510 



b. Parallel statements hold for vectors, m and H, 

 in (52)2,3- 



Also, it is convenient to decompose the assigned 

 fields, f and H , into two parts, one of which repre- 

 sents the three-dimensional body force acting on the 

 continuum which is assumed to be derivable from a 

 potential function, n(r,i}i), and the other which 

 represents the effect of applied surface loads on 

 the major surfaces of the fluid sheet. Thus, we 

 write 



f = - 



dr -c 



a = (- 



an 



3<j) 



+ i 



(53) 



With the foregoing definitions of the various 

 field quantities and with reference to the present 

 configuration, the conservation laws for a restricted 

 theory of a Cosserat surface [different from the re- 

 stricted and constrained theories discussed previ- 

 ously by Naghdi (1972) and by Green and Naghdi (1974)] 

 are: 



dt Jl 



pda = , 



■r— p (v + kwb)da = pfdo + _^ N ds , 



dtjp- - J F ~ J3P~ 



Sfj.^'^ 



J (p£-m)da +/gp 



+ kwb)wda = b[ {pZ-m)da + Mds] 



+ b X [ (pc-s)da 

 J p ~ ~ 



•J. 



dt 



/, 



+ I gpSds] , 



p [r X v + k(r x wb + d x v)]da 



= p [r X f + d X (b X c) ]da 

 J P 



/ 



+ J p[r X N + d X (b X s)]ds , 



-T- I p[e+n+l2(v'v+ 2kv • wb + kw2)]da 

 dt J p - - - . 



= [ p(r + f •v+ «._w)da 

 J P 



/a: 



+ p (N • v + Mw - h)ds 



(54) 



In the above equations (54) j is a statement of con- 

 servation of mass, (54)2 the conservation of linear 

 momentimi, (54) 3 that of the conservation of the 

 director momentiom, (54) 4 the conservation of moment 

 of momentum and (54)5 represents the conservation 

 of energy. It should be noted that the quantities, 

 M and l ...c no contributions to the moment of momen- 

 tum equation , and the quantities , c and S , make no 

 contribution to the equation for conservation of 

 energy in the present restricted theory. 



Under suitable continuity assumptions, the curve 

 force, N, the director force, M, and the heat flux, 

 h, can be expressed as 



N = n"v , 

 ~ a 



a 

 M V 



S = s"v 



a a a 



h = qv , q=q-a 



(55) 



where q is the heat flux vector and the fields , N , 

 SO',M°',q", are functions of e"'^,t. The five conserva- 

 tion equations in (54) then yield the local equa- 

 tionsT 



h Y 



pa^ = Y(e') , (56) 



(a^") + Yf = Y(v + kwb) . (57) 



(a^ ) + Y^l = ma" + Y(kv • b + kw) , 



(a^S ) + YC = a s - b X ykv , (58) 



X N + d 



(b X s) 



+ d 



(b 



s") 



,(59) 



pr - div q - pe + N 



V +mw+Mw =0 ,(60) 



~ ,a ,a 



where "div " is the surface divergence operator de- 

 fined by divg q = q,j^ • a'^ and a comma denotes par- 

 tial differentiation with respect to the surface 

 coordinates, 9^. It should be noted that the vector 

 fields, S*^ and s, are workless and do not contribute 

 to the reduced energy equation (50) . 



The above results include (60) , which is derived 

 from (54)5. For the purely mechanical theory in 

 which the law of conservation of energy is excluded, 

 the appropriate conservation laws are the first four 

 of (54) . In the context of the purely mechanical 

 theory, it is worth recalling that the rate of work 

 by all contact and assigned forces acting on P and 

 on its boundary, 3P , minus the rate of increase of 

 the kinetic energy in P can be reduced to [see 

 Naghdi (1972,1974) ] : 



;. 



p(f • V + H • w)da + 



3P 



(N 



V + M • w) ds 



dt 



where 



h{v • V + 2kv • w + kw^)da = Pda , (61) 



/. 



P=N 'V +mw+Mw 



~ ,a ,a 



is the mechanical power. 



Before closing this section, we also note that 

 the restriction imposed on the motion of the medium 

 by the condition of incompressibility , in the context 

 of the restricted theory under discussion, reduces 

 to§ 



"I"!!! line with a remark made at the end of the previous 

 section, we note that equations (56) -(60) can also be 

 derived by suitable integration across the thickness of 

 the sheet, respectively, from the three-dimensional equa- 

 tions (9)1 2 3 ^^^ the three-dimensional energy equation. 



In general, there are two conditions of incompressibility 

 in the theory of incompressible directed fluid sheets; for 

 a discussion of these, see Naghdi (1974, Section 3). In 

 our present discussion, since d is assumed to have the form 

 (51) 1, the second condition is satisfied identically and 

 the corresponding pressure (arising from the constraint 

 response) is a part of the response functions for S and s. 

 The specification (62) is motivated from an examination of 

 the incompressibility condition in the three-dimensional 

 theory when the position vector is approximated by (21) . 



