Thermal Stresses in Elastic Plates

SOKOLNIKOFF 1. Introduction.Thermal changes in an elastic body are accompanied by shifts in the relative positions of the particles composing the body.Such shifts, in general, cannot proceed freely, and thermal stresses are set up in the body.The analytical basis for the determination of such stresses was provided by Duhamelf and NeumannJ who, starting from certain assumptions, have modified the stress-strain relations of Hooke.The theory based on the law formulated by Duhamel has not been very much developed!because of the complicated character of the partial differential equations satisfied by the stress components.In particular, there is a notable lack of a careful formulation of the differential equations governing the deflection of elastic plates subjected to nonuniform distribution of temperatures.Starting with the usual assumptions of the thin plate theory, Nádai|| has developed the differential equation for deflection of a thin elastic plate subjected to a linear distribution of temperature in the direction of the thickness of the plate.In two recent papers Marguerre^ has considered some related problems, and, on the basis of reasoning essentially similar to that of Nádai, was led to a somewhat more general equation.In the present paper the differential equation governing the deflection of an elastic plate is derived without using the objectionable assumption of the thin plate theory.It will be seen that the differential equations obtained by Nádai and Marguerre are special cases of the more general equation here given.Furthermore, the hypotheses of Nádai and Marguerre that a heated plate will be in a state of plane or generalized plane stress (which form a focal point in their discussion) are not used in this paper.The abrogation of these hypotheses enormously complicates the analysis, but inasmuch as there has been some considerable doubt regarding the validity of the generalized stress assumption, it appears necessary to sacrifice simplicity.The complexity of the situation lies in the nature of the problem itself.