501 



(1972) , which also contains additional historical 

 remarks relevant to oriented continua and to the 

 theory of thin elastic shells. A parallel develop- 

 ment in the theory of a Cosserat curve with two 

 deformable directors begins with a paper of Green 

 and Laws (1966) whose derivation is carried out 

 mainly from an appropriate energy equation, together 

 with invariance requirements under superposed rigid 

 body motions. A related theory of a directed curve 

 with three deformable directors at each point of the 

 curve, in the context of a purely mechanical theory 

 and with the use of a virtual work principle, is 

 given by Cohen (1966) . A further development of 

 the basic theory of a Cosserat curve along with 

 certain general developments regarding the construc- 

 tion of nonlinear constitutive equations for elastic 

 rods is given by Green et al. (1974a, b) . 



In general, two entirely different approaches may 

 be adopted for the construction of one-dimensional 

 and two-dimensional theories of mechanics pertain- 

 ing to certain motions and (three-dimensional) media 

 responses which are effectively confined, respec- 

 tively, to one-dimensional and two-dimensional re- 

 gions. For example, the theory of slender rods and 

 that of fluid jets are both one-dimensional theories; 

 and, similarly, the theory of thin shells and that 

 of fluid sheets are both two-dimensional theories 

 in the context of the particular classes of three- 

 dimensional bodies mentioned earlier. 



Of the two approaches just mentioned, one starts 

 with the three-dimensional equations of the classi- 

 cal continuum mechanics and by applying approxima- 

 tion procedures strives to obtain one-dimensional 

 (in the case of jets and rods) and two-dimensional 

 (in the case of sheets and shells) field equations 

 and constitutive equations for the medium under 

 consideration. In the other approach, the particu- 

 lar medium response mentioned above is modelled as 

 a one-dimensional and a two-dimensional directed 

 continuum, namely a Cosserat curve and a Cosserat 

 surface introduced earlier; and one then proceeds 

 to the development of the field equations and the 

 appropriate constitutive equations. If full inform- 

 ation is desired regarding the motion and deforma- 

 tion of the continuiom under study in the context of 

 the classical three-dimensional theory, then there 

 would be no need to develop a particular one- 

 dimensional and a two-dimensional theory. In fact, 

 the aim of one-dimensional and two-dimensional theo- 

 ries of the type mentioned above is to provide only 

 practical information in some sense: for example, 

 in the case of fluid sheets information concerning 

 quantities which can be regarded as representing 

 the medium response confined to a surface or its 

 neighborhood as a consequence of the (three- 

 dimensional) motion of the body, or the determina- 

 tion of certain weighted averages of quantities 

 resulting from the (three-dimensional) motion of 

 the body. A parallel remark may be made, of course, 

 in the case of fluid jets. The desire for obtain- 

 ing limited or partial information if the basic 

 motivation for the construction of such one- 

 dimensional and two-dimensional theories as those 

 for slender rods and thin shells and for fluid flow 

 problems of jets and sheets. 



The nature of difficulties associated with the 

 development of both the shell theory and the theory 

 of water waves on the one hand, and that of rods and 

 jets on the other, from the full three-dimensional 

 equations is well known and has been elaborated upon 



on various occasions.* In view of these, it is rea- 

 sonable to attempt to formulate one-dimensional and 

 two-dimensional theories of the types described above 

 by replacing the continuum characterizing the (three- 

 dimensional) medium in question with an alternative 

 model which would reflect the main features of the 

 response of the three-dimensional medium and which 

 would then permit the formulation of appropriate 

 one-dimensional and two-dimensional theories by a 

 direct approach and without the appeal to special 

 assumptions or approximations generally employed in 

 the derivation from the three-dimensional equations. 



Of course, the introduction of an alternative 

 model and formulation of one-dimensional and two- 

 dimensional theories by the direct approach do not 

 mean that one ignores the nature of the field equa- 

 tions in the three-dimensional theory. In fact, 

 some of the developments of the field equations by 

 direct procedures are materially aided or influenced 

 by available information which can be obtained from 

 the three-dimensional theory. For example, the inte- 

 grated equations of motion from the three-dimensional 

 equations provide guidelines for a statement of one 

 and two-dimensional conservation laws in conjunction 

 with the one and two-dimensional models, and also 

 provide some insight into the nature of inertia terms 

 and the kinetic energy in the direct formulation of 

 the one-dimensional and two-dimensional theories. 



Inasmuch as most of the difficulties associated 

 with the derivation of the one-dimensional and two- 

 dimensional theories from the three-dimensional equa- 

 tions occur in the construction of the constitutive 

 equations, it is in fact here that the direct ap- 

 proach offers a great deal of appeal. This construc- 

 tion, as well as the entire development by the 

 direct approach, is exact in the sense that they 

 rest on (one-dimensional and two-dimensional) pos- 

 tulates valid for nonlinear behavior of materials 

 but clearly they cannot be expected to represent all 

 the features that could only be predicted by the 

 relevant full three-dimensional equations. Theories 

 constructed via a direct approach necessarily sat- 

 isfy the requirements of invariance under superposed 

 rigid body motions that arise from physical consider- 

 tions and, of course, they are also consistent and 

 fully invariant in the mathematical sense. More- 

 over, the development by the direct approach is con- 

 ceptually simple and does not have the difficulties 

 involving approximations usually made in the devel- 

 opment of the theory of thin shells and the theory 

 of water waves (or the theories of slender rods and 

 jets) from their corresponding three-dimensional 

 equations. 



Following some general background information 

 and definitions of jet-like and sheet-like bodies 

 in Section 2, the remainder of the paper is arranged 

 in two parts which can be read independently of each 

 other: one part (Part A) is concerned with the 

 theory of fluid jets and the other (Part B) is de- 

 voted to the theory of fluid sheets and its applica- 

 tion to water waves. In our discussion of the 

 direct formulation of these two topics, considerably 



The nature of these difficulties with particular reference 

 to shells is discussed by Naghdi (1972, Sees. 1,4,19,20,21). 

 Some of the difficulties associated with both nonlinear and 

 linear theories of water waves are noted by Naghdi (1974) and 

 are also discussed in the first and final sections of the 

 paper of Green et al . (1974c). 



t 



See the remarks following Eqs . (26) and (50). 



