MATHEMATICAL ELASTICITY VOLUME III: THEORY OF SHELLS
VOLUME
III: T H E O R Y
OF SHELLS
Part A. Linear shell theory Chapter
1. Three-dimensional linearized elasticity and Korn's inequalities in curvilinear coordinates
Chapter Chapter
2. Inequalities of Korn's type on surfaces 3. Asymptotic analysis of linearly elastic shells: Preliminaries and outline 4. Linearly elastic elliptic membrane shells 5. Linearly elastic generalized membrane shells 6. Linearly elastic flexural shells 7. Koiter's equations and other linear shell theories
Chapter Chapter Chapter Chapter
Part B. Nonlinear shell theory Chapter
8. Asymptotic analysis of nonlinearly elastic shells: Preliminaries Chapter 9. Nonlinearly elastic membrane shells Chapter 10. Nonlinearly elastic flexural shells Chapter 11. Koiter's equations and other nonlinear shell theories
MATHEMATICAL ELASTICITY V O L U M E III: T H E O R Y OF S H E L L S
PHILIPPE G. CIARLET Institut Universitaire de France Universitd Pierre et Marie Curie, Paris, France
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31 f i g u r e s b y t h e a u t h o r
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MATHEMATICAL PREFACE 1
ELASTICITY:
GENERAL
This treatise, which comprises three volumes, is intended to be b o t h a thorough introduction to contemporary research in elasticity and a working textbook at the graduate level for courses in pure or
applied mathematics or in continuum mechanics. During the past decades, elasticity has become the object of a considerable renewed interest, both in its physical foundations and in its m a t h e m a t i c a l theory. One reason behind this recent attention is that it has been increasingly acknowledged that the classical linear equations of elasticity, whose mathematical theory is now firmly established, have a limited range of applicability, outside of which they should be replaced by the genuine nonlinear equations that they in effect approximate. Another reason, similar in its principle, is that the validity of the classical lower-dimensional equations, such as the two-dimensional yon Ks equations for nonlinearly elastic plates or the twodimensional Koiter equations for linearly elastic shells, is no longer left unquestioned. A need has been felt for a better assessment of their relation to the corresponding three-dimensional equations that they are supposed to "replace". Thanks to the ever-increasing power of available computers, sophisticated mathematical models that were previously intractable by approximate methods are now amenable to numerical simulations. This is one more reason why these models should be established on firm grounds. This treatise illustrates at length these recent trends, as shown by the main topics covered: A thorough description, with a pervading emphasis on the nonlinear aspects, of the two existing mathematical models of threedimensional elasticity, either as a boundary value problem consisting of a system of three quasilinear partial differential equations of -
1This "General preface" is an updated excerpt from the "Preface" to the first edition (1988) of Volume I.
Mathematical Elasticity: Generalpreface
vi
the second order together with specific boundary conditions, or as a minimization problem for the associated energy over an ad hoc set of admissible deformations (Vol. I, Part A);
A mathematical analysis of these models, comprising in particular complete proofs of all the available existence results, relying either on the implicit function theorem, or on the direct methods of the calculus of variations (Vol. I, Part B); -
-
A mathematical justification of the well-known two-dimensional
linear Kirchhoff-Love theory of plates, by means of convergence theorems as the thickness of the plate approaches zero (Vol. II, Part A); Similar justifications of mathematical models of junctions in linearly elastic multi-structures and of linearly elastic shallow shells -
(Vol. II, Part A); - A systematic derivation of two-dimensional plate models from nonlinear three-dimensional elasticity by means of the method of formal asymptotic expansions, which includes a justification of wellknown plate models, such as the nonlinear Kirchhoff-Love theory and the yon Kdrmdn equations (Vol. II, Part B); - A derivation of the large de]ormation, frame-indifferent, nonlinear planar membrane and flexural theories by means of the method of formal asymptotic expansions and a justification of nonlinear planar membrane equations by means of a convergence theorem (Vol. II, Part B);
A mathematical analysis of the two-dimensional, linear and nonlinear, plate equations, which includes in particular a review of the existence and regularity theorems in the nonlinear case and an introduction to bifurcation theory (Vol. II, Parts A and B); -
- A mathematical justification by means of convergence theorems of the two-dimensional membrane, flexural, and Koiter equations of a linearly elastic shell (Vol. III, Part A); - A systematic derivation of the two-dimensional membrane and flexural equations o] a nonlinearly elastic shell by means of the method of formal asymptotic expansions and a justification of nonlinear membrane shell equations by means of a convergence theorem (Vol. III, Part B).
- A mathematical analysis of the two-dimensional, linear and nonlinear, shell equations, with a particular emphasis on the existence theory (Vol. III, Parts A and B).
Mathematical Elasticity: General preface
vii
Although the emphasis is definitely on the mathematical side, every effort has been made to keep the prerequisites, whether from mathematics or continuum mechanics, to a minimum, notably by making this treatise as largely self-contained as possible. Its reading only presupposes some familiarity with basic topics from analysis and functional analysis. Naturally, frequent references are made to Vol. I in Vol. II, and to Vols. I and II in Vol. III. However, I have also tried to render each volume as sel~-contained as possible. In particular, all relevant notions from three-dimensional elasticity are (at least briefly) recalled wherever they are needed in Vols. II and III. References are also made to Vol. I regarding various mathematical notions (properties of domains in ]Rn, differential calculus in normed vector spaces, Sobolev spaces, weak lower semi-continuity, etc.). This is a mere convenience, reflecting that I also regard the three volumes as forming a coherent whole. I am otherwise well aware that Vol. I is neither a text on analysis nor one on functional analysis. Any reader interested in a deeper understanding of such notions should consult the more standard texts referred to in Vol. I. Each volume is divided into consecutively numbered chapters. Chapter m contains an introduction, several sections numbered Sect. re.l, Sect. m.2, etc., and is concluded by a set of exercises. Within Sect. m.n, theorems are consecutively numbered, as Thin. re.n-l, Thm. m.n-2, etc., and figures are likewise consecutively numbered, as Fig. re.n-l, Fig. m.n-2, etc. Remarks and formulas are not numbered. The end of the proof of a theorem, or the end of a remark, is indicated by the symbol m in the right margin. In Chapter m, exercises are numbered as Ex. re.l, Ex. m.2, etc. All the important results are stated in the form of theorems (there are no lemmas, propositions, or corollaries), which therefore represent the core of the text. At the other extreme, the remarks are intended to point out some interpretations, extensions, counter-examples, relations with other results, that in principle can be skipped during a first reading; yet, they could be helpful for a better understanding of the material. When a term is defined, it is set in boldface if it is deemed important, or in italics otherwise. Terms that are only given a loose or intuitive meaning are put between quotation marks. Special attention has been given to the notation, which so often has a distractive and depressing effect in a first encounter with elasticity. In particular, each volume begins with special sections, which
viii
Mathematical Elasticity: General preface
the reader is urged to consult first, about the notations and the rules that have guided their choice. The same sections also review the main definitions and formulas that will be used throughout the text. Complete proofs are generally given. In particular, whenever a mathematical result is of particular significance in elasticity, its proof has been included. More standard mathematical prerequisites are presented (usually without proofs) in special starred sections, scattered according to the local needs. The proofs of some advanced, or more specialized, topics, are sometimes only sketched, in order to keep the length of each volume within reasonable limits; in this case, ad hoc references are always provided. These topics are assembled in special sections marked with the symbol b, usually at the end of a chapter. Exercises of varying difficulty are included at the end of each chapter. Some are straightforward applications of, or complements to, the text; others, which are more challenging, are usually provided with hints or references. This treatise would have never seen the light had I not had the good fortune of having met, and worked with, many exceptional students and colleagues, who helped me over the past three decades decipher the arcane subtleties of mathematical elasticity; their names are listed in the preface to each volume. To all of them, my heartfelt thanks! I am also particularly indebted to Arjen Sevenster, whose constant interest and understanding were an invaluable help in this seemingly endless enterprise! Last but not least, this treatise is dedicated to Jacques-Louis Lions, as an expression of my deep appreciation and gratitude.
August, 1986 and October, 1999
Philippe G. Ciarlet
PREFACE
TO VOLUME
11
A fascinating aspect of three-dimensional elasticity is that, in the course of its study, one naturally feels the need for studying basic mathematical techniques of matrix theory, analysis, and functional analysis; how could one find a better motivation? For instance: - Both common and uncommon results from matrix theory are often needed, such as the polar factorization theorem (Thm. 3.2-2), or the celebrated Rivlin-Ericksen representation theorem (Thm. 3.6-1). In the same spirit, who would think that the inequality [tr A B I < ~ i vi(A)vi(B), where vi(A) and vi(B) denote the singular values, arranged in increasing order, of the matrices A and B, arises naturally in the analysis of a wide class of actual stored energy functions? Incidentally, this seemingly innocuous inequality is not easy to prove (Thm. 3.2-4)! - The understanding of the "geometry of deformations" relies on a perhaps elementary, but "applicable", knowledge of differential geometry. For instance, my experience is that, among those of my students who had been previously exposed to modern differential geometry, very few could effectively produce the formula da~' = ICofVT~ nlda relating reference and deformed area elements (Thm. 1.7-1). - The study of geometrical properties (orientation-preserving character, injectivity) of mappings in IR3 naturally leads to using such basic tools as the invariance of domain theorem (Thms. 1.2-5 and 1.2-6) or the topological degree (Sect. 5.4); yet these are unfortunately all too often left out from standard analysis courses. - Differential calculus in Banach spaces is an indispensable tool which is used throughout this volume, and the unaccustomed reader should quickly become convinced of the many merits of the Frdchet derivative and of the implicit function theorem, which are the keystones to the existence theory developed in Chap. 6. - The fundamental Cauchy-Lipschitz ezistence theorem for ordinary differential equations in Banach spaces, as well as the conver1This "Preface to Volume I )' is an updated excerpt from the "Preface)' to the of Volume I.
K~st e d i t i o n ( 1 9 8 8 )
x
Preface to Volume I
gence of its approximation by Euler's method, are needed in the analysis of incremental methods, often used in the numerical approximation of the equations for nonlinearly elastic structures (Chap. 6). Basic topics from functional analysis and the calculus of variations, such as Sobolev spaces (which in elasticity are simply the "spaces of finite energy"), weak convergence, existence of minimizers for weakly lower semi-continuous functionals, pervade the treatment of existence results in three-dimensional elasticity (Chaps. 6 and 7). Key results about elliptic linear systems of partial differential equations, notably sufficient conditions for the W2'p(f~)-regularity of their solutions (Thm. 6.3-6), are needed preliminaries for the existence theory of Chap. 6. As a result of John Ball's seminal work in three-dimensional elasticity, convexity and the subtler polyconvezity play a particularly important rSle throughout this volume. In particular, we shall naturally be led to finding nontrivial examples of convex hulls, such as that of the set of all square matrices whose determinant is > 0 (Thm. 4.7-4), and of convex functions of matrices. For instance, functions such as F --~ ~i{)~i(FW-~)} a/2 with a > 1 naturally arise in the study of Ogden's materials in Chap. 4; while proving that such functions are convex is elementary for c~ = 2, it becomes surprisingly difficult for the other values of c~ >_ 1 (Sect. 4.9). Such functions are examples of John Ball's polyconvex stored energy functions, a concept of major importance in elasticity (Chaps. 4 and 7). - In Chap. 7, we shall come across the notion of compensated compactness. This technique, discovered and studied by Fran~;ois Murat and Luc Tartar, is now recognized as a powerful tool for studying nonlinear partial differential equations. -
-
-
Another fascinating aspect of three-dimensional elasticity is that it gives rise to a number of open problems, for instance: - The extension of the "local" analysis of Chap. 6 (existence theory, continuation of the solution as the forces increase, analysis of incremental methods) to genuine mixed displacement-traction problems; - "Filling the gap" between the existence results based on the implicit function theorem (Chap. 6) and the existence results based on the minimization of the energy (Chap. 7); An analysis of the nonuniqueness of solutions (cf. the examples given in Sect. 5.8); -
Preface to Volume I
xi
A mathematical analysis of contact with friction (contact, or self-contact, without friction is studied in Chaps. 5 and 7); -
- Finding reasonable conditions under which the minimizers of the energy (Chap. 7) are solutions of the associated Euler-Lagrange equations; While substantial progress has been made in the study of statics (which is all that I consider here), the analysis of time-dependent elasticity is still at an early stage. Deep results have been recently obtained for one space variable, but formidable difficulties stand in the way of further progress in this area. -
This volume will have fulfilled its purposes if the above messages have been conveyed to its readers, that is, if it has convinced its more application-minded readers, such as continuum mechanicists, engineers, "applied" mathematicians, that mathematical analysis is an indispensable tool for a genuine understanding of three-dimensional elasticity, whether it be for its modeling or for its analysis, essentially because more and more emphasis is put on the nonlinearities (e.g., injectivity of deformations, polyconvexity, nonuniqueness of solutions, etc.), whose consideration requires, even at the onset~ some degree of mathematical sophistication; -
if it has convinced its more mathematically oriented readers that three-dimensional elasticity~ far from being a dusty classical field, is on the contrary a prodigious source of challenging open problems. -
Although more than 570 items are listed in the bibliography, there has been no attempt to compile an exhaustive list of references. The interested readers should look at the extensive bibliography covering the years 1678-1965 in the treatise of Truesdell & Noll [1965], at the additional references found in the books by Marsden & Hughes [1983], Hanyga [1985], Oden [1986], and especially Antman [1995], and in the papers of Antman [1983] and Truesdell [1983], which give short and illuminating historical perspectives on the interplay between elasticity and analysis. The readers of this volume are strongly advised to complement the material given here by consulting a few other books, and in this respect~ I particularly recommend the following general references on three-dimensional elasticity (general references on lower-dimensional theories of plates, shells and rods are given in Vols. II and III):
xii
Preface to Volume I
In-depth perspectives in continuum mechanics in general, and in elasticity in particular: The treatises of Truesdell & Toupin [1960] and Truesdell & Noll [1965], and the books by Germain [1972], Gurtin [1981b], Eringen [1962], and Truesdell [1991]. - Classical and modern expositions of elasticity: Love [1927], Murnaghan [1951], Timoshenko [1951], Novozhilov [1953], Sokolnikoff [1956], Novozhilov [1961], Landau & Lifchitz [1967], Green & Zerna [1968], Stoker [1968], Green & Adkins [1970], Knops & Payne [1971], Duvaut & Lions [1972], Fichera [1972a, 1972b], Gurtin [1972], Wang & Truesdell [1973], Villaggio [1977], Gurtin [19Sla], Ne~as & Hlav&~ek [1981], and Ogden [1984]. Mathematically oriented treatments in nonlinear elasticity: Marsden & Hughes [1983], Hanyga [1985], Oden [1986], and the landmark book of Antman [1995]. - The comprehensive survey of numerical methods in nonlinear three-dimensional elasticity of Le Tallec [1994]. -
-
In my description of continuum mechanics and elasticity, I have only singled out two azioms: The stress principle of Euler and Cauchy (Sect. 2.2) and the axiom of material frame-indifference (Sect. 3.3), thus considering that all the other notions are a priori given. The reader interested in a more axiomatic treatment of the basic concepts, such as frame of reference, body, reference configuration, mass, forces, material frame-indifference, isotropy, should consult the treatise of Truesdell & Noll [1965], the books of Wang & Truesdell [1973] and Truesdell [1991], and the fundamental contributions of Noll [1959, 1966, 1972, 1973, 1978]. At the risk of raising the eyebrows of some of my readers, and at the expense of various abus de Iangage, I have also ignored in this volume the difference between second-order tensors and matrices. The readers disturbed by this approach should look at the books of Abraham, Marsden & Ratiu [1983] and, especially, of Marsden & Hughes [1983], where they will find all the tensorial and differential geometric aspects of elasticity explained in depth and put in their proper perspective. Likewise, Vol. III should be also helpful in this respect. This volume is an outgrowth of lectures on elasticity that I have given over the past 15 years at the Tata Institute of Fundamental Research: the University of Stuttgart, the Ecole Normale Sup~rieure,
Preface to Volume I
xiii
and the Universit4 Pierre et Marie Curie. I am particularly indebted to the many students and colleagues I worked with on that subject during the same period; in particular: Michel Bernadou, Dominique Blanchard, Jean-Louis Davet, Philippe Destuynder, Giuseppe Geymonat, Hu Jian-wei~ Srinivasan Kesavan, Klaus Kirchgiissner, Florian Laurent, Herv4 Le Dret, Jind~ich Ne~as, Robert Nzengwa, Jean-Claude Paumier, Peregrina Quintela-Estevez, Patrick Rabier, and Annie Raoult. Special thanks are also due to Stuart Antman, Irene Fonseca, Morton Gurtin, Patrick Le Tallec, Bernadette Miara~ Francois Murat, Tinsley Oden, and G4rard Tronel, who were kind enough to read early drafts of this volume and to suggest significant improvements. For their especially expert and diligent assistance as regards the material realization of this volume, I very sincerely thank H41~ne Bugler, Monique Damperat, and Liliane Ruprecht. August, 1986 and October, 1999
Philippe G. Ciarlet
This Page Intentionally Left Blank
PREFACE
TO VOLUME
II 1
Lower-dimensional plate, shell, and rod, theories that rely on a priori assumptions of a mechanical or geometrical nature have been proposed by A.-L. Cauchy, Sophie Germain, G. Kirchhoff~ T. yon Ks163 A.E.H. Love, E. Reissner~ Jakob Bernoulli, C.-L.-M.-H. Navier, L. Euler, S.-D. Poisson, E. and F. Cosserat, L.H. Donnell, W. Fliigge, S.P. Timoshenko, V.V. Novozhilov, I.N. Vekua, A.E. Green, W.T. Koiter~ J.G. Simmonds~ P.M. Naghdi, and others. There are two reasons why these lower-dimensional theories are so often preferred to the three-dimensional theory that they are supposed to "replace" when the thickness, or the diameter of the crosssection, is "small enough". One reason is their simpler mathematical structure, which in turn generates a richer variety of results. For instance, the existence, regularity, or bifurcation, theories, and more generally the "global analysis", are by now on firm mathematical grounds for nonlinearly elastic rods (see Antman [1995] for a scholarly and comprehensive exposition) or for nonlinearly elastic yon K~rm~n plates (see Ciarlet & Rabier [1980]). By contrast, these theories of global analysis are still partly in their infancies for nonlinear three-dimensional elasticity (see Marsden & Hughes [1983] and Vol. I for comprehensive surveys): After the fundamental ideas set forth by Ball [1977], who was able to establish the existence of a minimizer of the energy for a wide class of realistic nonlinearly elastic materials, there indeed remain manifold challenging open problems; for instance, there is no known set of sufficient conditions guaranteeing that such a minimizer satisfies the equilibrium equations even in the weak sense of the principle of virtual work (another existence theory, based on the implicit function theorem~ does not share this drawback, but it is restricted to problems with smooth data and to special boundary conditions~ unrealistic in practice; see Vol. I and the comprehensive treatment of
V l nt [19ss]). 1A substantial portion of this preface is an excerpt from the "Introduction" in Ciarlet & Lods [1996b].
xvi
Preface to Volume II
Another virtue of lower-dimensional theories is their far better amenability to numerical computations. For instance, directly approximating the three-dimensional displacement field of a cooling tower seems out of reach at the present time, even in the linearly elastic realm: The existing codes use two-dimensional equations, such as those of W.T. Koiter; see Bernadou [1994] for a comprehensive account. Likewise, although substantial progress has recently been achieved for directly approximating the "three-dimensional" displacement field of a linearly elastic rectangular plate, current codes are almost invariably based on two-dimensional equations, such as those of the Kirchhoff-Love or Reissner-Mindlin theories, whose numerical approximation is by now on essentially safe theoretical grounds; see, e.g., Ciarlet [1978, 1991], Glowinski [1984], Hughes [1987], Robert & Thomas [1991], Brezzi & Fortin [1991], Brenner & Scott [1994], Destuynder &: Salaun [1996]. Be that as it may, the locking phenomenon~ and the proper handling of boundary layers, in the numerical approximation of twodimensional plate or shell equations still pose challenging problems; for plates, see notably Arnold [1981], Bathe & Brezzi [1985], Brezzi & Fortin [1986], Hughes & Franca [1988], Pitk~iranta [1988], Bathe, Brezzi & Fortin [1989], Arnold & Falk [1989, 1990, 1996], Brezzi, Fortin & Stenberg [1991], Arnold & Brezzi [1993], Lyly, Stenberg & Vihinen [1993], Chenais & Paumier [1994], Schwab [1994, 1996], Schwab & Suri [1994], Schwab & Wright [1995], Suri, Sabu~ka & Schwab [1995], Pitk~iranta & Suri [1996]. Lower-dimensional models being thus widely used, two essential, and in fact intimately related, questions arise: Given a 'flower-dimensional" elastic body, together with specific loadings and boundary conditions, how to choose between the mani]old lower-dimensional models that are available? For instance, given a linearly elastic shell, which theory should be preferred, among those of Koiter, Naghdi, Novozhilov, Budiansky-Sanders, etc.? This question is of paramount practical importance, for it makes no sense to devise accurate methods ]or approximating the solution of a "wrong" model! Consequently, before approximating the exact solution of a given lower-dimensional model, we should first know whether it is "close enough" to the exact solution of the three-dimensional model it is intended to approximate. This observation leads to the second question:
Preface to Volume H
xvii
How to mathematically justify in a rational fashion a lower-dimensional model from the three-dimensional model? This question has been answered through three different approaches (only scant references are given here to these approaches, as many additional ones are provided throughout the text). The first approach consists in directly estimating the difference between the three-dimensional solution and the solution of a given, i.e., "known in advance", lower-dimensional model (this difference makes sense once the three-dimensional solution is properly averaged or the lower-dimensional one is extended in some fashion to a three-dimensional field). For linearly elastic plates, the first such estimate seems to be due to Morgenstern [1959], who cleverly used the Hellinger-Reissner variational principle of the linear theory; see also Morgenstern & Szab6 [1961], Goldenveizer [1969], Nordgren [1971, 1972], Simmonds [1971a], Ladev~ze [1976, 1980], Shoiket [1976], and Kohn & Vogelius [1985]. This approach was likewise successfully applied to linearly elastic shells by Koiter [1970], Simmonds [1971b], and Koiter & Simmonds [1973]. The second approach, essentially due to Naghdi [1972] for plates and shells, consists in using a hierarchic method, whose governing principle is an a priori assumption that the admissible displacement fields are restricted to a specific form. For a plate (to fix ideas), such "test functions" are finite sums of products of unspecified functions of the in-plane variables times given linearly independent functions of the "transverse" variable. The functions of the in-plane variables are then determined by inserting these test functions into the threedimensional equations or into the three-dimensional energy, a process that leads to the solution of a finite number of two-dimensional boundary value problems. Increasing the number of linearly independent functions of the transverse variable thus yields a "hierarchy" of models, which may be deemed two-dimensional, as they are determined by solving two-dimensional problems. References to this approach are numerous. For plates, see notably Naghdi [1972], Destuynder [1980, Chap. 5], Miara [1989], Sabu~ka & Li [1991, 1992], Schwab [1994, 1995, 1996], Alessandrini, Arnold, Falk & Madureira [1999], and Madureira [1999]; for rods, see Antman
[1972],
bur
[ 992], M sca e.has
T abur
[ 992],
Figueiredo & Trabucho [1993], and Antman [1995]; for shallow shells, see Figueiredo & Trabucho [1992]; for a general analysis, see Antman [1976] and Antman & Marlow [1991]. See also the related "con-
xviii
Preface to Volume H
straint method", advocated by Podio-Guidugli [1989, 1990] for modeling plates and shells. These two approaches nevertheless rely on some a priori assumptions of a mechanical or geometrical nature, intended to account for the "smallness" of a geometrical parameter and intended to be more effective as this parameter approaches zero. Hence the need arises to mathematically justify these a priori assumptions, together with
the lower-dimensional theories they engender, directly from threedimensional elasticity. Otherwise, these assumptions and theories can be thought of as being "handed down by some higher power (a Hungarian wizard, say)", to quote Truesdell [1978]. This direct justification is achieved by the third approach, which consists in applying an asymptotic method. It has recently received considerable attention, as exemplified by the books of Destuynder [1986] and Ciarlet [1990] for plates; Le Dret [1991] for plates and beams (straight rods); Trabucho & Viafio [1996] for beams; and Ciarlet [1990], Le Dret [1991], Kozlov, Maz'ya & Movchan [1999] for multi-structures. In a formal asymptotic method, the three-dimensional solution (the displacement field and, in some cases, the stress field) is first "scaled" in an appropriate manner so as to be defined on a fixed domain, then expanded as a formal series expansion in terms of a "small" parameter e, which is the "dimensionless" half-thickness of a plate or a shell, or the "dimensionless" diameter of the cross-section of the rod. "Dimensionless" means that e measures the ratio between the thickness or diameter and some "characteristic" dimension. For a cooling tower, for instance, where common values for the average thickness and height are 0.3m and 150m, the ratio 2e is thus equal to 1/500. It is worthwhile to keep in mind this order of magnitude. The formal series expansion of the scaled three-dimensional solution is then inserted into the three-dimensional problem, and sufficiently many factors of the successive powers of ~ found in this fashion are equated to zero until the leading term of the expansion can be computed and, presumably, identified with the scaled solution of a known lower-dimensional problem. Such a method is "formal" in that the series is not expected to converge (as an infinite series in powers of e); in fact, the successive terms of the expansion, except the leading one, cannot usually satisfy the boundary conditions of the three-dimensional problem! This situation is typical of such singular perturbation problems; see in this respect the comprehensive
Preface to Volume H
xix
treatments given in Lions [1973] and Eckhaus [1979]. The fundamental contributions of Priedrichs & Dressler [1961] and Goldenveizer [1962, 1964] for plates, Rigolot [1972, 1976] for rods, Goldenveizer [1963, 1964] for shells, are among the first successful attempts to apply formal asymptotic methods in linearized elasticity. Some restrictions or a priori assumptions were, however, still needed. Another shortcoming is the lack of convergence theorems of the scaled three-dimensional solution to the leading term of its formal expansion as ~ --+ 0, essentially because the asymptotic method is applied in these works to the partial differential equations of the three-dimensional problem; in this case, convergence results usually rely on a maximum principle (see Eckhaus [1979]), which does not hold for the system of linearized three-dimensional elasticity. Ciarlet & Destuynder [1979a, 1979b] applied instead the formal asymptotic method to the variational, or weak, formulation of the three-dimensional boundary value problems of linearly and nonlinearly elastic plates. Without making any a priori assumption of a mechanical or geometrical nature, they justified in this fashion the linear and nonlinear KirchhofJ-Love plate theories (only the magnitudes of the components of the applied loads and of the Lam4 constants must behave as appropriate powers of the thickness, but, as shown in a systematic way by Miara [1994a, 1994b], such asymptotic behaviors are unavoidable). This approach was extended to yon Kdrmdn plates by Ciarlet [1980], to Marguerre-von Kdrmdn shallow shells by Ciarlet & Paumier [1986] and Busse [1997], to general nonlinear constitutive equations by Davet [1986], to nonlinearly elastic plates with varying thickness by Quintela-Estevez [1989], and to nonlinear elastodynamics by Raoult [1988] and Karwowski [1993]. By allowing a larger class of behaviors on the applied loads, Fox, Raoult & Simo [1993] were also able to justify in this fashion twodimensional nonlinear "planar membrane" and "planar flezural " theories that are valid for "large" deformations and frame-indifferent, in that they share the same invariances as the three-dimensional theory (while Miara [1994b] assumed at the outset that the nonlinear twodimensional models found by the formal asymptotic method have to reduce to the classical ones once linearized, this assumption was not made by Fox, Raoult & Simo [1993], who were thus able to consider other classes of behaviors).
xx
Preface to Volume H
The one-dimensional equations of a nonlinearly elastic beam (a beam is a straight rod) were likewise justified by Cimeti~re, Geymonat, Le Dret, Raoult & Tutek [1988] and Karwowski [1990]. Nonlinear beam theory has also been related to the three-dimensional theory by Mielke [1988, 1990], who justified St Venant's principle by a remarkable use of the center manifold theorem. Various classes of lower-dimensional equations modeling "shallow" or "arbitrarily curved" nonlinearly elastic rods have been similarly identified by Karwowski [1996a, 1996b]. The most noticeable virtue of the asymptotic method applied to the weak formulation of linear elasticity problems is its amenability to a rigorous asymptotic analysis, which shows that the threedimensional scaled solution converges in some Hilbert spaces (H 1 or L 2) to the leading term of the formal asymptotic expansion as the "small" parameter approaches zero. Such convergence theorems have been established by Destuynder [1980, 1981], CaiUerie [1980], Ciarlet g~ Kesavan [1981], Kohn gz Vogelius [1984, 1985, 1986], Raoult [1985], Blanchard & Francfort [1987], Paumier [1991], Cioranescu & Saint Jean Paulin [1995, 1999], Destuynder & Gruais [1995], Aganovi(3, Maru~iS-Paloka & Tutek [1995], Dauge gz Gruais [1996, 1998a, 1998b], Paumier & Raoult [1997], Aganovid, Jurak, Maru~id-Paloka & Tutek [1998], Dauge, Djurdjevic & RSssle [1998a, 1998b], Andreoiu [1999b], Dauge, Djurdjevic, Faou & RSssle [1999], Dauge, Gruais 8r RSssle [1999], Djurdjevic [1999], RSssle [1999a, 1999b], and RSssle, Bischoff, Wendland & Ramm [1999] for linearly elastic plates; Ciarlet g~ Miara [1992a] and Busse, Ciarlet gz Miara [1996] for linearly elastic shallow shells; Bermudez & Viafio [1984], Aganovi~ & Tutek [1986], Geymonat, Krasucki & Marigo [1987], Trabucho & Viafio [1987], Raoult [1988], Veiga [1995], and Le Dret [1995] for linearly elastic beams (see also the comprehensive survey of Trabucho gc Viafio [1996] and the works cited therein). Special mention must also be made of the approach of Mielke [1995], who keeps the thickness fixed, but lets the lateral boundary of the plate "go away to infinity". In these works, the proofs essentially rely on the ideas and methods described and developed in Lions [1973] for analyzing "abstract" linear variational problems that contain a small parameter. Convergence theorems can also be obtained from F-convergence theory, as in Bourquin, Ciarlet, Geymonat g~ Raoult [1992] and Anzellotti, Baldo g~ Percivale [1994] for linearly elastic plates. A remarkable feature of F-convergence theory is that it also led to the
Preface to Volume H
xx2
first convergence result for planar nonlinearly elastic bodies, due to Le Dret & Raoult [1995a] who themselves based their approach on that of Acerbi, Buttazzo & Percivale [1991] for strings. After the earlier formal attempts of A.L. Goldenveizer cited supra, a first major step for linearly elastic shells was achieved by Destuynder [1980] in his doctoral dissertation (see also Destuynder [1985]), where a convergence theorem for membrane shells was "almost proved"; another major step was achieved by Sanchez-Palencia [1990], who clearly delineated the kinds of geometries of the middle surface and boundary conditions that yield either two-dimensional membrane, or two-dimensional flezural, equations when the method of formal asymptotic expansions is applied to the variational equations of three-dimensional linearized elasticity (see also Caillerie & SanchezPalencia [1995b] and Miara & Sanchez-Palencia [1996]). Then Ciarlet & Lods [1996b, 1996d] and Ciarlet, Lods & Miara [1996] carried out an asymptotic analysis of linearly elastic shells that covers all possible cases: Under three distinct sets of assumptions on the geometry of the middle surface, the boundary conditions, and the order of magnitude of the applied forces, they established convergence theorems in H 1, in L 2, or in ad hoc completion spaces, that justify either the linear two-dimensional equations of an "elliptic membrane shell", or those of a "generalized membrane shell", or those of a "flezural shell". Combining these convergences with results of Destuynder [1985] and Sanchez-Palencia [1989a, 1989b, 1992] (see also Sanchez-Hubert & Sanchez-Palencia [1997]), Ciarlet & Lods [1996c] have also justified the well-known two-dimensional Koiter equations of a linearly elastic shell (Koiter [1970]), again in all possible cases. For nonlinearly elastic shells, a first noteworthy achievement is due to John [1965, 1971], who showed that, in the absence of surface loads and "away from the edge", the state of stress is "approximately planar" and the stresses "parallel to the middle surface" vary "approximately linearly" across the thickness if the thickness is sufficiently small. These remarkable results laid the ground for the two-dimensional, linear and nonlinear, shell theories of Koiter [1966, 1970] and Koiter & Simmonds [1973]. However, in spite of their elegance and depth, John's results hold only for special cases of loadings; besides, they do not provide information "up to the boundary" (of the middle surface of the shell), let alone about the boundary conditions of the associated two-dimensional problem.
xxii
Preface to Volume H
What is especially remarkable in John's analysis (based on exceedingly delicate a priori estimates) is that the constitutive equation is of the most general form. His results were therefore the first indication that general stress-strain laws could also be successfully handled by an asymptotic analysis. Formal asymptotic methods can be applied to nonlinearly elastic shells as well. In this direction, the earlier attempts of Green [1962] and Green & Naghdi [1965], also described in Green & Zerna [1968, Chap. 16]~ are particularly worthy of interest. A major step was then achieved by Miara [1998] and Lods & Miara [1998], who showed in this fashion that the leading term of the asymptotic expansion of the scaled three-dimensional displacement~ again in terms of the thickness as the "small" parameter~ can be identified with the solution of nonlinear two-dimensional "membrane", or "flexural", shell equations, according to the geometry of the middle surface and the boundary conditions as in the linear case. See also Rao [1994] for spherical shells. Another approach has been proposed by Ge, Kruse & Marsden [1996] for justifying time-dependent, nonlinear Cosserat shell theories. Based on the Hamiltonian structure of the equations of threedimensional nonlinear elastodynamics, this approach combines the features of both the hierarchic and asymptotic methods; see also Kirchg/issner & Djurdjevic [1997]. Another major step is due to Le Dret & Raoult [1996], who established the first convergence theorem for nonlinearly elastic shells. To this ends they used r-convergence theory for justifying a nonlinear "membrane" shell model (which coincides with that obtained by Miara [1998] only for specific classes of deformations). Linear and nonlinear shell theories constitute the themes of Volume III. The objective of this volume is to show how asymptotic methods, with the thickness as the "small" parameter~ indeed provide a powerful means of justifying two-dimensional plate theories. More specifically~ without any recourse to any a priori assumptions of a geometrical or mechanical nature, it is shown in the linear case that the three-dimensional displacements~ once properly scaled, converge in H i towards a limit that satisfies the well-known twodimensional equations of the linear Kirchhoff-Love theory; the convergence of the stresses is also established.
