SOME INVERSE PROBLEMS IN PENETRATION MECHANICS

Mechanics Based Design of Structures and Machines, 38(4): 468 48, 21 Copyright Taylor & Francis Group, LLC ISSN: 1539-7734 print/1539-7742 online DOI: 1.18/15397734.21.51274 SOME INVERSE PROBLEMS IN PENETRATION MECHANICS 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 Many empirical and semi-empirical models that are widely used in impact dynamics employ the dependencies between the impact (initial) velocity and the depth of penetration (DOP). These dependencies are often obtained by statistical processing of experimental results and are not based on the theoretical models of penetration. In this study we develop an approach that allows using such phenomenological correlations for determining the dependencies of forces acting on a penetrator versus the instantaneous DOP and the instantaneous velocity. The developed approach can be also useful in other fields of mechanics. Keywords: Concrete; Granular media; High speed; Impact; Penetration. INTRODUCTION Inverse problems often occur in various fields of science and are a subject of numerous studies (see e.g., Denisov, 1994; Ramm, 25 and references therein). Solution of the inverse problems for ordinary differential equations (ODEs) implies determining either the right-hand side of ODEs or their coefficients (including the case when they are functions of the independent variable), using the known solution. However, inverse problems of different type also occur in applied sciences, and particular in penetration mechanics (Ambroso et al., 25; Eisler et al., 1998; Katsuragi and Durian, 27; Kennedy, 1976; Pilyugin, 24). Comprehensive theoretical foundations for investigating these problems have not been developed as yet. It is convenient to describe the essence of these problems without dwelling on a particular application. Let y = x y be a solution of the first-order ODE with the initial condition y = y for x =. Let us consider for y =ỹ the set of x 1 and y, which satisfy the equation ỹ = x 1 y. It is assumed that the empirical dependence x 1 y x 1 y ỹ = is known. Consider the problem of recovering the ODE, which includes the unknown functions such that the solution of this ODE implies the dependence, x 1 y =. In the case when ODE describes penetration of a striker Received April 16, 21; Accepted June 4, 21 Communicated by N.Banichuk. Correspondence: Tov 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; E-mail: elperin@.bgu.ac.il 468

469 BEN-DOR ET AL. into a semi-infinite shield, x and y are the instantaneous depth of penetration (DOP) and the instantaneous velocity of the striker, respectively, and formula x 1 y = relates the initial velocity of the striker, y, and the maximum penetration depth, x 1, whereby the velocity of the striker vanishes (ỹ =. The goal is determining the dependence of the drag force acting at the striker as a function of the instantaneous velocity and the instantaneous DOP. In the present study we develop an approach for investigating inverse problems of this type, which was suggested by Ben-Dor et al. (29) for modeling penetration of strikers into various media. FORMULATION OF THE PROBLEM Consider normal penetration of a rigid projectile into a semi-infinite shield along the axis H, where the coordinate H determines the instantaneous location of the penetration and is defined as the distance between some reference point at the impactor and the front surface of the shield; the effects associated with the stage of the incomplete immersion of the nose of the impactor into the shield are neglected. Assuming that the total force acting at the impactor, R, depends on the velocity of the impactor, V, and the instantaneous coordinate of the impactor, H, Newton s second law describing the motion of the impactor with mass m can be written as follows: mv dv/dh = R H V (1) If V = V imp H is the solution of Eq. (1) with the initial condition V = V imp (V imp is the impact velocity) then the DOP, H max (a depth whereby the velocity of the penetrator vanishes), is determined by the following formula: V imp H max =. Assume that the empirical dependence, H max = F V imp, is known and F = F V imp > F V imp > V imp > (2) The problem is to determine a function R that implies this empirical dependence. Using the dimensionless variables Eq. (1) can be rewritten as follows: where h = H H v = V V w = v 2 = V 2 dw/dh = D h w (3) V 2 D h w = 2H R H mv 2 h V w (4) V is some reference velocity and H is a characteristic length. The empirical dependence between the squared dimensionless impact velocity, w imp, and the dimensionless DOP, h max, can be expressed in the following two forms: h max = F w w imp F w = or w imp = F h h max F h = (5)

