On the formulation of contact problems of shells and plates

The equations of motion in terms of resultants and the constitutive equations in the theory of shells and plates are ordinarily derived from the three-dimensional equations relative to an interior surface, often taken to be the middle surface in the reference configuration of the shell-like body. This usual formulation is not, in general, applicable to contact problems of shells for which one of the major surfaces, e.g., the upper surface, is specified as the reference surface. The present paper is concerned with the transformation relations between the relevant variables (and hence also the response functions) of the traditional formulation and one which can be obtained relative to one of the major surfaces. Our derivations, carried out both from the three-dimensional equations and by a direct approach, utilize only the conservation laws and the field equations without recourse to constitutive equations. Hence, the results are applicable to any shell-like medium and their validity is not limited to elastic shells alone.