Preface to Volume H
xxiii
In the nonlinear case, again after ad hoc scalings have been performed, it is shown that the leading term of a formal asymptotic expansion of the three-dimensional solution satisfies well-known twodimensional equations, such as those of the nonlinear Kirchhoff-Love theory, or the yon Kdrmdn equations. Special attention is also given to the first convergence result obtained by Le Dret & Raoult [1995a] in this case, which leads to two-dimensional large deformation, ~ameindifferent, nonlinear planar membrane theories. It is also shown that asymptotic methods can likewise be used for justifying other "lower-dimensional" theories, some known, some new, such as the two-dimensional equations of elastic shallow shells, and the coupled "pluri-dimensionar' equations of elastic multi-structures, i.e., structures with junctions. In each case, the existence, uniqueness or multiplicity, and regularity of solutions to the "limit" equations obtained in this fashion are also studied. Although I have chosen here the viewpoint of asymptotic methods, I have simultaneously tried to provide reasonable introductions, and references, to other approaches, such as the Reissner-Mindlin theory, Naghdi's theory, hierarchic plate theories, theories derived by the constraint method, etc. Fortunately, there remains an abundance of challenging open problems; for instance: -
Finding a rigorous justification of the Reissner-Mindlin equa-
tions; - Justification of the nonlinear Kirchhoff-Love theory by a convergence theorem as the thickness approaches zero; Existence of solutions of three-dimensional nonlinear plate problems obtained through a proper extension of the two-dimensional solutions (known to exist); -
- Existence theory for two-dimensional plate equations, without any restrictions on the boundary conditions or on the magnitude of the applied forces; Numerical comparison (essentially lacking at the present time, even in the linear case) between three-dimensional and two-dimensional solutions of plate equations; etc. -
xxiv
Preface to Volume H
For the reader's convenience, this volume is written in such a way that, to a large extent, each chapter can be read independently of the others" For instance, a devotee of the yon Ks equations may proceed directly to Chap. 5, without having necessarily mastered Chaps. 1 to 4, although I obviously do not wish that this always be the case! To this end, each chapter begins with a substantial introduction detailing the scalings and assumptions on the data, and expounding the main ideas and results. A reader in a hurry may thus get a quick idea of the content of this volume by reading the introductions of the five chapters; consulting the preliminary sections titled "Plate equations at a glance" and "Shallow shell equations at a glance" should also be helpful in this respect. As in Vol. I, I have tried to provide a "reasonably complete" bibliography, but in view of the formidable existing literature on plates, I am also well aware that the appended list of references is far from being exhaustive. I apologize for any significant reference that I may have inadvertently overlooked. This volume is an outgrowth of series of lectures that I have given during the past twenty years at Tel Aviv University, Fudan University, the University of Stuttgart, the University of Bucharest, the Ecole Polytechnique F6d6rale de Lausanne, Neresheim (Seminar der Deutschen Mathematiker-Vereinigung), the EidgenSssische Technische Hochschule (Ziirich), the Chinese University of Hong Kong, and the Universit6 Pierre et Marie Curie. Substantial portions of the manuscript were also completed during stays at many other places, notably at New York University (Courant Institute of Mathematical Sciences), Cornell University (Mathematical Sciences Institute), Brown University, the Istituto Mauro Picone (Roma), the University of Texas at Austin, Kyoto University, Stanford University, the Universidade de Santiago de Compostela, and the Istituto di Analisi Numerica (Pavia). I am in this respect particularly indebted to my hosts in all these institutions, as their kind hospitality greatly contributed to the completion of this enterprise! The support of the project "Junctions in Elastic Multi-Structures" of the European Cooperation "S.C.LE.N.C.E." Programme is also gratefully acknowledged. This volume is also an updated, completely re-organized, and considerably expanded version (about twice as long) of my earlier
Preface to Volume H
xxv
monograph "Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis"~ co-published in 1990 by Masson, Paris, and Springer-Verlag, Berlin. These lecture notes were based on a course taught over the years at the Universit4 Pierre et Marie Curie, Paris, as part of our Doctoral School "D.E.A. d'Analyse Num4rique". This volume is for the most part the result of joint efforts. I am deeply indebted in this respect to Philippe Destuynder~ Herr6 Le Dret, Patrick Rabier, and Annie Raoult, whose fundamental contributions and kind cooperations were essential to the success of this enterprise. I am also very grateful to the many other students and colleagues who either collaborated with me on, or brought their own contributions to, the theory of plates: Martial Aufranc, Michel Bernadou, Dominique Blanchard, Fr6d4ric Bourquin, Monique Dauge, Jean-Louis Davet, Giuseppe Geymonat, Isabelle Gruais, Fr4d6ric d'Hennezel, Srinivasan Kesavan, Christophe Lebeltel, Bernadette Miara, Robert Nzengwa, Paula Oliveira, Jean-Claude Paumier, Peregrina QuintelaEstevez, Jos6 M. Rodr/guez, Luis Trabucho de Campos, Juan M. Viafio Rey, Xiang Yan. Special thanks are also due to Daniel Coutand, Karine Genevey, Herr6 Le Dret, Cristinel Mardare, Bernadette Miara, Arnaud Montenay, V6ronique Lods, and Sebastian Slicaru, who were kind enough to read preliminary versions of the manuscript and to propose many improvements. I express my particular appreciation to Stuart Antman for the manifold "grammatically elastic" advices he provided me with over the years; he is in particular responsible for suggesting the convenient terminologies "nonlinearly elastic" and "linearly elastic" that I so often use. I also thank Genevieve Raugel and Alice Traynard, who kindly translated for me the seminal article of yon Ks163 [1910]. A reproduction of p. 350 of this article, where the celebrated "yon Kgrm~n equations" appeared for the first time in print, is shown on page lxiii. Last but not least, I express my heartfelt gratitude to Mathieu Ciarlet, who greatly helped me through the arduous task of compiling the bibliography. January, 1997 and October, 1999
Philippe G. Ciarlet
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PREFACE
TO VOLUME
III 1
The objective of this volume is to lay down the proper mathematical foundations of the two-dimensional theory of shells. To this end, it provides, without any recourse to any a priori assumptions of a geometrical or mechanical nature, a mathematical justification of two-dimensional nonlinear and linear shell theories, by means of asymptotic methods, with the thickness as the "small" parameter. A major virtue of this approach is that it naturally leads to precise mathematical definitions of "membrane" and "flezural" shells, be they nonlinearly or linearly elastic. Another noteworthy feature is that it automatically provides in each case the "limit" two-dimensional energy, together with the function space over which it should be minimized. This process highlights in particular the rSle played by two fundamental tensors, each associated with a displacement field of the middle surface, the change of metric and change of curvature tensots (either in eztenso in the nonlinear theories, or in their linearized version in the linear theories). More specifically, under fundamentally distinct sets of assumptions bearing on the geometry of the middle sur]ace, on the boundary conditions, and on the order of magnitude o] the applied forces, it is shown t h a t in the linear case, the three-dimensional displacements, once properly scaled, converge (in H I, or in L 2, or in ad hoc completions) towards a "two-dimensional" limit that satisfies either the
linear two-dimensional equations of a "membrane" shell (themselves divided into two subclasses) or the linear two-dimensional equations
of a "flezural" shell. Under the same assumptions, the two-dimensional linear Koiter equations are also justified for all the above classes of linearly elastic shells. The existence and uniqueness of solutions to each of these linear 1The preceding "Preface to Volume Ir' should be read in conjunction with the present one, since most of its content is as relevant to shell theory as it is to plate theory.
xxviii
Preface to Volume III
equations are also studied in detail, essentially by means of crucial inequalities of Korn's type on surfaces. In the nonlinear case, again under fundamentally distinct sets of assumptions on the geometry of the middle surface, on the boundary conditions, and on the order of magnitude of the applied forces, it is shown that the leading term of a formal asymptotic expansion of the three-dimensional solutions, once properly scaled, satisfies either the nonlinear two-dimensional equations o] a "membrane" shell, or the nonlinear two-dimensional equations of a "flexural" shell. Another class of two-dimensional equations for a nonlinearly elastic "membrane" shelly this time justified by means of a convergence theorem, is also discussed. Finally the existence of solutions to the "limit" nonlinear equations obtained in the above fashions is also studied. Thanks to the timely availability of S.S. Antman's admirable book "Nonlinear Problems of Elasticity" (Springer-Verlag, 1995), I do not dwell, however, on the approach that consists in "directly" viewing a shell as a two-dimensional deformable body, in the spirit of the Cosserat shell theories. Fortunately, there remains an abundance of challenging open problems; for instance: A justification, by a convergence theorem as the thickness approaches zero, of the nonlinear membrane and flexural theories obtained by a formal method in Chaps. 8 to 10; - A clarification of the respective applicabilities of the two nonlinear membrane shell theories described in Chap. 9; A higher-order asymptotic analysis, including error estimates, for linearly elatic shells (such an analysis has been successfully developed for linearly elastic plates; cf. Vol. II, Sect. 1.12); A "local" theory of shells, accounting for the possible coexistence of regions with a "membrane-dominated" behavior with ones with a "flexural-dominated" behavior (as the theory presented in this volume is a "limit" one, it gives rise to a single, "global", behavior); - An analysis of singularities "away from the boundary" (i.e., other than boundary layers) that appear in some "generalized membrane" shells; - An asymptotic justification of Naghdi's shell equations; A refined classification of nonlinearly elastic shells (the definitions proposed in this volume still do not account for some examples -
-
-
-
Preface to Volume III
xxix
and do not seem to be entirely coherent with the definitions of the linear theory; see the commentaries in Sects. 9.1 and 10.2); A numerical comparison between three- and two-dimensional solutions, so as to test the "range of validity" (in terms of data) of the limit two-dimensional equations; etc. -
Naturally, this volume relies heavily on elementary notions from differential geometry in I~3. However, I do not assume any a priori knowledge of this subject, whose needed prerequesites are expounded at length in the first two chapters. This volume is thus entirely selfcontained in this respect. Be that as it may, this volume is neither a treatise on differential geometry nor one on tensor analysis: The incursions into these fields have been kept to the minimum required here (for instance, the definitions of a "tensor" is nowhere to be found in this volume). In the same spirit, I have consistently given explicit, but occasionally lenghty, formulas rather than more condensed, intrinsic, ones. This patti pris may irritate some differential geometers or shell aficionados, but it should render the book more accessible to browsers and non-experts; for the same reason, I have systematically avoided using the (otherwise quite convenient!) terminology "immersion". At the expense of intentional repetitions, most chapters are written in such a way that they are as sel]-contained as possible. I apologize to those readers, brave enough to read the entire volume, who might be annoyed by such a procedure. As in Vol. II, each chapter begins with a substantial introduction expounding its main results. A reader in a hurry may thus get a quick idea of the contents of this volume by reading the introductions of the eleven chapters. Consulting the preliminary sections titled "Twodimensional linear shell equations at a glance" and "Two-dimensional nonlinear shell equations at a glance" should also be helpful in this respect. I have tried to provide a "reasonably complete" bibliography about "mathematical" shell theory (only scant incursions are otherwise provided into the engineering or computationally-oriented literature), but I am also well aware that the appended list of references is far from being exhaustive: Shell theory has offered so many challenges during its long history that it has generated an immense literature.
xxx
Preface to Volume III
As a complement, I advise the readers of this volume to consult great classics, such as Fliigge [1934], Pogorelov [1956, 1966, 1967], Vlasov [1958], Novozhilov [1959, 1970], Goldenveizer [1961], Mushtari
a C limo [ 961],
a
[1968], Timo he o
Womow ky-
Krieger [1970], Fliigge [1973], Rutten [1973], as well as more recent treatments, such as Vinson [1974], Pietraszkiewicz [1977, 1979], Lukasiewicz [1979], Dikmen [1982], Calladine [1983], Sasar & Kr~itzig [1985], Niordson [1985], Vekua [1986], Axelrad [1987], Gould [1987], Pogorelov [1988], Destuynder [1990], Dym [1990], Sernadou [1994], Antman [1995], Stolarski, Selytschko & Lee [1995], Valid [1995], Sanchez-Hubert & Sanchez-Palencia [1997], Libai & Simmonds [1998], Vorovich [1999]. I also recommend the short, but illuminating, accounts by Sanchez-Palencia [1995] and Chapelle & Bathe [1998a] of the often elusive behavior of shells and of the difficulties inherent in their finite element analysis. Substantial portions of the manuscript were completed during stays at the Romanian Academy (Bucharest), the University of Bucharest, the Istituto di Analisi Numerica (Pavia), Fudan University, the University of Stuttgart, and the Liu Bie-ju Centre for Mathematical Sciences of the City University of Hong Kong. I am deeply indebted to my hosts in these institutions: Viorel Barbu, Marius Iosifescu, George Dinc~, Franco Brezzi, Li Ta-tsien, Klaus Kirchg~ssnet, and Roderick Wong. In particular, the first three chapters are an outgrowth of a series of lectures that I delivered at the University of Stuttgart, thanks to the Alexander yon Humboldt-Stiftung, whose support is gratefully acknowledged. These lectures, which greatly beneficiated from the critical comments of Mariana Haragus-Courcelle, Ivica Djurdjevie, and Andreas RSssle, later became a monograph "Introduction to Linear Shell Theory", co-published in 1998 by Gauthier-Villars, Paris and North-Holland, Amsterdam; cf. Ciarlet [1998c]. The support of the project "Shells: Mathematical Modeling and Analysis, Scientific Computing" by the "Human Capital and Mobility" Programme of the Commission of the European Communities is also gratefully acknowledged. This project culminated in an "International Conference on Shells", organized with the highest maestria by Juan Manuel Viafio in Santiago de Compostela in 1997. I also wish to express my deep appreciation to the far-sightedness of those who created the Institut Universitaire de France. Without
Preface to Volume III
xxxi
this remarkable institution, the writing of this volume would have required several additional years! This volume is for the most part the result of joint efforts. I am deeply indebted in this respect to Michel Bernadou, Philippe Destuynder, Herv~ Le Dret, V~ronique Lods, Cristinel Mardare, Bernadette Miara, Annie Raoult, and Evariste Sanchez-Palencia, whose fundamental contributions and kind cooperations were essential to the success of this enterprise. I am very grateful to the many other students or colleagues who also contributed in some way to the content of this volume: Georgiana Andreoiu, Adel Blouza, St~phane Busse, Denis Caillerie, Ma'it~ Carrive, Dominique Chapelle, Christophe Collard, Daniel Coutand, Monique Dauge, Isabel de Figueiredo, Karine Genevey, Giuseppe Geymonat, Patrick Giroud, Liliana Gratie, Oana Iosifescu, Srinivasan Kesavan, Fran~;ois Larsonneur, Patrick Le Tallec, Arnaud Montenay, Jean-Claude Paumier, Paolo Podio-Guidugli, Olivier Ramos, Rao Bopeng, Anne Roquefort, Sebastian Slicaru, Xiao Li-ming. Special thanks are also due to Elena Baderko, Adel Blouza, Dominique Chapelle, Daniel Coutand, Liliana Gratie, Herv~ Le Dret (in particular, for stimulating discussions about the nonlinear membrane theories!), Cristinel Mardare, Qin Tie-hu, Anne Roquefort, and Karim Trabelsi, who kindly read preliminary versions of the manuscript and suggested many improvements. I renew my particular appreciation for Stuart Antman, whose kind semantic assistance has been an invaluable help over so many years. October 1999
Philippe G. Ciarlet
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TABLE OF CONTENTS'
Mathematical Elasticity: General plan
...........
ii
......... v Preface t o Volume I . . . . . . . . . . . . . . . . . . . . . . . ix xv Preface t o Volume I1 . . . . . . . . . . . . . . . . . . . . . . xxvii Preface t o Volume 111. . . . . . . . . . . . . . . . . . . . . Differential geometry at a glance . . . . . . . . . . . . . . xxxix Mathematical Elasticity: General preface
Three-dimensional elasticity in curvilinear coordinates at a glance . . . . . . . . . . . . . . . . . . . . . . . xlvii Two-dimensional linear shell equations at a glance
...
li
.
lvii
Two-dimensional nonlinear shell equations at a glance
PART A.
LINEAR SHELL THEORY
Chapter 1. Three-dimensional linearized elasticity and Korn's inequalities in curvilinear coordinates Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1. Three-dimensional linearized elasticity in Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Curvilinear coordinates and metric tensor in a threedimensional domain . . . . . . . . . . . . . . . . . . . 1.3. The variational equations of three-dimensional linearized elasticity in curvilinear coordinates . . . . . . . 1.4. Covariant derivatives and Christoffel symbols in a threedimensional domain . . . . . . . . . . . . . . . . . . . 1.5. Linearized change of metric tensor in curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. The boundary value problem of three-dimensionallinearized elasticity in curvilinear coordinates . . . . . .
'
3 3
5 12
22 32 35 37
'The symbol indicates a section where most results are stated without proof.
xxxiv 1.7.
Table of contents
A l e m m a of J. L. Lions; t h r e e - d i m e n s i o n a l K o r n ' s inequalities a n d infinitesimal rigid d i s p l a c e m e n t l e m m a in curvilinear coordinates . . . . . . . . . . . . . . . .
40
Existence a n d uniqueness t h e o r e m in curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . .
50
1.9 ~. Complement" Recovery of a t h r e e - d i m e n s i o n a l manifold from its metric tensor field . . . . . . . . . . . . .
54
1.8.
Exercises
Chapter
. . . . . . . . . . . . . . . . . . . . . . . . .
2. I n e q u a l i t i e s o f K o r n ' s t y p e o n s u r f a c e s Introduction
. . .
.......................
56 61 61
2.1.
Curvilinear coordinates a n d metric tensor on a surface
63
2.2.
C u r v a t u r e tensor on a surface
73
2.3.
Covariant derivatives a n d Christoffel s y m b o l s on a surface . . . . . . . . . . . . . . . . . . . . . . . . . .
85
2.4.
Linearized change of metric tensor on a surface . . . .
90
2.5.
Linearized change of c u r v a t u r e tensor on a surface . .
93
2.6.
Inequalities of K o r n ' s t y p e a n d infinitesimal rigid disp l a c e m e n t l e m m a on a general surface . . . . . . . . .
100
I n e q u a l i t y of K o r n ' s t y p e a n d infinitesimal rigid disp l a c e m e n t l e m m a on an elliptic surface . . . . . . . .
117
2.8 ~. C o m p l e m e n t - Recovery of a surface from its m e t r i c a n d c u r v a t u r e tensor fields . . . . . . . . . . . . . . .
130
2.7.
Exercises
Chapter
. . . . . . . . . . . . . . . . . . . . . . . . .
132
3. A s y m p t o t i c a n a l y s i s o f l i n e a r l y e l a s t i c s h e l l s : Preliminaries and outline ............ 137 Introduction
3.1.
.............
.......................
T h e t h r e e - d i m e n s i o n a l e q u a t i o n s of a linearly elastic shell . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137 141
3.2.
T h e t h r e e - d i m e n s i o n a l equations over a d o m a i n independent ofe .......................
149
3.3.
G e o m e t r i c a l a n d mechanical preliminaries . . . . . . .
154
3.4.
T h e t w o - d i m e n s i o n a l equations of linearly elastic " m e m b r a n e " a n d "flexural" shells derived by m e a n s of a f o r m a l a s y m p t o t i c analysis . . . . . . . . . . . . . . . 161
3.5.
S u m m a r y of the convergence t h e o r e m s . . . . . . . . .
183
Exercises
190
. . . . . . . . . . . . . . . . . . . . . . . . .
xxxv
Table of contents
Chapter 4. Linearly elastic elliptic membrane shells Introduction
. . 193
.......................
193
4.1.
L i n e a r l y elastic elliptic m e m b r a n e shells: Definition, example, a n d a s s u m p t i o n s on the data; the threed i m e n s i o n a l e q u a t i o n s over a d o m a i n i n d e p e n d e n t ofr . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.
Averages w i t h respect to the t r a n s v e r s e variable
4.3.
A t h r e e - d i m e n s i o n a l inequality of K o r n ' s t y p e for a family of linearly elastic elliptic m e m b r a n e shells . . . 205
4.4.
C o n v e r g e n c e of the scaled displacements as r --4 0
4.5.
T h e t w o - d i m e n s i o n a l e q u a t i o n s of a linearly elastic elliptic m e m b r a n e shell; existence, uniqueness, a n d regu l a r i t y of solutions; f o r m u l a t i o n as a b o u n d a r y value problem . . . . . . . . . . . . . . . . . . . . . . . . . .
223
J u s t i f i c a t i o n of the t w o - d i m e n s i o n a l e q u a t i o n s of a linearly elastic elliptic m e m b r a n e shell; c o m m e n t a r y a n d refinements . . . . . . . . . . . . . . . . . . . . . . . .
230
Exercises
. . . . . . . . . . . . . . . . . . . . . . . . .
235
Chapter 5. Linearly elastic generalized membrane shells . . . . . . . . . . . . . . . . . . . . . . . . .
241
4.6.
5.1.
196
. . . 201
. . 209
Introduction ....................... 241 L i n e a r l y elastic generalized m e m b r a n e shells: Definition a n d a s s u m p t i o n s on the data; the t h r e e - d i m e n s i o n a l e q u a t i o n s over a d o m a i n i n d e p e n d e n t of r . . . . . . . 245
5.2.
A n a l y t i c a l preliminaries . . . . . . . . . . . . . . . . .
248
5.3.
A t h r e e - d i m e n s i o n a l inequality of K o r n ' s t y p e for a family of linearly elastic shells . . . . . . . . . . . . .
258
G e n e r a l i z e d m e m b r a n e shells of the first a n d second kinds . . . . . . . . . . . . . . . . . . . . . . . . . . .
261
A d m i s s i b l e applied forces . . . . . . . . . . . . . . . .
264
5.4. 5.5. 5.6.
C o n v e r g e n c e of the scaled displacements as ~ --~ 0
5.7.
T h e t w o - d i m e n s i o n a l e q u a t i o n s of a linearly elastic generalized m e m b r a n e shell; existence a n d u n i q u e n e s s of solutions . . . . . . . . . . . . . . . . . . . . . . . .
287
J u s t i f i c a t i o n of the t w o - d i m e n s i o n a l e q u a t i o n s of a linearly elastic generalized m e m b r a n e shell; examples, c o m m e n t a r y , a n d refinements . . . . . . . . . . . . . .
291
Exercises
297
5.8.
. . . . . . . . . . . . . . . . . . . . . . . . .
. . 266
xxxvi
Table of contents
Chapter 6. Linearly elastic flexural shells . . . . . . . . . Introduction
299
.......................
299
6.1.
Linearly elastic flexural shells: Definition, examples, a n d a s s u m p t i o n s on the data; the t h r e e - d i m e n s i o n a l equations over a d o m a i n i n d e p e n d e n t of ~ . . . . . . .
6.2.
Convergence of the scaled displacements as ~ --~ 0
6.3.
T h e two-dimensional equations of a linearly elastic flexural shell; existence a n d uniqueness of solutions . 317
6.4.
Justification of the two-dimensional equations of a linearly elastic flexural shell; c o m m e n t a r y a n d refinements . . . . . . . . . . . . . . . . . . . . . . . . . . .
323
Exercises
328
302
. . 308
.........................
Chapter 7. Koiter's equations and other two-dlmensional linear shell theories . . . . . . . . . . . . . . . . 333 Introduction 7.1.
7.2. 7.3.
.......................
333
T h e two-dimensional Koiter equations for a linearly elastic shell: Existence, uniqueness, a n d r e g u l a r i t y of solutions; formulation as a b o u n d a r y value p r o b l e m . 335 Justification of Koiter's equations for all types of linearly elastic shells . . . . . . . . . . . . . . . . . . . . 345 Koiter's equations: Additional c o m m e n t a r y a n d bibliographical notes . . . . . . . . . . . . . . . . . . . .
360
7.4.
T h e two-dimensional N a g h d i equations for a linearly elastic shell; existence a n d uniqueness of solutions . . 363
7.5.
O t h e r linear shell theories . . . . . . . . . . . . . . . .
367
7.6.
Linear shallow shell theories
369
Exercises
PART
B.
..............
.........................
NONLINEAR
SHELL
372
THEORY
Chapter 8. A s y m p t o t i c a n a l y s i s o f nonlinearly elastic shells: Preliminaries . . . . . . . . . . . . . . . . . . . . 381 Introduction 8.1. 8.2.
.......................
381
T h r e e - d i m e n s i o n a l nonlinear elasticity in C a r t e s i a n coordinates . . . . . . . . . . . . . . . . . . . . . . . . .
386
T h r e e - d i m e n s i o n a l nonlinear elasticity in curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . .
392
Table of contents
8.3.
xxxvii
T h e three-dimensional equations of a nonlinearly elastic shell . . . . . . . . . . . . . . . . . . . . . . . . . .
403
T h e three-dimensional equations over a d o m a i n indep e n d e n t of s . . . . . . . . . . . . . . . . . . . . . . .
407
G e o m e t r i c a l a n d mechanical preliminaries . . . . . . .
411
8.6.
T h e m e t h o d of formal a s y m p t o t i c expansions . . . . .
413
8.7.
T h e leading t e r m is of order zero . . . . . . . . . . . .
415
8.8.
Identification of a two-dimensional variational problem satisfied by the leading t e r m . . . . . . . . . . . .
424
Exercises
.........................
430
Chapter 9. Nonlinearly elastic membrane shells . . . . .
433
8.4. 8.5.
Introduction 9.1. 9.2.
.......................
433
Nonlinearly elastic m e m b r a n e shells: Definition, examples, a n d a s s u m p t i o n s on the d a t a . . . . . . . . .
436
T h e two-dimensional equations as a variational problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
443
9.3.
T h e two-dimensional equations as a m i n i m i z a t i o n problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
9.4.
T h e two-dimensional equations of a nonlinearly elastic m e m b r a n e shell derived by means of a formal asymptotic analysis; c o m m e n t a r y . . . . . . . . . . . . . . .
447
9.5 b. T h e two-dimensional equations of a nonlinearly elastic m e m b r a n e shell derived by means of r - c o n v e r g e n c e theory; c o m m e n t a r y . . . . . . . . . . . . . . . . . . .
453
Exercises
.........................
465
Chapter 10. Nonlinearly elastic flexural shells . . . . . .
469
Introduction
.......................
469
10.1. Identification of a two-dimensional variational problem satisfied by the leading t e r m when there are nonzero admissible inextensional displacements . . . . . . . . 472 10.2. Nonlinearly elastic flexural shells: Definition, examples, a n d a s s u m p t i o n s on the d a t a . . . . . . . . . . .
502
10.3. T h e two-dimensional equations as a variational problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
507
10.4. T h e two-dimensional equations as a m i n i m i z a t i o n problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
Table of contents
xxxviii
10.5. T h e t w o - d i m e n s i o n a l e q u a t i o n s o f a n o n l i n e a r l y e l a s t i c f l e x u r a l shell d e r i v e d b y m e a n s o f a f o r m a l a s y m p t o t i c analysis; c o m m e n t a r y
. . . . . . . . . . . . . . . . . .
521
10.6. E x i s t e n c e o f s o l u t i o n s to t h e m i n i m i z a t i o n p r o b l e m
. 526
Exercises
. . . . . . . . . . . . . . . . . . . . . . . . .
539
C h a p t e r 11. Koiter's equations and other t w o - d i m e n s i o n a l nonlinear shell theories . . . . . . . . . . . . . 545 Introduction
. . . . . . . . . . . . . . . . . . . . . . .
545
11.1. T h e t w o - d i m e n s i o n a l K o i t e r e q u a t i o n s for a n o n l i n -. . . . . . . . . . . . . . . . .
545
11.2. O t h e r n o n l i n e a r shell t h e o r i e s . . . . . . . . . . . . . .
e a r l y elastic shell . . . .
549
11.3. N o n l i n e a r s h a l l o w shell t h e o r i e s
552
References Index
............
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
557 583
DIFFERENTIAL
1. 2. 3. 4.
GEOMETRY
AT A GLANCE
~
General conventions Differential geometry of three-dimensional domains in IR3 Differential geometry of surfaces in R 3 Korn's and other inequalities in curvilinear coordinates
I. G E N E R A L
CONVENTIONS
(i) Latin indices and exponents: i, j, p, . . . , take their values in the set {1, 2, 3}, unless otherwise indicated as when they are used for indexing sequences. (ii) Greek indices and exponents: c~, ~, or,... , except s and u in the outer n o r m a l derivative operator 0~, take their values in the set {1, 2). (iii) T h e repeated index summation convention is systematically used in conjunction with conventions (i) and (ii). (iv) T h e symbol %" designates a p a r a m e t e r t h a t is > 0 and approaches zero.
2. D I F F E R E N T I A L GEOMETRY OF THREE-DIMENSIONAL DOMAINS
I N IR8
a . b: Euclidean inner product of a C ]~3 and b E IR3. a A b: exterior p r o d u c t of a E IR3 and b C IR3. la I: Euclidean n o r m of a E IR3. iOnly the notations and definitions that are specific to shell theory are listed in this section and the next ones, which otherwise complement the section "Main notation, definitions, and formulas" in Vol. I.
xl
Differential geometry at a glance
f~: domain in R z (open, bounded, connected subset of R z with a Lipschitz-continuous boundary, the set f~ being locally on one side of its boundary). x - (mi)" generic point in f~. din" volume element in f~. 0 Ozi F" boundary of f~. dr" area element along r . (ni)" along r .
unit outer normal vector (defined dP-almost everywhere)
(9 9~ C I~3 --+ R z" injective and smooth enough mapping such that the three vectors 0il!i)(m) are linearly independent at each point mEf~. gi = OiO" vectors of the covariant bases in the set (9(f~); this means that, at each x E f~, the three vectors gi(m) - 0 i O ( z ) form the covariant basis at the point (9(x). 9 i" vectors of the contravariant bases in the set (9(~); the vectors 9i(m) are defined at each m E ~ by the relations gi(m) 99j(m) - 6j. covariant components of the metric tensor of the set
g i j -- O i ' O j "
g -- det(gij). gij _ g i . g j , set O(f~).
contravariant components of the metric tensor of the
r ip = gP. Ojgi: Christoffel symbols. Villi -- Ojvi - - ~ i jPV p . 9 covariant derivatives of a vector field r i g i with covariant components vi " f~ -+ IR. oiJ Ilk - Ok crij + Fpko'PJ + P{q ~
covariant derivatives of a tensor .