SOME INVERSE PROBLEMS IN PENETRATION MECHANICS 47 where F h and F w are the known, mutually inverse functions, w imp = V imp /V 2 h max = H max /H (6) We consider expression for D h w that allows solving the equation of motion by quadratures. EQUATION WITH SEPARABLE VARIABLES General Solution Consider the case when the function D can be written as follows: D h w = D h h D w w (7) where D h and D w are some functions. Separating the variables in Eq. (3) and using initial conditions, w = w imp for h =, we obtain the equation of motion of the impactor. w w imp d w h D w w = D h h d h (8) Equation (8) can be written for the end point of the trajectory substituting w = and h = h max. wimp d w D w w = hmax D h h d h (9) Considering Eq. (9) as a definition of the functional dependence h max versus w imp and differentiating both sides of Eq. (8) with respect to w imp yields D w w imp D h h max dh max /dw imp = 1 (1) Substituting h max = F w w imp into Eq. (1) reduces Eq. (1) to the identity, D w w imp D h F w w imp F w w imp = 1, which must be satisfied for arbitrary w imp.itis convenient to rewrite this identity using the variable w instead of w imp. D w w D h F w w F w w = 1 (11) If function D h h in Eq. (7) is known then Eq. (11) implies the following expression for the unknown function, D w w : D w w = 1/ D h F w w F w w (12) Using the dependence between w imp and h max in the second form given by Eq.(5), Eq.(12) can be rewritten after the change of variables, F w x = F 1 h x, asd w F h h D h h F h h =. If D w w is known then D h h = F h h /D w F h h (13)

471 BEN-DOR ET AL. Equations (12) and (13) imply that the function D h w cannot be determined uniquely on the basis of the known correlation between the impact velocity and the DOP. Consequently, one can select either the function D w w or the function D h h. Substituting D w w from Eq. (12) or D h h from Eq. (13) into Eq. (7) we obtain two expressions for D. D h w = D h h D h F w w F w w D h w = D w w F h h (14) D w F h h In order to write the equation of motion of the impactor using only one unknown function, we substitute D w w from Eq. (12) into Eq. (8). wimp w h D h F w w F w w d w = D h h d h (15) After change of variable in the left-hand side, F w w = h, Eq. (15) can be rewritten as follows: Fw w imp F w w D h h d h= h D h h d h (16) Equation (16) implies that F w w imp F w w = h (17) where z = z D h h d h (18) Functions D h z and D w z can be replaced by the function z. Clearly (see Eq. (18)), and Eqs. (12) and (19) imply that D h z = z (19) D w z = 1/ F w z F w z (2) Therefore the instantaneous DOP as a function of the instantaneous velocity of the impactor as well as the instantaneous drag force can be represented using function in the following form: h w = 1( F w w imp F w w ) (21) h D h w = (22) F w w F w w where is an arbitrary function that satisfies the following conditions: = z > z > z > (23)

