Dynamic Transverse Deflection of a Free Mild-Steel Plate

Abstract

The problem analytically investigated is that a thin free plate of mild-steel struck at normal incidence by a flat ended rigid rod moving at high velocity. As in quasi-static deformation by extended slip, the strain-rate tensor is solenoidal and under dynamic loading conditions the Tresca yield criterion is modified so that the solenoidal property replaces the hypothesis of a viscoplastic overstress. Overstress then arises from inertial body forces and the high magnitudes found, in the following, for these forces are due to the influence of the propagating boundary. Two new theorems are proven. These theorems show that the deflection in the plate is entirely transverse, even in the case of indefinitely large punch deflections, and that the lines of equal transverse deflection in the plate are also principal lines of stress and strain-rate, as are the lines of steepest descent. A formula is obtained giving the inertial force opposing the punch as a function of the time and the theoretical deflection profile on a plate deformed by a flat-ended punch of circular section is presented. The stresses in the plate are then analyzed and it is shown that the stress inside the boundary in the direction of propagation, equals ρc2where ρ is the mass density of the plate material and the boundary wave propagates at speed c which, it is shown, is equal to one-half of the velocity of elastic waves of rotation in the solid concerned.

Share and Cite:

R. Bish, "Dynamic Transverse Deflection of a Free Mild-Steel Plate," World Journal of Mechanics, Vol. 3 No. 9, 2013, pp. 339-349. doi: 10.4236/wjm.2013.39037.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] T. Karman and P. Duwez, “The Propagation of Plastic Deformation in Solids,” Journal of Applied Physics, Vol. 21, No. 10, 1950, pp. 987-994. http://dx.doi.org/10.1063/1.1699544 [2] K. A. Rakhmatulin, “Propagation of a Wave of Unloading,” Prikl. Mat. Melch., Vol. 9, 1945, pp. 449-462. [3] L. Efron and L. E. Malvern, “Electromagnetic Velocity-Transducer Studies of Plastic Waves in Aluminium Bars,” Experimental Mechanics, Vol. 9, No. 3, 1969, pp. 255-262. http://dx.doi.org/10.1007/BF02325157 [4] G. I. Taylor, “The Plastic Wave in a Wire Extended by an Impact Load,” (The Scientific Papers of G. I. Taylor, Vol. I, Mechanics of Solids, Edited by G. K. Batchelor), Cambridge University Press, Cambridge, 1958. [5] V. V. Sokolovsky, “Propagation of Elastic-Viscoplastic Waves in Bars,” Prikl. Mat. Mekh., Vol. 12, 1948, pp. 261-280. [6] L. E. Malvern, “The Propagation of Longitudinal Waves of Plastic Deformation in a Bar of Material Exhibiting a Strain Rate Effect,” Journal of Applied Mechanics, Vol. 18, No. 2, 1951, pp. 203-208. [7] P. Perzyna, “Fundamental Problems in Viscoplasticity,” Advances in Applied Mechanics, Vol. 9, Academic Press, New York, 1966. [8] R. L. Bish, “Tri-Axial Deformation of a Plastic-Rigid Solid,” Acta Mechanica, Vol. 223, No. 3, 2012, pp. 655-668. http://dx.doi.org/10.1007/s00707-011-0580-1 [9] R. L. Bish, “Transverse Deflection of a Clamped Mild-Steel Plate,” Acta Mechanica, Vol. 223, No. 11, 2012, pp. 2411-2423. http://dx.doi.org/10.1007/s00707-012-0712-2 [10] R. L. Bish, “Plastic Shear Deformation of a Thin Strain-Hardening Disc: Variational Principles,” Philosophical Magazine, Vol. 91, No. 25-27, 2011, pp. 3343-3357.http://dx.doi.org/10.1080/14786435.2011.580285 [11] R. L. Bish, “Rotation-Rate Continuity in Bi-Axial Deformation,” Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 84, No. 4, 2004, pp. 266-279. http://dx.doi.org/10.1002/zamm.200310098 [12] R. L. Bish, “Rotational Continuity in Thin Transversely Deflected Iron Plates,” International Journal of Mechanical Sciences, Vol. 43, No. 3, 2001, pp. 817-829. http://dx.doi.org/10.1016/S0020-7403(00)00028-X [13] T. Børvik, O. S. Hopperstad, M. Langseth and K. A. Malo, “Effect of Target Thickness in Blunt Projectile Penetration of Weldox 460 E Plates,” International Journal of Impact Engineering, Vol. 28, No. 4, 2003, pp. 413-464. http://dx.doi.org/10.1016/S0734-743X(02)00072-6 [14] G. I. Taylor and H. Quinney, “The Plastic Distortion of Metals,” Proceedings of the Royal Society: London A, Vol. 230, 1931, pp. 323-362. [15] A. H. Cottrell, “Theory of Dislocations,” Progress in Metal Physics, Vol. 4, Pergamon Press, London, 1953. [16] J. J. Gilman, “The Plastic Resistance of Crystals,” Australian Journal of Physics, Vol. 13, No. 2, 1959, pp. 327-346. http://dx.doi.org/10.1071/PH600327a [17] D. Hull, “Introduction to Dislocations,” 2nd Edition, Pergamon Press, Oxford, 1975. [18] J. D. Campbell and W. G Ferguson, “The Temperature and Strain Rate Dependence of the Shear Strength of Mild-Steel,” Philosophical Magazine, Vol. 21, No. 169, 1970, pp. 63-82. http://dx.doi.org/10.1080/14786437008238397 [19] J. D. Campbell, “Dynamic Plasticity of Metals,” Springer Verlag, Wien, New York, 1972. [20] R. L. Bish, “A Method of Revealing Deformation in Mild Steel,” Metallography, Vol. 11, No. 2, 1978, pp. 215-218. http://dx.doi.org/10.1016/0026-0800(78)90039-3 [21] G. W. C. Kaye and T. H. Laby, “Tables of Physical and Chemical Constants and Some Mathematical Functions,” 8th Edition, Longmans Green and Co., London, 1936. [22] A. N. Lowan, “Tables of Sine, Cosine and Exponential Integrals,” Vol. 1, National Bureau of Standards (Edited by L. J. Briggs), 1940. [23] H-M. Wen and N. Jones, “Low Velocity Perforation of Punch-Impact-Loaded Metal Plates,” Pressure Vessel Technology, Vol. 118, No. 2, 1996, pp. 181-187. http://dx.doi.org/10.1115/1.2842178