.
field with contravariant components a zJ 9fl ~ I~. g o ( v ) " covariant components of the metric tensor of the set (o + k eillj(V) -- 1Cgij(V ) - gij] lin : 1
89
" g i + Oiv " g j ) P
9covariant components of the linearized change of metric tensor associated with a =
( ,llJ +
Jll,) -
+
-
Differential geometry at a glance
xli
displacement field ~ - - r i g i of the set | the notation [.-.]tin denotes the linear part with respect to v in [... ]. E Ijj(
) -
- g j)
__
1
-- ~(Vill/+ Villi + gmnvmlliVnllj ) . covariant components of the change of metric tensor, also called the Green-St Venant strain tensor, associated with a displacement field rig i of the set (!i)(~).
3.
DIFFERENTIAL
GEOMETRY
OF
SURFACES
IN
R3
w: domain in I~2 (open, bounded, connected subset with a Lipschitz-continuous boundary, the set w being locally on one side of its boundary). y-
(ya)" generic point in ~.
dy" area element in w. 0 02 - ~, 0~ 0,~ Oy,~ OyaOy~ 7: boundary of the set w. dT: length element along 7. (va): unit outer normal vector along 7. (7"a) with
T 1 --" - - V 2 ,
T 2 --" V l :
unit tangent vector along 7.
Or8 -- vaOaS: outer normal derivative of 8 along 7. 0~-8 -- 7"aOaS: tangential derivative of 8 along 7. 8 9~ C R 2 -+ R 3" injective and smooth enough mapping such that the two vectors OaS(y) are linearly independent at each point yE~. S - 8(~): surface. vectors of the covariant bases of the tangent planes; this means that, at each y E ~, the two vectors OaS(y) form the covariant basis of the plane tangent to S at the point 8(y) E S. aa
-
OaS:
aa: vectors of the contravariant bases of the tangent planes; at each y E ~, the vectors aa(y) of the plane tangent to S at 8(y) are defined by the relations aa(y) 9a~(y) - ~ . a3
--a
3 =
al
Aa2
xlii
Differential geometry at a glance
covariant components of the metric tensor of the
aa~ = a a . a o : surface S.
a = det (aao). a a13 - -
a a.
a ~"
contravariant components of the metric tensor
of S. bao - a 3" O~aa" covariant components of the curvature tensor of S. b~a - aOabaa: mixed components of the curvature tensor of S. F~ 71al~
- a ~ . O~aa" Christoffel symbols. -
-
O ~ r l a - F ~ r / ~ and ~31~ -- O~r/3: covariant derivatives of a
vector field yia / with covariant components r/i : ~ -+ JR. yal~ - 0 ~ a + r ~ r / ~ , covariant derivatives of a tangential vector field ~ T a a a with contravariant components r/a : ~ -+ ]R. flail ~ - - rlal~ -- ba~rl3
and ~311t3- r/31~ - b~r/~.
rla~l~ - O~n a~ + r~nZ" + r{~n ~ . covariant derivatives of a tensor field with contravariant components n a ~ 9 ~ JR.
~1~ -
0~
+ r~bX - r ~ .
co~i~=t derivatives of the
curvature tensor, defined here by means of its mixed components ~ 9~ - + ]R. aa~(r/): covariant components of the metric tensor of the surface
(o + ,7~a~)(~). -~,~(,) - 89[ a ~ ( n ) - aa~] "'~ -- 89(0~,) 9a,~ + 0 ~ 0 " a ~ ) -- ~-(~o1~ + ~ , ~ ~)89 covariant components of the linearized change of metric tensor associated with a displacement field ~ - y i a ' of the surface S; the notation [... ]tin denotes the linear part with respect to r~ in [... ].
~(~)
-
89
. a~ + o~
.
~).
bat,(r/): covariant components of the curvature tensor of the surface (0 + yia')(-~) (only defined at those points in U where the two vectors Oa(O + ~7iai) are linearly independent).
~(~)
-
[ b ~ z ( . ) - b~Z] ~" - (O~Z# - r x ~ o ~ )
9a3
= ~ I ~ - b ~ b ~ + b ~ l ~ + b;~,~ + b ; l ~ = 0~.~
-
r ~ 0~ ~
-
b,~ ~,,.~,7~
+~g(0~.~ - r;~.~) + b;(O~ +(0~b; + r~b~
- r~.~)
- r a~b~)~Tr ~ " covariant components of
Differential geometry at a glance
xliii
the linearized change of curvature tensor associated with a displacement field ~ - rlia i of the surface S; the notation [... Izin denotes the linear part with respect to r / i n [..-]. ~(0)
-
(0~
- r~o~O)
. as.
1 G a ~ ( r l ) - -1~ ( a a ~ ( u ) - - a a ~ ) - ~(rla lf3 +rl~l a +amnrimll,arlnl ~), w h e r e a mn = a m 9an: covariant components of the change of metric tensor associated with a displacement field ~Tiai of the surface S.
Raf~ (r/) - baf~(r/) - baf~" covariant components of the change of metric tensor associated with a displacement field rlia i of the surface S (only defined if the two vectors Oa(O+rlia i) are linearly independent in w).
4. K O R N ' S A N D O T H E R I N E Q U A L I T I E S CURVILINEAR COORDINATES 1
IN
Given a domain a in R n, the norm in L~(a) or 1,2(fl) is noted 10,a and the norm in Hm(fl) or Hm(fl), m >_ 1, is noted II" I[m,a. T h r e e - d i m e n s i o n a l Korn's inequality "without boundary conditions" in curvilinear coordinates (Thm. 1.7-2):
i
i,j
for a11 v - ( v i ) E Hl(f~). T h r e e - d i m e n s i o n a l Korn's inequality in curvilinear coordinates (Thm.
1.7-4): }1/2 IlVlll, a <_ C
~ [eil j(v)[2,a for a11 v e V(fl), where i,j V ( f ~ ) -- {V -- (vi) E H l ( f ~ ) ; v - 0 o n r 0 ) .
1The definitions of the sets f~, w, etc., and of the functions eilij(v), 7a~(r/), etc., used in the subsequent inequalities are given in the preceding subsections.
xliv
Differential geometry at a glance
Uniform positive definiteness of the three-dimensional elasticity tensor (Thm. 1.8-1)"
i,j for all x C 12 and all symmetric matrices (tij).
Inequality of Korn's type "without boundary conditions" on a general surface (Thm. 2.6-1)"
2 }112 Ol
a,~
a
for all n -
a,~
(~i) e Hi(w) • Hi(w) • H2(w).
Inequality of Korn's type on a general surface (Thin. 2.6-4)"
{ ~ [JTlalJ2~,~ +
2 ) 1/2
IIn~ll2,~
for all rl - (yi) e VK (w), where
v ~ ( ~ ) - {n -(n~) e Hl(~) • Hl(~) • H~-(~); ~i - 0~3 - 0 on 70}. Inequality of Korn's type on a general surface with little regularity (Thin. 2.6-6)"
2 )1/2 C{ ~ JSa.(O)[O2,w+ ~ Jpa~(O)[~,w}1/2 for a,~
a11 0 E ~rK(w ),
~,~
where ~rK(w ) -- {0 C I-I~(w); 0 ~ 0 " a3 C L2(w)}.
Differential geometry at a glance
xlv
Second inequality of Korn's type "without boundary conditions" on a general surface (Thm. 2.7-1)"
a
i
a,fl
for all r / - (r/i) e H 1 (w) x H 1(w) x L 2 (w).
Inequality of Korn's type on an elliptic surface (Thm. 2.7-3)"
for all r / - (r/i) C V M ( W ) - H~(w) x H~(w) x L2(w).
Uniform positive definiteness of the scaled two-dimensional elasticity tensor of a shell (Thin. 3.3-2)"
a,13 for all y E ~ and all symmetric matrices (tar3).
Uniform positive definiteness of the scaled three-dimensional elasticity tensor of a shell (Thm. 3.3-2)" ItijJ 2 < CeAiJhl(e)(x)tkltij i,j for all 0 < e < e0, all x E f~, and all symmetric matrices (tij).
Three-dimensional inequality of Korn's type for a family of linearly elastic shells (Thm. 5.3-1)"
[ivjJl, f~
__<,.,=C{EJeiJJJ (~; v)[20,n)1/2 i,j
for a l l O < s < s l
and a l l v E V ( ~ ) .
xlvi
Differential geometry at a glance
Three-dimensional inequality of Korn's type for a family of linearly elastic elliptic membrane shells (Thm. 4.3-1)"
{~lvofl ~1,a +
~}
I/2
2
[V3[o,a
i,j fo~ ~n o < ~ < ~ ~ d
~11 v - ( ~ )
6 V(~).
THREE-DIMENSIONAL ELASTICITY IN CURVILINEAR COORDINATES AT A GLANCE
1. Notations common to linear and nonlinear three-dimensional elasticity 2. Three-dimensional linearized elasticity in curvilinear coordinates 3. Three-dimensional nonlinear elasticity in curvilinear coordinates Frequent use is made in this section of notations whose definitions, not repeated here, are found in the preceding subsection "Differential geometry of three-dimensional domains in ~3,,.
1. N O T A T I O N S NONLINEAR
COMMON TO LINEAR THREE-DIMENSIONAL
AND ELASTICITY
~: domain in It~3 with boundary I~. P - Po U rl" partition of the boundary of ~ with area ro > O. O 9~ C R 3 -+ R 3" injective and smooth enough mapping such that the three vectors 0 i 0 are linearly independent at all points in ~. O ( ~ ) C R 3" reference configuration of an elastic body. AiJkl _ Agijgkl § #(gik gjl § gil gjk). contravariant components of the three-dimensional elasticity tensor.
)~ ) 0 and # > 0" Lam~ constants of a homogeneous, isotropic, elastic material, whose reference configuration is a natural state. fi E L2(g2) 9 contravariant components of the applied body force density. h i C L2(p1) 9 contravariant components of the applied surface force density. u - (ui)" unknown vector field, whose components ui " ~ -+ ]R are the covariant components of the displacement field; this means that u i ( x ) g i ( x ) is the displacement vector of the point O (x) e O (~).
xlviii
Three-dimensionalelasticity in curvilinear coordinates at a glance
2. T H R E E D I M E N S I O N A L LINEARIZED IN CURVILINEAR COORDINATES
ELASTICITY
Minimization problem (Thm. 1.8-2):
u e v(a):=
J(u) = J(.) -
{ . - (~,) e H ~ ( a ) ; ~ - o o n r o ) ,
inf J(v), where ,,~v(n)
A~Jkt~kljt(v)~ltj(.) ~ d~
- { ~ f i v i v ~ d x + fr hiVix/~dr }.
Variational problem (Thm. 1.3-1):
u e V(12) and, for all v e V(~2),
1
Boundary value problem (Thm. 1.6-1):
_aij[[j _ fi in 12, where crij - AiJkleklll(U), ui = 0 on 1"o, ~iJ nj -- h i on r l .
Three-dimensional elasticity in curvilinear coordinates at a glance 3. T H R E E - D I M E N S I O N A L NONLINEAR IN CURVILINEAR COORDINATES
xlix
ELASTICITY
Only St Venant-Kirchhoff materials are considered in this subsection.
Minimization problem (Thm. 8.2-1):
{. -
c w(~).-
J(u)-
J(v) -
inf
(.,) e wx.4(~);.
- o o11 r o ) .
J(v),
lfn AiJktEklll(V)Eillj(v)x/~
dx
Variational problem (Thm. 8.2-3) u E W(w) and, for all v E W(f~),
a AijklEk It(u)(E~llJ(U)v)v/~ dx hiviv/gdF, where
--
1
1 E'illJ ( u ) v - ~ (Villi § Villi § gmn{UmlliVnllJ § UnlljVmlli)) .
Boundary value problem (Thm. 8.2-4): - ( c rij + crkjgilutllk)[lj -- fi in f~, where crij - AiJktEkllt(u),
ui = 0 on to, (orij + crkjgitutllk)n j -- hi on r l .
This Page Intentionally Left Blank
TWO-DIMENSIONAL AT A GLANCE 1
1. 2. 3. 4. 5.
General Linearly Linearly Linearly Koiter's
LINEAR
SHELL
EQUATIONS
notations for shell equations elastic elliptic membrane shells elastic generalized membrane shells elastic flexural shells equations for a linearly elastic shell
All the equations listed in this section (save the last minimization problem) are justified by a convergence theorem. In each case (save the last minimization problem), the shell is subjected to a homogeneous boundary condition of place along the portion O(70 x [-~, ~]) of its lateral face, i.e., the displacement field vanishes there. Frequent use is made of notations whose definitions, not repeated here, are found in the preceding subsection "Differential geometry of surfaces in IR3''.
1. G E N E R A L N O T A T I O N S F O R SHELL E Q U A T I O N S
w: domain in ~2 with boundary 7. 7 = 70 tA 71: partition of the boundary of w with length 7o > O. 0 E CZ(W; IRa) 9 injective and smooth enough mapping such that the two vectors a a = OaO are linearly independent at all points of W. S = O(W): middle surface of the shell. 2~: thickness of the shell. ~=w•
r~=70•
r~=w•177
1Linear shallow shell equations in curvflinear coordinates are not reproduced here, as their justifications rely on asymptotic analyses similar to those found in Vol. II, Chap. 3; they are nevertheless reviewed in Sect. 7.6.
lii
Two-dimensional linear shell equations at a glance
O(y, xl) -- O(y) + x l a 3 ( y ) for all (y, xl) e ~e.
{~(~e) C ]~3. reference configuration of the shell. a~~, ~
4A~# ~ aa~a ~r + 2#e(aa~a ~r + a a r a ~ ) 9 contravariant Ae + 2# ~ components of the two-dimensional elasticity tensor of the shell.
--
Ae > 0 and #e > 0: Lam~ constants of the material constituting the shell.
fi, ~ E L 2(f~e)" contravariant components of the applied body force density acting in the interior O(fl 6) of the shell. hi, e( ., :ke) E L2(w) 9contravariant components of the applied surface force density acting on the "upper" and "lower" faces O(P~_) and O ( I ~ ) of the shell. pi, e _ ~
f
fZ'edx i + hi'e( ., +e) + hi'e( ., - e ) . e
- f' eai.
~e _ ( ~ ) . unknown vector field, whose components ~ 9~ -~ ]R are the covariant components of the displacement field ~ a i of the middle surface S. This means that ~ ( y ) a i ( y ) is the displacement vector of the point O(y) E S.
~e
_
~ a i " unknown displacement field of the middle surface S.
r - (~:,i)" unknown vector field for the linear and nonlinear Koiter equations; the components ~ , i " ~ -+ IR are the covariant components of the displacement field of S.
2. L I N E A R L Y SHELLS
ELASTIC
ELLIPTIC
MEMBRANE
Definition (Sect. 4.1)" A shell is a linearly elastic elliptic membrane shell if its middle surface S is elliptic and the displacement field vanishes on its entire lateral face 0(3' • [-e, e]), i.e., ~'0 - 7.
Two.dimensional linear shell equations at a glance
liii
M i n i m i z a t i o n problem (Thm. 4.5-2):
r
e v~(~)
j~(~)-
- Ho~(~) x Ho~(~) x L~-(~),
inf j~(17) , where nEVM(W)
1Z
/.
Variational problem (Thm. 4.5-2): ~.e E V M ( W ) and, for all ,7 ~ VM(o~),
B o u n d a r y value problem (Thin. 4.5-2): _ n a # , e l # _ pa,~ in w, where n a#,e - eaa~Zr'e7zr(r _ba[3na#, e _ p3, s in w,
~ -- 0 on 3"
3. L I N E A R L Y E L A S T I C G E N E R A L I Z E D MEMBRANE SHELLS Definition (Sect. 5.1): A shell is a linearly elastic generalized m e m brane shell if the space
VF(w) -- { r l - (Yi) C H I ( w ) x Hi(w) x H2(w); ~7i -- 0,~73 -- 0 on 70, 7 ~ ( n )
- 0 in w}
reduces to {0}, but the shell is not a linearly elastic elliptic membrane shell (in which case the space VF(w) also r e d u c e s to {o}). Note: Only linearly elastic generalized membrane shells "of the first kind" (Sect. 5.4) are considered here.
lip
Two-dimensional linear shell equations at a glance
Preliminary definitions (the functions ~pa13 E L2(w) found in the linear form L ~ originate from the assumed "admissibility" of the applied forces; cf. Sect. 5.5 and Thm. 5.6-1):
V(w) - { r / - (r/i) E Hi(w); r / - 0 on V0}, V~M(W) -- completion of V(w) with respect to [. ]M~, where
I.l
- {E
B~((~, rl) - e ~ aa~'~r'e7,~r((~)Vaf3(rl)~dy for ~, r/C V(w), L6M(rl) -- e ~ qoa~')'af3(rl)x/~dy for r/E V(w), B M 9 unique continuous extension of the bilinear form B ~ from V(w)to V~M(W), L M
"
unique continuous extension of the linear form L ~ from V(w)to V~M(W).
Minimization problem (Thm. 5.7-2):
r E V~M(W) and j ~ ( ~ e ) _ 1 ~e n)j~(,)- ~BM(n,
L~
inf
.~v~(~)
j ~ (r/), where
(~).
Variational problem (Thm. 5.7-2):
~s E V~M(W) and
BM(
~)
(T/) for
E V~M(W .
Two-dimensional linear shell equations at a glance
4. L I N E A R L Y
ELASTIC
FLEXURAL
lv
SHELLS
Definition (Sect. 6.1): A shell is a linearly elastic flexural shell if the space v~(~)-
{,1-
(,~) e H~(~) • H~(~) • H~(~); r/i - oq~,v/3 - 0 on 3'0, 7af3(v/) -- 0 in w }
contains nonzero elements.
Minimization problem (Thm. 6.3-2): ~ E VF(w) and j ~ ( ~ ) -
J~(~l) - -6
inf
J~(~l), where
P~r(~I)Pa~(~I)~ dY -
p" e~Ii~ dy.
Variational problem (Thm. 6.3-2): ~.e C V F ( w ) and, for all r / E V F ( w ) ,
-~
P~r( )paf3(~l)v~dy- p"eqiv~dy.
5. K O I T E R ~ S E Q U A T I O N S ELASTIC SHELL
FOR A LINEARLY
Minimization problem (Thm. 7.1-1): ~
E Vg(w) - { v / - (7/i) E Hi(w) x Hi(w) x H2(w); 7/i = Ova?3 = 0 on 3'o},
j~(4~)-
inf j~(~/), where ,lcvK(.,) 1
Two-dimensional linear shell equations at a glance
lvi
Variational problem (Sect. 7.1):
~
C VK(w) and, for all 17 e VK(W), ~a~"~.(r
+ -
a~rT~
= f pi'e~liv~dy for all v/= (yi) ~ Vg(w), Jw Boundary value problem (Thm. 7.1-2 and Ex. 7.2): ma~,ela;3 _ b~b~fjmafJ, e _ baf3na~,e _ pa, e in w, - ( n '~,~ + b~m~r/3,~)l~ - b~(m~,~[~) - f ' ~ in w, ~i~,K
- - 0 u ~ 3~, K
- 0 on 70,
ma~'eUaU~ - 0 on 71, (ma~'~la)~'~ + O.(m~'~'~,-~) - 0 on 7~, (n af3'e + 2ba~m~/J'e)u~ -- 0 on 71,
where e3 aa/3ar, e na~, e eaa/3ar, ma~' e = -~ po.r ( ~eK) and ~7~r(~:).
Minimization problem for a simply supported shell whose middle surface has little regularity (Thm. 7.1-2):
Jk(r
inf
jk(,~), where
n~v~c(~,) 1
e3 aa~Cr~",efSar(O)Pa~ (0) } v/a dy H--~-
fw ~e
" ~ l ~ dy.
TWO-DIMENSIONAL NONLINEAR EQUATIONS AT A GLANCE 1
SHELL
1. Nonlinearly elastic membrane shell (equations derived by means of a formal asymptotic analysis) 2. Nonlinearly elastic flexural shells (equations derived by means of a formal asymptotic analysis) 3. Nonlinearly elastic membrane shells (equations derived by means of r-convergence theory)
The equations listed in Subsects. 1 and 2 are justified by a formal asymptotic method for nonlinearly elastic St Venant-Kirchhoff materials. Those in Subsect. 3 are justified by a convergence theorem for a more general class of hyperelastic materials, which includes St Venant-Kirchhoff ones. In each case, the shell is subjected to a boundary condition of place along the portion 0(70 • [-e, e]) of its lateral face, i.e., the displacement field vanishes there. Frequent use is made of notations whose definitions, not repeated here, are found in the preceding subsections "Differential geometry of surfaces in I~3" and "General notations for shell equations".
1Nonlinear shallow shell equations in cuxvilinear coordinates are not reproduced here, as their justifications rely on asymptotic analyses similar to those of Vol. II, Sects. 4.14 and 5.12; they are nevertheless reviewed in Sect. 11.3.
Two-dimensional nonfinear shell equations at a glance
lviii
1. N O N L I N E A R L Y E L A S T I C M E M B R A N E SHELL ( E Q U A T I O N S D E R I V E D B Y M E A N S OF A F O R M A L ASYMPTOTIC ANALYSIS) Definition (Sect. 9.1): A shell is a nonlinearly elastic m e m b r a n e shell if the manifold ~o(~)
- {n - (w) e w~,~(~);
o - o on 7o,
aa~(rl) -aa~3 = 0 in w} reduces to {0}. M i n i m i z a t i o n problem (Thm. 9.4-1): ~e C WM(W) -- {v/ C w l ' 4 ( w ) ; v/ -- 0 on 3'o}, j~(~)-
inf j~(v/), where new.(~)
e f ~ a a ~ r ' e (a~r(~) -- a~r)(aaf3(~7) -- aa~)v/a dy j~M(~?) -- -~
-- ~ P i ' e~Ti V/-d d Y "
Variational problem (Thm. 9.4-1):
~.e E WM(w) and, for all ~/E WM(w),
B o u n d a r y value problem (Thm. 9.4-2):
_ba~(naf3,e + nZ/3,eaar~ll~)_ (naf~,el~lla)lf 3 _ p3, e in w, ~ = 0on3'0, (n~' ~ + n~'
e aar "e
~b'
%11~) ~3 - 0 on 7t
n ~ r 3llaUl3 ~ = 0 on 71,
where n a~,e - ea~,~E~lll3(~e ).
lix
T w o - d i m e n s i o n a l nonlinear shell equations at a glance
2. N O N L I N E A R L Y E L A S T I C F L E X U R A L S H E L L S ( E Q U A T I O N S D E R I V E D B Y M E A N S OF A F O R M A L ASYMPTOTIC ANALYSIS)
Definition (Sect. 10.2): A shell is a nonlinearly elastic flexural shell if the manifold ]9~ F ( W )
-- {71 E
W2'4(W); ~ --
O v ~ -- 0 Oil ~0,
a~(~)
- a~
= 0 in
~)
contains nonzero elements and, at each ~" C .h4F(W), the tangent space to the manifold .h/rE(w) contains nonzero elements.
Minimization problem (Thm. 10.5-1):
~ C .h4F(w) and j ~ ( ~ ) -
inf
j ~ ( y ) , where
g3 f aa/3crr'e JF(~/) -- -6- d~ (b~r(r/) - b~r)(ba~(~/) -
ba~)v~dy
-- ~ Pi' erliv~ dY"
Variational problem (cf. Thm. 10.5-1; the functions R~f3(~) are defined in Thm. 10.3-1):
~ E ~4F(W) and
--~
eRz~(~e)((R~) ( ~ e ) n ) ~ d y -
for ~11,1 ~ vr ~ F ( ~ )
pZ,eTii~dy
-- ( ~ ~ W 2 ' 4 ( ~ ) ; ~ -- 0 ~ a' r
-- 0 o . ~0,
lx
Two-dimensional nonlinear shell equations at a glance
3. N O N L I N E A R L Y E L A S T I C M E M B R A N E S H E L L S ( E Q U A T I O N S D E R I V E D B Y M E A N S OF F-CONVERGENCE THEORY)
The shell is made of a homogeneous hyperelastic material characterized by a stored energy function l~" M 3 --+ IR satisfying ad hoc growth assumptions "governed" by an exponent 1 < p < oc. The shell is subjected to a boundary condition of place along its entire lateral face, i.e., "70 - 7. The minimization problem is expressed in terms of the usual curvilinear coordinates (those of the points of w). However, the unknowns are no longer the covariant components, but instead the Cartesian components, of the displacement field ~6 of the middle surface.
Minimization problem (Thm. 9.5-1): ~e _ ~ a i e W o1 , p (w) and j ~ ( ~ ) j~(r/) -- 2e f~ Ql~0(y, (al + 01~;
inf
,)cw~"(~) a2
~ j~r162 where
+ 02r
- f P~ "fTv/ady, l~o(y, (bl; b 2 ) ) -
inf l~((bl; b2; b3)(al(y), a2(y), a3(y))),
baEI~ s
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PART A
LINEAR SHELL THEORY
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CHAPTER 1 THREE-DIMENSIONAL ELASTICITY AND KORN'S INEQUALITIES COORDINATES
LINEARIZED IN CURVILINEAR
INTRODUCTION The equations of three-dimensional linearized elasticity have been extensively studied, but mostly in Cartesian coordinates. It is well known for instance that the existence and uniqueness of their solution depend in a crucial way on Korn's inequality, again expressed in Cartesian coordinates. The linear shell theories justified in this volume from three-dimensional elasticity require, however, that these equations be rather expressed in terms of curvilinear coordinates that "follow the geometry" of the shell in a most natural way. Accordingly, the purpose of this preliminary chapter is to provide a thorough derivation and a mathematical treatment of the equations of linearized three-dimensional elasticity in terms of arbitrary curvilinear coordinates. This chapter includes in particular a direct proof of the three-dimensional K o r n inequality in curvilinear coordinates. The treatment is entirely self-contained, in that no a priori knowledge of differential geometry is assumed. More specifically, we begin by reviewing (Sect. 1.2) the basic definitions and properties arising when a three-dimensional set of the form { ~ } - - O(~), where ~2 is a three-dimensional domain in I~3 and O is a smooth injective mapping with other ad hoc properties, is equipped with the coordinates of the points of fl as its curvilinear coordinates. Of particular importance is the metric tensor of the domain { ~ } - , whose covariant and contravariant components gij - gji " ~ -+ I~ and gZ3 _ g3Z. ~2 --+ I~ are given by (Latin indices or exponents take their values in {1, 2, 3)): gij -- gi "gj and gij _ gi . gj, where gi : OiO and gJ "gi : (~.
Further properties of the m~etric tensor are also reviewed in Sect. 1.9.
4
Three-dimensional linearized elasticity
[Ch. 1
We then examine how the equations of three-dimensional linearized elasticity expressed in terms of the Cartesian coordinates of the points in the set { ~ } - (this formulation is reviewed in Sect. 1.1) are transformed when they are expressed in terms of curvilinear coordinates, i.e., in terms of the coordinates of the points in the set - {9-1 ( { ~ } - ) . We begin by showing, by means of a direct (but not so obvious!) computation (Thm. 1.3-1), that the variational, or weak, formulation of these equations takes the form: ~, - ( ~ ) e v ( ~ ) : =
{ . - (,~) e H ~ ( ~ ) ; ~ = o on r 0 ) ,
fa AiJk'ekll,(u)eilli(v)~/~dx - ~ fivi~ dX + ~r hiviv~ dr for all v - (vi) C V ( ~ ) , where uig s is the unknown displacement vector field inside the set { ~ ) - , r0 is a subset of 0~2 with area F0 > 0,
A ~ j k t _ )~gij gkt + D(gik gfl + git gjh), and # are the Lam~ constants of the constituting elastic material, g - det(gij), the functions fi C L2(12) and h i E L2(rl), where r l -- 0 ~ - r0, account for the applied body and surface forces, and the linearized strains in curvilinear coordinates eillj(v) E L2(~) are defined for each v - (vi) e H I ( ~ ) by 1
ei[Ij(V ) -- ~(OjVi + OiVj) - Fijvp, p P -- gP " Oigj. where Fij
The interest of this calculation is that it naturally introduces essential notions, such as the covariant derivatives vijlj - - O j V i - - rijV of a vector field rig i and the linearized change of metric tensor associated with this vector field, found here by means of its covariant components 1
~llJ(~) - ~(~llJ + ~Jlli).
These notions are then studied in greater details for their own sake (Sects. 1.4 and 1.5). We also describe in detail (Thm. 1.6-1) the equivalent boundary value problem when it is likewise expressed in terms of curvilinear coordinates. Finally, we show how a fundamental lemma of J.L. Lions (Thm. 1.7-1) can be put to use for directly establishing Korn's inequality in
Linearized elasticity in Cartesian coordinates
Sect. 1.1]
5
curvilinear coordinates: This inequality, which asserts the existence of a constant C such that (Thm. 1.7-4)
}1/2
Ilvlll, n <_ c
X~ le'ilJ(v)lN, n
for all v E V(f~),
i,j in turn yields (Thm. 1.8-2) the emistence and uniqueness of a solution to the variational formulation of the equations of three-dimensional linearized elasticity, again directly in curvilinear coordinates. THREE-DIMENSIONAL LINEARIZED ELASTICITY IN CARTESIAN COORDINATES
1.1.
Throughout this volume, Latin indices and exponents vary in the set {1, 2, 3) (except if otherwise indicated, as when they are used for indexing sequences), and the summation convention with respect to repeated indices or exponents is systematically used in conjunction with this rule. Thus for instance, 3
xie.i--~~,ie, 3
~(~,)
- ~
3
i,
~ivi--~fivi,
i=1
~,(~,1,
i=1
3
~,j(~,)~,j(~) : ~
p=l
~,J(~,l~,J(~),
i,j:l
"eiltj(v ) -- 0 in f~" means "eillj(V ) -- 0 in f~ for i, j -- 1, 2, 3,
,,Aijkl~kl(~)~Z j : ~i
on rl"
means 3
"
~
A~Jk~k~(i,)~j - h ~ on fl~ for i - 1, 2, 3," etc.
j, k, l-=l As a model of the three-dimensional "physical" space E s, we take a r e a l t h r e e - d i m e n s i o n a l af[ine E u c l i d e a n s p a c e , i.e., a set in which a point 0 has been chosen as the o r i g i n and with which is associated a r e a l t h r e e - d i m e n s i o n a l E u c l i d e a n s p a c e , denoted E 3, endowed with an o r t h o n o r m a l b a s i s consisting of three vectors ~i _ ~i; "orthonormar' means that e i " e.j - 6ij where a . b - aibi denotes the E u c l i d e a n i n n e r p r o d u c t of a -- ai &i E E 3
6
Three-dimensional linearized elasticity
[Ch. 1
and b -- bi ~j C ]~3, and 5ij designates the Kronecker symbol. The definition of E 3 as an affine Euclidean space means that with any point x E ~3 is associated a uniquely defined vector in E 3, denoted 0 ~ . The origin 0 E E 3 and the vectors ei E E 3 together constitute a Cartesian f r a m e in E 3 and the three components xi of the vector 0~, over the basis formed by the vectors ~i are called the Cartesian coordinates of ~ C ~3 or the Cartesian c o m p o n e n t s of O~ C E 3. Once a Cartesian frame has been chosen, any point ~ E E 3 may thus be identified with the vector O~ ~i ~i ]~3, Or with the vector (xi) E JR3; we then let Oi "-- O/O~i. -
E
Remark. The notion of affine space affords the definition of " ~ + a " as a point in E 3 whenever ~) is a point in E 3 and a is a vector in E 3 in such a way that & - O + 0 ~ for all & C E3; we refer to Schwartz [1992, Chap. 3] for an excellent introduction to affine Euclidean spaces. II Finally, we let lal := ~/a. a denote the E u c l i d e a n n o r m of a E E 3 and we let a A b := ~iJhajbk~i denote the exterior p r o d u c t of a = ai ~i C E 3 and b = bie, i E :E 3, where gijk _ +1 if (i, j, k) is an even permutation of (1, 2, 3), g~jk = - 1 if (i, j, k) is an odd permutation of (1, 2, 3), and gij~ = 0 otherwise. A d o m a i n in E 3 is a bounded, open, and connected subset (] of ~3 with a Lipschitz-continuous boundary I', the set ~ being locaUy on one side of r (this definition holds verbatim in I ~ ; cf. Sect. 1.7). Let ~ be a domain in E a, let d& denote the volume element in ~, let d r denote the area element along r , and let ~ ni ~i denote the unit ( l ~ i - 1) outer normal vector along r (d~ is well defined and is defined dr-almost everywhere since I' is assumed to be Lipschitzcontinuous). FinaUy, let r - r0 U r t be a dr-measurable partition (r0 N r t - r of the boundary I' that satisfies -
area ro > O.