SOME INVERSE PROBLEMS IN PENETRATION MECHANICS 472 Using Eq. (21), D could be represented as a function of single argument, h or w, but corresponding expressions are cumbersome and are not presented here. Some Classes of Models Model z = z p If z = z p, where p>1 is a constant, then 1 z = z 1/p and Eqs. (21) and (22) can be written as follows: h w = F w w imp p F w w p 1/p (24) D h w = (25) F w w p 1 F w w Substituting h from Eq. (24) into Eq. (25) we obtain expression for D as a function of w. h p 1 D w = F w w imp p F w w p p 1 /p F w w p 1 F w w (26) Equations (24) and (25) remain valid for p = 1 after setting p = 1, h p 1 = F w h p 1 = 1. Formula (19) implies that q = 1 corresponds to the case D h w = 1 in Eq. (7). then Model z = F h z q F w w F w w z =F 1 w If z = F h z q = Fw 1 z q, where q>1is a constant, z = d F w w dw z =F 1 w z (27) = dwq dw = qwq 1 1 z = F w z 1/q and Eqs. (21) and (22) can be written as follows: Equation (27) implies that and the expression for D as a function of h reads ( h w = F w w q imp wq 1/q) (28) D h w = F h h q 1 F h h w1 q (29) w h = w q imp F h h q 1/q (3) D h = dw/dh = w q imp F h h q 1 q /q F h h q 1 F h h w1 q (31) Equations (28) and (31) remain valid for q = 1 after putting q = 1 and w 1 q = F h h q 1 = 1. Equations (14) and (27) imply that q = 1 corresponds to the case D w w = 1 in Eq. (7).

473 BEN-DOR ET AL. Particular Case Assume that the dependence between w imp and h max is given in an implicit form: w imp = G h max (32) where and G are positive, increasing functions for positive arguments and = G = (33) As previously we consider Eq. (9) as a definition of the functional dependence h max versus w imp. Equations (9) and (32) imply that wimp d w D w w = w imp hmax D h h d h = G h max (34) and, consequently, Using Eqs. (18), (32), and (35) we find that D w w = 1/ w D h h = G h (35) D h w = G h / w (36) z = G z F w w = G 1( w ) (37) Combining Eqs. (21) and (37) we arrive at the following relation: h w = G 1( w imp w ) (38) Note that Eq. (34) yields only one of the all possible models of penetration. The complete set of models can be obtained using the general method outlined in Section 3.1 and applying it to the dependence, F w w = G 1 w. High Speed Penetration into Semi-Infinite Concrete Shield NDRC Formula Consider widely known modified National Defence Research Committee (NDRC) formula (Kennedy, 1976), associated with penetration of a rigid projectile into a semi-infinite concrete shield which can be represented in the form given by Eq. (32) with w imp = wimp n n = 9 (39) { h G h max = max /2 2 if h max 2 (4) h max 1 if h max > 2 where the characteristic length H = d, d is the diameter of a projectile, the reference velocity, V, depends on d, the shape of the impactor s nose and mechanical properties of the shield.

SOME INVERSE PROBLEMS IN PENETRATION MECHANICS 474 Equation (32) allows us to write the following expression for the functions F w and F h : F w z = G 1 z = { 2z n/2 if z 1 z n + 1 if z>1 (41) { F h z = Fw 1 z z/2 2/n if z 2 z 1 1/n if z>2 (42) Model z = z p In the case of the model z = z p Eqs. (24) and (22) after substituting F w from Eq. (41) yield 2 w pn/2 imp wpn/2 1/p if w w imp for w imp < 1 {[ h w = w n imp + 1 p w n + 1 p]1/p if 1 w w imp (43) [ for w imp 1 w n imp + 1 p 2w n/2 p]1/p if w 1 h/2 p 1 w 2 pn /2/n if w 1 D h w = h p 1 w 1 n (44) if w>1 n w n + 1 p 1 where 1 p<2/n. Model z = F h z q If z = F h z q Eqs.(24) and (25) after substituting F w from Eq. (41) yield the following relations: 2 w { q imp wq n/ 2q if w w imp for w imp < 1 h w = 2 w q imp wq n/ 2q if w q imp w w 1 1/q imp (45) w q imp wq n/q + 1 if w w q imp for w imp 1 1 1/q { h/2 2q n /n w 1 q /n if h 2 D h w = (46) h 1 q n /n w 1 q /n if h>2 where n/2 <q 1. In the particular case q = n the model recovers the model suggested by Kennedy (1976) who derived it using the approach, which is considered in the Section 3.2.3. Results of Calculations Figures 1(a) and (b) and 2(a) and (b) illustrate dependences w h and D h for models described in Eqs.(43) and(44) (solidcurves) and Eqs.(45) and(46) (dashcurves);correlationsd h are obtained by excluding w from Eqs. (43) (44) and (45) (46). Inspection of these figures shows that a large variety of different dependencies can be obtained by selecting the function.

