On the plastic theory of plates

This paper presents a theory of the small deformations of a thin uniform plate under transverse load. The plate is made of non-hardening rigid-plastic material obeying the Tresca yield condition and associated flow rule. The basic assumptions are similar to those made in the conventional engineering theory of thin elastic plates, and the effects of transverse shear strain and rotatory inertia are neglected. Hitherto, the theory has been developed only under conditions of circular symmetry, and the object of the present paper is to remove this restriction. Attention is confined here to the derivation and classification of the field equations. The field equations involve the stress moments and the middle-surface curvature rates as the associated generalized stresses and strain rates. These equations are first referred to Cartesian co-ordinates. The condition of isotropy requires the coincidence of the directions of principal stress moment and curvature rate. One of these two families of directions is characteristic for the equations appropriate to certain plastic régimes. The field equations are therefore referred to curvilinear co-ordinates taken along these directions. A detailed study is made of discontinuities in the field quantities. The field equations are either parabolic or elliptic for the principal plastic régimes.