The set { ~ ) - is the reference configuration occupied by an elastic b o d y in the absence of applied forces. We assume that ( ~ } is a natural state, i.e., that the body is stress-free in this configuration. We also assume that the constituting elastic material is i s o t r o p i c and h o m o g e n e o u s (details about these notions are found
Sect. 1.1]
Linearized elasticity in Cartesian coordinates
7
in Vol. I t, Sects. 3.1, 3.4, and 3.6). Under these assumptions, the behavior of the elastic material for "small strains" is entirely governed by two constants, called the L a m ~ c o n s t a n t s of its constituting material (Vol. I, Sect. 3.8). Experimental evidence shows that the Lam~ constants, denoted )~ and #, of usual elastic materials such as steel or aluminum satisfy (Vol. I, Fig. 3.8-4) )~ > 0 a n d # > 0. The body is subjected to a p p l i e d b o d y forces in its interior and to a p p l i e d s u r f a c e forces on the portion r l of its boundary, given by their densities (]i) " ~ --> R 3 and (hi) " r I --> I~3 (Vol. I, Sect. 2.1). For definiteness, we henceforth assume that
]i C L2(~) and hi E L2(F1).
Remark. In order to avoid any confusion with the notation gi used later for the vectors of the covariant basis (Sect. 1.2), the components of the applied surface force density are denoted hi in this volume; otherwise the notation gi would have been consistent with that used in Vols. I and II 2. [] The unknown is the vector field/~ - (~i)" { ~ } - --->I~3, where the three functions ui " { ~ } - ~ IR are the C a r t e s i a n c o m p o n e n t s of the d i s p l a c e m e n t field ~ i e . i 9 { f i } - --} S 3 that the body undergoes when it is subjected to the applied forces. This means that "~i(~g)ei is the displacement 4 the point ~ E { ~ } - (Fig. 1.1-1). As E 3 may be identified with IR~, we may identify Ui(X)e i E E 3 with the vector ~(~) E IR3 and write accordingly the displacement field as
Note that this identification has been implicitly made throughout Vols. I and II. l"Vol, r' stands for "Ciaxlet, P.G. [1988]: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, North Holland, Amsterdam." ~"Vol. Ir' stands for "Ciaxlet, P.G. [1997]: Mathematical Elasticity, Volume II: Theory of Plates, North-Holland, Amsterdam."
8
Three-dimensional linearized elasticity
.
[Ch. 1
!
@
2c)
Fig. 1.1-1: Three-dimensional elasticity in Cartesian coordinates. Let there be given an elastic b o d y whose reference configuration is the closure of a domain 1~ in the "physical space", modeled as a real three-dimensional affme Euclidean space Es. The displacement vector ~2i($)~ i of the point ~ E {1~}- is defined by its Cartesian components fii($) in a Cartesian frame of C s, consisting of an origin 0 and an orthonormal basis formed by three vectors ~i.
It is further assumed that the displacement field vanishes on the set to, i.e., that it satisfies the b o u n d a r y c o n d i t i o n of place
=0on 0. Note that, unless otherwise specified, a "boundary condition of place" is always meant in this volume to be homogeneous. Then in linearized elasticity, the unknown it = (~ti) satisfies the following variational problem, which constitutes the v a r i a t i o n a l , or weak, f o r m u l a t i o n , of the e q u a t i o n s of t h r e e - d i m e n s i o n a l linearized e l a s t i c i t y in C a r t e s i a n c o o r d i n a t e s (Vol. I, Sects. 6.2 and 6.3):
Sect. 1.1]
Linearized elasticity in Cartesian coordinates
/, e v ( h ) . -
f
9
{/, - ( ~ ) ~ n x ( h ) ; ~ - o on/~o},
()~gpp(/t)gaa(fi) + 2#gij(it)gij(i~)} d~
]zi~)idr for a11/~ e V(fi),
= 1 .
.
.
.
where H I ( ~ ) denotes the space of vector fields 6 - (~i) with components ~3i in the Sobolev space H I ( ~ ) , and the functions 1
L2
denote the components of the l i n e a r i z e d c h a n g e of m e t r i c t e n s o r
e(~) .= (~j(~)) in C a r t e s i a n c o o r d i n a t e s associated with an arbitrary displacement field /~ - ~i& i - (~i) C V(~). The functions ~ij(v) are also called the l i n e a r i z e d s t r a i n s in C a r t e s i a n c o o r d i n a t e s and the tensor &(6) is also called the l i n e a r i z e d s t r a i n t e n s o r . In view of its importance and of its subsequent generalization to curvilinear coordinates (Sect. 1.5), we briefly recall the significance of this tensor. Let ~ 9( ~ ) - --~ R 3 be an arbitrary displacement field of the reference configuration { ~ } - and let
be the associated d e f o r m a t i o n . Then the "new" metric in the associated d e f o r m e d c o n f i g u r a t i o n ~ ( ( ~ } - ) i s given by the associated (right) C a u c h y - G r e e n s t r a i n t e n s o r defined by (Vol. I, Sect. 1.8) ~ T ~
_ I + Vi~ T + Vi~ + ~ i ~ T v i ~ ,
where the matrices ~ 6 - (0jgi) and V~b - (Oj~bi) (i is the row index and j is the column index) respectively denote the corresponding
10
[Ch. 1
Three-dimensional linearized elasticity
displacement and deformation gradients. Hence 1 [~r(bT~r~ _ i]li -
where [... ]tin means that only the linear part with respect to ~ is retained in the expression [... ]. In other words, the tensor ~(i~) measures (half of) the linearized difference between the "new" metric in ~ ( { ~ } - ) and the "old" metric in { ~ } - (whose right Cauchy-Green strain tensor is I), both expressed in terms of the Cartesian coordinates ~/. This is why ~(9) is called the linearized "change of metric" tensor in Cartesian coordinates. It is well known that the proof of the existence and uniqueness of a solution to this variational problem relies on Korn's inequality "in Cartesian coordinates" (see, e.g., Vol. I, Sect. 6.3, or Vol. II, Sects. 1.1 and 1.2). In fact, we shall provide in this volume a direct proof of a more general Korn's inequality "in curvilinear coordinates" (Thm. 1.7-4), which contains Korn's inequality in Cartesian coordinates as a special case. As is easily seen by means of Green's formula (Vol. I, Thm. 6.3-1), the above variational problem is, at least formally, equivalent to the following boundary value problem (hij denotes the Kronecker symbol):
~i -
{)~pp(iL)~ij Jr 2/z~ij(/L))r
0
-- hi
on
r0,
on
rl"
Either formulation constitutes the e q u a t i o n s of t h r e e - d i m e n sional linearized elasticity in C a r t e s i a n c o o r d i n a t e s for a linearized displacement-traction problem (cf. Vol. I, Sect. 6.2; "displacement-traction" refers to the kind of boundary conditions considered here). A solution of the variational problem, which as such may be only in H I ( ~ ) , is called a weak solution, while a solution of the boundary value problem (e.g., in C2({~}-)) is called a classical solution. Except in some very special cases, such as when the boundary F is smooth and F1 - r or F0 and F1 are two connected components of r , "weak solutions are seldom classical" (Vol. I, Sect. 6.3).
Sect. 1.1]
Linearized elasticity in Cartesian coordinates
11
Remark. This boundary value problem is the linearization about /, - 0 of the corresponding boundary value problem of nonlinear elasticity (considered later; cf. Sect. 8.1). This linearization is detailed in Vol. I, Sect. 6.2. II Let us also denote by jij the Kronecker symbol and by /i :__ /i and h i := hi the components of the applied forces, and let the components ~ijkl of the t h r e e - d i m e n s i o n a l e l a s t i c i t y t e n s o r in C a r t e s i a n c o o r d i n a t e s be defined by
~ijkl
:=
)k~ij~kl q_ iz(~ik~jl _+.~il~jk).
Then the above variational equations may also be written as
for all ~ E V ( h ) , and the above boundary value problem may also be written as
~2i -- 0
on
-
P0,
o n
If we introduce the l i n e a r i z e d stresses"
~rij
:=
Aijkl~kl(~l),
the variational equations take the even shorter form
/f~ ~iJeij(O ) dx -- f(l /ioi dx + f~ hivi d~
for all
~ E V(~),
1
which constitutes the l i n e a r i z e d p r i n c i p l e of v i r t u a l w o r k in C a r t e s i a n c o o r d i n a t e s . The equations
-Oj /J - / / &ijhj -- hi
in on
rl,
12
Three-dimensional linearized elasticity
[Ch. 1
constitute the l i n e a r i z e d e q u a t i o n s of e q u i l i b r i u m in C a r t e s i a n coordinates. The above principle of virtual work and equations of equilibrium are special cases of more general relations "in curvilinear coordinates" (Sects. 1.3 and 1.6). Thanks to the new notations ~ij, ] i and h i, they now obey the proper rules concerning "covariant indices" and "contravariant exponents". We also note that the three-dimensional elasticity tensor satisfies
the inequality i,j for all symmetric matrices (tij) and that its components satisfy the symmetries
These properties are again special cases of more general properties "in curvilinear coordinates" (Thms. 1.3-1 and 1.8-1).
Remark. Should we let instead ~ijkl _ )~ij~kl ~_ 2#~ik~jl, the left-hand side of the variational equations would be unaltered, but the symmetries ~iijkl - ~ljikx would be lost. Yet these are crucially needed for establishing the relation
used
in
turn
__~j{~ijkl~kl(~t))
1.2.
for
deriving __ /i in ~.
the
partial
differential
equations II
CURVILINEAR COORDINATES AND METRIC TENSOR IN A THREE-DIMENSIONAL DOMAIN
The theme of this volume is the study of elastic shells, i.e., of elastic bodies whose reference configuration { ~ e ) - C C3 consists of all points within a distance < ~ from a given surface S C C3 and e > 0 is thought of as being "small" (but this last property is irrelevant at this stage); cf. Fig. 1.2-1. Assume that the surface S is defined as the image 0(~) of the closure of a domain w in ]R2 where 0 9~ ~ E 3 is a smooth injective
Curvilinear coordinates and metric tensor
Sect. 1.2]
13
a3(x,,~:~)
2& 3
~e
x,J ~ ........~
Z
~
Fig. 1.2-1" A shell and its "natural" curvilinear coordinates. A shell is a tl~eedimensional elastic b o d y occupying a reference configuration { ~ } - of a specific shape, consisting of all points within a distance ~ e from a surface S, where > 0 is given. The surface S is defined as S = O(~), where w C R 2 and 0 : ~ ---> E s is a smooth injective mapping. Each point $" of { ~ ' } - is thus of the form ~" - | z2, z~) - O(zx, zn) + z ~ a 3 ( z l , z~.), where as(z1, z2) is a unit vector normal to S at the point 8(Zl, z2) and (zl , ~2, ~ ) E ~" = ~ x [-~, el. For e > 0 small enough, the mapping @ : = ~ • [-e, e] is also injective and the curvilinear coordinates of any point $~ E {fl~} - are then defined as the coordinates of the unique point ~ - (Xl, z~., ~ ) E ~ such that $~ = |
mapping.
Let a3 denote a continuously
varying unit normal
a l o n g S', a n d let
~.-
~•
~, ~[.
vector
14
Three-dimensional linearized elasticity
[Ch. 1
Hence the set ( ~ e } - is given by
-
e
where the mapping O 9~e C I~s --+ E 3 is defined by
(~) ($gl, X2, ;gl):: O(;gl, Lg2)-{-xla3(xl, ;g2) for aU (Xl, x2, x~) E ~ . If the mapping @ 9~ E 3 is injective (as is the case if ~ > 0 is small enough; cf. T h m . 3.1-1), each point ~e _ ( ~ , xl, x~) e { ~ e } is the image &e _ | of a unique point x e - (xl, x2, x~) e ~ , the three coordinates xl~ x2~ x~ of which are called the ~ n a t u r a P ~ e u r v i l i n e a r c o o r d i n a t e s of $~.
Remark. Later on, the curvilinear coordinates xa will be denoted Ya~ so as to afford the short notation y for a generic point in ~. m If a linearly elastic shell is subjected to applied forces and to a b o u n d a r y condition of place, the displacement field inside {~e } - satisfies the equations of three-dimensional linearized elasticity in Cartesian coordinates described in Sect. 1.1. As we shall see t h r o u g h o u t this volume, it turns out that an essential preliminary to the asymptotic analysis of such a shell problem when e approaches zero consists in rewriting these equations in terms of the curvilinear coordinates xl, x2, x~ (the "new" variables) instead of the Cartesian coordinates x~, xl, xl (the "old" variables). As a preparation to this rewriting~ we review in this section basic definitions and properties of curvilinear coordinates in a "general" three-dimensional domain ~ C Ea; thus the exponents e have no longer any raison d'etre. Let there be given a three-dimensional affine Euclidean space E a, a Cartesian frame i n E a a n d a d o m a i n ~ C ~a as in Sect 1.1 In addition, let there be given a three-dimensional vector space in which three vectors e i - ei form a basis; this space will accordingly be identified with I~3. We let xi denote the coordinates of a point x in this space and we let Oi := O/Oxi and Oij := 02/OxiOxi. Assume that there exist a domain ~ in I~3 and an injective mapping O 9 ~ -+ E a such that O ( ~ ) - { ~ } - . Hence each point
Sect. 1.2]
~ 3 ~.
Curvilinear coordinates and metric tensor
.,....
,,.*
~'.
e~.~.........'.'.'.'.'.': ....i""
15
-.. .....
e
Fig. 1.2-2: Curvilinear coordinates and domain. T h e three coordinates z l , z2, zs of $ - | G {(1}-. W h e n e v e r t h e y are g~(z) -- 0 i | form the covariant basis t h e coordinate lines passing t h r o u g h ~.
covariant bases in a three-dimensional of z E 12 are the curvilinear coordinates linearly independent, the three vectors at $ -- | t h e y are t h e n t a n g e n t to
& E {l~}- can be unambiguously written as
-
O(z),
x c 12,
and the three coordinates Xi of x are called the c u r v i l i n e a r c o o r d i n a t e s of ~ (Fig. 1.2-2). Naturally, there are infinitely many ways of defining curvilinear coordinates in a given domain ~, depending on how the domain 12 and the mapping O are chosen! Examples of curvilinear coordinates include the well-known cylindrical and spherical coordinates (see Fig. 1.2-3 and also Ex. 1.1). Another instance is provided by the curvilinear coordinates xl, x2, x~ used for defining a shell (Fig. 1.2-1); note that, in this instance, each set of curvilinear coordinates ~1, x2 on the surface S gives rise to a set of curvilinear coordinates in {l~e} -. Assume that the mapping O - | i 9~ C R s ~ {l~}- C g 3 is differentiable at a point x E 12. If ~z is such that (~ + 6 z ) C 12, we
Three-dimensional linearized elasticity
16
[Ch. 1
!
Fig. 1.2-3" Two familiar ezamples of curvilinear coordinates. Let 1~ be a threedimensional domain. The cylindrical coordinates of $ E ~ are ~o, p, z. The spherical coordinates of $ E ~ are ~o, ~, r. Special care must of course be exercised in order that such eurvilinear coordinates be unambiguously defined, since the same point is defined by (~o, p, z), or (~o + 2k~r, p, z), or (~o + k~r, - p , z), k E Z; in addition, neither ~o nor ~ are defined at the origin r As is customary, these coordinates are appended directly to the set 1"~, but they are in fact coordinates in a different domain f~, not represented here!
thus have
e(~ + ~ ) - e ( ~ ) + v e ( ~ ) o ~ + o ( ~ ) , w h e r e t h e m a t r i x V O ( x ) is g i v e n b y
VO(x):=
/
0t| 02Or 03
~t
0102 0202 0302 0t| 0203 0303
(x).
L e t t h e t h r e e v e c t o r s g i ( x ) C IR3 b e d e f i n e d b y
g~(~) .- o , |
/Oi@l/
[0,02 \o~03
(~),
Sect. 1.2]
Curvilinear coordinates and metric tensor
17
so t h a t gi(x) is the i-th column vector of the matriz V O ( ~ ) . Let ~ e = fi~iei; then the expansion of ~9 about ~ may also be written as
e(~ + 0~) = e ( ~ ) + 0~a~(~) + o ( ~ ) . If in particular ~ z is of the form ~a~ = ~tei, where ~t E Ii~ and e i is one of the basis vectors in I~s, this relation reduces to
e ( ~ + ~t~) = e ( ~ ) + ~tg~(~) + o(~). We henceforth assume that the three v e c t o r s gi(X) are linearly independent, in which case they constitute the e o v a r i a n t b a s i s at the point ~ - O(x). The last relation thus shows that, in this case, each vector gi(x) is tangent to the i-th c o o r d i n a t e line passing through - O(x), defined as the image by {9 of the points of ~ that lie on the line parallel to ei passing through x; cf. Fig. 1.2-2 (there exist to and tl with to < tl and 0 E [to, tl] such that the equation of the i-th coordinate line is t E [to, tl] --+ fi(t) : - O ( x + tei) in a sufficiently small neighborhood of ~; hence f~(0) - Oi| - gi(x)). Naturally, we are committing here a convenient abus de langage: The vector tangent to the i-th coordinate line at $ = | is in fact t h a t vector in the affine space E 3 that is parallel to gi(x) and has as its origin. R e t u r n i n g to a general increment ~ e = ~xiei, we also infer from the expansion of O about x that (recall that we use the s u m m a t i o n convention)"
te(~ + ~)-
e(~)l 2 - ~ T v e ( ~ ) T v e ( ~ ) ~
+ o (16~12)
In other words, the principal part of the length between the points | + ~z)and | {~xigi(x) 9gi(x)~xJ}l/2. This observation suggests the introduction of the symmetric matriz (gij (x)) of order three, whose elements
Three-dimensional linearized elasticity
18
[Ch. 1
are the c o v a r i a n t c o m p o n e n t s of the m e t r i c t e n s o r at ~ - O(x). Note that the matrix V O ( x ) is invertible and that the matrix (gij(x)) is positive definite, since the vectors gi(x) are assumed to be linearly independent. A w o r d of c a u t i o n . We refrain here from following the common, but improper, usage of calling "metric tensor" the matrix (gij(x)) itself. In spite of its convenience, it is a confusing and flagrant abus de langage, for the "genuine" metric tensor also has "contravariant components" gij (x) (see the next theorem); it even has "mixed comi ponents" gj(x) (which are simply the Kronecker symbols ~ ) . See, e.g., Antman [1996, Chap. 11]. II In the next theorem, we review fundamental formulas showing how volume, area, and length elements at a point a~ = O(x) in the set { ~ } - can be expressed either in terms of the matrix V O ( x ) or in terms of the matrix (gij(x)) or of its inverse matrix (gii (x)); see also Fig. 1.2-4. Parts (a) and (b) win be immediately put to use in the next section; part (c) provides the essence of the metric tensor. Otherwise, we refer to Vol. I, Sects. 1.5 to 1.8, for comments and references. If A is a square matrix, C o f A denotes the cofactor matrix of A. If A is invertible, we thus have C o f A - (det A ) A -T (details about the cofactor matrix may be found in Vol. I, Sect. 1.1). Also note that the assumptions made on the mapping O guarantee that { ~ } - - O ( ~ ) and that the boundaries F of l~ and r of ft are related by F - O(F) (Vol. I, Thin. 1.2-8 and Ex. 1.7). T h e o r e m 1 2 - 1 Let f~ be a domain in R 3 let | "-~--+ E 3 be an injective and smooth enough mapping such that the three vectors gi(x) := OiO(x) are linearly independent at all points x e f~, and let -
(a) The volume element d$ at ~, - O(x) E ~ is given in terms of the volume element dx at x E f~ by d~ - [det V O ( x ) [ dz - v / g ( x ) d x ,
where g(X) : : det(gij(x)) and gii(x)"= g i ( x ) , gj(x).
Curvilinear coordinates and metric tensor
Sect. 1.2]
19
)
f2
,g~
a?..+
fi: \
I .......
E
Fig. 1.2-4: Volume, area, and length elements in curvilinear coordinates. The elements d~, d r ( S ) , and di($) at $ = | E {1~}- are expressed in terms of dz, d r ( z ) , and 6~ at z E II by means of the covariant and contravariant components of the metric tensor; el. Thin. 1.2-1. The corresponding relations are used for computing the volume of a subdomain ~r = O(V) C { h } - , the area of a surface A = | C 0 ~ , and the length of a curve C = | C { ~ } - , where C - :f(I) and I C R.
( b ) The area e l e m e n t d r ( ~ ) at ~ -
|
E O h is given in t e r m s
of the area e l e m e n t d F ( z ) at z E Of~ by
d~(~) - ICofVO(x)n(x)l dr(x)
where n ( z ) ' - n i ( z ) e
i denotes the u n i t outer n o r m a l vector at z E Of~,
20
Three-dimensional linearized elasticity
[Ch. 1
and the matrix (gij(x)) is the inverse of the matrix (gij(x))"
, ~
The components g '3 (x) of this matrix are called the contravariant c o m p o n e n t s o] the m e t r i c tensor a t x .
(r The te,~gth ele,~,~t di(~) ~t ~ - 0 (~) e { h } - i, gi~,~ by
~[(~)- { ~ v o ( ~ ) ~ v o ( ~ ) ~ } ~ / ~ -
- {~%(~)~J}~/~,
where $o~ = (~xiei . Proof. The relation d~ = [ det V | dx between the volume elements is well known. The relation g(x) = [det ~70(x)[ 2 follows from the relation (gij(x)) V(~)(x)Tv~)(X). Indications about the proof of the relation between the area elements dF(&) and dr(x) given in (b) are found in Vol. I, Thin. 1.7-1 (naturaUy, n(x) - ni(x)e i is identified here with the column vector in ]Ra with ni(x) as its components). Using the relations C o f ( A T) - ( C o f A ) T and C o f ( X B ) - ( C o f A ) ( C o f B ) , we next have: -
-
[CofVO(x)n(x)] 2 - n(x)TCof (VO(x)Tvo(x))n(x) . .
= g(~)~(~)g'~(~)~(~). Either expression of the length element given in (c) simply recalls that dl($) is by definition the "principal part" with respect to Sa~ of the length 1(9(x + Sa~) - O(x)[, whose expression precisely led to the introduction of the matrix (gij (x)). m The relations found in Thm. 1.2-1 are used for computing volume integrals, areas, and lengths inside {~}-" Let V be a subdomain of ~, let V "- O(V), and let ] " V --+ ~ be a d&-measurable function. Then
Sect. 1.2]
In particular, the
Curvilinear coordinates and metric tensor
21
volume of V is given by
Next, let A C 0fl be a surface, let 2{ := O(A) C 0~, and let ]z" 2{ -+ IR be a dr-measurable function. Then
fA h(&) dF(&) - fA In particular, the
(h o O)(x)v/g(x)~//ni(x)giJ(x)nj(x) dr(x).
area of 2{ is given by
a~a A "- f:~dF(&) = / A
v/g(x) ~/ni(x)giJ (x)nj(x) dF(x).
Finally, consider a curve C - f(I) in ~, where I is a compact interval of I~ and if - j:Zei 9I -+ ~ is a smooth enough injective mapping. Then the length of the curve C := O(C) C { ~ } - is given by
length C "-- f / [-~ d (0 o f)(t)l dt
~g
d/i
df j (t) dr. (t)-~
Let (f~ be yet another way of designating the Kronecker symbol. Given a point x E f~, the nine relations i
unambiguously define three linearly independent vectors gi(x), which form the e o n t r a v a r i a n t basis at the point ~ = O(x). To see this, let ~ p~io~i g ' ( ~ ) -
xik(~)gk(~)
i . the ~ e l ~ t i o . s g ~ ( ~ ) . 0 ~ ( ~ ) -
~j.
xik(x)gkj(x) = 5j; consequently, xik(x) = gik(x), where (gij(x)) "- (gij(x)) -1 (Thm. 1.2-1 (b)). This also shows that
This gives
= g~k(~)gj~(~)gk~(~ ) _ g ~ k ( ~ ) ~
_ g~j(~),
22
[Ch. 1
Three-dimensional linearized elasticity
and thus the vectors gi(x) are linearly independent since the matrix (g~J(x)) is positive definite. We would likewise establish that g~(~) - g~j(~)gJ(~). Let us record for convenience these fundamental relations between the vectors of the covariant and contravariant bases and the covariant and contravariant components of the metric tensor: gij(X) -- g i ( x ) "
gj(x)
and
gij (z)
- g i ( z ) 9gJ (z),
gi(~) - gij(~)g j(~) ~nd g~(~) - g~J(~)gj(~). In Sect. 1.9, we shall briefly discuss the reciprocal question of
recovering a three-dimensional manifold 0(~) from its metric tensor field: Given a positive definite symmetric matrix field (gij) on ~, find conditions under which there exists a mapping O : ~ --~ E 3 such that
OiO . OjlD = gij
in f~.
Indications about the meaning of the adjectives "covariant" and "contravariant" used in the definitions of the components of the metric tensor are provided in Ex. 1.3. Otherwise they can be simply seen as particularly effective (though possibly mysterious!) definitions. 1.3.
THE VARIATIONAL EQUATIONS OF THREE-DIMENSIONAL LINEARIZED ELASTICITY IN CURVILINEAR COORDINATES
Our point of departure is the variational formulation of the equations of three-dimensional linearized elasticity described in Sect. 1.1, which consist in finding fi = (ui) such that ~t e V ( h ) = {~ = (~3i) e H i ( h ) ; ~ = 0 Oil r 0 ) ,
ffi AiJkt~k,(it)~ij(~ ) d~ - f~ /igi d~ + ~1 hi~i d~ for all ~ - ( ~ i )
E V(h),
where ~ is a domain, with boundary r partitioned as r - F0 U r l ,
Sect. 1.3]
Variational equations in curvilinear coordinates
23
in the affine Euclidean space t; 3,
ft~jkl _ AS~jSkl + tt(Sik5 jl + 5ilSJ~),
a n d / i E L2(~) and hi e L 2 ( r l ) are given functions. These equations are expressed in terms of the Cartesian coordinates xi of the points & - (xi) C ( ~ ) - and of the Cartesian components ~ti, fz, h z of the displacement and force densities. As in Sect. 1.2, we assume that we are also given a domain ft, with b o u n d a r y F, in R 3 and a smooth enough injective mapping O 9 ~ --+ t; 3 such t h a t O ( ~ ) - { ~ ) - and the three vectors gi(x) - 0iO(x) are linearly independent at all points x E f~. Our objective consists in expressing these equations in terms of the curvilinear coordinates xi of the points & - O(x) E { ~ } - , where x - (xi) E f~. In other words, we wish to carry out a change of variables, from the "old" variables xi to the "new" variables xi, in each one of the integrals appearing in the above variational equations, which we thus wish to write as ^ .
^
~
/h...d~c-f...dx
f~l...dF-frl...dF,
and
where F1 C F and O (F1) -- F1. Because the "old" unknowns ~i 9 { ~ } - -+ I~ are the components of a vector field, some care must evidently be exercised in the definition of the "new" unknowns, which must be related to the old ones by means of an intrinsic quantity, i.e., having "physical invariance". Observing that the displacement vector ~i(~)&i at each point & E { ~ } - possesses this property, we define three new unknowns ui " gt --+ I~ by requiring that (Fig. 1.3-1)
~i(&).&i =: ui(x)gi(x) for all & - O(m), m e ~, where the three vectors gi(x) form the contravariant basis at ~ - O (x) (Sect. 1.2). Using the relations g i ( x ) . g j ( x ) - ~ and &i. &j _ 5~, we immediately find how the old and new unknowns are related, viz.,
-
-
24
Three-dimensional linearized elasticity
[Ch. 1
OCt.
Fig. 1.3-1: Three.dimensional elasticity in curvilinear coordinates. In curvilinear coordinates, the displacement vector "tl,i(~)gi(~,) of the point $ = | E (l is de-
fined by its covariant components ui(z) over the contravariant basis vectors gi (z) defined by gi(z).gj(z) = J~, where gj(z) = OjO(z). In Cartesian coordinates, the same displacement vector ~i($)~ i is defined by its Cartesian components fii($) over the vectors ~i of the Cartesian frame chosen in Es; cf. Fig. 1.1-1. Let
[gj(x)] i := g j ( x ) . &i a n d [gJ(x)]i ' - g J ( x ) " &i, i.e., [gj(x)] i denotes the i-th c o m p o n e n t of the vector g j ( x ) a n d [gJ(x)]i denotes the i-th c o m p o n e n t of the vector g J ( x ) over the basis {&l, ~2, &3} _ {&l, &2, &3} of the E u c l i d e a n space E 3 associated w i t h the affine space E3; cf. Sect. 1.1. In terms of these n o t a t i o n s , the preceding relations thus become
uj(x) - ~i(ir
i a n d ui(x) - uj(x)[g j (x)]i, $ - |
T h e three c o m p o n e n t s ui(x) are called the e o v a r i a n t c o m p o n e n t s of the d i s p l a c e m e n t v e c t o r a t ~, a n d the three functions
Sect. 1.3]
25
Variational equations in curvilinear coordinates
m
ui " ~ -+ IR defined in this fashion are called the c o v a r i a n t c o m p o n e n t s of the d i s p l a c e m e n t field uig i 9 -+ E s.
A w o r d of c a u t i o n . The "old" unknown/,(&) - (fii(&)) e IR3 could be justifiably identified with the displacement vector ~i(&)~ i itself since the space E 3 may be identified with the space IR3 once the basis {~1, ~2, ~3} is given in E 3. However, this is no longer true -
e e 3 , sinr
its
ompo.r
u i ( x ) now represent the components of the displacement field over the basis {gt(x), g2(x), g3(x)}, which varies with x e -ft. m m
We likewise associate "new" functions vi" fl --+ IR with the "old" functions ~)i 9{(~}- --+ IR appearing in the variational equations by letting ~i(~)& i - " va(x)ga(x) for all & - O(x), x e ~.
R e m a r k . As the functions 72i belong to the Sobolev space HI((~), they are in fact equivalence classes of functions, and any function in a class is in fact only defined almost everywhere. Accordingly, the relations defining the (equivalence classes of) functions ui (which likewise belong to the Sobolev space HI(fl) under ad hoc assumptions on the mapping O; cf. Thm. 1.3-1) need hold only for almost all E O(fl). But in order to avoid cumbersome statements, we blithely omit such mentions, m
We begin the change of variables by considering the integrals found in the right-hand side of the variational equations, i.e., those corresponding to the applied forces. With the Cartesian components ]i . (~ _ O ( ~ ) -+ IR of the applied body force density, let there be associated its c o n t r a v a r i a n t c o m p o n e n t s f i . fl _+ IR, defined by
/ i ( ~ ) e i --" f i ( x ) g i ( X ) for a11 ~ - O(x), x e fl. This definition shows that
and consequently that f i E L2(f~) if ]J C L2(~). It also implies that
Three-dimensional linearized elasticity
26
[Ch. 1
]i(x)~i(~ ) = (]i(~,)~i) . (~)j(X)eJ) :
for all & = |
]i(&)r
(fi(x)~i(X))"
(Vj(X)OJ(X)) : ]i(x)Vi(X)
x E f~. Hence
- fi(x)vi(x)v/g(xi dx for all ~ - O(x),x e f~,
since d& - v/g(x)dx (Thm. 1.2-1 (a)), and thus
f . fivi d& - / ~ fivi ~ dx.
