Estimation of a crater volume formed by impact of a projectile on a metallic target

Size: px

Start display at page:

Download "Estimation of a crater volume formed by impact of a projectile on a metallic target"

Jacob Todd
5 years ago
Views:

1 Int J Mech Mater Des (28) 4: DOI 1.17/s Estimation of a crater volume formed by impact of a projectile on a metallic target K. Kline Æ I. Sevostianov Æ R. Parker Received: 31 October 27 / Accepted: 5 February 28 / Published online: 9 March 28 Ó Springer Science+Business Media B.V. 28 Abstract The crater volume has been an important factor in ballistics and has many influences such as material strength, initial projectile velocity, angle of incidence, and nose shape. The goal of this research is to predict the resulting crater volume of a long rod penetration based on the initial projectile velocity and mass. Mooney s (Bull Seism Soc Am 64(2):473, 1974) displacement equations were used to calculate the elastic crater volume for a given impulse force, P, varying as a delta function in time on the surface of an isotropic, semi-infinite solid. This estimated elastic volume, V elastic is linearly related to the experimental ballistic volume, V experimental by an energy dissipation factor, k. V elastic = kv experimental. The energy dissipation factor lumps together elastic and plastic deformation mechanisms. Terminal ballistic data for a steel long-rod projectile into semiinfinite steel or aluminum target will be compared to the crater volume calculated through the use of k. Keywords Crater volume Terminal ballistics Long-rod projectile Dynamic contact problem K. Kline (&) R. Parker Los Alamos National Laboratory, PO Box 1663, Los Alamos, NM 87545, USA kkline@lanl.gov I. Sevostianov Mechanical & Aerospace Engineering, New Mexico State University, PO Box 31/MSC 345, Las Cruces, NM , USA 1 Introduction In this paper we are interested in predicting the resulting target crater volume of a normal, long-rod penetration using the initial projectile s velocity and mass. The crater volume is calculated from the vector displacement of a point on the impact surface. The point s displacement vector is estimated from algebraic equations used to calculate seismological displacements for Lamb type problems. The solutions to Lamb s problem determine the elasto-dynamic response of an isotropic, elastic, semi-infinite half space to a time dependent concentrated force. This concentrated force is usually a pulse of short time duration which varies in time as a unit step function or a Dirac delta function. Much research has been done in determining the displacement fields for various loads beginning with Lamb s landmark paper in 194. The following papers are not inclusive of the work contributed to impact and material displacement calculations. Lamb (194) showed the displacement solutions for a uniform force vertically applied along a line of the surface of a semiinfinite or in an infinite, isotropic, elastic solid. Pekeris (1955) determined exact and closed expressions for vertical and horizontal surface displacements for a semi-infinite, uniform, isotropic elastic solid under a normal, Heaviside step load pulse by use of a Laplace- Hankel integral transform and contour integration. In 1966, Eason extended Pekeris s (1955) work to include exact solutions for displacements at any internal point

2 376 K. Kline et al. in the solid due to a normal uniform or non-uniform circular load. Mitra (1964) presented a displacement field solution in the form of definite integrals in response to a finite, uniform disk load varying as the Dirac delta function in time. Mooney (1974) provided response solutions in closed-form to Lamb s problem and assumed an axi-symmetric configuration. Specifically, Mooney considered response solutions for a time-dependent normal, surface loading in the form of an ideal step function, delta function, or an arbitrary function onto a uniform half-space. Then, Mooney presented analytical solutions for surface displacements, velocity, acceleration, and horizontal strain. The radial and horizontal displacement was dependent upon the target material s Poisson ratio and shear modulus. In this paper, Mooney s equations will be used to determine displacement in the depth and radial directions relative to an input force. The input force is specified so that it varies in time as a Dirac delta function. From displacement field, the elastic crater volume can be calculated. Next, using experimental terminal ballistic data, an energy dissipation factor, k, can be constructed using the ratio, k = V elastic / V experimental. With k calculated, an energy dissipated crater volume can be predicted and compared to the experimental crater volume. 2 Mooney s equations for displacement We assume an axi-symmetric configuration as Mooney did. The sign convention used is the vertical displacements are positive along the negative z-axis. The vertical load is also applied along the negative z- axis at the plane (Fig. 1). The input force is specified so that it varies in time as a Dirac delta function. The magnitude of the applied force, P, is calculated from the work energy principle. The work done on the target is assumed to be equivalent to the kinetic energy of the projectile. Therefore, P dz ¼ 1=2mv 2 where P is assumed to be constant over the distance traveled into the target, dz. m is the mass of the projectile and v is the projectile s velocity. All units are assumed to be in SI. Fig. 1 Geometry of an impulse force on a semi-infinite plane. The receiver is location (z =, r) away from the impulsive source and moves to point (dz, dr) 2.1 Vertical displacement equations The vertical displacement, W(t), for a force varying in time as a unit step function or Heaviside function acting on a semi-infinite half-space is calculated by WðtÞ ¼Z GðsÞ where Z ¼ P p 2 lra and s ¼ C st 2 r is the dimensionless time. r is the radial distanceofthe 2¼ receiver to the source, t is time, and a 2 ¼ C s C L 1 2t 2ð1 tþ : C L is the longitudinal wave velocity. The target is assumed to have material properties as follows: the Shear wave velocity is C s = 1; the density is q ¼ 1; and therefore the Shear Modulus of the target is l = 1. The calculation of G(s) is as follows and it is assumed t =.25: G ¼ for s\a; G ¼ P 1 2t p p 2 lra " # A 3 A 2 A 1 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 3 s 2 s 2 x 2 s 2 x 1 for a\s\c s ; ¼ p pffiffiffi pffiffi ð3 3 þ 5Þ 1=2 ð3 3 5Þ 1=2 (6 96 ðx 3 s 2 Þ 1=2 ðs 2 x 2 Þ 1=2 ) ð3þ1=2 ðs 2 x 1 Þ 1=2

