Scientia Sinica

ON THE EQUILIBRIUM AND VIBRATION OF A TRANSVERSELY ISOTROPIC ELASTIC BODY*

Hu HarCuane (#8##5)

(Institute of Mathematics, Academia Sinica)

I. INtRopuUcTION

In the year 1940, S. G. Lehnitzky™ obtained a stress function for the axisymmetric deformation of a transversely isotropic elastic body. His result has been generalized by A. Moisil®*!, W. Nowacki®! and the author™® 7! independently. In papers [6] and [7], the general solutions of the equations of equilibrium without body forces are expressed in terms of two stress functions. This method is applied later to the equilibrium of a spherically isotropic body’ *!. In this paper, we shall apply the same method to the investigations of the equilibrium under body forces, the thermal stresses, and the rimecen of a transversely isotropic elastic body.

II. Generat Equations

Consider a transversely isotropic body. Take rectangular coordinate axes x, y, and z, such that the xy-plane is parallel to planes of isotropy of the body. Let w, v, and w be the components of displacement, ¢,, %,,-::, Ty be the components of stress and T be the temperature. In this system of coordinates, the generalized Hooke’s law including thermal expansion may be written in the following form:

0 Ov Ow o 0 oz = Ay ae ea : ae eee ae ct ds, Tyz = Ags Syl oe), Ou Ow Oo Ou oy = Ano + dn 1 Ais ae et ray = An (SH + | oe, (1)

0 Ow a2 = Ars + 435 Asse Ota a Try = Age ane 5

*First published in Chinese in Acta Physica Sinica, Vol. XI, No. 3, pp. 219—238, 1955.