Remarks. (1) What has just been proved is in effect the invariance of the number fi(x)vi(x) with respect to changes of curvilinear coordinates, provided one vector appears by means of its "contravariant" components (i.e., on the "covariant" basis) and the other by means of its "covariant" components (i.e., on the "contravariant" basis). Naturally, this number is nothing but the Euclidean inner product of the
two vectors! (2) The adjectives "covariant" and "contravariant" used in the definitions of the components of the displacement and force vector fields are given a proper interpretation in Ex. 1.2. II W i t h the Cartesian components h / 9 ~1 - O ( r l ) --+ R of the applied surface force density, let there be likewise associated its cont r a v a r i a n t c o m p o n e n t s h / . r l -~ ]~, defined by
hi(&)&idr(& ) -" hi(x)gi(x)v/g(x)dr(x) for all ~ - O(x), x C where the area elements d~(&) at & - | are related by (Thm. 1.2-1 (b))
e r l and d r ( x ) at x e r l
This definition shows that
for all ~ = |
x E rl,
Variational equations in curvilinear coordinates
Sect. 1.3]
hence that h i E
L~(r~) if h i
27
E L2(rl), since
hi(x) -- ~/nk(x)gkt(x)nt(x) hJ(F~)[gi(x)]j.
The factor {nkgmnt} -1/2 is introduced in the definition of the functions h i in order that
frl ~i~)idr - f r l hiviv/g dr,
i.e., in order that the same factor v/~ appears in both integrals fn fiviv/'g dx and fr~ hiviv/~ dr. It in turn gives rise to a more "natural" boundary condition on F1 when the variational equations are transformed into a boundary value problem (Thm. 1.6-1). Transforming the integrals appearing in the left-hand side of the variational equations seems to be a similarly innocuous enterprise, simply requiring in addition applications of the chain rule, since firstorder derivatives occur in the integrands. In fact, carrying out this transformation in a finite time is a subtle task, which relies in particular on the notion of covariant differentiation of a vector field; cf. part (iii) of the next proof. T h e o r e m 1.3-1. h~ e L ~ ( ~ ) b~ g i , ~ the weak solution of Cartesian coordinates
Let ~ be a domain in E 3, let ]i C L2(~) and Inactions, ~ d t~t ~ (~i) c V(fi) d ~ o t ~ the associated linearized elasticity problem in (Sect. 1.1).
Let ~2 be a domain in IRa and let O be a s of-~ onto { ~ } - - O ( ~ ) , so that the vectors gi(x) - OiO(x) are linearly independent at all points x E -~. Let the vectors g~(x) be defined by the relations g i ( x ) , gj(x) - 5j and let g(x) - det(gi(x ) 9gi(x)) and 9 ~j(~) - a~(~) 9a j(~), 9 e ~. Then the vector field u - (ui) "-~ -+ I~3 defined by
~ti(&)~ i =: ui(x)gi(x) for all & -- O(x), x 6 ~,
Three-dimensional linearized elasticity
28
[Ch. 1
satisfies the following variational problem: U e V(~):=
{~ --
(Vi) e
HI(~);
"v = 0 011 r 0 ) ,
fa A~Jk'ekll,(u)eill,(v)~/~dz - f y'v~/~ dz + fr h%~v~dr 1
fo~ all ~ = (~1 e v ( a ) ,
where ro . - o - l C P o ) , r l := o-l(:P1), the ]unctions h i E L 2 ( r l ) are defined by: ]i($)&i d$ : : v / g ( x ) f i ( x ) g i ( x ) d x ,
~- |
fi
x e f~,
hi(~)~i dr(~) =: v/g(~)hi(x)gi(x) dF(x), ~ - | the functions A ijkt - A jikt - A kui
AiJkl
:__
E cl(~)
L2(f~) and
e
x E rl,
are defined by:
Agijgkt + iz (gik gj! q_ gil gjk),
and finally, the functions eillj(v) - ejlli(~ ) E L2(~) are defined for all v E HI(~) by:
P eillj(v) := ~1 (0j,~ + 0 ~ j ) -rijvp, where
r~5 . - g~.O~gj : rj~i e c~
Proof. The following convention holds throughout this proof: The simultaneous appearance of ~ and x in an equality means that they are related by & = O(x) and that the equality in question holds either for all x E (2 or for almost all x E (2. (i) A,othe~ ~ , e ~ i o ,
of [g~(~)]k "- g~(~)" ek.
Let O(x) - | and (9(&) = Oi(~)ei, where 19" { ~ } - ~ _ denotes the inverse mapping of | Since O ( O ( x ) ) - x for all x e f~, the chain rule (see, e.g., Vol. I, Thm. 1.2-1) shows that the matrices
Sect. 1.3]
Variational equations in curvilinear coordinates
29
VO(x)
"-- (0jOk(:c)) (the row index is k) and ~ r 6 ( ~ ) : = (~k~)i(~)) (the row index is i) satisfy
vo(~)ve(~)
- ~,
or equivalently,
ojos(~) i = 5j.
The components of the above column vector being precisely those of the vector gj(x), the components of the above row vector must be those of the vector gi(x) since gi(x) is uniquely defined for each exponent i by the three relations g i ( x ) . g j ( r . ) - ~ , j - 1, 2, 3. Hence the k-th component of gi(x) over the basis {&l, e2, &3) has the following expression in terms of the inverse mapping {b"
[g~(~)]k - & ~ ( e ) . (ii) Definition of the Christoffel symbols. We next compute the derivatives Otgq(x) (the fields gq = gqrg r are of class C1 on ~ since O is assumed to be of class C2) as they will clearly be needed (see (iii)) for expressing the derivatives Oj~i(a~) as functions of x (recall that u i ( x ) - uk(x)[gk(x)]i). The vectors gk(x) forming a basis, we may write a priori
0~g~(~) - -r~k (~)gk (~), thereby unambiguously defining functions r~k 9~ -+ I~, Which are called the Christoffel symbols. To find their expressions in terms of the mappings O and ~), we observe that
r~(~) - r~(~)5~
- rT~(,)g~(,),
g~(~) - - 0 , g ~ ( ~ ) . g~(~),
and, noting that Ot(gq(x) 9gk(x)) -- 0 and [gq(X)]p - OpE)q(~), we obtain
rTk(~) - g~(~). 0~gk(~) - 8 ~ q ( ~ ) 0 ~ k o p ( ~ )
- r~,(,).
30
[Ch. 1
Three-dimensional linearized elasticity
Since O ~ C2(~;~ 3) and O e C~({fi}-; ~3) by assumption, the last relations show that r~t ~ C~ (iii) Let ~ -- (~)i) be given in the space V(~).
Then the vector field v - (vi) defined by ~)i(~)e i -- V i ( x ) g i ( x ) iS in the space V(f~); moreover,
~(~)
- ,kll~(~)[g~(~)]~[g~(~)]~,
where
~11~(~) . - o ~ ( ~ 1
- r~(~)~(~)
denotes a "covariant derivative" of the vector field vkg k at x, [gk(x)]i denotes the i-th component of gh(x) over the basis {&l, &2, &3), and r~k(x) - g q ( x ) . Orgy(x) are the Christoffel symbols introduced in (ii).
By a classical result about composite mappings (see, e.g., Ne~as [1967, Chap. 2, Lemma 3.2] or Adams [1975, Thm. 3.35]), a function o (9 is in H1(~2)if ~ e H I ( ~ ) a n d | ~ -+ { ~ } - - | a bijection such that both O and its inverse mapping O are Lipschitzcontinuous; consequently, the functions ~j o O are in H l(f~). Since the functions [gi]1 are in C1(~), the functions vi - (6j o O)[gi]J are thus in Hl(fl); besides, they satisfy vi - 0 on r0 since ~3j - 0 on r0. We next compute the partial derivatives vSj~3i(~) as functions of z by means of the relation ~3i(~) - Vk(X)[gk(x)]i 9 To this end, we first note that a differentiable function w" fl --+/~ satisfies
~j~(6(e)) - 0 ~ ( ~ ) ~ j ~ ( e ) -
o~(~)[g~(~)]~,
by the chain rule and by (i). In particular then,
= 0~k(~l[g~(~l]j[gk(~)]~ + ~(~1 (0~[g~(~)]i)[g~(~l]j :
(0~,~(~1 - r T ~ ( ~ ) , ~ ( ~ ) ) [ g k ( x ) ] i [ g l ( x ) ] j ,
since O t g q ( x ) - - r I ~ ( ~ ) g k ( ~ )
by
(ii).
(iv) The integrand AiJkl~kl(iz)~ij(i~ ) 9 ~ -+ IR appearing in the left-hand side of the variational equations over ~ satisfies
(x),
Variational equations in curvilinear coordinates
Sect. 1.3]
31
where (covariant derivatives such as villi have been introduced in (iii))" AiJkl :_ Agij gkl + i~(gik gjl + gil gjk) __ AJikl _ Aklij E C1(~), 1 P - ~Jtl~(") e L 2 ( a ) . ~JlJ(") "= ~("~llJ + vJlr~) - ~1 (Ojvi + Oivj) - r~j,,p We first have, by (iii), 1 1
Since the Christoffel symbols satisfy r ~ - r ~ (c~. (ii)), the functions ek[ll(v) likewise satisfy ekl[t(V ) -- el[ih(v ). Recalling that a ~ ( ~ ) , aJ (=) - g~J (~), we ~e~t h ~
~iJ~ij(iJ)($) - ~ p p ( ~ ) ( $ ) - (ekllt(v)[gk]p[gt]p) ( x ) -
(ekl[t(v)g kI) (x),
and thus
Likewise,
1 (~.~, + ~ . ~ ) ~.(a)(~)~.~(~)(~)- ~.~(~)(~1~.~(~1(~1 : (~ll,(~)~ll.(~)[~],[~']~[~m],[~"]~)(~1 -
(~"~'"~ll,(,,)~mll,,(,,))(~,) 1
since e~ll~(u ) - e~ll~(u ).
(~)
Co,~a~on,.
shows that
Si~c~ d~ - v / ~ ( x i d~ ( T h e .
1.~-1
(~)), part (~)
Three-dimensional linearized elasticity
32
[Ch. 1
on the one hand. At the beginning of this section, it was also shown that the definition of the functions fi and h i implies that
~ ]iv i d~, -- /fi fivi~/r~ dx, f
1
hi~)idr -- ~ hi~i~/Cgd~, 1
on the other, and thus the proof is complete.
m
Naturally, if ~s is identified with I~3 and | -- ida8, each vector gi(x) is equal to ei and thus gi(x) = ~i, g(x) -- 1, giJ(x) - 6 ij, and PiPj(x) - 0 for all m E ~; in addition, the fields (ui) and (fii), (fi) and (]i), (h i) and (hi), and (A ijkl) and (~lijkl) coincide. The variational problem found in Thm. 1.3-1 constitutes the varia t i o n a l , or w e a k , f o r m u l a t i o n of the e q u a t i o n s of t h r e e - d i m e n sional l i n e a r i z e d e l a s t i c i t y in e u r v i l i n e a r c o o r d i n a t e s . The functions A ijkl" -~ -~ I~ introduced in Thm. 1.3-1 are called the e o n t r a v a r i a n t c o m p o n e n t s of the t h r e e - d i m e n s i o n a l elast i c i t y t e n s o r in c u r v i l i n e a r c o o r d i n a t e s ; they thus generalize to arbitrary curvilinear coordinates the components ~ijkt . { ~ } - _+ of the elasticity tensor in Cartesian coordinates (Sect. 1.1). The boundary condition u - (ui) - 0 on r0, or the equivalent relation uig' -- 0 on O(r0), constitutes a (homogeneous) b o u n d a r y c o n d i t i o n of place. o
Remark. The interpretation of the adjective "contravariant" attached to the components of the elasticity tensor is given in Ex. 1.4. m
1.4.
COVARIANT DERIVATIVES AND CHRISTOFFEL SYMBOLS IN A THREE-DIMENSIONAL DOMAIN
We now record as a theorem several important definitions, relations, or properties all originating from the proof of Thm. 1.3-1. Together with those of Sect. 1.2, these constitute our first encounter with differential geometry "in a three-dimensional manifold in IR3 ". Other related notions are treated later, such as the covariant derivatives of
Covariant derivatives and Christoffel symbols
Sect. 1.4]
33
a tensor field (Thin. 1.6-1) and the recovery of a three-dimensional manifold from its metric tensor field (Sect. 1.9).
For further details and complements, see classical texts such as Malliavin [1972], Choquet-Bruhat, Dewitt-Morette & Dillard-Bleick [1977], or Abraham, Marsden & Ratiu [1983]. The books by Green & Zerna [1968], Marsden & Hughes [1983], and Simmonds [1994] provide treatments of differential geometry and tensor analysis that are essentially motivated by, and thus well adapted to, three-dimensional elasticity. T h e o r e m 1.4-1. Let the assumptions on the mapping 19 and the definitions of the vector fields gi and gJ be as in Thm. 1.3-1, and let there be given a vector field rig 2 on f~ with smooth enough covariant components vi 9f~ --~ ~. (a) The components of the vectors gi(x) and gJ(x) are given by
=
In other words, the components of g i ( z ) are those of the i-th column of the matriz V O ( z ) while those of gJ(z) are those of the j-th row of the matriz V 0 ( ~ ) . (b) The f i r s t - o r d e r covariant derivatives Villi " f~ --~ I~ of the vector field rig', which are defined by ^
^
p ._ gp Villi : : Ojvi - r ipj v p, where Fij "Oigj,
can be also defined by the relations
(c) The Christoffel symbols satisfy the relations
::
g
.Oig
OigP - - r pij gJ and 0j gq - rjqgi" i
-
.
Three-dimensional linearized elasticity
34
(d) Let r
[Ch. 1
i be the vector field defined on {fi}- - - |
by
~i(~)~ i : - vi(x)gi(x) for all ~ - O(x), x e ~, and let [gk(x)]i denotes the i-th component of gk(x) over the basis {el, e2, &z}. Then, for all x E f~,
~j~)i(~,) - (Vklll[gk]i[gl]j) (X), X -- O(X). II
Proof. It remains to verify that the covariant derivatives Villi, defined in part (iii) of the proof of Thm. 1.3-1 by p
Villi -- Ojvi -- rijv p, may be equivalently defined by the relations
Oj(vig i) --
Villjg i,
which unambiguously define the functions the vectors gi are linearly independent at tion. To this end, simply note that, by symbols satisfy Oigp - - r P .3g j (eL part 1.3-1); hence
oj(,ig i) - ( o j , i ) g i +
Villi -- ( O j ( v k g k ) } "gi since all points of ~ by assumpdefinition, the Christoffel (ii) of the proof of Thin.
~ia~g ~ - (aj V i)g i - ~ i r j ki g k -~illjg'."
i note that To establish the other relations Ojgq - r jqgi, 9
0 - Oi(g p . g q ) - -F~ig~.gq + g P . Oig q
_
-r~j + gp .Ojg~; p
hence
Ojgq -- (Ojgq.gP)gp -- rpqjgp" II If the affine space C3 is identified with IR3 and 19 - id, the relation COj(vigi)(x) -- (Villjgi)(x) found in Thm. 1.4-1 (b) reduces to
~j(~)i(x)e i)
-
-
(Oj~)i(x))e, i. In this sense, a covariant derivative of the
Sect. 1.5]
Linearized change of metric tensor in curvilinear coordinates
35
first order constitutes a generali~,ation of a partial derivative of the first order in Cartesian coordinates. Remarks. (1) The relation between the functions 0jvi and Vklll (Tam. 1.4-1 (d)) can be inverted; cf. Ex. 1.5. (2) The formula
established in part (iv) of the proof of Thm. 1.3-1, is the expression of the divergence of a vector field in curvilinear coordinates. Formulas likewise expressing the gradient, curl, and Laplacian operators in curvilinear coordinates are given in Ex. 1.6. m 1.5.
LINEARIZED CHANGE OF METRIC CURVILINEAR COORDINATES
TENSOR
IN
Let there be given an arbitrary d i s p l a c e m e n t field of the set O(fl), i.e., an arbitrary vector field rig z defined by means of its cov a r i a n t c o m p o n e n t s vi" ~ ~ R; this means that vi(x)gi(x) is the displacement of the point ~ = O (z). We then show that the matrix 1
p
vi + Oiv~) - ri~vp),
introduced in Thm. 1.3-1, generalizes to arbitrary curvilinear coordinates the linearized strain tensor
in Cartesian coordinates (Sect. 1.1). More specifically, we show that the matriz (eillj(V)) likewise measures (half of) the linearized difference between the "new" metric in the deformed configuration ( 0 + vigi)(-~) and the "old" metric in the reference configuration O(~), but now expressed by means of their covariant components in terms of the curvilinear coordinates xi. We also record for future reference the relation between the functions ~j(~) and ekllt(v) that was established in part (iv) of the proof of Thm. 1.3-1 (this relation can be inverted; cf. Ex. 1.5).
Three-dimensional linearized elasticity
36
[Ch. 1
T h e o r e m 1.5-1. Let f~ be a domain in I~3, let 19 :-0 --> E 3 be a smooth enough injective mapping such that the three vectors gi = Oi| are linearly independent at all points of-0, and let the vectors gi be defined by g i . g j = ~ . Given an arbitrary displacement field
V" :--
V
ig i
w
of the set | with smooth enough covariant components vi" f~ ~ I~, let the e o v a r i a n t c o m p o n e n t s of the l i n e a r i z e d s t r a i n , or line a r i z e d c h a n g e of m e t r i c , t e n s o r associated with this vector field be defined by 1 eillj(V ) := ~[ffij(V) -- ffij] fin
where 9ij and 9ij(v) denote the covariant components of the metric tensors respectively attached to the sets 19(-0) and (19 + vigi)(-~), and [...]tin denotes the linear part with respect to v := (vi) in the ezpression [...]. Then
eillj(v)
-
~
" gi
"gj
(v)
1 = ~(vill j + Villi) = 2
- Pv.._--_~.,,3
P p " - O ~ I ~ and Fij p 9-where the functions Villi - O j v i - Fijv f~ ~ R are respectively the first-order covariant derivatives of the vector field rig i and the Christoffel symbols of the surface S (Thin. 1.4-1). The functions eillJ(V ) are also called the l i n e a r i z e d s t r a i n s in e u r v i l i n e a r coordinates. The linearized strains in Cartesian and curvilinear coordinates are related by:
.=
:
-
Sect. 1.6]
Boundary value problem in curviIinear coordinates
37
Proof. The covariant components gij (v) are defined in ~ by (Sect. 1.2)
g,j(~) : o~(| + ~). oj(|
+ ~).
Note that both the reference configuration O(~) and the deformed configuration (| + ~)(12) are equipped with the same curvilinear coordinates xi. The relations o~(| + ~) : g~ + O~ then show that
= g~j + oj~ 9g~ + 0 ~ 9gj + o ~ . oj~, hence that 1
- ~l ( 0 j ~ "g, + o ~
. g j .)
The other expressions of the functions eillj(V) follow from the relations 0j~ - vklljg k and vkllj - Ojvk - r j kpvp (Thm. 1.4-1). m
1.6.
THE BOUNDARY VALUE PROBLEM OF THREE-DIMENSIONAL LINEARIZED ELASTICITY IN CURVILINEAR COORDINATES
While deriving the boundary value problem that is (at least formally) equivalent to the variational equations of three-dimensional linearized elasticity in Cartesian coordinates simply amounts to applying the fundamental Green formula, doing so in curvilinear coordinates is more subtle. As we next show, it relies in particular on the notion of covariant differentiation of a tensor field. We recall that ni(x)e i denotes the unit outer normal vector at x E r and that f i E L2(~) and h i C L2(r~) by assumption.
Three-dimensional linearized elasticity
38
[Ch. 1
T h e o r e m 1.6-1. Let the notations and assumptions be as in Thm. 1.3-1. If the solution u = (ui) to the variational problem: E V(~):=
{V = (Vi) E Hl(fl); v = 0 on r 0 } ,
f A'Jk~ekll,(u)e~ll~(v)v/~dz=fi'vi~dz+frh%~dr for all v = (vi)E V(f~), 1
is smooth enough, it also satisfies the following b o u n d a r y v a l u e p r o b l e m of t h r e e - d i m e n s i o n a l l i n e a r i z e d e l a s t i c i t y in e u r v i linear coordinates: -~ilj-f
~ in
f~,
on
r0,
ui -- 0
where the functions oriJ :--- AiJktekllt(u )
are the c o n t r a v a r i a n t c o m p o n e n t s sor field, and the functions
of the l i n e a r i z e d s t r e s s t e n -
are f i r s t - o r d e r e o v a r i a n t d e r i v a t i v e s of this tensor field (naturally, r jjq - ~j:~3 rjq~ when k - j according to the summation convention)
Proof. (i) We first establish the relations (needed in part (ii)): q
To this end, we recall that x/~ - I det V{91, that the column vectors of the matrix V | are 01, 09., g3 (in this order), and that the vectors gi are linearly independent at all points in f~. Assume for instance that det V | > 0, so that V~ : det V O : det(gl, g2, g3) in f~.
Boundary value problem in curvilineav coordinates
Sect. 1.6]
39
Then
OjX/~- det(0jgl, g2, g3)-1-det(gl, Ojg2, g 3 ) + det(gl, g2, Ojg3) - r~j aet(g,, g2, g~) + r~j aet(gl, g~, g~) + r[j aet(g~, g~., g~) P (Thm. 1.4-1 (c)); hence since 0jgq - rqjgp
0 j @ - (r b + r~j + r~j)det(ffl,
if2, if3) -
r~j ~4~.
The proof is similar if v / ~ - - d e t ~ ' O in f~. (ii) Assume that the functions a ij are in Hl(f~) (as is the case if u E H2(f~) and the assumptions on the mapping O are those of Thm. 1.3-1), so that we can apply the following Green formula in Sobolev spaces (see, e.g., Vol. I, Thm. 6.1-9)"
ff(Ojv)wdx - - ff vOjwdx+ frVWnjdF f~ all v, w E Hl(f~)" Taking also into account the symmetries crij - orJi (which themselves follow from the symmetries A ijkl - AJikl; cf. Thm. 1.3-1) and q (part (i)), weobtain: the relations 0j x/~ - V~ rqj
fn AiJkleklll(u)eillj(v)v/g dx - / ~ aiJeillj(v)v~ dx a =
.. 1
p _
rijvpdx
-- - ~ Oj (~criJ)vi dx + fr v~aiJnjvi dr - ~ ~apJripjvi dx
----ff~ %//g(Ojo'iJ-~- FipjaPJ+ r~qaiq)vidx+ fr %//-goriJnjVidr
for all v - (vi) C Hl(f~). Hence the variational equations imply that
fnr
(~J [Ij +/*) v, dx - fr 4~ (~iJnj - h*) v~ dr 1
Three-dimensional linearized elasticity
40
[Ch. 1
for all (vi) E V(f~). Letting the functions Vi vary in ~ ( f t ) shows t h a t crijiij + f i _ 0 in f~ and letting the functions (vi) vary in V(ft) in the remaining equations shows that aiJnj - h i - 0 on F l, m Naturally, the same b o u n d a r y value problem in curvilinear coordinates can be directly obtained from its Cartesian counterpart: __~j~iJ
=
~2i --
&ij hj
-
]i
in ~,
0
on on
hi
I'o, FI,
where &ij _ f~ijkl~kl(~t ) (Sect. 1.1); but again this requires some care; cf. Ex. 1.7. The variational equations
aiJeillJ(v)~/~ dx -- ffl fivi~/rg dx -f- f r h i v i v ~ d r 1
for all v = (vi) E V ( ~ ) constitute the linearized principle of virtual w o r k in c u r v i l i n e a r c o o r d i n a t e s . The equations
- ~ i ~ l l j - f ~ in cri] nj -- h i
on
f~, F1,
constitute the linearized equations of equilibrium in c u r v i l i n e a r coordinates. 1.7.
A LEMMA
O F J . L. L I O N S ;
THREE-DIMENSIONAL KORN'S INEQUALITIES AND INFINITESIMAL RIGID DISPLACEMENT LEMMA
IN CURVILINEAR
COORDINATES
We first review some essential definitions and notations, together with a fundamental lemma of J.L. Lions (Thm. 1.7-1); we borrow here the beginning of Sect. 1.1 from Vol. II. A domain ft in IRn is an open, bounded, connected subset of IRn with a Lipschitz-continuous boundary F - Of~, the set ft being locally
Korn's inequalities in curvilinear coordinates
Sect. 1.7]
41
on one side of F. As F is Lipschitz-continuous, a measure elf can be defined along F and a u n i t o u t e r n o r m a l v e c t o r v = (vi) ni--1 ("unit" means that its Euclidean norm is one) exists dF-almost everywhere along F (see, e.g., Vol. I, Sect. 1.6). For the same reason, the classical spaces Cm(f~) or Cm' a (f~), 0 < c~ < 1, can be unambiguously defined for any integer m > 0; see, e.g., Ne~as [1967, Chap. 2], Adams [1975, Chap. 1], Stein [1970]. Let f~ be a domain in I~n. For each integer m > 1, Hm(f~) and H~ n(f~) denote the usual S o b o l e v s p a c e s ; in particular,
HI(~) "- {~ E L2(~); Oi~ E L2(~), 1 < i <_~}, H2(~) .- {~ ~ HI(~); 0~j, E L2(~), 1 < i, j < ~}, where Oiv and Oiiv denote the partial derivatives of the first and second order in the sense of distributions, and
H~(~)
:= {~
e
m(~); v
-
o o~ r},
H2(f~) := {v E H2(f~); v - Ovv - 0 on r } , where the relations on r are to be understood in the sense of traces, and O~v " - ~i"=1 viOiv denotes the outer normal derivative of v along F. Boldface letters denote vector-valued or matrix-valued functions, also called vector fields or matrix fields, and their associated function spaces. The norm in L2(f~) or L2(f~) is noted I" Io, n and the norm in H m(f~) or H m(~), m > 1, is noted Ii" lira, n. In particular then, Ivl0,n " -
(Io
Ivl 2 dx
}1/2
{~ Ivlo, a " -
Ilvll~,~ "-
if v E L2(f~), if V - - ( V i ) i n l E L2(a),
~ [vil2, n i=i
2 Ivlo,~+
IOivl~, ~
i f v E H 1(a),
i--1
Ilvll~,a "-
~ Ilvill~,a
if V - - ( V i ) i n l E I-II(~'~),
i=1
11~112,~ " -
Ioivt2~ +
Ivl2,n + i:1
IOijvl2,n
i f v E H2(f~).
i,j:l
Most properties needed in this volume about the Sobolev spaces gm(f~), and also about the Sobolev spaces Wm'P(f~), 1 G p G oo, are
42
[Ch. 1
Three-dimensional linearized elasticity
recalled in Vol. I, Sect. 6.1. Detailed treatments are found in Ne~as [1967], Lions & Magenes [1968], Dautray & Lions [1984, Chaps. 1-3], Adams [1975]. An excellent introduction is given in Brezis [1983]. In this section, we also consider the Sobolev space H-~(n)
:=
dual
space of Hl(f~).
Another possible definition of the space H~(f~) being H](f~) -
closure of 9(f~) with respect to I1" [1~..,
where T~(f~) denotes the space of infinitely differentiable real-valued functions defined over f~ whose support is a compact subset of f~, it is clear that v E L2(f~) =~ v E H-i(f~) and Oiv E H-i(f~), 1 < i < n, since (the duality between the spaces 29(f~) and 29'(f~) is denoted < ", 9>)] < v, qo > l - l fnvqodxl <_ Ivl0,nllqolli,n,
I<
Oiv,
>1-!-
< v, oil, > 1 -
- f. vOi dx I <- Ivlo,nll
'llx,n
for all qo E 79(f~). It is remarkable, but also remarkably difficult to prove, that the converse implication holds: T h e o r e m 1.7'-1 (lemma of J.L. Lions). Let f~ be a domain in ]~n and let v be a distribution on f~. Then {v E H-i(f~) and Oiv E H-i(f~), 1 < i < n} =~ v E L2(f~). II This implication was first proved by J.L. Lions, as stated in Magenes & Stampacchia [1958, p. 320, Note (27)]. Its first published proof for domains with smooth boundaries appeared in Duvaut & Lions [1972, p. 111]; another proof was also given by Tartar [1978]. Various extensions to "genuine" domains, i.e., with Lipschitz-continuous boundaries, are given in Bolley & Camus [1976], Geymonat & Suquet
Sect. 1.7]
Korn's inequalities in curvilinear coordinates
43
[1986], and Botchers gr Sohr [1990]; Amrouche & Girault [1994, Prop. 2.10] even proved that the more general implication {v E T~'(a) and Oiv E Wm'P(a), 1 <_ i <_ n} ~ v E Wm+l'P(f~) holds for arbitrary integers m E Z and real numbers 1 < p < oo. Some minimal regularity of the boundary is anyway required: Geymonat & Gilardi [1998] have shown that the lemma of J.L. Lions does not hold if the open set f~ satisfies only the "segment property".
Remark. Although Thm. 1.7-1 shall be referred to as "the" lemma of J.L. Lions in this volume, there are other results of his that bear the same name in the literature, such as his "compactness lemmas" (Lions [1961, Prop. 4.1, p. 59] or Lions [1969, Sect. 5.2, p. 57]) or his "singular perturbation lemma" (Lions [1973, Lemma 5.1, p. 126]). I Proving the existence and uniqueness of a solution to the variational problem of three-dimensional linearized elasticity found in Thm. 1.3-1 amounts to showing that the associated bilinear form is V(f~)-elliptic, as all the other assumptions of the Lax-Milgram lemma are immediately verified. To this end, an essential step consists in establishing a threedimensional Korn inequality in curvilinear coordinates (Thm. 1.7-4) as in Ciarlet [1993]; see also Ciarlet [1998a, 1998b]. In essence, this inequality asserts that, given a domain f~ C IR3 with boundary r and r0 C I' with area ro > O, the L2(f~)-norm of the matrix field (eillj(v)) is equivalent to the Hl(f~)-norm of all vector fields v E Hl(f~) that vanish on Fo (the V(f~)-ellipticity of the bilinear form also relies on the positive definiteness of the three-dimensional elasticity tensor introduced in Sect. 1.3, a property that will be established in Thm. 1.8-1). We recall that the functions eiIli(V) are the covariant components of the linearized strain tensor associated with a displacement field rig i (Thm. 1.5-1). Such a Korn inequality is obtained in three stages (Thm. 1.7-2, 1.7-3, and 1.7-4). The first one (Thin. 1.7-2) consists in establishing, as a consequence of the lemma of J.L. Lions, a Korn inequality valid for all vector fields v - (vi) E Hl(f~), i.e., that need not satisfy any boundary condition on r . As its Cartesian special case, this inequality is truly remarkable, since only siz different combinations of first-order partial derivatives, viz., 89 + Oivj), occur in its right-hand side, while all nine par-
44
[Ch. 1
Three-dimensional linearized elasticity
tial derivatives Oivj occur in its left-hand side! A similarly striking observation applies to part (ii) of the next proof. Note that we again adhere to our rule governing the usage of Latin indices.
Theorem 1.7-2 (Korn's inequality "without boundary conditions" in curvilinear coordinates). Let f~ be a domain in ~3 and let 0 be a C2-diffeomorphism of-~ onto { ~ } - -- O ( ~ ) , so that the three vectors gi -- Oi{~) are linearly independent at all points of f~. Given v - ( v ~ ) ~
Hl(f~), let 1
eillj(~) " - {-~(Oj vi 4- Oivj)
L2(~
P
denote the covariant components of the linearized change of metric tensor associated with the displacement field rig'. Then there exists a constant Co = Co (f~, 19) such that .
2 +~ II~llx,. ~ Co{ ~ IVilo,. i
leillj
(v)l 2,f~ }1/2
i,j for all v = (vi) E I-II(f~).
Proof. The proof given here is essentially an extension of that given in Duvaut & Lions [1972, p. 110] for proving Korn's inequality without boundary conditions in Cartesian coordinates. (i) Define the space w(n)
.-
e L2(n);
e
Then, equipped with the norm I[" lift defined by 1/2 i
i,j
the space W ( ~ ) is a Hilbert space. Note that the relations "e~llJ(V) E L2(~) '' are understood in the sense of distributions, i.e., they mean that there exist functions in
Sect. 1.7]
Korn's inequalities in curvilinear coordinates
45
L2(f~), denoted eillj(V), such that
eiltj(v)qo dr -- -
f.{1
-~(viOj qo for all qo E 79(f~).
