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This paper deals with some kinematical aspects of shells of revolution whose torsionless axisymmetrical deformation may be finite or one with small strain accompanied by large or moderately large rotation. The results are summarized in the form of three theorems. 1. Preliminary remark. The purpose of this paper is to state and prove certain purely kinematical results for torsionless axisymmetrical deformation of shells of revolution. The deformation of the shell may be finite or one with small strain accompanied by large or moderately large rotation. Since the results obtained are purely kinematical, they hold for all materials and are not necessarily limited to elastic shells. We utilize a direct formulation of the theory of shells based on a 2-dimensional continuum model as a Cosserat surface with a single director [1,2], but the corresponding kinematical developments can be also effected from an appropriate approximation for the position vector in the 3-dimensional theory (see, e.g., [2, Eq. (2.25)]). For readers' convenience, we recall that a Cosserat surface is a body comprising a material surface embedded in a Euclidean 3-space, together with a single deformable vector field—called director—attached to every material point of Sf. The director, which is not necessarily along the unit normal to the surface surface Sf, has in particular the property that it remains unaltered in length under superposed rigid body motions.1 Clearly, a Cosserat surface is not just a 2-dimensional surface; but is, in fact, endowed with some structure in the form of an additional primitive kinematical vector field. The magnitude of the director along the normals to the reference surface may be regarded as •Received June 21, 1983. The results reported here we obtained in the course of research supported by the U.S. Office of Naval Research under Contract N00014-75-C-0148, Project NR 064-436 with the University of California, Berkeley. 1 As in the paper of Naghdi [2], the director here is chosen to have the physical dimension of length. This differs from an earlier choice in Naghdi [1], where the director was assumed to be dimensionless. For further discussion on this point, see Naghdi [2, Sec. 3]. ©1985 Brown University 24 P. M. NAGHDI AND L. VONGSARNPIGOON representing the thickness of the shell-like body 38 and similarly the material surface Zfol may be identified with the reference surface (e.g., the middle surface) in the shell-like body. Related remarks pertaining to the identification of various constitutive results, the assigned fields and the inertia coefficients in the theory of a Cosserat surface are made in Naghdi [1] and [2, Sec. 5]. Also, following the procedure of Casey and Naghdi [3], in the kinematical development of a Cosserat surface we remove the local rotation and the relative displacement at one point of the surface ^resulting in a configuration k* so as to render all kinematical and associated kinetical results invariant under arbitrary finite superposed rigid body motion (for details, see Naghdi and Vongsarnpigoon [4, Sec. 4-5] and [5, Sec. 2]). By way of additional background and in the context of a 3-dimensional theory, consider a relative displacement2 u* = r* R* and the relative displacement gradient H* = F* I = Gradu*, where r* and R* are, respectively, the position vectors of a material point in the deformed and undeformed configurations, F* is the deformation gradient tensor and I is the identity tensor. Recall that through the polar decomposition theorem, we have F* = R*U*, where R* is the (local) rotation tensor and U* is the right stretch tensor which are related to the relative strain tensor E* through the expression 2E* = (U*)2 — I = p*rF* I. If H* is infinitesimal, then the strain E* and the rotation R* are also infinitesimal. But if only E* is infinitesimal of order e the relative deformation u* is not necessarily small and the rotation tensor R* may be large or moderately large of order e1/2 (see Naghdi and Vongsarnpigoon [4]). Keeping the above background in mind, consider now a motion of a Cosserat surface # characterized by the two functions r and d corresponding to the position vector of a material point on the material surface Sfand the director at taht point. The deformation graident F and director gradient G can be defined in terms of suitable components of Gradr and Gradd (see Eqs. (Al)12 of Appendix A) and these, in turn, give rise to the strain tensor 2E = FrF — I and the curvature tensor F7G defined on the material surface ^in the current configuration. Further, through the polar decomposition theorem, we again have F = RU, where R is the (local) rotation tensor and U the right stretch tensor. Given F, the position vector r and the director d in the deformed configuration k can be determined (see (Al)t) and thus the deformation is known. A deformation of a shell-like body is said to be infinitesimal if the deformation gradient F differs only infinitesimally from the unit tensor I and, consequently, the