475 BEN-DOR ET AL. Figure 1 (a) and (b). Instantaneous squared velocity of impactor versus instantaneous depth of penetration(dimensionless variables, penetration into concrete) for models described by Eqs.(43) - (46) LINEAR EQUATION General Solution Assume that the expression for the resistance force D can be written as follows: D h w = D 1 h w + D h (47) where D h and D 1 h are some functions. The solution of Eq.(3) with the initial condition w = w imp reads w h = ( w imp h [ D h / h ] ) d h h ( h = exp h ) D 1 h d h (48)

SOME INVERSE PROBLEMS IN PENETRATION MECHANICS 476 Figure 2 (a) and (b). Instantaneous resistance force versus instantaneous depth of penetration of impactor (dimensionless variables, penetration into concrete) for models described by Eqs.(43)-(46) Setting w = and h = h max in Eq. (48) we conclude that this model implies the following relationship between the DOP, h max, and the impact velocity: w imp = hmax D h / h d h (49) Considering w imp as a known function of h max, w imp = F h h max, differentiating Eq. (49) with respect to h max and replacing h max by h yields Equation (48) implies that D h = h F h h (5) D 1 h = h / h (51)

477 BEN-DOR ET AL. Therefore functions D h and D 1 h can be determined if one selects the function h, where h > = 1 (52) Substituting D h and D 1 h from Eqs. (5) and (51) into Eqs. (47) and (48) we obtain D h w = h / h w + h F h h (53) w h = w imp F h h h (54) Substituting w from Eq. (54) into Eq. (53) we obtain D h = h w imp F h h + h F h h = dw/dh (55) Low-Speed Penetration into Granular Media Consider a low-speed normal penetration of a rigid ball into a granular medium, and assume that the empirical dependence between the impact velocity and the DOP reads H max d b = k ( ) V 2 imp + H max (56) 2gd b d b where d b is the diameter of the ball and g is the acceleration of gravity. Parameters k and are determined by the choice of a particular model. In the model suggested by Uehara et al. (23), k = 14/ b / g, = 1/3, where g and b are densities of the grains and of the impacting ball, respectively, = tan is the grain grain friction coefficient, is the repose angle of the grains. In the model considered by Seguin et al. (28) which takes into account the influence of the confinement on penetration, k = A A b / g, =, where parameters A > and < < 1 correspond to the unbounded case, <A < 1 and < < 1 are the correction for the confinement and >1 is a parameter. We consider the case when the expression for the total resistance force, R, reads R H V = R 1 H V 2 + R H (57) where R and R 1 are some functions. Semi-empirical models of penetration into granular media are usually selected among the models of this type. Note that similar models are also widely used in high-speed penetration dynamics (Ben-Dor et al., 26). Then Newton s second law, Eq. (1), can be written in the form given by Eqs. (3) and (47) where D h = R H h D mg 1 h = 2H m R 1 H h (58)