Let there be given a Cauchy sequence (vk)~=l with elements v k - (v/k) e W(f~). The definition of the norm I1" Ila shows that there exist functions vi E L2(f~) and e/llj C L2(I]) such that
v~ -+ vi in L2(f~) and eillj(v k) -+ eillj in L2(f~) as k --+ c~, since the space L2(f~) is complete. Given a function qo E :D(f~), letting k --+ c~ in the relations
p k } dx, k >_ 1, eillj(vk)qo dx - _ ff~ {-~l (vki Oj q~ + vjkOiqo) + rijvpqo shows that eillj - eillj(v ). (ii) The spaces W(f~) and H 1(f~) coincide. Clearly, Hl(f~) C W(f~). v - ( v i ) e W(f~). Then
%(v)
To establish the other inclusion, let
v := ~1 (Ojvi + Oivj) - (el IIJ(v) + rijvv}
since eillj(v ) C L2(f~), r ip C C~
~ L2(f~),
and vp C L2(f~). We thus have OkVi E H - I ( ~ ) ,
o j ( o k ~ ) - {oj~/k(~) + oh~/j(~) - o ~ j k ( ~ ) } e H - ~ ( ~ ) ,
since w E L2(f~) implies Okw e H-l(f~). Hence Okvi C L2(f~) by the lemma of J.L. Lions (Thm. 1.7-1) and thus v e Hl(fl). (iii) Korn's inequality without boundary conditions. The identity mapping t from the space H I ( ~ ) equipped with I1" II1,• into the space W ( G ) equipped with I1" I1~ is injective, continuous (there clearly exists a constant c such that [[v[[~ < cllvlll, n for all v C Hi(G)), and surjective by (ii). Since both spaces are complete (cf. (i)), the closed graph theorem (see, e.g., Brezis [1983, p. 19] for
46
Three-dimensional linearized elasticity
[Ch. 1
a proof) then shows that the inverse mapping 5-1 is also continuous; this continuity is exactly what is expressed by Korn's inequality without boundary conditions. II Our next objective is to "get rid" of the norms ]vilo, n in the righthand side of the Korn inequality established in Thm. 1.7-2 when the fields v - (vi) E t t l ( f l ) are subjected to the boundary condition v = 0 on r0 C r and area r0 > o. As a preliminary, we establish the weaker property that the semi-norm v--+
i,j
becomes a n o r m for such fields, by generalizing to curvilinear coordinates the well-known infinitesimal rigid displacement l e m m a in Cartesian coordinates (see, e.g., part (ii) of the proof of Thm. 6.3-4 in Vol. I); "infinitesimal" reminds that if eillj(v) - 0 in fl, i.e., if only the linearized part of the change of metric tensor vanishes, the corresponding displacement field rig i is likewise only the linearized part of a genuine rigid displacement (for more details in Cartesian coordinates, see Vol. I, Ex. 6.2). Part (a) in the next theorem is an infinitesimal rigid displacement l e m m a "without boundary conditions"~ while part (b) is an infinitesimal rigid displacement lemma "with boundary conditions". 1.7-3 ( i n f i n i t e s i m a l rigid d i s p l a c e m e n t l e m m a in e u r v i l i n e a r c o o r d i n a t e s ) . Let the assumptions be as in Thm. 1.7-2. (a) Let v = (vi) e I'II(f~) be such that Theorem
eillJ(V ) - 0 in ft. Then the vector field r i g i iS an i n f i n i t e s i m a l r i g i d d i s p l a c e m e n t , in the sense that there ezist two vectors ~, tl C N 3 such that
vi(x)gi(x) - ~-4-d A O(x) for all x E ~, (b) Let ro be a dF-measurable subset of r = Of~ that satisfies area r o > O.
Sect. 1.7]
Korn's inequalities in curvilinear coordinates
47
Then v--
(vi) E
HI(f~), v =0 onr0, } ~ v = 0 inf,. eillj(V )
--0 in f~
Pro@ In part (iv) of the proof of Thm. 1.3-1, we established the relation ~ii(@)(&) - (ekll,(V)[gk]i[gt]j)(x)
for all ~ - |
x E ~2,
where gij(i~) : 89162 + Oivj) and the vector fields ~ -- (~3i) E I-II(h) and v - (vi) E H I ( ~ ) are related by
~i(&)~ i - vi(x)gi(x) for all & - |
x e f~.
Hence and the identity (the same as in the proof of Thm. 1.7-2) c~i(vSk~i) - 0i~ik(v) + 0keii(v) - c~ieik(v) in T~'(~) further shows that eij(~)) -- 0 in (2 ::ff Oj(Ok~)i) -- 0 in ~)'((2).
By a classical result from distribution theory (Schwartz [1966, p. 60]), each function Oi is therefore a polynomial of degree < 1 (recall that the set ~ is connected). In other words, there exist constants ci and dii such that
~i(~.) - c.i + dij~j for all ~ - (~i) C ~. But eij(v) - 0 also implies that dii vectors ~, d E IR3 such that
-dii; hence there exist two
i~i(~.)~ i - ~. + (/A O ~ for aU ~
E ~,
hence such that
vi(x)gi(x) - & + (~ A O ( x ) for all x C C~.
Three-dimensional linearized elasticity
48
[Ch. 1
Since the set where such a vector field Vi ~i vanishes is always of zero area unless ~ -- d = 0 (as is easily proved; see, e.g., Vol. I, T h m . 6.3-4), it follows that ~ - 0 when area ro > O. I Remarks. (1) Since the fields gi are of class C 1 on ~ by assumption, the components vi of a field v = (vi) E Ht(f~) satisfying eillj(v) = 0 in f~ are thus automatically in C1(~) since vi - (vjg j) "gi.
(2) Remarkably, the field rig i - fi + d A O inherits in this case even more regularity, as it is of class C2 on ~ (a similar p h e n o m e n o n will be encountered in Thm. 2.6-3)! (3) The assertion in part (a) illustrates a key idea from differentim geometry' An "intrinsic" property (the vector field rig z is an infinitesimal rigid displacement) is derived from relations (the assumptions eilij(v) - 0 in f~) written in a particular system of curvilinear coordinates. 1 We are now in a position to prove a K o r n inequality "with boundary conditions", which plays a fundamental r61e in three-dimensional linearized elasticity in curvilinear coordinates (Thm. 1.8-2). T h e o r e m 1.7-4 ( K o r n ' s i n e q u a l i t y in c u r v i l i n e a r c o o r d i n a t e s ) . Let f~ be a domain in I~3 and let 0 be a C2-di~eomorphism of-~ onto { ~ } - - O ( ~ ) , so that the three vectors gi - OiO are linearly independent at all points of f~, let ro be a dP-measurable subset of F = Of~ that satisfies
area I'0 > 0,
and let the space V(f~) be defined by
V(fl)
:= {v -
(v,) c
H~(fl);
v - o
on
r0}.
Then there exists a constant C = C(f~, t o , O) such that
'}
{
IlVlll,a ~ C ~ ]eilij(v)10,a i,j
~/z for all v C V(f~).
Sect. 1.7]
Korn's inequalities in curvilinear coordinates
49
Proof. Given v --(vi) e HI(f~), let 2
I~1.'= ~le, llj(~)lo,.
}1/2
i, j
If the announced inequality is false, there exists a sequence (v k ) koo= 1 of elements v h E V ( ~ ) such that ""llvklll,~ = 1 for all k and
Since the sequence
(vk)~~
lira [vki~ = 0.
k--+oo
is bounded in Hi(12), a subsequence
(Vl)~~ 1 c o n v e r g e s in L2(~/) by the Rellich-Kondragov theorem (see,
e.g., Vol. I, Thm. 6.1-5); furthermore, since liml~oo Ivt[n - 0, each sequence (eilj(vl))~l also converges in L2(f~) (to 0, but this information is not used at this stage). The subsequence (v l ) ~ l is thus a Cauchy sequence with respect to the norm }1/2
-(vi) -+ ~ [~il02. + ~ leijjj(~)l~o,. i
i,j
hence with respect to the norm II. IIx, n by Korn's inequality without boundary conditions (Thm. 1.7-2). The space V(f~) being complete as a closed subspace of Hl(f~), there exists v E V(f~) such that v I --+ v in Hi(12), and the limit v satisfies leillj(V)]0,~ = limt~oo le~ll~(Vt)10,~ - 0; hence v - 0 by Thin. 1.7-3. But this contradicts the relations IIv~llx,~ - 1 for all I > 1, and the proof is complete, m Identifying E 3 with I~3 and letting O - idR8 shows that Thms. 1.7-2, 1.7-3, and 1.7-4 contain as special cases the Korn inequalities and the infinitesimal rigid displacement lemma in Cartesian coordinates (see, e.g., Duvaut & Lions [1972, p. 110], Vol. I, Sect. 6.3, or Vol. II, Sect. 1.1).
50
[Ch. 1
Three-dimensional linearized elasticity
1.8.
EXISTENCE AND UNIQUENESS CURVILINEAR COORDINATES
THEOREM
IN
As a preliminary to showing that the bilinear form found in Thm. 1.3-1 is V(f~)-elliptic, we first need to establish the uniform positive definiteness of the three-dimensional elasticity tensor ("uniform" means with respect to points in f~ and to symmetric matrices of order three). Incidentally, the proof of this property has a flavor typical of differential geometry: Although the proof is seemingly innocuous, it may not be that innocuous to find (the reader should verify this assertion)! T h e o r e m 1.8-1. Let the assumptions on the mapping (9 be as in Thm. 1.3-1, let the contravariant components A ijkl 9-~ -+ I~ of the
elasticity tensor be defined by A i J k t _ Agijgkt + #(gik gfl + gil gjk),
and assume that )~ >_ 0 a n d # > O.
Then there exists a constant Ce = Ce(f~, (9, #) > 0 such that [tijl z <_ C~AiJkt(x)tkttij i,j
for all x E f~ and all symmetric matrices (tij). Proof. We first note that
u
for all x E ~2 and all matrices (tij). Using the symmetries tij - t j i and the relations
Ezistence and uniqueness theorem in curvilinear coordinates
Sect. 1.8]
51
we next have -
1 2
(#ik(z)gjl(:~)+ git(z)gjk(x,) ) tkltij -- 9ik(x,)gfl (x)tkttij
= [gi(X)]m[gJ(X)]nto[gk(X)]m[gt(X)]ntkt = E
([gi(X)]m[gJ(~')]ntij) 2 ~-0
for all x ~ fi and all symmetric matrices (t 0). We recall t h a t the vectors gi and the mappings 19 - - Omem and ~9 - ~3ie~ satisfy the relations (both established in part (i) of the proof of Thin. 1.3-1):
[gi(x)]m -- OmE)i(~,) and (gpOm(x,)om@i(~,)
-
for all & - O(x), x ~ f~. These relations therefore imply that _> 0,
-
where We have thus proved that gik (x)gfl(x)tkttij is always ~ O and vanishes if and only if tmn(x) - 0 for all m, n. But the relation between the components ~mn(~) a n d tij can be inverted as
= opoqt j
-
tpq,
and thus
im,,(~) - bm(gi(~)On(gJ(~,)tij
- 0 for all m , n ~ tij - 0 for all i, j.
To sum up, we have shown that for all z C f~ and all symmetric matrices (to),
g ik ( z ) g j~(z)tktt 0 > 0 unless t 0 - 0 for all i, j. Let g3 denote the space of all symmetric matrices. mapping
(X, (tij)) E K
:= ~X
{(tij)E ~3; E ['ijl2--1}
i,j
--~
Since the
gik(x)gJl(X)tkltij
52
[Ch. 1
Three-dimensional linearized elasticity
is continuous and its domain of definition is compact, we infer that C "=
g~k ( ~ ) F ~ ( ~ ) ~ k ~ j > O. inf (z, (tij))EK
Hence
ltij[ 2 ~ i,j
a-lgik(m)gJl(m)tkl~ij,
and thus
Itijl 2 ~
C~Ai~kl(x)tkttij
i,j
for aU x C ~ and all symmetric matrices (tij) with Ce : : (2#c) -1 since A _ 0 and # > 0 by assumption, m Remarks. (1) Another proof is suggested in Ex. 1.8.
(2) The elasticity tensor retains its uniform positive definiteness "even if )~ is slightly negative"; cf. Ex. 1.9. m Combined with Korn's inequality, the positive definiteness of the elasticity tensor leads to the existence and uniqueness of a w e a k s o l u t i o n , i.e., a solution to the variational equations of three-dimensional linearized elasticity in curvilinear coordinates. T h e o r e m 1.8-2. Let ~ be a domain in I~a and let 0 be a C 2diffeomorphism of ~ onto its image O ( ~ ) , so that the three vectors g i(x) - OiO(x) are linearly independent at all points x C ~. Let Fo be a dF-measurable subset of r - O~ that satisfies area r0 > o. Finally, let there be given constants )~ ~_ 0 and p > 0 and ]unctions f i e L6/5(~) and h i E where r l : - ! ' - t o .
L4/3(r~),
Then there is one and only one solution u - (ui) to the variational problem:
~ v(~):=
{~ -
(,~) e H ~ ( ~ ) ; ~ - o o . ro},
1
for a11 v
-(vi)
e V(f~),
Ezistence and uniqueness theorem in curvilinear coordinates
Sect. 1.8]
53
where AiJkt _ Agijgkt + # (gik gjt + git gjk) ,
--
1
rijvp,
rij
. Oigj.
The field u is also a solution to the minimization problem: u e V(f~) and J ( u ) -
J(v) "-
lfn
inf J(v), where ,,~v(n)
aiJktekllt(v)eillj(V)V~ dx
-{f
hi vi ~/'g dr } .
+ fr 1
Proof. As a closed subspace of Hl(l~), the space V(f~) is a Hilbert space. The assumptions made on the mapping O ensure in particular that the functions A ijkl, rij,p and g are continuous on ~. Hence the
bilinear form (u, v) ---+ /ft Aijkleklll(u)eillj(v)v/g dx is continuous over I-I1(f~). The continuous imbedding H I ( I l) r L6(f~) and the continuity of the trace operator tr" Hl(f~) -+ L4(r) (see, e.g., Vol. I, Thms. 6.1-3 and 6.1-7) imply that the linear form
V ---+ { / f f i v i ~ d x + f p l h i v i v ~ d r
}
is continuous over H l(fl). Since the symmetric matrix (gij(x)) is positive definite for all x C f2 and f~ is compact by assumption, there exists a constant go such that
g(x) - det(gij(x)) >_ go > 0 for aU x E f~. Then Korn's inequality in curvilinear coordinates (Thin. 1.7-4) and the uniform positive definiteness of the elasticity tensor (Thin. 1.8-1) together imply that
f AiJkleklll(v)eillj(v)v~dx>_C~Ic-2v~o[lv]]~,n for all v C V(f~).
54
Three-dimensional linearized elasticity
[Ch. 1
Hence the bilinear form is V ( ~ ) - elliptic. The bilinear form being also symmetric since A ijkt = A ktij, all the assumptions of the Laz-Milgram lemma in its "symmetric" version (see, e.g., Vol. I, Thm. 6.3-2) are satisfied: The variational equations have one and only one solution, which may be equivalently characterized as the solution of the minimization problem stated in the theorem, m An immediate corollary with a more "intrinsic" flavor to the above result is the existence and uniqueness of a displacement field uig i, whose covariant components ui E H i ( n ) are thus obtained by finding the solution u = (ui) to the variational problem of Thm. 1.8-2. Since the vector fields 9i of the contravariant bases belong to the space C1(~) by assumption, the displacement field uig i also has its Cartesian components in H 1(fl). Naturally, the existence and uniqueness result of Thm. 1.8-2 implies the same result in Cartesian coordinates (to see this, identify E 3 with I~a and let O - idR3). The converse implication also holds; cf. Ex. 1.10.
Remarks. (1) When I'0 = F and the boundary r is smooth enough, regularity results can be obtained, showing that the weak solution obtained in Thm. 1.8-2 is also a "classical solution", i.e., a solution of the corresponding boundary value problem (Thin. 1.6-1); cf. Ex. 1.11. (2) The assumptions ]i E L2(~) and ]~i e L2(?l) made for definiteness in Sect. 1.1 can be slightly weakened as in Thm. 1.8-2, to ]i e L6/5(~) and hi e L4/3(rl). m 1.9 b.
C O M P L E M E N T : R E C O V E R Y OF A THREE-DIMENSIONAL MANIFOLD METRIC TENSOR FIELD
FROM ITS
Let ~ be an open subset in IR3 and let O : ~ -+ E 3 be a thrice continuously differentiable mapping such that the three vectors g i = 0 i 0 are linearly independent everywhere in ~. As before, let gij - 9 i ' g j , let the three vectors gJ be defined by the relations
gJ "gi - ~ , let gij _ 9 i . 9j and finally, let the Christoffel symbols be defined by Fi~ - 9P. ~jgi. It is easily found that the Christoffel
Recovery of a three-dimensional manifold
Sect. 1.9b]
55
symbols satisfy the following compatibility conditions (Ex. 1.12):
o~r ~ - o~ r~, + r,"~r ~ , - r ~ r~k - o i~ ~. These conditions are in effect relations between partial derivatives of the first, second, and third order of the mapping | They are also relations between the functions gij and their derivatives, since the Christoffel symbols r i~ themselves may be directly defined in terms of the functions gij as follows (Ex. 1.12):
r,~ - F ~ r , j ~ , where
rijq
:=
1
-~(Ojgiq + Oigjq -
Oqgij).
The functions rijq a r e called the Christoffel symbols of the first kind and the functions r pi are called in this context the Christoffel symbols of the second kind. The above compatibility conditions may also be expressed in terms of the Christoffel symbols of the first kind as (Ex. 1.12): O l r i k j -- o k r i l j -4- gmn(rit~rjkm - r i k , r j t m ) = 0 in f~.
Remarkably, these necessary conditions (in either one of their equivalent forms) are also sufficient for the existence of a mapping {9 : f~ C R 3 --> E 3 whose metric tensor field is given on f~: T h e o r e m 1.9-1. Let f~ be a simply connected open subset of ~3, and let there be given a twice continuously differentiable, symmetric, and positive definite matrix field (gij) on f~ that satisfies
otrikj - okrilj + g~"(ri~.rjk~
- rik.rj~m)
- o i n f~, w h e r e
1
Then there exists a mapping | C C3(f~; 1~3) such that Oi| . Oj | = gij in ft. Furthermore, this mapping is unique "up to rigid deformations in R 3 ": This means that any other solution is necessarily of the form
56
Three-dimensional linearized elasticity
[Ch. 1
x E f~ --+ e + Q| where e is a vector in I~3 and Q is an orthogonal matrix of order three. This result is a consequence of the compatibility relations satisfied by the Christoffel symbols, of the simple connexity of f~, and of a deep existence result of Thomas [1934]; see Blume [1989]. EXERCISES
1.1. Compute the vectors of the covariant and contravariant bases, the volume, area, and length elements, the Christoffel symbols~ and the covariant and contravariant components of the metric tensors (Sect. 1.2) corresponding to cylindrical and spherical coordinates (Fig. 1.2-3). 1.2. This exercise explains in what sense the components of a vector may be "covariant" or "contravariant". Let f~ and ~ be two domains in I~a and let @ 9~ ~ E a and @ 9 { ~ ) - ~ E s be two Cl-diffeomorphisms such that | - @(~) and such that the vectors g i ( x ) " - 0i@(x) and ~)i(&) - 0i@(~) of the covariant bases at the same point @(x) - ~)(~) - & e { ~ } are linearly independent. Let gi(x) and ~i(~) be the vectors of the corresponding contravariant bases at the same point &. (1) Show that gi(x)-
where X -
0X i g~ ~-~X/(X)gj(X) and ( x ) -
(X i) :=
~-x
0;~~ ~(~)~i(~)
o @ 9~ -~ { ~ } - (hence ~ - X(X)) and
( ~ ) . - x - ~ . {fi}- -+ ~.
(2) Let vi(x) and vi(x) be the covariant components and let vi(x) and ~i(~) be the contravariant components of the same vector at ~, i.e., ~(:)g~(:)
- ~i(~W(~)
- ,~(:)g~(~)
-
r
Show that
~(~)-
~OXi (x)6i(~ ) and v i ( ~ ) -
0 ~ ~ (~)~i(~)
In other words, the components vi(x) "vary like" the vectors 9 i ( x ) of the covariant basis under a change of curvilinear coordinates, while
Ezercises
57
the components vi(x) of the same vector "vary like" the vectors gi(x) of the contravariant basis: This is why they are respectively called "covariant" and "contravariant". 1.3. This exercise shows why the "covariant" and "contravariant" components of the metric tensor are so named. The notations and assumptions are those of Ex. 1.2. Let gij (x) and gij (&) be the covariant components and let gij (x) and ~z3(&) be the contravariant components of the metric tensor (Sect. 1.2) at the same point @(x) - @)($) - ~ e { ~ } - . Show that .
.
0x k
0x ~
g i j ( x ) - -~xi(x) ~
9 0~ ~ 0~J )~k~ (x).qkl(~) and g,3 ( x ) - ~ ( x ) - ~ l (x (x)"
These formulas explain why the components gij(x) and gij (x) are respectively called "covariant" and "contravariant": Each index in gij (x) "varies like" that of the corresponding vector of the covariant basis under a change of curvilinear coordinates (see Ex. 1.2 (1)), while each exponent in gZ3(x) "varies like" that of the corresponding vector of the contravariant basis (see ibid.).
Remark. What is exactly the "second-order tensor" hidden behind its covariant components gij(x) or its contravariant exponents gij (x) is beautifully explained in the gentle introduction to tensors given by Antman [1995, Chap. 11, Sects. 1 to 3]; it is also shown in ibid. that the same "tensor" also has "mixed" components g~(x), which turn out to be simply the Kronecker symbols (f~! Exhaustive treatments of tensor analysis, particularly as regards its relevance to elasticity, may be found in Boothby [1975], Marsden & Hughes [1983, Chap. 1], or Simmonds [1994]. 1.4. The assumptions are those of Ex. 1.2. 1 (1) Let ~iJk(x) if {i, j, k} is an even permutation of {1, 2, 3}, let eiJk(x) -
1
V/g(x ) if {i, j, k) is an odd permutation of
{1, 2, 3), and let eiJh(x) - 0 otherwise. Show that each exponent in the functions eiJk(x) "varies like" that of the corresponding vector of the contravariant basis under a change of curvilinear coordinates. (2) Show that each exponent in the "contravariant" components AiJkt(x) of the three-dimensional elasticity tensor in curvilinear co-
58
Three-dimensional
[Ch. 1
linearized elasticity
ordinates introduced in Sect. 1.3 again "varies like" that of the corresponding vector of the contravariant basis under a change of curvilinear coordinates.
Remark. See again Antman [1995, Chap. 11, Sects. 1 to 3] to decipher the "third-order tensor" and "fourth-order tensor" hidden respectively behind their contravariant components s ijk (x) and AiJkt(x). 1.5. In part (iii) of the proof of Thm. 1.3-1, it is shown that
Show that, conversely, each covariant derivative vilfj(x) can be expressed as a linear combination of the partial derivatives 0tvk(x). 1.6. This exercise provides the expression of the gradient, Laplaclan, and curl operators in curvilinear coordinates (that of the divergence operator is given at the end of Sect. 1.4). The notations and assumptions are those of Thm. 1.3-1. (1) Given a function 9" 12 -4 1~, let gra~"d~ be the vector field with 029 components Oil), let ~ "-0-~, and let the function v" f~ -4 R
-,~
b e defined by v(x) - 6($) for all ~ - @(x) C ~. Show that
(gr."--a ~)(~) -
A~(~) -
((o,~)g')
(.),
(1~o~(~g',o~) .. ) (~).
(2) Given a vector field ~ - (~i) 9 fi -~ N 3, let e u r l ~ be the vector field with components gOkOj~i, where gob _ +1 if {i, j, k} is an even permutation of {1, 2, 3}, gOk = - 1 if {i, j, k} is an odd permutation of {1, 2, 3}, and g i j k _ 0 otherwise. Show that
cur~l,~(~) - (~Jk~jll~gk)(~), 1 where eiJk(x):= ~ g i j k .
Jg(.)
1.7. Show that the boundary value problem of three-dimensional linearized elasticity in curvilinear coordinates (Thm. 1.6-1), viz., -aO[[j
_
fi
in 12,
ui - 0 on r0,
"" a2Jnj
--
h i on r l ,
Ezercises
59
where a ij - AiJhtekllt(U), can be directly derived from its Cartesian counterpart (Sect. 1.1), viz.,
_3j&ij _ ]i in h, ui - 0 on r0, &iJhj - ~i on F1, where &ij _ ~ijkt~kt(i, ) ("directly" means without recourse to the variational equations as in Thm. 1.6-1). 1.8. The notations and assumptions are those of Thm. 1.8-1. Letting t i ( x ) : = tijg j(x) and [ti(x)]J := ti(x). ~J, show that gik( )gJl( )tk t j
I[ti( )]Jgi( )12
--
J Then infer from this equality another proof of this theorem. 1.9. (1) Show that the three-dimensional elasticity tensor in Cartesian coordinates, defined in Sect. 1.1 by its components
remains positive definite "even if )~ is slightly negative", in the following sense: There exists a constant (7 > 0 such that
~ijkt tkttij ^ ^ >__0 ~
Itijl 2 for all symmetric matrices (tij)
i,j
if and only if/z 3> 0 and 2A + 3/, > 0. (2) Show that a similar property holds for the elasticity tensor defined in curvilinear coordinates by its contravariant components
Aijkl _ Agij gkt + lz(gik gjt + gil gjk). 1.10. Combining the relation (Thm. 1.3-1)
d~ - fn Aijktekllt( u )eillj ( v ) v/g dx with the three-dimensional Korn's inequality in Cartesian coordinates (see, e.g., Vol. I, Sect. 6.3 or Vol. II, Sect. 1.1), show directly that the bilinear form
.) e v(n) • v(n) -+ fn A ijktekllt(u)eillj(v)~/~ dx
Three-dimensional linearized elasticity
60
[Ch. 1
is V(f~)-elliptic, thus providing another proof to Thm. 1.8-2.
Hint: Use a classical result about composite mappings in Sobolev spaces (see, e.g., Ne~as [1967, Chap. 2, Lemma 3.2] or Adams [1975, Thm. 3.35]). 1.11. (1) Show that, if the boundary r and the mapping 19 are sufficiently smooth, if r0 - r , and if fi E LP(f~), p > 6 the weak solution u E V(f~) - H01(f~) found in Thm. 1.8-2 is in the space (2) Show that u satisfies the equations (Thm. 1.6-1)
--AiJklekllt(u)llj
_ fi
in LP(f~).
Hint: Use the regularity of the weak solution of the corresponding boundary value problem of three-dimensional linearized elasticity in Cartesian coordinates (Vol. I, Thin. 6.3-6). 1.12. (1) Show that the Christoffel symbols of the second kind satisfy the relations m j
m j
(2) In the text, the Christoffel symbols of the second kind are defined as r,5 : 0jg, (Thm. 1.4-1), i.e., by means of the vectors of the covariant and contravariant bases. It is remarkable that they can also be defined in terms of the covariant and contravariant components of the metric tensor. More precisely, show that
-
where
rij
1 .=
(Ojgiq q- Oigjq - -
Oqgij).
(3) Show that the relations satisfied by the Christoffel symbols of the second kind (cf. (1)) may be equivalently expressed in terms of the Christoffel symbols of the first kind as
olrikj - okrilj + gm~ (ritnrjkm --
riknrj )
- o in ~.
CHAPTER 2 INEQUALITIES
OF KORN'S
TYPE
ON SURFACES
INTRODUCTION We shall see in the next chapters that the theory of linearly elastic shells leads to two-dimensional equations that are "posed on the middle surface S of the shell", i.e.~ that are expressed in terms of curvilinear coordinates of the surface S. The purpose of this chapter is to study such equations per themselves. To this end, we first provide all the necessary preliminaries from the di~erential geometry of surfaces in It~3 (Sects. 2.1 to 2.3). We then provide complete proofs of the existence and uniqueness of the solutions to three fundamental classes of linear shell equations, the elliptic membrane, flexural, and Koiter equations (Sects. 2.6 and
2.7). As in Chap. 1, the treatment is entirely sel]-contained, in that no a priori knowledge of differential geometry is assumed. More specifically, recall that a "three-dimensional manifold" 0(12) in IR3, where 12 is a three-dimensional domain in I~3 and O 9~ -+ I~3 is an ad hoc injective mapping, is unambiguously defined (up to rigid deformations) by a single tensor field~ the metric tensor field~ whose covariant components gij - gji " 12 ~ R are given by gij - 0 i O . 0 j O (cf. Sects. 1.2 and 1.9). Consider instead a surface S - 8(~) in I~3 where w is a two-dimensional domain in I~2 and 8 9 ~ -+ I~~ is a s m o o t h injective mapping with other ad hoc properties. T h e n by contrast, such a "two-dimensional manifold" requires two tensor fields for its definition (again up to rigid deformations)~ the metric tensor field and the curvature tensor field, whose covariant components aafj = af~a : w -+ I~ and bar3 = bf3a : w -+ IR are respectively given by (Greek indices or exponents take their values in {1, 2}):
aa~ = aa " aft and bail = a3 9Oaa~, where a a - 0 a e and a3 -
a l Aa2 la 1 A a21" These two tensors, which are
also called the first and second ]undamental forms of the surface, are studied in Sects. 2.1 and 2.2; see also Sect. 2.8.
62
[Ch. 2
Inequalities of Korn's type on surfaces
In Sects. 2.4 and 2.5, we introduce two other fundamental tensors, which play a key rble in the two-dimensional theory of linearly elastic shells, the linearized change of metric tensor and the linearized change of curvature tensor, each one being associated with a displacement vector field ~Tiai of the surface S = O(~). where the vectors a i are defined by the relations a i . a j = ~ . The covariant components of these tensors are given by 1 p.~(.)
-
oo~0~ - r ~o~o~0~ - b~b~,~ + b;(O~,~ - r ~ , ~ )
+b~(0~.~ - r ~ . ~ )
+ (0~b~ + r ~ b 3 - r ~
~
where F~f3 - aa.Oaaf~ are the Christoffel symbols of S (Sect. 2.3) and /fin are the mixed components of the curvature tensor of S (Sect. 2.2). The functions 7af3(rl) and paf3(r/) may be also written in more condensed manners, either by means of covariant derivatives or through the introduction of the vector field yia* (Thms. 2.4-1 and 2.5-1). Note that
7o~(~) c L2(~)
i~ ~ -
( ~ ) c H~(~) • H~(~) • L~(~),
An inequality of Korn's type "on a general surface" is then established (Thin. 2.6-4): Given arty subset 70 of 7 = Ow satisfying 0 < length 70 < length 7, define the space
vx(~)
= {~ - (0~) e H ~ ( ~ ) • H ~ ( ~ ) • g 2 ( ~ ) ; ~ = 0 ~
= 0 o . ~0}.
Then this inequality asserts the existence of a constant c such that 2
x,,,, + fir/3 ]12,,,, O~
a,f3
2
}1/2
a,f~
fo~ an ~ = (~) e v x ( ~ ) . The existence and uniqueness of a solution to the two-dimensional Koiter equations for a linearly elastic shell and to the two-dimensional equations of a linearly elastic flexural shell follow from this inequality. We also show how this inequality of Korn's type has been recently extended to surfaces "with little regularity" (Thm. 2.6-6). If 70 = 7 and the surface S is elliptic, i.e., both principal radii of curvature are everywhere of the same sign, it is remarkable that an
Sect. 2.1]
63
Curvilinear coordinates and metric tensor on a surface
inequality of Korn's type "on an elliptic surface" that only involves the linearized change of metric tensor in its right-hand side can be established (Thm. 2.7-3). This inequality asserts the existence of a constant CM such that
{
ll,oll + (:It
}1/,.
~/2 O~lf ~
for all ~ / - VM(W) -- H i ( w ) x H01(w) x L2(w). The existence and uniqueness of a solution to the two-dimensional equations of a linearly elastic elliptic membrane shell then foUow from this inequality. Note that Chaps. 1 and 2 together provide a unified presentation of inequalities of Korn's type "in curvilinear coordinates", whether in a three-dimensional domain in IR3 or on a surface in IRa, notably by showing that aU these inequalities of Korn's type hinge cruciaUy on the lemma of J.L. Lions recalled in Thm. 1.7-1 and on ad hoc infinitesimal rigid displacement lemmas. 2.1.