3 Estimation of a crater volume formed by impact of a projectile on a metallic target 377 " # G ¼ P 1 2t p p 2 lra A 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 3 s 2 ( ¼ p pffiffiffi ) ð3 3 þ 5Þ 1= ðx 3 s 2 Þ 1=2 p for C s \s\ ffiffiffiffi x 3; G ¼ sign A 3 inf pffiffiffi ¼ sign ð3 3 þ 5Þ 1=2 inf pffiffiffiffi for x 3 ¼ s; G ¼ P 1 2t p p 2 lra ¼ p p ffiffiffiffi for x 3\s\2: 8 where A i ¼ ð1 2x iþ 2 ða 2 x i Þ 1=2 ðx i x j Þðx i x k Þ : x 3 is the largest, real root of the Rayleigh function, 1-8x + 8x 2 (3-2a 2 )- ffiffi 16x 3 (1-a 2 ) =, or for m ¼ :25; x 1 ¼ 1 4 ; x 2 ¼ ð3 p 3 Þ ffiffi 4 ; and x 3 ¼ ð3þ p 3 Þ 4 : Though an arbitrary t can be used, the displacement equations shown in this paper are valid only for t \ At t =.2631, all the roots of the Rayleigh function become real, beyond which two roots will become complex (Kausel 26). Now, we are interested in the displacement due to a Dirac delta time impulse. The displacement equation due to a delta impulse force is the derivative with respect to s of the displacement solution in response to a force varying as a unit step function. Therefore, for a vertical displacement, W ¼ Z GðsÞ due to a Heaviside function, the vertical displacement due to a Dirac delta Impulse function is W ¼ Z C s dgðsþ r ds : The derivative of G(s) or R(s) with respect to s is calculated numerically using the central difference formula with Ds ¼ :2: f ðx þ DsÞ fðx DsÞ 2Ds Figure 2 shows the vertical displacement function, W ðtþ; due to a delta forcing function. The function s discontinuity at approximately t = 1.87 s is not significant because we are interested in the displacement when the impulse is applied, approximately t =.5776 s. 2.2 Radial displacement equations The radial displacement, Q(t), for a force varying in time as a unit step function or Heaviside function Vertical Displacement (m) acting on a semi-infinite half-space is calculated by Q ¼ Z RðsÞ: The radial displacement for a Delta Impulse source function is the derivative of Q with respect to s; Q ¼ Z C s r drðsþ ds : The calculation of R(s) for the radial displacement equations involves the use of elliptical integrals of the first and third kind. The elliptical integrals of the first kind is KðkÞ ¼ R p=2 7.E+6 5.E+6 3.E+6 1.E E+6 Time (s) Fig. 2 Vertical displacement function, W (t), versus t. P = 1N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh and the elliptical integrals of the third 1 k 2 sin 2 h kind is Z p=2 dh Pðh; n; kþ ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ð1 þ n sin 2 hþ 1 k 2 sin 2 h The calculation of R(s) is as follows: R ¼ for s\a; s R ¼ 4 ffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2tÞ p 2KðkÞ B1 Pðn 1 k 2 ; kþ 1 a 2 2 B 2 Pðn 2 k 2 ; kþ B 3 Pðn 3 k 2 ; kþ ¼ 3s p 16 ffiffi 2KðkÞ B 1 Pðn 1 k 2 ; kþ 6 B 2 Pðn 2 k 2 ; kþ B 3 Pðn 3 k 2 ; kþ for a \ s \ C s, where k 2 ¼ ðs2 a 2 Þ b 2 pffiffiffiffi s k R ¼ 2 4 ffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 2tÞ p 2KðkÞ B1 Pðn 1 ; kþ 1 a 2 2 B 2 Pðn 2 ; kþ B 3 Pðn 3 ; kþ ¼ 3sk p 16 ffiffi 2KðkÞ B 1 Pðn 1 ; kþ B 2 Pðn 2 ; kþ 6 B 3 Pðn 3 ; kþ for C s \s ¼ p ffiffiffiffi x 3; where k 2 ¼ b2 ðs 2 a 2 Þ