SOME INVERSE PROBLEMS IN PENETRATION MECHANICS 478 Table 1 Three models that imply Eq. (59) for = 3. Model h Value of the parameter 1 exp c 1 h c 1 = 86 2 1/ 1 + c 2 h 3/2 c 2 = 1 38 3 1 c 3 h 2 c 3 = 3 H = d b k 1/ 1 is the initial collapse depth of the ball prior to the beginning of its motion from the state of rest (V imp =, variables h, v, w, w imp, h max are determined by Eqs. (4) and (6) and V = 2gH. Using the dimensionless variables Eq. (56) can be rewritten as follows: w imp = F h h max F h z = z z = 1/ (59) and Eqs. (53) and (55) imply the following dependencies for the drag force and squared instantaneous velocity versus the instantaneous DOP: D h = w imp h + h h + h h 1 1 (6) w h = ( w imp h + h ) h (61) Since the considered problem has a non-unique solution, it is feasible, as a rule, to generate different models having the same accuracy by selecting different h. To illustrate this, several versions of h are selected (see Table 1, columns 1 and 2) including Model 1 by Ambroso et al. (25) that can be readily obtained in the framework of the suggested approach. In comparison with the experiment, we use the approach suggested in Ambroso et al. (25) that is based on the following relationship that can be Figure 3 Comparison of models with experimental data for penetration into granular media.

479 BEN-DOR ET AL. obtained from Eqs. (59) and (61): ( ) V 2 ( = V imp 1 H 1 H 1 1 ) H/ (62) where H = h/h max = H/H max 1, = 1/h max = H /H max 1. Following Ambroso et al. (25), we determine parameters of the Models 2 and 3 using the relationship (62) on the basis of the same experimental data (495 points) for the same value = 3 and average value = 36. The best one-parameter fits to Eq. (62) for the Models 2 and 3 are shown in Fig. 3, as well as, for comparison, the curve corresponding to the Model 1 suggested by Ambroso et al. (25); the values of the parameters are presented in Table 1 (column 3). Inspection of this figure demonstrates that the experimental data are faithfully approximated by each of the models and the models have the same accuracy. CONCLUDING REMARKS We suggested an approach for solving some problems of penetration dynamics but the scope of application of the proposed method is considerably wider: this method can be used in modeling different phenomena, which are described by ODEs. The developed approach allows extension to other types of ODEs and other kinds of relationships that are assumed to be known. Non-uniqueness in the choice of the model, which implies the known empirical relationship, affords a wide range of possibilities for constructing the appropriate model of the considered phenomenon. We developed a general procedure for constructing a variety of models for selecting a particular model. The choice of the particular model is another problem, which is outside the scope of this study. It must be emphasized that the suggested method on its own does not guarantee the adequacy of the obtained equations. REFERENCES Ambroso, M. A., Kamien, R. D., Durian, D. J. (25). Dynamics of shallow impact cratering. Physical Review E 72:4135. Ben-Dor, G., Dubinsky, A., Elperin, T. (26). Applied High-Speed Plate Penetration Dynamics. Dordrecht: Springer. Ben-Dor, G., Dubinsky, A., Elperin, T. (29). Modeling of penetration by rigid impactors. Mechanics Research Communications 36(5):625 629. Denisov, A. M. (1994). Introduction to Theory of Inverse Problems [Vvedenije V Teoriju Obratnykh Zadach]. Moscow: Moscow State University. Eisler, R. D., Chatterjee, A. K., Burghart, G. H., Loan, P. (1998). Simulates the Tissue Damage from Small Arms Projectiles and Fragments Penetrating the Musculoskeletal System. Fountain Valley CA: Mission Res. Corp., Final Report. Katsuragi, H., Durian, D. J. (27). Unified force law for granular impact cratering. Nature Physics 3(6):42 423. Kennedy, R. P. (1976). A review of procedures for the analysis and design of concrete structures to resist missile impact effects. Nuclear Engineering and Design 37(2):183 23.

SOME INVERSE PROBLEMS IN PENETRATION MECHANICS 48 Pilyugin, N. N. (24). A simulation of the shape of a crater in an organic-glass target under high-velocity impact. High Temperature 42(3):481 488. Ramm, A. G. (25). Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering. New York: Springer. Seguin, A., Bertho, Y., Gondret, P. (28). Influence of confinement on granular penetration by impact. Physical Review E 78:131. Uehara, J., Ambroso, M., Ojha, R., Durian, D. (23). Low-speed impact craters in loose granular media. Physical Review Letters 9:19431.