strain tensor E and rotation tensor R are both infinitesimal. However, if E is small, the displacement vector u = r — rR, where rR is the position vector in the reference configuration, is not necessarily infinitesimal; and the deformation gradient tensor F could differ significantly from I and may be accompanied by large or moderately large R. Our main results are summarized as three theorems in Sec. 2 and the proofs are supplied in Sec. 3. Briefly, with reference to torsionless axisymmetrical deformation of a shell of revolution, we first show in Theorem 1 that the knowledge of the relative strain tensor at every point of the reference surface (e.g., the middle surface) is sufficient to 2 We attach an asterisk to the kinematical quantities associated with the 3-dimensional theory to differentiate them from corresponding quantities employed in the direct (2-dimensional) theory of a Cosserat surface. AXISYMMETRICAL DEFORMATION OF SHELLS OF REVOLUTION 2 5 determine the change in the second fundamental form and hence the deformed configuration of the shell. Furthermore, if the strain and its first and higher order gradients are assumed to be small, then by our Theorem 2 the only kind of deformation accompanied by large rotation (or large deflection)—excluding, of course, the rigid body translation—of a nonshallow shell is that which involves an inversion of the shell. Indeed, it is a consequence of Theorem 2 that if the possibility of an inversion of the (nonshallow) shell is not admitted, a deformation with small strain accompanied by large rotation can take place only by allowing the strain gradient to be large. On the other hand, for a shallow shell of revolution (which includes an initially flat circular plate as a special case), while a deformation with large rotation (or large deflection) is not possible when the strain and strain gradients are assumed to be small, according to Theorem 3 the case in which the shell is subjected to a deformation with moderate rotation is admissible. Throughout the paper, we use the usual summation convention over repeated indices with Greek indices taking the values 1, 2 and Latin indices having the ranges 1, 2, 3. Also, in the main text and the Appendix A, we employ both a coordinate-free notation and the component form of various results and equations. 2. Some background information. A summary of the main results. Let e, (/ = 1,2, 3) be a set of orthonormal base vectors associated with a system of rectangular Cartesian coordinates (x, y, z) and let 8" (a = 1,2) designate a convected coordinate system on the material surface yof a Cosserat surface e€. Further, with reference to a cyclindrical polar coordinate system (r, 8, z), let e, = er(8) and e0 = eg(8) be the unit base vectors defined by er = cos^! + sin#e2, e9 = -sintfej + cos 0e2. (2.1) Without loss in generality we may identify the convected coordinate 62 with the angle 8 of the cyclindrical polar coordinate system and write 81 = 8. (2.2) Let SfR, a two-dimensional region of space occupied by the material surface Sf of #in the reference configuration k0, be a surface of revolution with its position vector specified by tr = ro(01)*r + Zo(0l)*3(2.3) Then, from (2.3), the surface base vectors AQ associated with the convected coordinates 6" and the outward unit normal A 3 to £fR are calculated to be Ai = roer + zoe3> A2 = r0efl, A3 = (r0'e3 z'0er)/a0, (2.4) where prime denotes partial differentiation with respect to 81 and where a0 = ao(0') is defined by «o = (rof +(zo)2(2.5) The coefficients of the first and second fundamental forms of the material surface yin the reference configuration are ^11 = «0> ^22 = r0 ' ^12 = ^2 ^ •®11 = (r0Z0 ~ z0r0')/a0< ^22 ~ r0Z0/a0> ^12 = 026 p. M. NAGHDI AND L. VONGSARNPIGOON Let D denote the director at rR in the reference configuration k0 of the Cosserat surface and, for covenience, introduce the notation D„ = Aa, D3 = D (2.7) In what follows, whenever desirable, the notations D, = (Dx, D2, D3) and the set (2.7), i.e., (Aa,D), will be used interchangeably depending on the particular context. Since D are linearly independent, as in Naghdi [2, Sec. 3], we may also introduce a set of reciprocal vectors D' such that D,. ■ D' = 8{, (2.8) where 8/ is the Kronecker symbol in 3-space. Without loss in generality, we choose the director in the reference configuration to be along the normal to every point of so that D = Z)A3, D = D{dl). (2.9) It follows from the choice (2.9) and the definition of D' in (2.8) that D° = A", D3 = -^A3, (2.10) where AQ are the surface reciprocal base vectors to AQ. Let the Cosserat surface be deformed axisymmetrically without torsion such that the reference surface £fR becomes another surface of revolution with the position vector r = t(6a, t) in the current configuration k specified by r = r(0\ t)er + z(6\ t)e3. (2.11) Since the deformation is assumed to be torsionless, the meridians 62 = const, and the parallels 61 = const, of the undefo

Some general results in the kinematics of axisymmetrical deformation of shells of revolution