4 378 K. Kline et al. Radial Displacement (m) 7.E+6 5.E+6 3.E+6 1.E E+6 Time (s) displacement of interest is when the impulse is applied, approximately t =.5776 s. As can be seen in Figs. 2 and 3, the displacements are quite large (*1 6 m) for an applied force of 1 N. Mooney s displacement equations assume the shear modulus is unity. In metals, the shear modulus is usually far from unity; therefore the equations need to be reduced by the shear modulus of the target material. Fig. 3 Radial displacement function, Q (t), versus t. P = 1N pffiffiffiffi s k R ¼ p 2 4 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a 2 ð1 2tÞ 2KðkÞ B1 Pðn 1 ; kþ 2 B 2 Pðn 2 ; kþ B 3 Pðn 3 ; kþ þ ps 24 ðs2 x 3 Þ 1=2 ¼ 3sk p 16 ffiffi 2KðkÞ B 1 Pðn 1 ; kþ B 2 Pðn 2 ; kþ 6 B 3 Pðn 3 ; kþš þ ps 24 ðs2 x 3 Þ 1=2 p ffiffiffiffi for x 3\s\2; where k 2 b 2 ¼ ðs 2 a 2 Þ : where B i ¼ ð1 2x iþð1 x i Þ ðx i x j Þðx i x k Þ for x i =x j =x k and n i ¼ b2 ða 2 x i Þ with b 2 = 1-a 2. Figure 3 shows the radial displacement function, Q ðtþ; due to a delta forcing function. The radial 3 The calculation of k Once the displacement field has been determined, the elastic crater volume can be calculated. The crater volume is assumed to have a right cylindrical shape, V elastic = pq 2 W. This is a valid assumption as the velocity increases for metal penetrating into metal (Hohler and Stilp 198). Next, using experimental terminal ballistic data, an energy dissipation factor, k, can be constructed using the ratio, k = V elastic /V experimental. k incorporates the deformation mechanisms into one term. The energy dissipation factor can include the following mechanisms: target melting, projectile deformation, elastic target deformation, plastic-flow in the target, and target plastic strain energy. All mechanisms of energy dissipation are assumed to depend on the projectile material/target material pair. k is calculated for each Fig. 4 The average k ratio for a steel projectile into various steel targets 6E E+12 Average k-factor 4E+12 2E E E E E E E+1 34 Stnl St37/52 Ger Arm St HzB,A Steel 12 Steel 13 Steel 414 Target material

5 Estimation of a crater volume formed by impact of a projectile on a metallic target 379 Fig. 5 The average k ratio for a steel projectile into various aluminum targets 8E E+12 6E E+12 Average k-factor 4E+12 2E E E E E+1 11-O 224-T3 661-T651 Al 224S Al 224 T-4 Aluminum Target Material Fig. 6 The elastic crater volume prediction plotted with the experimental data (Billington and Carley 1969) for k = e8 for a steel projectile into an aluminum targets Volume (m3) 8.E-5 6.E-5 4.E-5 Experimental Predicted 2.E-5.E+ 5 1 Velocity (m/s) energy level and then averaged over the range for the projectile/target material. The experimental steel into steel data for the calculation of k comes from four sources: Christman and Gehring (1963), Christman and Gehring (1965), Hohler and Stilp (1977, 1984, 1987), Herrmann and Jones (1961). The experimental steel into aluminum data for the calculation of k comes from five sources: Christman and Gehring (1963), Christman and Gehring (1965), Forrestal et al. (1991), Luk and Piekutowski (1991), and Herrmann and Jones (1961). k is calculated for hardened steel projectiles into various steel and aluminum targets as shown in Figs. 4 and 5, respectively. k is highly material-pair dependent. With k calculated, an energy dissipated crater volume can be predicted and compared with the experimental crater volume. 4 Prediction of experimental data Crater volume was predicted for steel into aluminum (Billington and Carley 1969) with k = e8 and for steel RC-66 into steel 414 (Herrmann and Jones 1961) with k = e11. The elastic crater volume prediction and experimental data for the

