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Edge impact of an elastic-plastic semi-infinite plate

A near-field solution of the two-dimensional equations is compared with the one-dimensional description and experimental observation

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Abstract

The edge impact of an elastic-plastic semiinfinite plate subject to conditions of plane strain is investigated analytically and experimentally. The theoretical analysis is based on the strain-rate-independent theory of plastic-wave propagation. The plate is initially unstressed; the boundary condition for the edge of the plate corresponds to constant-velocity longitudinal impact except that the step in velocity has a finite rise time. Calculations are carried out according to both the approximate one-dimensional theory and the two-dimensional theory for an elastic-plastic isotropic work-hardening material. The rise time for both solutions is chosen so as to optimize the agreement between theoretical and experimental strain-time profiles. A numerical solution of the two-dimensional equations is obtained by using a difference method developed by Clifton;1 the onedimensional approximation is solved by the well-known method of characteristics.

The problem was approximated experimentally by the axial collision of 4-in.-diam annealed aluminum thick-walled cylinders with a diameter-to-wall-thickness ratio of ten. For impact velocities of 90, 130 and 160 ips (corresponding to maximum strains of 0.12, 0.22 and 0.36 percent, respectively), dispersive characteristics and maximum strain amplitudes of the strain wave are found to be in good agreement with the theoretical predictions of both solutions. However, the two-dimensional solution indicates that the stresses, strains and velocities in regions of high strain-rate are highly nonuniform across the plate thickness.

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Abbreviations

c :

plastic-wave speed

c 0 :

elastic-wave speed

d :

half-plate thickness

f :

yield condition

g(t) :

applied-velocity-boundary condition

r :

radius of wire

t :

time variable

u :

particle velocity inx-direction

x,y :

coordinate axes (Fig. 1)

E :

modulus of elasticity

T :

slope of stress-strain curve (dσ/dε)

ε x :

strain inx-direction

\(\dot \in _{ij} ^p \) :

plastic-strain-rate tensor (i,j=x,y,z)

Λ:

wavelength

ν:

Poisson's ratio

ρ:

density

σ x :

stress inx-direction

σ ij :

stress tensor (i,j=x,y,z)

σ1, σ2, σ3 :

principal stresses

References

  1. Clifton, R. J., “A Difference Method for the Dynamic Elastic-Plastic Equations under Conditions of Plane Strain,” Proc. Fifth U. S. Natl. Congr. Appl. Mech., Minneapolis, Minn., 1966, 546; See also, Clifton, R. J., “Analysis of Dynamic Deformation of Elastic/Plastic Solids under Conditions of Plane Strain,” Ph.D. Dissertation, Carnegie Institute of Technology (1965).

  2. von Karman, Th., “On the Propagation of Plastic Deformation in Solids,” NDRC Report No. A-29, OSRD No. 365 (1942).

  3. Taylor, G. I., “The Plastic Wave in a Wire Extended by an Impact Load,” “The Scientific Papers of G. I. Taylor,”Vol. 1, Mechanics of Solids, edited by G. K. Batchelor, University Press, Cambridge, England, 467–479 (1958).

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  4. Kolsky, H., “Stress Waves in Solids,”Dover Publication, New York, 2nd edition, 54–65 (1963).

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  5. Simmons, J. A., Hauser, F., andDorn, J. E., “Mathematical Theories of Plastic Deformation under Impulsive Loading,”University of California Press, Berkeley, Calif.,5,7 (1962).

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  6. Koiter, W. T., “Stress-Strain Relations, Uniqueness and Variational Theorems for Elastic-Plastic Materials with a Singular Yield Surface,”Quart. Appl. Math.,11,350–354 (1953).

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  7. Bell, J. F., “An Experimental Study of the Unloading Phenomenon in Constant Velocity Impact,”Jnl. Mech. Phys. Solids, London,9,5 (1961).

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Author information

Authors and Affiliations

  1. National Bureau of Standards, Washington, D. C.

    L. R. Hettche (recipient of a NAS-NRC Postdoctoral Research Associateship)

  2. Carnegie Institute of Technology, Pittsburgh, Pa

    T. Au (Professor of Civil Engineering)

Authors
  1. L. R. Hettche
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  2. T. Au
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Additional information

The research reported in this paper was conducted in the Civil Engineering Department, Carnegie Institute of Technology, and supported by a grant from the National Science Foundation.

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Hettche, L.R., Au, T. Edge impact of an elastic-plastic semi-infinite plate. Experimental Mechanics 7, 302–308 (1967). https://doi.org/10.1007/BF02327136

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