Large deformation, deep penetration theory for a compressible strain-hardening target material

Abstract

Hypervelocity impact of a rigid spherical projectile with a compressible target material is investigated. Analytic solutions obtained for this problem are based on dynamic cavity expansion and Goodier deep penetration theories and are not restricted, as in previous work, to an incompressible medium. Target compressibility is introduced through a locking approximation for real material behavior under hydrostatic stress. In particular, values of locking density are obtained from Hugoniot curves characterizing homogeneous, isotropic material behavior at different impact velocities. Results of the theory indicate that terminal penetration depth is a function of projectile diameter and mass, and initial density, locking density, yield stress, and strain-hardening modulus of the target material. Using results of the theory, curves are plotted to show the comparison between compressible and incompressible penetration theory for the hypervelocity impact of steel projectiles on an aluminum target. NomenclatureA,D,M = cross section area, diameter, and mass of projectile, respectively a = radius of spherical cavity Bi,Bz = constants related to dynamic pressure, see Eq, (71) b = radius of locked-elastic, locked-plastic spherical shock front Ci = constant of integration E,G = modulus of elasticity and shear modulus, respectively Et = tangent modulus for linear strain-hardening e -bulk strain / = limit of integration f,g = functions of integration k = radius of stress-free, locked-elastic spherical shock front = static pressure term given by Eq. (71) p(t) = dynamic pressure applied to spherical cavity surface as a function of time <?><?># = depth, velocity, and acceleration, respectively, of projectile in target material q Q) v Q = penetration depth and velocity of projectile after completion of shallow penetration phase q t = final penetration depth of projectile r = Eulerian radial coordinate t = time v = outward particle velocity in radial direction Y = yield stress a = material constant given by Eqs. (16), (24) /3 = material constant given by Eq. (40) 8 = material constant given by Eq. (66) ei,e2 = small quantities in the mathematical sense ei = locking strain, ei = -ei = normal strains in radial and circumferential directions, respectively = series expression given by Eq. (54) 0 }( p = equatorial and meridional spherical coordinates, respectively P)PI = density and locking density of target material, respectively o>,<70 = normal stresses inr adial and circumferential directions, respectively Downloaded by UNIVERSITY OF MICHIGAN on February 20, 2015 | http://arc.aiaa.org |

The erosion transition of tungsten-alloy long rods into aluminum targets

“…While one might expect the resistance to radial crater expansion to be of a similar magnitude to the resistance to axial penetration, there is reason to suspect that the two values are not identical. For example, the classical indentation literature (Tabor, 1951) indicates that both steel and Al's resistance to deep indentation (penetration) is from 2.8 to 3· the yield strength, while the derived expressions presented by others (Hopkins, 1960;Goodier, 1965;Hanagud and Ross, 1971) for spherical cavity expansion can range from 3.5 to 4.1· the yield strength, depending on material stiffness and strength, for the range of steels and alloys of Al in the current study. On this basis, one might expect the ratio H LAT /H to lie in the range 1.17-1.46.…”

Computational Aspects of Penetration Mechanics

“…Computer programs for the analysis of impact in solids have been in existence since 1958, when Bjork [28] developed a hydrodynamic computer program for hypervelocity impact. Until recently, all of the computer programs available for impact calculations were based on two-dimensional description of the phenomenon.…”

“…Neither the eroding rod model nor any of the experiments described above gives any consideration to the dynamics of hole growth, although some attention has been devoted to this problem in the literature, e.g., Hanagud and Ross [28] or Ravid and Bodner [14]. As the penetrator moves through the target, target material must flow to the sides around the penetrator.…”

“…Lamb [28] has presented a corresponding states plot of the real and imaginary parts of the shear impedance as functions of frequency for a variety of large molecules. The frequency-dependent viscosity derived from these curves deviates substantially from the Maxwellian model, but resembles much more closely [4] corresponding calculations carried out by Evans [29] for a simple central-force model at much higher frequencies.…”

A simplified analytical method for soil penetration analysis

“…Substituting the shock front co-ordinate r' = h and equation (7) into equation ( 5 ) yields Thus, the interaction pressure on the internal boundary is expressed as a function of the radial values of displacement, velocity and acceleration at the internal boundary, and of the material properties. The expression of the interactive pressure is a function of the motion of a system having a single degree of freedom, where the independent co-ordinate is R(t).…”