flat, and ogive-shaped noses. Børvik et al. (22a,b) studied experimentally and numerically perforation of the thick steel plate by the projectiles of the same mass and diameter with flat, semispherical, and conical noses. It was found that the BLV of the cylinder is consideray less than the BLVs of two other impactors. Dey et al. (24) puished the results of experiments and computer simulations on perforation of shields manufactured from three structural steels by projectiles with flat, conical, and ogival noses. It appears that the cylinder is the impactor with the minimum BLV. Gupta et al. (27) conducted the experimental and numerical investigations on perforation of aluminum plates by unt, ogive, and semispherical nosed projectiles. Certainly reference to the latter study in the context of our
124 Ben-Dor et al. investigation is not completely justified since only thin plates were used in Gupta et al. (27). However it must be noted that for the plate with the maximum width of 2.5 mm, the BLV of the cylinder is consideray smaller than for the impactors with other shapes. Clearly, the proposed approach can be applied for functionals that differ from the functional in Eq. (19). Consequently, the suggested modification of the method of local variations can be employed for solving optimization proems involving complicated functionals that are encountered in various design proems. REFERENCES Banichuk, N. V., Petrov, V. M., Chernous ko, F. L. (1969). The method of local variations for variational proems involving non-additive functionals. Comp. Math. and Math. Phys. 9(3):66 76. Ben-Dor, G., Dubinsky, A., Elperin, T. (22). Optimal nose geometry of the impactor against FRP laminates. Composite Struct. 55(1):73 8. Ben-Dor, G., Dubinsky, A., Elperin, T. (23). Numerical solution for shape optimization of an impactor penetrating into a semiinfinite target. Comput. Struct. 81(1):9 14. Ben-Dor, G., Dubinsky, A., Elperin, T. (25). Ballistic impact: recent advances in analytical modeling of plate penetration dynamics. A review. Appl. Mech. Rev. 58:(6):355 371. Ben-Dor, G., Dubinsky, A., Elperin, T. (26). Applied High-Speed Plate Penetration Dynamics. Netherlands: Springer. Ben-Dor, G., Dubinsky, A., Elperin, T. (27). Modification of the method of local variations for shape optimization of penetrating impactors using the localized impactor/shield interaction models. Mechanics Based Design of Structures and Machines 35(2):113 125. Børvik, T., Langseth, M., Hopperstad, O. S., Malo, K. A. (22a). Perforation of 12 mm thick steel plates by 2 mm diameter projectiles with flat, hemispherical, and conical noses part I: experimental study. Int. J. Impact Eng. 27(1):19 35. Børvik, T., Langseth, M., Hopperstad, O. S., Malo, K. A. (22b). Perforation of 12 mm thick steel plates by 2 mm diameter projectiles with flat, hemispherical, and conical noses part II: numerical simulations. Int. J. Impact Eng 27(1):37 64. Chernous ko, F. L., Banichuk, N. V. (1973). Variational Proems of Mechanics and Control. Moscow: Nauka (in Russian). Dey, S., Børvik, T., Hopperstad, O. S., Leinum, J. R., Langseth, M. (24). The effect of target strength on the perforation of steel plates using three different projectile nose shapes. Int. J. Impact Eng. 3(8 9):15 138.
Shape Optimization of Impactors 125 Goldsmith, W., Finnegan, S. A. (1971). Penetration and perforation processes in metal targets at and above ballstic velocities. Int. J. Mech. Sci. 13(1):843 866. Gupta, N. K., Iqbal, M. A., Sekhon, G. S. (27). Effect of projectile nose shape, impact velocity, and target thickness on deformation behavior of aluminum plates. Int. J. Solids & Str. 44(1):3411 3439. Korn, G. A., Korn, T. M. (1968). Mathematical Handbook for Scientists and Engineers. New York: McGrawHill Book Comp. Recht, R. F. (199). High velocity impact dynamics: Analytical modeling of plate penetration dynamics. In: Zukas, J. A., ed. High Velocity Impact Dynamics. New York: Wiley and Sons, pp. 443 513. Sagomonyan, A. Ya. (1988). Dynamics of Barriers Perforation. Moscow: Moscow Univ. Pu. (in Russian).