SHAPE OPTIMIZATION OF HIGH VELOCITY IMPACTORS USING ANALYTICAL MODELS
G. Ben-Dor, A. Dubinsky and T. Elperin
The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O.Box 653, Beer-Sheva, 84105, Israel
Tel: 972-7-6472104; Fax: 972-7-6472990
The high velocity penetration of a 3D rigid body with a length L into a ductile target with a finite thickness b is considered. Assume that the shape and the initial velocity of the body are such that they result is a normal penetration. The relevant notations are presented in Fig. 1.
Figure 1. Coordinates and notations.
The coordinate , the depth of the penetration, is defined by the position of the nose of the impactor. The cylindrical coordinates and are associated with the impactor. Its shape is determined by the function . The equation of motion of the impactor can be written in the form
(1)
where is a mass of the impactor, is the drag force and the current velocity of the impactor v is considered as a function of . The ballistic limit velocity is defined as the initial velocity of the impactor which results in perforation with zero final velocity, i.e., where is a solution of the Eq. (1) with the initial condition
(2)
Although some models, describing the impactor-target interactions, enable one to obtain analytical solutions of Eqs. (1) and (2), the resulting formulas for the ballistic limit as the function of the impactor’s shape are quite complex [1]. These formulas cannot be used for analytically optimizing the shape of an impactor. In this note we outline an approach which enables to reduce the problem of the minimization of the ballistic limit velocity to solving relatively simple variational problem.
The following relation between the drag force acting on the penetrating impactor and impactorױs velocity was proposed and validated in [2]:
. (3)
The parameter depends, primarily, on the properties of the target material, , the parameter depends also on the shape of the impactor. The impact, residual and ballistic limit velocities calculated with Eq. (3) are in good agreement with the experimental results obtained for spherical impactors [2-4]. However the authors themselves pointed out the shortcomings of their model. In particular, it generally yields wrong non-zero values of the drag at the beginning of the penetration and at the perforation moment because the influence of the extent of the penetration of the impactor into the target on the drag is not taken into account. These shortcomings can be eliminated by improving this model using the localized interaction approach [1, 5] whereby the overall effect of the target on the impactor can be represented as a superposition of the independent local effects on the impactor’s surface elements. Every local interaction between the impactor and the target is determined by the local impactor’s velocity, the angle between the velocity vector and the normal vector to the impactor's surface at this point, and by some global parameters which account for the properties of the target.
Thus, the impactor-target interaction at a given location at the impactor’s surface which is in contact with the target can be represented by the following equation
(4)
where is the force acting on the surface element of the impactor; unit vectors , , and are directed along the inner normal, the tangent and the velocity at a given location at the impactor’s surface, respectively; andare functions which determine the impactor-target interaction model and depend on the impactor’s velocity , the parameter is the cosine of the angle between and at a given location at the surface and the global parameters determining the properties of the target are not listed as the arguments. The total force is determined by integrating the local force over the impactor-target contact surface that lies between the cross-sections and where (see Fig. 2)
, (5)
Using formulae of differential geometry
(6)
the expression for the drag force can be written as
(7)
Different phenomenological localized interaction models with different functions andare widely used in modeling the penetration into ductile media [6,7]. The proposed generalized model employs a relation similar to Eq. (3) to describe the local impactor-target interaction, i.e.,
(8)
where and are functions which can be determined from experimental data. Using Eqs. (2), (7) and (8) the solution of Eq. (1) can be written as
(9)
Figure 2. The area of integration with respect to x and h.
After changing the order of integration over x and over h (the integration domain is shown in Fig. 2) the solution of Eq. (9) can be rewritten as follows:
(10)
where the mass of the impactor can be expressed in terms of its volume and density
(11)
Remarkably, the shape of a minimum ballistic limit impactor is independent of the thickness of the target and it coincides with the shape of a minimum drag projectile moving in a target with a constant velocity.
The shape of the impactor given by Eq. (10) yield a power-law dependence of the ballistic limit upon the parameter b/m. Notably, similar dependence appears in the experimental curves presented in [6] (p. 472) and [7] (p.772).
Eqs. (10) and (11) allow to pose and analyze various optimization problems for 3D impactors and impactors having the form of the body of revolution taking into account various geometrical restrictions. The minimum of the ballistic limit or the minimum of the thickness of the protective barrier can be considered as objective functions. Construction of specific models in the form given by Eq. (8) and investigation of these variational problems are the subject of our ongoing investigation.
REFERENCES
[1] G. Ben-Dor, A. Dubinsky and T. Elperin,Theoretical and Applied Fracture Mechanics (1997), in press.
[2] S. T. Mileiko and O. A. Sarkisyan, J. Appl. Mech. and Tech. Phys. 5 (1980) 711-713
[3] S. T. Mileiko, O. A. Sarkisyan and S. F. Kondakov, Theoretical and Applied Fracture Mechanics 21 (1994) 9-16.
[4] V. Muzuchenko and V. Postnov, J. Appl. Mech. and Tech. Phys. 5 (1984) 774-776
[5] A. I. Bunimovich and A. V. Dubinsky, Mathematical Models and Methods of Localized Interaction Theory, World Sci. Publ. (1995)
[6] J. A. Zukas (Ed.), High Velocity Impact Dynamics, Wiley Interscience Publ. (1990)
[7] I. F. Langrov and O. A. Sarkisyan, J. Appl. Mech. and Tech. Phys., 5 (1983) 771-773
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