CURVILINEAR COORDINATES TENSOR ON A SURFACE
AND METRIC
In addition to the rules governing Latin indices that we set in Sect. 1.1, we henceforth require that Greek indices and exponents, except e and v and ~- in the notations 0v and Or, vary in the set {1, 2} and that the summation convention be systematically used in conjunction with these rules. For instance, "Ta/3(r/) = 0 in w" means "Tab(r/) = 0 in w for a, t3 = I, 2", the relation Oa(~Tia i) -- (r//3la -- balg~13)a/3 --k (~73la -4- b/3ar//3)a 3
means that 3
2
2 a3
i--1
for a = 1, 2, etc.
,~--1
,~--1
64
Inequalities of Korn's type on surfaces
[Ch. 2
Fig. 2.1-1: Curvilinear coordinates on a surface and covariant and conteavariant bases of the tangent plane. Let S = 0(~) be a surface in Cs. The two coordinates yx, y2 of y G ~ are the cuxvilinear coordinates of ~1 = O(y) E S. If they are linearly independent, the two vectors a,~(y) = O,~O(y), which are tangent to the coordinate lines passing through ~, form the covariant basis of the tangent plane to S at ~ = 0(y); the two vectors a~(y) defined by a~(y) 9at3(y ) = ~ form its contravariant basis. Let t h e r e be given as in Sect. 1.1 a real t h r e e - d i m e n s i o n a l affine E u c l i d e a n space s e q u i p p e d w i t h a Cartesian f r a m e consisting of a n origin O t o g e t h e r w i t h t h r e e vectors ~i _ ei forming a n o r t h o n o r m a l basis, a n d let a . b, lal, a n d a/x b denote the E u c l i d e a n inner p r o d u c t , the E u c l i d e a n n o r m , a n d the vector p r o d u c t of vectors a , b in t h e a s s o c i a t e d space E s. In a d d i t i o n , let there be given a t w o - d i m e n s i o n a l vector space, in w h i c h two vectors e a = e a form a basis; this space will accordingly be identified with I~2. We let ya denote the c o o r d i n a t e s of a p o i n t y in this space a n d we let Oa "= O/Oya a n d 0a~ := 02/OyaOy~. Finally, let t h e r e be given a d o m a i n w in ]R2 (according to the definition given at the b e g i n n i n g of Sect. 1.7) a n d an injective m a p p i n g
Sect. 2.1]
C u r v i l i n e a r coordinates a n d m e t r i c t e n s o r o n a surface
65
O" ~ --4 E 3. The set s :: is called a s u r f a c e in E 3. Each point 9 E S may thus be unambiguously written as 9 - o(y),
y e
and the two coordinates Ya of y are called the c u r v i l i n e a r c o o r d i n a t e s of ~ (Fig. 2.1-1). Well-known examples of surfaces and of curvilinear coordinates and their corresponding coordinate lines (defined below) are given in Figs. 2.1-2 and 2.1-3.
Remark. According to the usual terminology, the set 0(w) is a two-dimensional manifold, while the surface 0(~) is a two-dimensional manifold "with boundary", in E 3. By contrast, a two-dimensional manifold "without boundary" in E 3, such as a sphere or a torus, requires for its description several mappings Op 9wp C I~2 ---> E 3, p - 1, 2, . . . , P, satisfying ad hoc compatibility conditions on the "overlapping submanifolds" Op(wp) N Oq(wq), p ~ q, of S. Excellent (and particularly readable!) introductions to finitedimensional manifolds are given in Schwartz [1992, Chap. 3, Sect. 9], Schwartz [1993, Chap. VI, Sects. 8-10], and do Carmo [1994, Chaps. 3 and 4]. ll Naturally, once a surface S is defined as S = 0(~), there are infinitely many other ways of defining curvilinear coordinates on S, depending on how the domain w and the mapping 0 are chosen. For instance, a portion S of a sphere may be represented by means of Cartesian coordinates, spherical coordinates, or stereographic coordinates (Fig. 2.1-3). Incidentally, this example illustrates the variety of restrictions that have to be imposed on S according to which kind of curvilinear coordinates it is equipped with! Assume that the mapping O - 0 ~ ~ . w c R 2 -~ O(w) - S c E 3
is differentiable at a point y E ~. If r thus have 0(y +
- 0 ( y ) + v0(y)
is such that (y + r
y +
E ~, we
66
Inequalities of Korn's type on aurfaces
[Ch. 2
~d
Fig. 2.1-2: Two familiar ezamples of surfaces, curvilinear coordinates, and coordinate lines. A portion S of a circular cylinder of radius R corresponds to a mapping O of the form (~, ~) -~ (R~os ~, R ~ h ~ , ~). A portion S of a torus corresponds to a mapping 0 of the form (~, x) -~ ((R + . ~o~ x) ~o~ ~, (R + . ~o~ x) s i . ~, R s h ~), with R > r. In each case, the corresponding coordinate lines axe represented on S with self-explanatory graphical conventions; see also Ex. 2.1.
where the matrix V0(y) is given by
VO(y) .-
I OiOi 01030203 0~02
0202
/
(Y).
Sect. 2.1]
Gurvilinear coordinates and metric tensor on a surface
67
# sSss
sD
__2]
Fig. 2.1-3: Several systems of curvilineav coordinates on a sphere. Let ~ be a sphere of radius R. A portion S C ~ contained in the "northern hemisphere" can be represented by means of Cartesian coordinates: (~, y) -+ (~, y, { R ~ _ (~, + y~)}~/~). A portion S C ~ that excludes a neighborhood of both "poles ') and of a "meridian" (to fix ideas) can be represented by means of spherical coordinates: (~v, r --+ (R cos r cos ~v, R cos r sin ~, R sin r A portion S C ~ that excludes a neighborhood of the "North pole" can be represented by means of stereographic coordinates: ( 2R2u 2R2v u ~ + v 2 - R ~) (u, v) ~ u2 + v2 + R2 , u2 + v2 + R2 , Ru~ + v2 + R2 . As in Fig. 2.1-2, the corresponding coordinate lines are represented in each case with self-explanatory graphical conventions (see also Ex. 2.1).
68
Inequalities of Korn's type on surfaces
[Ch. 2
A w o r d of c a u t i o n . The presentation in this section follows t h a t of Sect. 1.2, "the mapping 8 9~ C It~2 -+ E 3 replacing the m a p p i n g O 9fl C R 3 --+ 6 3". As such, there are strong similarities between the two presentations, such as the way the metric tensor is defined in each case (see below), but there are also sharp differences. For instance, the matrix V 0 ( y ) is not a square matrix, while the m a t r i x VO(a~) is one! m Let the two vectors aa(y) E IR3 be defined by
a~(y) .= 0~o(y) -
IO~Oi1 0~02
(y),
0~03
so t h a t aa(y) is the a-th column vector of the matrix V 0 ( y ) . Let 5y - 5yaea; then the expansion of 8 about y may be also written as
o(y + ~y) = o(y) + 5y"~,(y) + o(~y). If in particular ~y is of the form ~y - r where ~t E IR and ea is one of the basis vectors in IR2 this relation reduces to
0(y + ~te~) = 0(y) + ~ t ~ ( y ) + o(~t). We henceforth assume that the two vectors aa(y) are linearly independent. The last relation thus shows that in this case each vector aa(y) is tangent to the a - t h c o o r d i n a t e line passing through 9 - 0(y), defined as the image by 8 of the points of ~ t h a t lie on a line parallel to ea passing through y; cf. Fig. 2.1-1 (there exist to and tl with to < tl and 0 E [to, tl] such that the equation of the a - t h coordinate line is t E [to, tl] --+ f a ( t ) :-- O(y + tea) i n a sufficiently small neighborhood of Y; hence :f~(0) - OaO(y) - aa(y)). The vectors aa(y), which thus span the tangent plane to the surface S at = 0(y), form the c o v a r i a n t b a s i s o f t h e t a n g e n t p l a n e to S at Y; cf. Fig. 2.1-1. Remark. A differentiable mapping 0 9~ C R 2 -4 R 3 such t h a t the two vectors OaO(y) are linearly independent at all points y E
Sect. 2 . 1 ]
Curvilinearcoordinates and metric tensor on a surface
is called an immersion.
69 B
Naturally, we are committing here two convenient abus de fangage: The vector tangent to the a - t h coordinate line at ~ - 8(y) is in fact the vector in the affine space t; 3 that is parallel to ha(y) and has ~ as its origin. Likewise, the tangent plane to S at ~ is in fact the aJ~fine plane in the affine space E ~ that passes through ~ and is parallel to the vectors ha(y). Returning to a general increment ~y = 5yaea, we also infer from the expansion of 0 about y that
le(y + ~y) - e(y)l ~ - ~ y T v e ( y ) r v e ( y ) ~ y = 5y~(y).
+ o(l~yl ~)
~,(y)hy~ + o(l~y12).
In other words, the principal part of the length between the points 0(y + ~y) and O(y) is {hyaaa(y). a~(y)~y~} 1/2. This observation suggests the introduction of the symmetric matrix (aa~(y)) of order two, whose elements
are called called the Note that ha(y) are
the e o v a r i a n t c o m p o n e n t s of the m e t r i c t e n s o r , also first f u n d a m e n t a l f o r m , of the surface S at ~ = 8(y). the matrix (aa~(y)) is positive definite since the vectors assumed to be linearly independent.
In the next theorem, we review the fundamental formulas expressing the area and length elements at a point ~ - 8(y) of the surface S in terms of the matrix (aa~(y)); see also Fig. 2.1-4.
Inequalities of Korn's type on surfaces
70
[Ch. 2
2.1-1. Let w be a domain in ~2, let 8 : -~ -q E 3 be a smooth enough and injective mapping such that the two vectors ha(y) -- OaS(y) are linearly independent at all points y e ~, and let Theorem
s
=
(a) The area element dS(~) at ~ = O(y) E S is given in terms of the area element dy at y E -~ by
where a(y) := det(aa~(y)) and aa~(y) = ha(y)" a~(y). (b) The length element d[(9) at 9 - e(y) e S is given by
Proof. The relation (a) between the area elements is well known. In particular, it can also be deduced from the relation between the area elements dr(~) and dF(x) given in Thm. 1.2-1 (b) by means of an ad hoc "three-dimensional extension" of the mapping 8; cf. Ex. 2.2. The expression of the length element in (b) simply recalls that d[(~) is by definition the principal part with respect to ~y - 5yaea of the length [O(y § ~y) - 8(y)], whose expression precisely leads to the introduction of the matrix (aa~(y)). m Remark. The (otherwise natural) notation da(9) for the area element at ~ - 0(y) is avoided as it bears too much resemblance with the customarily used notation a(y) for det(aa~(y)), m The relations in Thm. 2.1-1 are used for computing surface integrals and lengths on the surface S (Fig. 2.1-4): Let A be a dymeasurable subset of ~, let .4 "- O(A), and let ] " A ~ ~ be a dS-measurable function. Then
](9)
-/A(/o 0)(y)
Sect. 2 . 1 ]
C u r v i l i n e a r coordinates a n d m e t r i c t e n s o r o n a surface
,
~;
I
71
R
"
Fig. 2.1-4: Area and length elements on a surface. The elements dS(fl) and d[(fl) at ~l = O(y) E S are related to dy and 6 ! / b y means of the covariant components of the metric tensor of the surface S; cs Than. 2.1-1. The corresponding relations are used for computing the area of a surface .~ = 8 ( A ) C S and the length of a curve C = O(C) C S, where C = 3e(I) and I C R.
I n p a r t i c u l a r , t h e a r e a o f .~ is g i v e n b y
Consider next a curve i n t e r v a l in I~ a n d f
-
faea
C
-
~(I)
in ~ , w h e r e I is a c o m p a c t
9 I -+ ~ is a s m o o t h e n o u g h i n j e c t i v e
Inequalities of Korn's type on surfaces
72
[Ch. 2
mapping. Then the length of the curve C "- O(C) is given by
length C "-
I-~(0 o ~)(t)idt =
aaf3(~(t))-d~ (t)-d~ (t)dt.
Remark. The first fundamental form is also used for computing angles between coordinate lines; cf. Ex. 2.3. m Let 5~ denote the Kronecker symbol. Given a point y E ~, the
four relations
unambiguously define two linearly independent vectors ha(y) in the tangent plane, which form the e o n t r a v a r i a n t basis of t h e t a n g e n t p l a n e (Fig. 2.1-1). To see this, let a priori h a ( y ) = YaZ(y)a~(y) in the relations aa(y).a~(y) - 5~. This gives YaZ(y)a~(y) - 5~; hence ya~(y) = aa~(y), where (recall that the symmetric matrix (aa~(y)) is positive definite by assumption):
(aa[3(y)) :--(aa~(y)) -1. The elements aa~(y) of the symmetric matrix (aa~(y)) are called the c o n t r a v a r i a n t c o m p o n e n t s of the m e t r i c t e n s o r , or first f u n d a m e n t a l f o r m , of the surface S at ~ - 8(y). The relations a
(y) =
how t h a t
=
a"
(y)a
-
-
and thus the vectors aa(y) are linearly independent since the matrix (a af3 (y)) is positive definite. Let us assemble for convenience these fundamental relations between the vectors of the covariant and contravariant bases of the tangent plane and the covariant and contravariant components of the metric tensor" and and
Sect. 2.2]
2.2.
CURVATURE
C u r v a t u r e t e n s o r on a surface
TENSOR
73
ON A SURFACE
While a "three-dimensional manifold" in t; 3, i.e., a set O(~t) associated with a mapping 0 9~2 C I~3 -+ $~, is well defined by its "metric" (uniquely up to rigid deformations in 1t~3) provided ad hoc compatibility conditions are satisfied by the covariant components gij " ~t ~ IR of its metric tensor (Sect. 1.9), a "two-dimensional manifold" O(w) in t; a, i.e., a set O(w) associated with a mapping 0 9w c ~2 _+ E3, cannot be defined by its "metric" alone. We shall not dwell into this fascinating question in this section, referring instead to Fig. 2.2-1 for an intuitive example and to Thm. 2.8-1 for a precise statement. The "missing information" is provided by the "curvature" of the surface, introduced in this section by means of the covariant components of its curvature tensor. Consider as in Sect. 2.1 a surface S - 0(~) in t; 3, where the injective mapping 0 9 ~ C II~2 -+ t; 3 is smooth enough and such that, for each y E ~, the two vectors ha(y) - OaO(y) are linearly independent. For each y E ~, the vector
.
-
is thus well defined, has Euclidean norm one, and is normal to the surface S at the point 9 - O(y). This is another abus de langage as the vector normal to S at ~ is in fact the vector in the aJfine space t; 3 that has ~ as its origin and is parallel to ha(y). Note that the denominator in the definition of ha(y) may be also written as
where a(y) = d e t ( a ~ ( y ) ) (Sect. 2.1). This relation, which holds in fact even if a(y) - O, will be established in the proof of Thm. 2.5-1. Fix y C ~ and consider a plane P normal to S at ~ - O(y), i.e., a plane that "contains the vector a3(y)"; by the same abus de langage as above, "a plane containing a3(y)" means in fact any plane in the ajfine space ~3 that contains the point ~ and is parallel to the vector a3(y). The intersection C - P N S is thus a planar curve on S.
74
Inequalities of Korn's type on surfaces
[Ch. 2
i/ ! t
i/ I
t //
/7
iJ
Fig. 2.2-1: A metric does not define a surface in ~s. A flat surface So may be deformed into a portion $1 of a cylinder or a portion $2 of a cone without altering the length of any curve drawn on it (cylinders and cones are instances of "developable surfaces"). Yet it should be clear that in general So and Sz, or So and Sz, or $1 and $2, are not identical surfaces modulo a rigid deformation of R3!
As shown in Thm. 2.2-1, it is remarkable that the curvature of at ~ can be computed by means of the covariant components aa~(y) of the first fundamental form of S introduced in Sect. 2.1, together with the covariant components ba~(y) of the "second" fundamental ]orm of S; the definition of the c u r v a t u r e of a planar curve is recalled in Fig. 2.2-2. If the algebraic curvature of C at ~ is ~ 0, it can be written 1 as ~ , and R is then called the algebraic r a d i u s of c u r v a t u r e of the curve C at ~; this means that the c e n t e r of c u r v a t u r e of the curve C at ~ is the point (~)§ Ra3(y)); cf. Fig. 2.2-3. While R is
Curvature tensor on a surface
Sect. 2.2]
75
!
,
p(s).RvCs) \
p(s)
Fig. 2.2-2: Curvature of a planar curve. Let 7 be a smooth enough planar curve, parametrized by its curvilineax abscissa s. Consider two points p(s) and p(s + As) with curvilinear abscissae s and s + As and let Ar be the algebraic angle between the two normals v(s) and ~(s + As) (oriented in the usual way) to 7 at those points. When As ~ 0 the ratio Ar '
has a limit, called the "cuxvature"
A8
of 7 at p(s). If this limit is non-zero, its inverse R is called the (algebraic) "radius of curvature" of 7 at p(s) (the sign of R depends on the orientation chosen on 7). The point p(s) + Rv(s), which is intrinsically deft.ned, is called the "center of curvature" of 7 at p(s): It is the center of the "osculating circle" at p(s), i.e., the limit as As --~ 0 of the circle tangent to 7 at p(s) that passes through the point p(s + As). The center of curvature is also the limit as As ~ 0 of the intersection of the normals v(s) and v(s + As); consequently, the centers of curvature of 7 lie on a curve (dashed on the figure), called "la d~velopp~e" in French, that is tangent to the normals to 7.
not intrinsically defined (its sign changes in any system of curvilinear coordinates where "a3(y) is replaced by -a3(y)"), the center of curvature is intrinsically defined. If the curvature of C at ~ is 0, the radius of curvature of the curve C at ~ is said to be infinite; for this 1 reason, it is customary to still write the curvature as ~ in this case. 1 Note that the real number -R is always well defined by the formula given in the next theorem, since the symmetric matrix (aa~(y)) is positive definite. This implies in particular that the radius of curvature never vanishes along a surface S = 0(-~) defined by a mapping O
76
Inequalities of Korn's type on surfaces
[Ch. 2
satisfying the assumptions of the next theorem, hence in particular of class C2 on -~. It is intuitively clear that the vanishing of R implies t h a t 0 cannot be "too" smooth: Think of a surface made of two portions of planes intersecting along a segment, which thus constitutes a fold on the surface; or think of a surface O(~) with 0 E w and O(yl, y2) - ]yll l+a for some 0 < a < 1, so that 0 E c l ( ~ ; s but 0 ~ C2(~; s The radius of curvature of a curve corresponding to a constant y2 vanishes at yl - 0 . T h e o r e m 2.2-1. Let w be a domain in ~2 and let 0 E C2(~; s be an injective mapping such that the two vectors a a ( y ) = OaO(y) are linearly independent at all points y E -~. Fix y E -~ and consider a plane P normal to S = 0(-~) at the point ~1 - O(y). The intersection P A S is a curve C on S, which is the image C - O(C) of a curve C in the set-~. A s s u m e that the curve C is represented in a sufficiently small neighborhood of y as the image f ( I ) of a closed interval I C R with a non-empty interior, where ~ = f a e a : I --+ R is a smooth enough injective mapping that satisfies -~ dfa ( t ) e a ~ 0 where t E I is such that y - 5(t) (Fig. 2.2-3). 1 Then the curvature -R of the planar curve C at ~t is given by the ratio
1 R t)
where aa~(y) are the covariant components of the metric tensor of S at y (Sect. 2.1) and
Proof. (i) We first establish a well-known formula for the curva1 ture ~ of a planar curve. Using the notations of Fig. 2.2-2, we note that -
+
-
-{,.,(s
+
,.,(8)}.
+
Curvature tensor on a surface
Sect. 2.2]
Fig. 2.2-3:
77
Curvature on a surface.
al (y) A as(y)
Let P be a plane containing the vector 1 which is n o r m a l to S. T h e algebraic c u r v a t u r e ~ of t h e
p l a n a r curve C - P N S - O(C) at 9 -- O(y) is given by the ratio
R
~ , (r
dff'
dr:3.,
~t~
~ ,~)
dff3(.
where a~f3(y) a n d b,~(y) are the covariant c o m p o n e n t s of the m e t r i c a n d c u r v a t u r e tensors (also called the first a n d second f u n d a m e n t a l forms) of the surface S at ~9 a n d
(t) are the c o m p o n e n t s of the vector t a n g e n t to the curve
1 C - $t(I) at y = ~(t) - j:'~(t)e,~. If ~ r 0, the center of c u r v a t u r e of the curve (~ at t) is t h e point (!) + Raa(y)), E u c l i d e a n space C a .
which is intrinsically defined in the afFme
Inequalities of Korn's type on surfaces
78
[Ch. 2
so that 1 R
lim Ar ~.~o As
dr(s) ds
lim sin Ar h,-+o As
r(s).
(ii) The curve (8 o ~)(I), which is a priori parametrized by t E I, can be also parametrized by its curvilinear abscissa s in a neighborhood of the point ~. There thus exist an interval I C I and a mapping p - J -~ P, where J C I~ is an interval, such that
(0 o f)(t) - p ( s ) and (as o f)(t) - v(s) for all t e i, 8 e J, 1 By (i), the curvature ~ of C is given by 1
dv
-R =
ds (S) " r(s)'
where
dv (s) - d(az o f ) ( t ) d t dfdta dt d---s dt -~s = Oaa3(f(t)) (t) ds' dp d(O o f ) ( t ) d t r ( s ) - -~s(S)dt -~s df ~. dt ~~ dt = O~O(f(t))-~ (t)-~s = a~(f(t)) (t) d--~ Hence 1 _ R
(t)
(t) 1
To obtain the announced expression for ~ , it suffices to note that a,(f(t))
- b.,(I(t)),
by definition of the functions ba~ and that (Thm. 2.1-1)
dr. m
Remark. The knowledge of the curvatures of curves contained in planes normal to S suffices for computing the curvature of any
Sect. 2.2]
79
Curvature tensor on a surface
curve on S. More specifically, the radius of curvature/~ at 9 of any smooth enough curve C (planar or not) on the surface S is given by cos ~ = --,1 where ~ is the angle between the "principal normal" to R at 9 and a3(y) and ~ is given in Whm. 2.2-1; see, e.g., Stoker [1969, Chap. 4, Sect. 12].
II
Precise information about the shape of the surface in a neighborhood of its point ~ - 0(y) can thus be gathered by "letting the plane P turn around the normal vector a3(y)" and by following in this process the variations of the curvatures at ~ of the corresponding planar curves C - P N C, as given in Thm. 2.2-1. As we now show, these curvatures span a compact interval of R, i.e., they "stay away from infinity". 2.2-2. (a) Thm. 2.2-1. For a fixed normal to S at ~1 = O(y). planar curves P N S, P Theorem
[
]
Let the assumptions and notations be as in y E -~, consider the set 79 of all planes P Then the set of curvatures of the associated C 79, is a compact interval of I~, denoted
o=o of
bo h,
bo 0)
R~(y)' R~(y) (b) Let the matrix ( ~ ( y ) ) , a being the row index, be defined by
where (aa~(y)) - (aaf3(y)) -1 (Sect. 2.1) and the matrix (baf3(y)) is defined as in Thin. 2.2-1. Then 1
1
= b~(y) + hi(y), R2(y) det(ba~ (y)) 1 = b~(y)b~(y) b~(y)b~(y) d~t(~.~(y))" R~(y)R2(y)
R~(y)
t
1 1 (c) If R1 (y) ~ R2(y)' there is a unique pair of orthogonal planes P1 E 79 and P2 C 79 such that the curvatures of the associated planar 1 1
Inequalities of Korn's type on surfaces
80
[Ch. 2
Proof. (i) Let A ( P ) denote the intersection of P E T~ with the tangent plane to S at ~, and let (7(P) denote the intersection of P with the surface S; hence A ( P ) is tangent to (7(P) at ~ C S. Since we assume as in Thm. 2.2-1 that C ( P ) - (0 o S(P))(I) for some mapping f ( P ) : I C I~ --+ ~, the line A ( P ) is given by
A ( p ) --
~+
Ad(O o 5(P))(t); A e IR} dt
where ~a ._ dfa(P) (t) and ~aea # 0 by assumption. Since the line
"dt {y + tt~aea; t t e I~} is tangent to the curve C ( P ) " - 0 - 1 ( C ( P ) ) at
y E ~ (the mapping 8" ~ -+ R 3 is injective by assumption) for each such parametrizing function se(p) : I -+ C and the vectors as(y) are linearly independent, there exists a bijection between the set of all lines A ( P ) C T, P C 7~, and the set of all lines supporting the nonzero tangent vectors to the curve C(P). Hence Thm. 2.2-1 shows that when P varies in T', the curvature of the corresponding curves C ' - C'(P) at ~ takes the same values as does the ratio bal3(Y)~a~ when ~ "= (~a) varies in IR2 - {0}. (ii) Let the symmetric matrices A and B of order two be defined by
A "- (aa~(y)) and B := (baf3(y)). Since A is positive definite, it has a (unique) square root C, i.e., a symmetric positive definite matrix C such that A - C 2 (for a proof, see, e.g., Vol. I, Thm. 3.2-1). Hence the ratio
ba[3(Y)~a~[3 -- ~TB~ -- r / T C - 1 B C - l r / , where r / - C~,
is nothing but the Rayleigh quotient associated with the symmetric matrix C - 1 B C -1. When r/varies in I~2 - { 0 } , this Rayleigh quotient thus spans the compact interval of I~ whose end-points are the small1 1 est and largest eigenvalue, respectively denoted and
R (y)
of the matrix C - 1 B C -1 (for a proof, see, e.g., Ciarlet [1982, Thm. 1.3-1]). This proves (a).
Curvature tensor on a surface
Sect. 2.2]
81
Furthermore, the relation C ( C - 1 B C - 1 ) C -1 = B C -2 _ B A - 1 shows that the eigenvalues of the symmetric matrix C - 1 B C -1 coincide with those of the (non-symmetric in general) matrix B A -1. Note that B A -1 - ( ~ ( y ) ) with ~ ( y ) ' - a~(y)ba~(y), c~ being the row index, since A -1 - (aaf3(y)); hence the relations in (b) simply express that the sum and the product of the eigenvalues of the matrix B A -1 are respectively equal to its trace and to its determinant, det (baf3(y)) -1 _ which may be also written as since B A (b~a(y)). det (aaf3 (y)) This proves (b). (iii) Let rlz - (r]~) - C~z and r12 - (r~2) - C~2, with ~z - ((~) and ~2 - ((~), be two orthogonal (rlTrl2 -- 0) eigenvectors of the 1 symmetric matrix C - 1B C - z, corresponding to the eigenvalues
n (v)
1
and R2t)'y" respectively. Hence
o
-
-
-
-
o,
since C T - C. By (i), the corresponding lines A(P1) and A(P2) of the tangent plane are parallel to the vectors ~ a a ( y ) and ~af3(y), which are orthogonal since
{ ~ a a ( y ) }" {~fl2af3(y) } - aa/3(y)~fl2 - ~TA~ 2. 1 1 If RI(y) # R2(y)' the directions of the vectors rll and 7/2 are uniquely determined and the lines A(P1) and A(P2) are likewise uniquely determined. This proves (c). II We are now in a position to state several fundamental definitions: The elements ba~(y) of the (symmetric) matrix (ba~(y)) defined in Thm. 2.2-1 and the element ~ ( y ) of the (non-symmetric in general) matrix ( ~ ( y ) ) defined in Thm. 2.2-2 are respectively called the c o v a r i a n t c o m p o n e n t s and the m i x e d c o m p o n e n t s of the c u r v a t u r e t e n s o r , also called the s e c o n d f u n d a m e n t a l f o r m , of the surface S at ~ ) - O(y).
82
[Ch. 2
Inequalities of Korn's type on surfaces
A w o r d of c a u t i o n . With this definition of "curvature tensor", I depart from other texts~ where "Curvature tensor" of S often means the "Riemann curvature tensor" of S (the covariant components of this other tensor are defined in the last exercise of this chapter), m 1 1 The real numbers Ri(y) and R2(y) (one or both of which being possibly 0) found in Thm. 2.2-2 are called the p r i n c i p a l c u r v a t u r e s of S at ~. 1 1 If Ri(y) - R2(y)' the curvatures of the planar curves P N S are the same in all directions, i.e., for all P E P. If
1
R (y)
----
1
R (y)
-- 0,
1 1 the point ~ -- 8(y) is called a p l a n a r point; if (y---~ Ri - R2(y) ~ 0, ~) is called an u m b i l i c a l point; cf. Exs. 2.4 and 2.5. Let ~ - 8(y) E S be a point that is neither planar nor umbilical; in other words, the principal curvatures at ~ are not equal. Then the two orthogonal lines tangent to the planar curves Pi N S and P2 A S (Thin. 2.2-2 (c)) are called the p r i n c i p a l d i r e c t i o n s at ~. A line of c u r v a t u r e on S is a curve on S that is tangent to a principal direction at each one of its points. It can be shown that a point that is neither planar nor umbilical possesses a neighborhood where two orthogonal families of lines of curvature can be chosen as coordinate lines; see, e.g., Klingenberg [1973, Lemma 3.6.6]. If both
1
R (y)
and
1
R (y)
are ~ 0, the real numbers Ri(y) and
R2(y) are called the (algebraic) p r i n c i p a l r a d i i of c u r v a t u r e of 1 S at ~). If~ e.g., Ri(y) = 0, the corresponding principal radius of curvature Ri(y) is said to be infinite. While the principal radii of curvature may simultaneously change their signs in another system of curvilinear coordinates (as already noted), the associated centers of curvature are intrinsically defined. ( The numbers
1 Ri(y)
i ) t R2(y) and Ri(y)R2(y)' identified as
the principal invariants of the matrix ( ~ ( y ) ) in Thm. 2.2-1, are respectively called the m e a n c u r v a t u r e and the G a u s s i a n , or total~ c u r v a t u r e of S at ~.
Sect. 2.2]
Curvature tensor on a surface
83
As shown in Fig. 2.2-4, the points on a surface can be classified as elliptic, p a r a b o l i c , or h y p e r b o l i c , according as their Gaussian curvature is > 0, = 0 but they are not planar, or < 0. An a s y m p t o t i c line is a curve on a surface that is everywhere tangent to a direction along which the radius of curvature is infinite; any point along an asymptotic line is thus either parabolic or hyperbolic. It can be shown that, if all the points of a surface are hyperbolic, any point possesses a neighborhood where two intersecting ]amilies of asymptotic lines can be chosen as coordinate lines; see, e.g., Klingenberg [1973, Lemma 3.6.12]. As intuitively suggested at the beginning of this section, a surface in I~3 cannot be defined by its metric alone, i.e., by its first fundamental form alone, since its curvature must be in addition specified through its second fundamental form; see Sect. 2.8. But quite surprisingly, the Gaussian curvature can also be expressed solely in terms of the functions ha/3 and their derivatives! This is the celebrated Theorema egregium ("outstanding theorem") of Gautl [1828]; see Sect. 2.8. Another striking result where the Gaussian curvature is used is the equally celebrated Gaufl-Bonnet theorem, so named after Gaufl [1828] and Bonnet [1848] (for a "modern" proof, see, e.g., Klingenberg [1973, Tam. 6.3-5] or do Carmo [1994, Chap. 6, Tam. 1]): Let S be a smooth enough, "closed", "orientable", and compact surface in ~3 (a "closed" surface is one "without boundary", such as a sphere or a torus; "orientable" surfaces, which exclude for instance Klein bottles, are defined, e.g., in Klingenberg [1973, Sect. 5.5]) and let K : S --+ I~ denote its Gaussian curvature. Then K ( 9 ) dS(9) = 27r(2 - 2g(S)), where the g e n u s g(S) is the number of "holes" of S (for instance, a sphere has genus zero, while a torus has genus one; cf. Ex. 2.6). The integer X(S) defined by x ( S ) : = ( 2 - 2 g ( S ) ) i s the E u l e r c h a r a c t e r istic of S. A d e v e l o p a b l e s u r f a c e is one whose Gaussian curvature vanishes everywhere (this is the definition of Stoker [1969, Chap. 5, Sect. 2]; developable surfaces are otherwise often defined as "ruled" surfaces whose Gaussian curvature vanishes everywhere, as in, e.g.: Klingenberg [1973, Sect. 3.7]). A portion of a plane provides a first example, in fact the only one of a developable surface all points of
84
Inequalities of Korn's type on surfaces
[Ch. 2
f
Fig. 2.2-4: Different kinds of points on a surface. A point is elliptic if the Gaussian curvature is > 0 or equivalently, if the two principal radii of curvature are of the same sign; the surface is then locally on one side of" its tangent plane. A point is parabolic if exactly one of the two principal radii of curvature is in_Finite; the surface is again locally on one side of its tangent plane. A point is hyperbolic if the Gaussian curvature is < 0 or equivalently, if the two principal radii of curvature are of different signs; the surface then intersects its tangent plane along two curves.