6 38 K. Kline et al. Fig. 7 The elastic crater volume prediction plotted with the experimental data (Herrmann and Jones 1961) for k = e11 for a steel RC-66 projectile into steel 414 targets Volume (m3) 5.E-7 4.E-7 3.E-7 2.E-7 Experimental Predicted 1.E-7.E+ 5 1 Velocity (m/s) aluminum target and steel target are plotted in Figs. 6 and 7, respectively. The prediction reasonably captures the trend of the experimental data though it tends to predict a lower crater volume. The oscillatory trend of the prediction crater volume for the steel target is considered hardening that is occurring but isn t account for in the prediction. 5 Conclusions An energy dissipation factor is not an unreasonable first estimate despite assuming t =.25 for the target material. Though highly material projectile/target pair dependent, it predicts the qualitative trend in the data. The next step is to investigate the relationship between the projectile s initial kinetic energy and the resulting crater volume. This relationship has shown to be linear for various metals (Partridge et al. 1958). We will also predict crater volumes for other material projectile/target pairs of interest and investigate the usefulness of the energy dissipation factor. Acknowledgements We would like to thank Dr. Karen Wells and Los Alamos National Laboratory for supporting for this research. References Billington, E.W., Carley, D.J.: Velocity dependence of impact parameters involved in long-rod penetrations into semiinfinite targets. Br. J. Appl. Phys. Ser 2. 2, 613 (1969) Christman, D.R., Gehring, J.W.: Semiannual report on penetration mechanisms of high-velocity projectiles. Report No. TR63-25, prepared for Ballistic Research Laboratories under Contract No. DA AMC-(R), GM Defense Research Laboratories, Santa Barbara, CA (1963) Christman, D. R., Gehring, J.W.: Final report on penetration mechanisms of high velocity projectiles. Report No. TR65-5 prepared for Ballistic Research Laboratories under Contract No. DA AMC-534(X), GM Defense Research Laboratories, Santa Barbara, CA (1965) Eason, G.: The displacements produced in an elastic half-space by a suddenly applied surface force. J. Inst. Math. Appl. 2, 299 (1966) Forrestal, M.J., Brar, N.S., Luk, V.K.: Penetration of strainhardening targets with rigid spherical-nose rods. J. Appl. Mech. 58, 7 1 (1991) Hohler, V., Stilp, A.J.: Penetration of steel and high density rods in semi-infinite steel targets. Proceedings of the 3rd International Symposium on Ballistics. Karlsruhe, FRG, March 1977 Hohler, V., Stilp, A.J.: Study of the penetration behavior of rods for a wide range of target densities. Proceedings of the 5th International Symposium on Ballistics. Toulouse, France, April 198 Hohler, V., Stilp, A.: Influence of the length-to-diameter ratio in the range from 1 to 32 on the penetration performance of rod projectiles. Proceedings of the 8th International Symposium on Ballistics. Orlando, FL, October 1984 Hohler, V., Stilp, A.: Hypervelocity impact of rod projectiles with L/D from 1 to 32. Int. J. Impact Eng. 5(1-4), (1987) Herrmann, W., Jones, A.H.: Survey of hypervelocity impact information. A.S.R.L. Report No October 1961 Kausel, E.: Fundamental Solutions in Elastodynamics: A Compendium. Cambridge University Press, New York (26) Lamb, H.: On the propagation of tremors over the surface of an elastic solid. Philos. Trans. R. Soc. Lond. Ser. A. 23, 1 (194)

7 Estimation of a crater volume formed by impact of a projectile on a metallic target 381 Luk, V. K., Piekutowski, A.J.: An analytical model on penetration of eroding long rods into metallic targets. Int. J. Impact Eng. 11(3), (1991) Mitra, M.: Disturbance produced in an elastic half-space by impulsive normal pressure. Proc. Camb. Phil. Soc. 6, 683 (1964) Mooney, H. R.: Some numerical solutions for lamb s problem. Bull. Seism. Soc. Am. 64(2), 473 (1974) Partridge, W.S., Vanfleet, H.B., Whited, C.R.: Crater formation in metallic targets. J. Appl. Phys. 29(9), 1332 (1958) Pekeris, C.L.: The seismic surface pulse. Proc. Natl. Acad. Sci. 41, 469 (1955)

Long-rod penetration: the transition zone between rigid and hydrodynamic penetration modes

Available online at www.sciencedirect.com ScienceDirect Defence Technology 10 (2014) 239e244 www.elsevier.com/locate/dt Long-rod penetration: the transition zone between rigid and hydrodynamic penetration