Scientia Sinica

No. 1 HU: EQUILIBRIUM & VIBRATION OF TRANSVERSELY ISOTROPIC ELASTIC BODY 3

III. Egurtnmrrum Unpver Bopy Forces PERPENDICULAR To PLANES OF IsoTROPY Let us consider first the case of equilibrium under body forces per-

pendicular to planes of isotropy. In this case, system (4) is simplified to the following:

Ou O2u Oru 62u 62w

Bizet B66 Oy? I Bas Be 1 TE ero 1 Bi; dxdz Ms Ou Ov Ou O*y Ow _

By Bx Oy + Bée ox + By Oy? + Bas az? hs OyOz 0; ™ O7u 67 Oew Ow Ow

Ubleres, Soyon |" eqe| (eye 20

The solution of this system consists of two parts, namely an arbitrary special solution corresponding to Z, and a general solution of the homogeneous system. The latter may be expressed in terms of two stress functions as shown in [6, 7]. Therefore in this paper we shall consider only the special solutions of system (7).

In order to satisfy the first two equations of (7), we may express 4, v, W in terms of a function F as follows:

oF oF Bu —2 Bag O?F = — = — = ~A— V ‘ Ox Oz? * yon Bg) am oer (8) where o? o? vi=s5 t+. 1 Ox? Oy? (9) Substituting expressions (8) into the last of the equations (7), we obtain OZ NC i 62 B wv? eo v2 eee 13 a8 ( rorya, OF) gee >) where g= B3,+By1 Bs3—Bis + V (B3,+Bi1 Bs3s—B3s)?—4 Br Bas Bly 2 B33 Bas i (11)

g= B2,+By; B3;—Bis — V (Bi,+B11 B3s—Bi3)*—4 Bis Bas Big 2 B33 Bay

Consequently our problem is reduced to the finding of a special solution of equation (10).