Scientia Sinica

No. 1 HU: EQUILIBRIUM « VIBRATION OF TRANSVERSELY ISOTROPIC ELASTIC BODY 13

O O7F OF OF OF | , De {—3u 5 Bil aga — Bu aa + Pop ee a\ = 0, (65)

s

8p Sp . By OD 0 &P — — (re 66 Qu? + Gy? + Bye Oz? Bes OF feo)

(66) is the equation satisfied by the function @. From (65), we get

OF O°F OF OF 2 apa Gare og ian roe ame

where / is an arbitrary function of x and y. It is evident that #, v will not be altered by adding an arbitrary function of x, y to F. Therefore we may put /=0. Thus we obtain

(2 ee (68) 3 13

This is the expression for the displacement component w. Substituting expressions (59) and (68) into the last equation of (58), we finally obtain the equation satisfied by F as follows:

By Bay 4 BZ+B11 Bss—Birs —2 OF B33By, O'F —_—_—— 7 Ws Se SS ES Ba ul BA 1g 1 BR Oz" 69 0 {By+Bss 2 , BastBiz 0? | OF Emon — ——_ V SS — 0 5 By3 { By3 1+ By3 Oz? Oz? +r Be oz* For the sake of simple writing, let us denote By B33 By B66 2 ———— —— Obi i 2 = 5 =S); Bi3 By3 By) ears (70) Bip By — Bee 2 e —S > _— ————_ = 2 — Say ——_ = kK By Ba Yo" “Bis

The results obtained above can be stated as follows. The solution of the system of equations (58) can be expressed in terms of two stress functions as follows:

i @F_ 8 ,_ OF ,% Ox Oz Oy ° Oy Oz Ox ” (71) 2 O?F O?F Wa NEY ez 1 rae zae:

where F and satisfy respectively the following equations: