## Jl

The second term in the first condition is 0(r)2s) for k2{aio) = 0 and 0(r/2~3'') fork2(aio) 40. The numbers p2,pi, and m are found by formulas (45.7) from the solutions to problems VIII—XI. The number Ji is calculated using equation 45.11, where we put i = \\

Jl J-oo where S"l2 has been found from the homogeneous equations of the antiplane problem with the following conditions at the edge = 0

(n, = r21 = rl3 = Dt = 0, a = 0, /? = I - .y, 6 = - I + 2q)

THE INTERNA L ELECTRO ELA STIC STA TE A ND THE IiO UN DA RY LAYER 217

A2 oa2

Here, J2 is found by equation 45.11 (/ = 2) from the solution of the antiplane problem with the following conditions at the edge = 0:

The numbers d\,q3, and t\ were introduced by equation 34.7. For the rigidly fixed edge a2 = a2o

V] = vi - i'3 - D2 = 0, a = — 1 + q — s, 0 = ~q, 6 = -l+q

After transformations, the conditions at the rigidly fixed edge a2 = a2a will be written as

«,=0, w = 0 72 +.dl5 \ - ^ <rf3l G, + d„G2) = 0 [V -"] «2 + (4T\ + if372) + W3iTi + di3T2) = 0 [T?1 -"] . (45.15)

In order to find /x,ai, and a2, we should solve the homogeneous system of equations for the plane boundary layer with the following conditions at the edge 6 = 0:

 XII. V2. +c = 0, V3* = 0, d2, = 0 XIII. V2» = 0, +1 = 0, D2t = 0 XIV. V2» = 0, V3» = 0, D2t + ç = o XV. >'2* +1 = 0, = 0, d2. = 0 XVI. >'2» = 0, + < = 0, d2. = 0 XVII. ''2. = 0, >'3« = 0, d2, + 1= 0.

Then we should find the resultants of the vertical forces N\2,N\3. and the bending moments M\2,M\3, and A/14 in problems XII, XIII, and XIV; and the horizontal forces rJ5, Fi6, and Fn in problems XV, XVI, and XVII at the edge £2 = 0. The numbers //, ¿1, and 62 are found by the formulas

 Nu Nu A = N12 N13 M13 M>4 M12 M, 3

For the hinge-supported edge a2 = a20

7-2=0, G2 = 0, u- = 0. (45.16) For the free edge a2 = O20

where

The number /ij is found from the solution of the plane problem with the following conditions at the edge 6 = 0-

The number J3 is found by equation 45.11 from the solution of the homogeneous equations of the antiplane problem (equations 40.6 and 40.7) with the conditions

46 BOUNDARY CONDITIONS IN THE THEORY OF SHELLS WITH TANGENTIAL POLARIZATION (FACES WITHOUT ELECTRODES)

On faces without electrodes, the electrical boundary condition has the form

For the rigidly fixed edge qj = Q|o without electrodes p6

 D\0) - pc-, b 1 0 2 A2 da2 r2 Ni Mb = 7T > » 1 MT THE INTERNAL ELECTRO ElA STIC STA TE A NO THE HO UN DA RYI A YER 219 where Ti and F2 are the horizontal resultants of the edge stresses found from the solution of the simultaneous homogeneous equations of the plane boundary layer with the following conditions at the edge St = 0: W3 and Nt, are the vertical resultants of the edge stresses found from the plane auxiliary problems III and IV with the following conditions at the edge Si = 0: For the hinge-supported edge o i = a)0 without electrodes Tt = 0, u2 = 0, vi' = 0, G, = 0 where ¡i% is found from the solution of the plane problem with the edge conditions (St = 0) For the free edge a \ = Q|o without electrodes li c)FJ A2 an2 A 2 oa2 Here, J4 is found by equation 45.11 (i = 4) from the solution of the antiplane problem in equation 39.6 with edge conditions in equation 45.12 and electrical conditions of the type in equation 46.1. For the rigidly fixed electrode-covered edge a, - a 10, .sf2r, + Tj + 2/ii/j|£j0> = 0 [?/', u-i = 0, u> = 0 3 /i7 Tlie numbers ¡i^ and /¿7 are the same as for the electrode-covered rigidly fixed edge a\ = aio without electrode. For the hinge-supported electrode-covered edge rvi = a-|o u2 = 0, vi' = 0, Ti= 0, Ci = 0, V(0) = V. (46.6) For the/ree electrode-covered edge a, = a io h dH2i
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