509 



where a superposed dot denotes differentiation with 

 respect to t holding 9" fixed. 



Let P, bounded by a closed curve, 3P, be a part 

 of J occupied by an arbitrary material region of 

 S in the present configuration at time, t, and let 



V = v"a = V a" 



(49) 



be the outward unit normal to 3 P. It is convenient 

 at this point to define certain additional quantities 

 as follows: The mass density, p = p(0',t), of the 

 surface,^, in the present configuration; the con- 

 tact force, N = N{6^,t;V), and the contact director 

 force''', M = M(9^,t;V), each per unit length of a 

 curve in the present configuration; the assigned 

 force, f = f(8^,t), and the assigned director force, 

 I. = £{e^,t) , each per unit mass of the surface,,/ ; 

 the intrinsic director force, m, per unit area of 

 J; the inertia coefficients, k = k(6^) and k = kO^), 

 which are independent of time; the specific internal 

 energy, e = e(6'V,t); the heat flux, h = h(9"'^,t;v) 

 per unit time and per unit length of a curve, 3P; 

 the specific heat supply, r = r(9Y,t), per unit time; 

 and the element of area, do, and the line element, 

 ds, of the surface, J . The assigned field, f, may 

 be regarded as representing the combined effect of 

 (i) the stress vector on the major surfaces of the 

 sheet-like body denoted by f^, e.g., that due to the 

 ambient pressure of the surrounding medium, and (ii) 

 an integrated contribution arising from the three- 

 dimensional body force denoted by fj-,, e.g., that due 

 to gravity. A parallel statement holds for the as- 

 signed field, l. Similarly, the assigned heat sup- 

 ply, r, may be regarded as representing the combined 

 effect of (i) heat supply entering the major surfaces 

 of the sheet-like body from the surrounding environ- 

 ment, denoted by r , and (ii) a contribution arising 

 from the three-dimensional heat supply, denoted by 

 rj^. Thus, we may write 



~b ~c 



~b ~c 



. (50) 



The various quantities in (50) are free to be speci- 

 fied in a manner which depends on the particular ap- 

 plication in mind and, in the context of the theory 

 of a Cosserat surface, the inertia coefficients, k, 

 k and the mass density, p, require constitutive equa- 

 tions. Indeed, fc«Jc ^■'"^ ^c' ^^ well as f-^,Z^ and 

 rj^, can be identified with corresponding expressions 

 in a derivation from the three-dimensional equations 

 [for details, see Naghdi (1972,1974)]. Likewise, p 

 and the coefficients, k,k, may be identified with 

 easily accessible results from the three-dimensional 

 theory. 



In terras of the above definitions, the conserva- 

 tion laws for a Cosserat surface can be stated in 

 fairly general forms. We do not record these here 

 since they are available elsewhere [Naghdi (1972) , 

 p. 482) or Naghdi (1974)]. Instead, we turn our 

 attention to the relatively simple theory of the 

 next section. 



It may be noted that the local field equations 

 in the mechanical theory of a Cosserat surface have 



the same forms as those that can be derived from the 

 three-dimensional field equations (9) j 2 3 by suit- 

 able integration between the limits, Cj and ^2> and 

 in terms of certain definitions for integrated mass 

 density and resultants of stress [for details, see 

 Naghdi (1972, Sections 11-12) or Naghdi (1974)]. 

 Moreover, given the approximation (21) , there is a 

 1-1 correspondence between the two-dimensional field 

 equations that follow from the conservation laws of 

 a Cosserat surface and those that can be derived from 

 (9)1,2,3 provided we identify the director, d, in 

 (21) with (46)2 and adopt the definitions of the re- 

 sultants mentioned above. As similar 1-1 correspon- 

 dence can be shown to hold between the two-dimensional 

 energy equation in the theory of a Cosserat surface 

 and an integrated energy equation derived from the 

 three-dimensional energy equation. 



6. A RESTRICTED THEORY OF A COSSERAT SURFACE 



Special cases of the general theory can be obtained 

 by the introduction of suitable constraints, thereby 

 resulting in constrained theories. Alternatively, 

 corresponding special cases can be developed in which 

 the kinematic and the kinetic variables are suitably 

 restricted a priori and then restricted theories are 

 constructed by direct approach. Such special cases 

 of the general theory have been discussed previously 

 by Naghdi (1972, Sections 10 and 15) and by Green 

 and Naghdi (1974) and are of particular interest in 

 the context of elastic shell theory. We provide here 

 an outline of a restricted theory developed by Green 

 and Naghdi (1977) mainly for application to problems 

 of fluid sheets. The resulting equations can also 

 be obtained as a constrained case of those given for 

 directed fluid sheets [Green and Naghdi (1976) ] , but 

 it is more convenient to restrict the kinematic and 

 the kinetic variables at the outset and construct a 

 corresponding restricted theory from an appropriate 

 set of conservation laws in integral form. 



Let the director, d, while deforming along its 

 length, always remain parallel to a fixed direction 

 specified by a constant unit vector, b. It should 

 be kept in mind that b is fixed relative to the body 

 and not relative to the space. Thus, recalling (45)2 

 and (48)2, "^ write 



>{9 ,t)b 



w = w(9 , t)b 



(51) 



Further, in view of the assumed form of (51) j for 

 the director, it is convenient to decompose M,m and 

 I into their components along and perpendicular to 

 the unit vector, b, i.e., 



M = M(9'V,t;v)b + b X S(9^,t;v) , S • b = , 



m = m(9"^,t)b + b x s(9''',t) , s • b = , 



£ = J.(0''',t)b + b X c(9'^,t) , c • b = , (52) 



The terminology of director couple is also used for M depend- 

 ing on the physical dimension assumed for the director, d. 

 Here we choose d to have the physical dimension of length so 

 that M has the same physical dimension as N. For further 

 discussion see Naghdi (1972, Ch. C) and Green and Naghdi 

 (1976) . 



where M,m and i are scalar functions and S,s,c are 

 vector functions of their arguments. According to 

 the decomposition (52) j the vector, M, is resolved 

 into two parts . One pa t is along b and the other 

 part is the perpendicular projection of M onto the 

 plane defined by S • b = which is perpendicular to 



