Dynamic Transverse Deflection of a Free Mild-Steel Plate

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.