The theory presented in this article was developed in an attempt to describe the observed motion and plastic deformation of clamped metal diaphragms used in certain underwater explosion experiments and in certain mechanical gauges. The theoretical attack on this problem enables one to set up certain equations of motion, which may be solved in finite form under certain conditions. The solutions enable one to specify, for instance, the final deformed diaphragm profile, the distribution of thickness after deformation, the swing-time, which is the total time for deformation to take place, and many other quantities. The simplest case, termed the ``elementary approximation,'' turns out, except for relatively minor details, to describe adequately for many purposes the motion and final shape of the diaphragm; it is found that the deformed diaphragm shape is conical, the thickness distribution shows a marked dimpling at the center of the diaphragm, and the swing-time ts is, to this order of approximation tS=a/c, where a is the radius of the diaphragm, and c is the square root of the ratio of the ``yield stress'' to the density. These results are all in good agreement with experimental facts.