CHARACTERIZATION OF THE WEAKLY RIGID MOTIONS IN LINEAR KINEMATICS OF COSSERAT SURFACES* Cesare Davini** SOMMARIO. Per stabilire esistenza ed unicitd di soluzioni deboli nei problemi di elastostatica lineare delle superfici di Cosserat ~ necessario studiare la classe dei moti corrispondenti a misure di deformazione nulle e soddisfacenti le condizioni geometriche al contorno. Si dimostra che rnoti di questo tipo sono regolari e corrispondono a moti rigidi in senso classico. SUMMARY. To state existence and uniqueness of weak solutions in linear elastostatics of Cosserat surfaces it is necessary to study the class of motions corrisponding to vanishing deformations and satisfying the Dirichlet boundary data. ICe prove those motions are regular and correspond to rigid motions in the classical sense. 1. INTRODU .CTION. In this paper we consider configuration changes of struc- tured bodies constituted by a material surface embedded in the three-dimensional Euclidean space to every point of which a material director is assigned. A change of configuration is a pair of displacement fields of points of the surface and directors for which some deformation measures are defined according to the linearized kinematics (see, for instance, Green - Naghdi - Wain Wright [ 1]). In a previous paper [2] we studied the existence of weak solutions for the static boundary value problem of a C- -surface and we found cases where the solution may be non- -univoquely determined or may lack at all if the data do not satisfy some suitable condition. Indeterminacies occur when there are possible motions corresponding to vanishing deformation measures and satisfying the Dirichlet boundary data. On the contrary, uniqueness is assured if the surface is sufficiently constrained at the boundary even though, as in three-dimensional elasticity (see S. Campanato [3]), for the formal character of the theory it is not trivial to get down this quality into terms of the geometric properties of the boundary We wish here to examine the relationship between the rigid motions defined in the weak sense and the rigid motions according to the classical definition. By means of a suitable sequence of smooth motions we show that the rotation field * This work was supported by the "Gruppo Nazionale per la Fisica Matematica ~ of C.N.R. ** Istituto di Scienza deUe Costruzioni, Urdversit~ di Pisa. Presently, at the Istituto di Elaborazione ddl'Informazione, Pisa. of weakly-rigid changes of configuration, locally not determi.. ned by strain-displacement relations, coincides almost every- where with a constant. Therefore they are classically rigid. The result gives evidence to the classification coming from the weak theory of existence, [2], and is useful when we wish to discuss existence and uniqueness of solutions for a specific problem (1). As an example we use the result in studying two problems in the class covered by [2] and obtain a complete geometric description of the cases where indeterminacies of the solution are axcluded. 2. STATEMENT OF THE PROBLEM. We consider a surface ~9 ° whose reference configuration is described by an iniective mapping × from an open bounded set/2 C R 2 into a Euclidean 3-space 8. We shall assume that × is as smooth on ~ as we need in the following. The result we obtain is also valid for surfaces constituted by pieces like 6,a, locally mapped by smooth functions onto open subsets of R 2. Let A a = 1,2 be the fields of coordinate vectors related to the map × and A 3 be the unit normal to the surface, and let D = DA3 be the field of the directors of our Cosserat surface that we suppose constant in magni- tude. We consider there the infinitesimal displacements u, of the material points and of the directors of ~9 ~ from the reference configuration (×, D). This change of configuration takes place with a strecht of the surface locally described by the deformation measures (2) 1 e- (Vu ru -- +~r ~), 2 ~ k-d,+DBa , (2,1) 1 1 where B is the curvature tensor in the reference configuration (1) An analogous problem in three-dimensional elasticity has been considered by Hlavfi~ek-Ne~as[4] who, by using the proved regularity of the weakly rigid motions, derived some sufficient conditions for uniqueness involving geometry and extent of the constrained part of the boundary. (~) For the deduction of these results in linearized kinematics see P, M. Naghdi [5]. Here we use the same symbols as in [2]. 38 MECCANICA