Sect. 2 . 3 ]
Covariantderivatives and Christoffel symbols on a surface
85
which are planar (Ex. 2.4). Any developable surface all points of which are parabolic can be likewise fully described: It is either a portion of a cylinder, or a portion of a cone, or a portion of a surface spanned by the tangents to a skewed curve. The description of a developable surface comprising both planar and parabolic points is more subtle (although the above examples are in a sense the only ones possible, at least locally; see Stoker [1969, Chap. 5, Sects. 2 to 6]). The interest of developable surfaces is that they can be, at least locally, continuously "rolled out", or "developed" (hence their name), onto a plane without changing the metric of the intermediary surfaces in the process. Remarks. (1) That such "continuously varying isometric surfaces" exist plays a key r61e in the theory of "flexural shells" (Chaps. 6 and 10). (2) Interesting connections between the theory of developable surfaces and the appearance of folds on plates and shells subjected to applied forces have been exhibited and studied by Sanchez-Palencia [1995], Pomeau [1995], Ben Amar & Pomeau [1997]. II In Sect. 2.8, we shall briefly discuss the reciprocal question of recovering a surface 8(w) from its metric and curvature tensor fields: Given a positive definite symmetric matrix field (aa~) on w and a symmetric matrix field (ba~) on w, find conditions under which there exists a mapping 8 :w -+ t; 3 such that OaS" 0~8 = aa~ in w, O~e A 028 9 O a ~ 8 = ba~ in w.
Io o A o ol 2.3.
COVARIANT CHRISTOFFEL
DERIVATIVES SYMBOLS
AND
ON A SURFACE
The content of Sects. 2.1 and 2.2 and of the present section constitute our first encounter with differential geometry "on a surface". Other related notions are treated later, such as covariant derivatives of a tensor field on a surface (Thm. 2.5-1) and the recovery of a surface from its metric and curvature tensor fields (Sect. 2.8).
86
Inequalities of Korn's type on surfaces
[Ch. 2
For further details and complements, also for the notion of "tensots on a surface", see classical texts such as Valiron [1950, Chap. 13], Struik [1961], Stoker [1969], Klingenberg [1973], Spivak [1975], do Carmo [1976], or Berger & Gostiaux [1992, Chaps. 10 and 11] and the more recent books of do Carmo [1994] and Hsiung [1997]; Sanchez-Hubert & Sanchez-Palencia [1997] provide a wealth of complements particularly well adapted to shell theory. More "intrinsic" approaches to the differential geometry of surfaces have also been advocated in shell theory, notably by Destuynder [1990, Chap. 2], Podio-Guidugli [1991], Carrive [1995], Carrive,
TaUer
Mou o [1995], Valid [1995], Delfou
Zolts o [1997].
We begin by some definitions. As in Sects. 2.1 and 2.2, consider a surface S = 0(~) in E a, where 8 : ~ C R 2 --+ IRa is a smooth enough injective mapping such that the two vectors c a ( y ) - OaO(y) are linearly independent at all y C ~ and let
Then the vectors c a ( y ) (which form the covariant basis of the tangent plane to S at 9 - 0(y); see Fig. 2.1-1) together with the vector as(y) (which is normal to S and has Euclidean norm one) form the covariant basis at ~. Let the vectors c a ( y ) of the tangent plane to S at ~) be defined by the relations c a ( y ) 9af3(y) - ~ . Then the vectors c a ( y ) (which form the contravariant basis of the tangent plane at ~); see again Fig. 2.1-1) together with the vector a s (y) form the c o n t r a v a r i a n t basis at 9; cf. Fig. 2.3-1. Note that the vectors of the covariant and contravariant bases at 9 satisfy hi(y), aj (y) -- 5~. Consider an arbitrary d i s p l a c e m e n t field of the middle surface S, i.e., an arbitrary vector field ~ia i defined by means of its covaria n t components ~i " ~ -4 R; this means that ~Ti(y)ai(y) is the displacement of the point 9 = O(y) C S (Fig. 2.3-1). As we shaU see, finding the explicit forms of the associated linearized change of metric tensor (Thin. 2.4-1) and linearized change of curvature tensor (Thin. 2.5-1) requires the explicit forms of the
Sect. 2.3]
Covariant derivatives and Christof]el symbols on a surface
87
O§
0
/
Fig. 2.3-1: Contravariant basis and displacement vector field along a surface. At each point ~ = O(y) E S, the thxee vectors ai(y), where a"(y) form the contravaxi-
~lCy) A ~,(y)
ant basis of the tangent plane to S at ~ (Fig. 2.1-1) and an(y) = lax(y ) A a2(y)l' form the contravariant basis at ~. An arbitrary displacement field of S may then be defined by its covariant components r/i : ~ --+ R: This means that ~7~(y)ai(y) is the displacement of the point ~. The mapping (0 + r/ia/) : ~ --+ gn defines a surface (0 + r/~ad)(5) in gn, which is thus equipped with the same cuxvilinear coordinates (the coordinates yl, y~ of y E ~) as those of the surface S = 0(~).
partial d e r i v a t i v e s Oa(~Tiai) of this vector field. These are found in the next theorem, as immediate consequences of two basic formulas, those of Gau~ and Weingarten. The Christoffel symbols "on a sur]ace" and the covariant derivatives "on a surface" are also naturally introduced in this process.
88
[Ch. 2
Inequalities of Korn's type on surfaces
Whenever no confusion should arise, we henceforth drop the explicit dependence on a particular point. For instance, the relation "Oaa 3 - - b a o a [3'' means that "Oaa3(y) - - b a o ( y ) a ~ ( y ) for all y E -~"; "Oaa 3 is in the tangent plane" means "Oaa3(y) is in the tangent plane to S at ~ - O(y) for all y C ~"; etc.
T h e o r e m 2.3-1. Let w be a domain in I~2 and let 0 E C2(~; s be an injective mapping such that the two vectors aa - OaO are linearly independent at all points of-~. (a) The derivatives of the vectors of the covariant and contravariant bases are given by the f o r m u l a s of Gauf$: 0~,
- r~,~
+ b~,~
and
O~a ~ -
~+
-r~
bl3aa3,
and Weingarten: Oaa3 - Oaa 3 - - b a o a [3 - - b ~ a a ,
where the covariant and mixed components bao and ~ of the curvature tensor of S are defined in Thms. 2.2-1 and 2.2-2 and r~.-
a ~ . o~ a~ - - o~ a ~ . a~ - r 3 ~
are the C h r i s t o f f e l s y m b o l s of the surface S. (b) Let there be given a vector field yia i with covariant components Yi E H i ( w ) . Then ~ia i E Hi(w) and the partial derivatives
0~(~)
~ ~2(~) ~
O ~ ( ~ a ~) -
g i ~ e . by
(0~
- r a[3~7~ ~ - ba/3~73)a/3+ (0a?']3 -~- bBa~/3)a3
-- (~7~]a - ba[3~73) ao -k (r/31a + b/3a~o)a3, where
tibia "-- 0a~//3 -F~O~/a and ~731a " - denote the f i r s t - o r d e r rlia i.
covariant
derivatives
OAT]3
of the vector field
Sect. 2.3]
Covariant derivatives and Christoffel symbols on a surface
89
P r o @ F i x a E {1, 2} throughout this proof. Since any vector c in the tangent plane can be expanded as c - (c-af3)a ~ - (c. a ~ ) a ~ , since Oaa 3 is in the tangent plane (Oaa 3 . a 3 - 8 9 3 -a 3) - 0), and since O a a 3 . a f ~ - -baf~ (Thm. 2.2-1), it follows that O a a 3 -- ( O a a 3 . a/3 )a/3 -- - b a / 3 a ~ ,
and thus the first formula of Weingarten is established. This formula, together with the definition of the functions ~ (Thm. 2.2-2), implies that o~a3 - ( o ~ a 3 . a ~ ) a ~ - - b ~ , ( a ~ . a ~ ) a ~ - - b ~ ~ ' ~ .
- -b;a~,
and thus the second formula of Weingarten is established. Any vector e can be expanded as e - (c. a i ) a i - ( e . a j ) a j. In particular, Oaaf3 -- (Oaa~o " a ~
. -~- ( O a a ~ " a 3 ) a 3
--
r ~ ~ + b~3,
by definition of F ~ all and bar3 Finally,
o.~
- (o.~'. ~)~
+ (0.~'. ~)~3 = _r~~
+ ~a3,
since O a a ~ . a 3 -- - - a ~ " O a a 3 -- b~acr " a ~ -
b/3a,
by the second formula of Weingarten; the formulas of Gaus are thus established. That rli a i C I-II(w) if rli C HI(w) is clear since a i C e l ( F ) if 0 E C2(~; E3). The formulas of Gaufl and Weingarten immediately lead to the announced expression o f Oa(~Tiai). m The definition of the covariant derivatives Ya[f3 - 0f3ya- F ~ "on a surface in C3'' given in Thm. 2.3-1 is highly reminiscent of the definition of the covariant derivatives Villi - 0ivi - r#vpP "in a threedimensional manifold in E ~'' given in Chap. 1. However, the former are more subtle to apprehend than the latter. To see this, recall that p the covariant derivatives villi - O j v i - F i j v p may be also defined by the relations (Thm. 1.4-1) ~
~llj~ ~ -
~
Oj(v~g').
90
[Ch. 2
Inequalities of Korn's type on surfaces
By contrast, even if only tangential vector fields ~Taaa on S are considered (i.e., vector fields yia z for which r/3 - 0), their covariant derivatives ~al/3 -- O ~ ? a - F 'ra~7~ only satisfy the more general relation ,
p
-
where P denotes the projection operator on the tangent plane in the direction of the normal vector (i.e., P ( c i a i) "= caaa), since -
+
3
for such tangential fields by Thm. 2.3-1. The reason behind this discrepancy is simple: While a "three-dimensional manifold in IR3'' has zero curvature (Sect. 1.9), a surface has in general a nonzero curvature, manifesting itself here by the "extra term" b~Taa 3. This term vanishes in ~ if S is a portion of a plane, since in this case b~ - ba~ - 0. Note that, again in this case, the formula giving the partial derivatives in Thm. 2.3-1 (b) reduces to =
We shall not dwell any further in this section into these aspects, referring instead to the references listed at the beginning of this section for more details. Meanwhile, the covariant derivatives Yila may be simply viewed as convenient notations. R e m a r k . The Christoffel symbols r ~
- a ~ . Oaa~ "on a surface
in d~3'' are to be carefully distinguished from the Christoffel symbols ri~ = g P ' O i g j "in a three-dimensional manifold in d~3'' introduced in Chap. 1. To avoid any confusion due to the identical notation when i, j, p E {1, 2}, specific notations shall always be introduced, should the two kinds of Christoffel symbols coexist in a given argument. II 2.4.
LINEARIZED CHANGE ON A SURFACE
OF METRIC
TENSOR
We recall that, given an arbitrary displacement field rig i of a three-dimensional manifold O(~) in E 3 defined by its covariant components vi 9 f~ --~ I~, the covariant components eillj(V) of the associated linearized change of metric tensor are defined by (see Thm.
Sect. 2.4]
Linearized change of metric tensor on a surface
91
1.5-1; also for the notations, not recalled here) 1 lin ~llJ(") - ~ [g~J(") - giJ] 9
It was then found that 1 e~llj(v) - ~(villj + vjlti ) = ~1 ( 0 j ~ + 0 ~ j )
- ri~vp.
Since a surface, i.e., a two-dimensional manifold in e s, also has a metric tensor (Sect. 2.1), it is natural to likewise define the covariant components of a "linearized change of metric tensor" associated with any displacement field defined on it; this is the object of this section. Notice the strong analogy between the next theorem and Thm. 1.5-1. T h e o r e m 2.4-1. Let w be a domain in R 2 and let 0 E C2(~; E a) be an injective mapping such that the two vectors aa = OaO are final A a 2 early independent at all points of-~, let as - la 1 A a2[' and let the vectors a i be defined by a i . a j = ~ . Given an arbitrary displacement field ~
of the surface S - 0(-~) with smooth enough covariant components ~li "-w --+ R, let the c o v a r i a n t c o m p o n e n t s of the l i n e a r i z e d c h a n g e of m e t r i c t e n s o r associated with this vector field be defined by
7~(~) .= ~1 [ ~ ( ~ ) - ~ ] lin where aaf3 and aa/3(Vl) are the covariant components of the metric tensor of the surfaces 8(~) and (0 + 71iai)(-~) (Fig. 2.3-1), and [... ]lin denotes the linear part with respect to ~ - (~i) in the expression [... ]. Then
1 1 - ~(rIalf3 -t- ~7/31a) - ba/3~73 1 = -~(o~, + o~)
- r ~ , 7 ~ - b~,Ts,
92
[Ch. 2
Inequalities of Korn's type on surfaces
where the Christoffel symbols F ~ and the covariant derivatives ~Tal~ are defined in Thin. 2.3-1 and the covariant components bad of the curvature tensor of S are defined in Thin. 2.2-1. Consequently,
r/a e Hi(w) and r/3 e L2(w) =~ 7af~(~7) e L2(w).
Proof. The covariant components aaf3(17) of the metric tensor of the surface (0 + yia')(-~) are by definition (Sect. 2.1) given by ao~(~) - o~(o + ~). o~(o +
#).
Note that both surfaces 0(~) and (0 + yiai)(-~) are thus equipped with the same curvilinear coordinates Ya. The relations
O~(o+#)=a~+O~# then show that a ~ ( ~ ) - (a~ + 0 ~ # ) . (a~ + 0~#) = aa/3 + O/3rl'aa + Oa*'l'af3 + OaO" 0/30,
hence that
7.z(n)
1
1
- ~ [ a o ~ ( n ) - aoz] "" - ~ ( 0 Z # 9~ . + 0 . 0 .
~Z).
The other expressions of 7of3(~7) immediately follow from the expression of 0a~/-- Oa(~Tiai) given in Thm. 2.3-1 (b). II While the expression of 7of3(r/) in terms of the covariant components rli of the displacement field is well known, that in terms of ~7 - ~?iaz seems to be less known; it was recently put to efficient use by Blouza & Le Dret [1994a, 1994b, 1999], who noticed that it has the advantage of still making sense under substantially weaker smoothness assumptions on the mapping 0; see Thms. 2.6-5 and 2.6-6. A w o r d of c a u t i o n . The vector fields ~/-- (~?i) and ~/=
~7ia*,
which are both defined on-~, must be carefully distinguished? While the latter has an intrinsic character, the former has not; it only provides a means of recovering the field ~/via its covariant components yi. II
Sect. 2.5]
2.5.
Linearized change of curvature tensor on a surface
93
LINEARIZED CHANGE OF CURVATURE TENSOR ON A SURFACE
As a surface S possesses a curvature tensor in addition to its metric tensor (Sect. 2.2), it is likewise natural to associate a "linearized change of curvature tensor" with any displacement field defined on S, through a process akin to that followed in Thm. 2.4-1 for defining the linearized change of metric tensor. Together, these two tensors play a major r~le in the theory of linearly elastic shells, as it shall be amply demonstrated in the next chapters. T h e o r e m 2.5-1. Let w be a domain in R 2 and let 0 E C3(~; s be an injective mapping such that the two vectors aa = OaO are linearly independent at all points of-~. Given a displacement field
of the surface S = 0(-~) with smooth enough and "small enough" covariant components 71i 9-~ -+ R, let the c o v a r i a n t c o m p o n e n t s of the l i n e a r i z e d c h a n g e of c u r v a t u r e t e n s o r associated with such a vector field be defined by
p.,(,)
:=
[ b . ~ ( , ) - b~,] "~,
where bar3 and bar3(rl) are the covariant components of the curvature tensors of the surfaces 0(-~) and (O+rliai)(-~) (Fig. 2.3-1), and [... ]tin denotes the linear part with respect to rI = (rli) in the expression [... ]. Then p.,(u)
= (0~
- r .~, 0 ~ )
9~3
p,.(u)
= v~l.~ - b ~ b ~ 3 + b~,~l~ + b~v.l. + b ~ l . ~ -" 0ctf~ ?73 -- r ~ f 30tr T]3 --
b~b~f3?73
94
Inequalities of Korn's type on surfaces
[Ch. 2
where the Christoffel symbols P ~ and the covariant derivatives ~lalf3 are defined as in Thm. 2.3-1,
Y31~ := Oa~r/3 - I'~130~7/3 and b~[a "- Oab~ +
- r x , b;
denote respectively a s e c o n d - o r d e r covariant d e r i v a t i v e of the vector field 7iia i and a f i r s t - o r d e r c o v a r i a n t d e r i v a t i v e of the curvature tensor of S, defined here by means of its mized components. Consequently,
71a e HZ(w) and r/3 E H2(w) =~ Pal3(~/) e L2(w).
The covariant derivatives br~[a satisfy the symmetry relations
Proof. For convenience, the proof is divided into five parts. In parts (i) and (ii), we establish elementary relations satisfied by the vectors ai and a i of the covariant and contravariant bases along S.
(i) The two vectors aa = OaO satisfy [al A a2l = v/-a, where a = det (ha/3). a l ((y)A y ) A a2(y) Let a3(y):= la a2(y)l and O(y, x3) :-- 8(y) + x3a3(y )
for
all y C ~ and z3 E I~. Using the notations of Sect. 1.2 (see notably Thm. 1.2-1), we have det V O = (gl A g2)'g3, where ga :-- Otto -- aa A- z3Octa3 and g3 :-- 0 3 0 = a 3 .
Assume for instance that det V O > 0 for Ix3] small enough, so that v/g - det V O for Ix3[ small enough, where g := det(gij) and gij := gi " gj. Hence V~[zs-O = d e t V O I z s = 0 = ( g l A g 2 ) ' g 3 [ z s - O = ( a l A a 2 ) . a 3 = t a l A a 2 ]
Sect. 2.5]
Linearized change of curvature tensor on a surface
95
on the one h a n d . Since
gl~3:0 - det
I gl'gl gl'g2 gl'g3 1 g2 gl g2"g2 g2"g3 g3 "gl g3 "g2 g3 "g3
a l . aall = det t a2
aa2.a2 l.a2 0
0
i~3=0
00 t 1
= det (aa~) - a on the other, the assertion is proved. T h e case where det V O for Ix31 small e n o u g h is t r e a t e d analogously.
< 0
(ii) The vectors ai and a a are related by al A a3 -- --v/aa 2 and a3/~ a2 - -v/-da 1. To prove t h a t two vectors c a n d d coincide, it suffices to prove t h a t c . a i - d . a i for i E {1, 2, 3}. In the present case, ( a l A a3) . a l = 0 a n d ( a l A a3) . a 3 = 0, (a~ A , ~ )
a~ 9 -
-(a~
A ,,~)
a~ 9 -
-v~,
since v/-da3 - a l A a2 by (i), on the one hand; on the other, -v~a~
9a l
-
-~a~
9a ~ -
o ~cl
since a i . a j - 5j. Hence a l A a3 similarly established.
- v~a~
9a ~ -
-~,
-~/-da 2. T h e other relation is
(iii) The covariant components ba~(~l) and ba~ satisfy bad(r/) - ba~ + ( 0 a ~ / -
F~0z~)
9a3 + h.o.t.,
where, here or in any other similar expansion found subsequently in this part of the proof, "h.o.t." stands for "terms of order higher than linear with respect to rI - (71i) ". Consequently, p~,(o)
:= [b~,(o)
- b o , ] ~'" -
(0o,~
- r;,0=~)
.~
-
p,o(o).
Since the vectors aa - Oa8 are linearly i n d e p e n d e n t in ~ a n d the fields r/ - (r]i) are s m o o t h enough by a s s u m p t i o n , the vectors
96
[Ch. 2
Inequalities of Korn's type on surfaces
Oa(O + ~Tia') are also linearly independent in ~ provided the fields are "small enough", e.g., with respect to the norm of the space C~(~; I~3). The following computations are therefore licit as they apply to a linearization around r / = 0. Let
as(r/) := Oa(8 + O) = aa + OaO and a3(r/):=
~(n) A a2(n)
where (cf. (i)) a(r/) '-- det(aaf~(r/)) and aa3(rl)"- aa(rl), afl(rl). Then
b~,(n) - o ~ , ( n ) . ~(n) 1 = --..~/a(17)(Oaafl + 0af3O)" (al A a2 + a l A 02~ + 01~ A a2 + h.o.t.)
1
= v/~(~) { ~(bo~ + oo~o. ~) } 1
+ v/~(n----5 { ( r o ~ + b o ~ ) . (~ A o~0 + o~0 A ~ ) + h.o.t.}, since bail - Oaafl.a3 and Oaafl - r ~afla~ + baf3a3 by the formula of Gaufl (Thm. 2.3-1 (a)). Next,
(r~a~ +
baf3a3). ( a l A ~20)
= r~,~. = ~
( ~ A o2~) - bo, O~O. (a~ A ~3)
( - r ~ o ~ o . ~ + bo~o~o. ~ ) ,
since, by (ii), a 2 . (al A 02~) -- - 0 2 ~ " (al A a2) -- - v ~ O 2 O . a 3 and a l A a3 - -v/-aa2; likewise,
( r 2 " ~ + b . ~ 3 ) . (o~0 A ~ ) - 4 ~ ( - r ~ , o ~ 0 . ~ + b.,o~n. ~). Consequently, bat3(r/) -
a a(r/) {bat3(1 + 0~0" a ~) + (0at30 - r ~ , o ~ ) 9~3 + h.o.t.}.
Sect. 2.5]
Linearized change of curvature tensor on a surface
97
There remains to find the linear term with respect to r/ - (r/i) 1 1 in the expansion v/a(,/) - ,/_d(1 + . . . ) . To this end, we use the Y
formule~ (see, e.g., Vol. I, Sect. 1.2) d e t ( A + H) - (det A)(1 + tr A - 1 H + o(H)), with A " - (aa~) and A + H : - (aa~(17)). In this case, H - ( O ~ O ' a a + OaO'a/3 + h.o.t.),
since [aa~(~) - a,~] fin -- O~O " aa + Oa~7 " a# (Thm. 2.4-1). Therefore, a(r/) - det(aa~(r/)) - det(aa~)(1 + 20aO" a a + h.o.t.), since A - 1 _ (aa~); consequently, 1
v/~(n)
1 ~(1-
O a i l . a a + h.o.t.).
Noting that there are no linear terms with respect to r/ - (yi) in the product ( 1 - OaO" aa)(1 + 0 ~ - a * ) , we find the announced expansion, viz., ba~(17) - ba~ + (Oa/30 - F ~ 0 ~ 0 )
9a3 + h.o.t.
(iv) The components pa~(rl) can be also written as
where the covariant derivatives ~731a/3 and br~]a are defined in the statem e n t o] the theorem.
By Thm. 2.3-1 (b),
0~
- (0~
- r ~ z ~ - b~,~3)~' + ( 0 ~ 3 + b ; ~ ) a 3.
Hence
since a i . a 3
-- ~ . Again by Thm. 2.3-1 (b),
Inequalities of K o r n ' s type on surfaces
98
o o , 0 . ~3 -
oo { ( o , ~
- r~.o~ - ~,~)~"
+(o,~3 + b~)~ = (0~
[Ch. 2
- r~~
}.~ - b,~3)Oo~.~
+(0~r/3 + b~7,)Oaa3.a3 = b~ ( o , ~ . - r ~ . , . )
- ~.,
~ + o.,~
+(0~b~),~ + b$0.,.,
since O a a ~ " a 3 -- ( - - r ~ r a Oaa 3 . a3 -
cr
r % baa
3
) . a3 -
cr
ba,
- b a ~ a ~ " a 3 -- O,
by the formulas of Gaug and Weingarten (Thm. 2.3-1 (a)). We thus obtain p~(r/) -
(0~0 - r~,~O~).a3
= b;(o,,. - r~,~)
- b~ba~3 + (gaf3rl3 + (Oab~)rlr + b~Oarlr
While this relation seemingly involves only the covariant derivatives Y31a~ and Yzl~, it may be easily rewritten so as to involve in addition the covariant derivatives Yrla and b~la. The stratagem simply consists in using the relation r ~ b ~ - r~b~ - 0! This gives Pot~ ( ~ )
--
(Ootl~ ?73 -- r~f3 (Or ?73 ) -- b~ b ~ ?73
+bX(oz~ - r~~)
+ b~(o~
- rx~n~)
(v) The covariant derivatives b~la are s y m m e t r i c with respect to the indices a and 13. Thanks to the formulas of Gaug and Weingarten (Thm. 2.3-1 (a)), we can write o - o o , ~ ~ - o , o ~ ~ = oo ( - r ~ ~ + b ~ ~) - o , ( - r : ~ ~ + b ~ ~) _- _ _ ( o o r ; ~ ) ~ ~ + ~r v - r ; ~ ba~ a ~ + (oob; )a 3 - b ; b o ~ ~ ~ ~ o .~~
Sect. 2.5] +(0,r~)~
Linearized change of curvature tensor on a surface ~ - r~r~.~.
+ r~b~
99
3 - (0,b~)~ 3 + b~b,~ ~
Consequently, o - (o~,~ ~ - o~~).
~
- oob~ - o , b ~ + r ~ b ~
- r~bX,
on the one hand. On the other, we immediately infer from the definition of the covariant derivatives b~l a that we also have
and thus the proof is complete.
m
Remarks. (1) Covariant derivatives basil~ can be likewise defined when the curvature tensor is defined by means of its covariant components ba~; see Ex. 2.7. (2) The functions ca~ := b~b~[3 - c~a appearing in the expression of Pat3 (r/) are the covariant components of the third fundamental f o r m of S. For details, see, e.g., Stoker [1969, p. 98] or Klingenberg [1973, p. 48]. (3) While the functions ba~(~/) are not always well defined (the vectors aa(y) must be linearly independent in ~), the functions pa~ (~/) are always well defined, m
While the expression of the components pat3 (r/) in terms of the covariant components r/i of the displacement field is fairly complicated but well known (see, e.g., Koiter [1970]), that in terms of ~ - yia i is remarkably simple but seems to be new. As its "linearized change of metric counterpart" (Thm. 2.4-1), it was efficiently put to use by Blouza & Le Dret [1994a, 1994b, 1999], who showed that its principal merit is to afford the definition of the components Pat3(r/) under substantially weaker regularity assumptions on the mapping 8; see Thms. 2.6-5 and 2.6-6. A w o r d of c a u t i o n . As already observed after Thm. 2.4-1, the vector fields r / - (r/i) and ~ / - yia ~, which are both defined on-~, must be carefully distinguished!
II
Inequalities of Kovn's type on surfaces
100
[Ch. 2
Remarks. (i) The symmetry Pa~ (rl) = P~a (rI) is an immediate consequence of the expression pa~(rl) - (Oa~O - F~OrO) "a3 found there. By contrast, deriving the same symmetry from the other expression of pa~(rl) requires proving first that the covariant derivatives b~! a are themselves symmetric with respect to the indices a and ~ (cf. part (v) of the proof of Thin. 2.5-1). While a proof of the symmetry b~l a = b~l~ is thus not essential here, it shall definitely be later in the proof of Thm. 2.6-2. (2) We assume that 8 E C3(~; E 3) in Thin. 2.5-1 (while we only assumed 0 E 62(~; E 3) in Thm. 2.4-1) in order that pa~(r/) C L2(w) if ~ / e Hi(w) x Hi(w) x H2(w) 9 The culprits are the functions b~la appearing in the functions pa/3(r/). As already noted, Blouza & Le Dret [1994a, 1994b, 1999] have shown that such regularity assumptions on 0 can be weakened if only the expressions of 7al3(r/) and pa/3(r/) in terms of the field ~ are considered. I
2.6.
I N E Q U A L I T I E S OF K O R N ' S T Y P E A N D INFINITESIMAL RIGID DISPLACEMENT LEMMA ON A GENERAL SURFACE
The two-dimensional Koiter equations for a linearly elastic shell, so named after Koiter [1970], take the following form: The unknowns are the covariant components ~i: K " ~ -'-4 I~ of the displacement field ~ie,K ai o f t h e middle surface S = 0(~) of the shell and r := (r ~,K) satisfies (0~ denotes the outer normal derivative operator along the boundary of w):
e
{n-
e
x
x
~7i - Ou~3 --0 on "[0}, f~ { caaf3ar' = f~ pi'e71iC'-ady for all r / - (~?i) E VK(W),
Sect. 2.6]
101
Inequalities of Korn's type on a general surface
where 3'0 is a subset of Ow with length 3'0 > 0, 2~ > 0 is the thickness of the shell, 4Ae#~
As + 2p ~
aa~a 'Tr + 2# e (a a'7af3"r + aara fh')
denote the contravariant components of the two-dimensional elasticity tensor of the shell ()t e and pe are the Lam~ constants of the elastic material constituting the shell), 7af3(r/) and pa/3(rl) denote the covariant components of the linearized change of metric and change of curvature tensors associated with a displacement field 71iai of S (Sects. 2.4 and 2.5; their definitions are recalled in Thm. 2.6-1 below), and the given functions pi,~ E L2(w) account for the applied forces; finally, the boundary conditions r/i = 0~7/3 = 0 on 3'0 express that the shell is clamped along the portion 0(3'0) of its middle surface (other boundary conditions are possible). These equations will be fully justified from three-dimensional linearized elasticity in Chap. 7, where various refinements and comments are also to be found. We shall later establish (Thm. 3.3-2) that there exists a constant ce = Ce(W, 0, p~) > 0 such that
Ita/3[2 < ceaa/3~r'e(y)t~rta~ a, f3
for all y existence means of existence
C~
_<
C ~ and all symmetric matrices (tar3). Establishing the and uniqueness of a solution to this variational problem by the Lax-Milgram lemma thus amounts to establishing the of a constant c such that
,,,oli 1,~, + Ilrull2,~, }
{
1/2 1/2
+
a,f3
for all r/E
VK(o)).
a,f~
The objective of this section consists in showing that such an "inequality of Korn's type" indeed holds on a general surface (Thm. 2.6-4). As is readily checked, the same inequality of Korn's type, together with the Lax-Milgram lemma, also provides an existence and uniqueness theorem for the two-dimensional equations of a linearly elastic flexural shell. These equations will be fully justified in Chap. 6
Inequalities of Korn's type on surfaces
102
[Ch. 2
through an asymptotic analysis of the three-dimensional solutions, under the assumption that the space Vv(w) := {r/E Vg(w); 7a#(r/) = 0 in w} does not reduce to {0}. The unknowns are again the covariant components, now denoted (~ 9~ -+ R, of the displacement field (~a i of the middle surface S of the shell and ~ := ((~) satisfies:
E vv(w), L
P~r(~e)Pa#(~l)v~ dY = Z Pi'eYiv~ dy
n=
E
In addition, such an inequality of Korn's type presents a mathematical interest per se: In Sect. 1.7, we established "three-dimensional" Korn inequalities, first without (Thm. 1.7-2), then with (Thm. 1.7-4), boundary conditions (the second one depending on a "three-dimensionar' infinitesimal rigid displacement lemma; ef. Thm. 1.7-3): Both inequalities involved the covariant components ei[ij(v ) of the "three-dimensional" linearized change of metric tensor. But while only one tensor, the metric tensor, is attached to a three-dimensional manifold in E 3, two tensors, the metric and curvature tensors, are attached to a surface in E 3 (Sects. 2.1 and 2.2). It is thus natural to likewise establish inequalities of Korn's type "on a surface", first without (Thm. 2.6-1), then with (Tam. 2.6-4) boundary conditions (the second one again depending on an infinitesimal rigid displacement lemma "on a surface"; cf. Thm. 2.6-3), such inequalities now involving the covariant components 7a~ (~/) and pa~(r/) of its linearized change of metric tensor and linearized change of curvature tensor, respectively defined in Sects. 2.4 and 2.5. We demonstrate that these inequalities are valid for a "general" surface S = 0(~), i.e., corresponding to a "general" injective mapping 0 : ~ --~ E 3 (save that 0 should be smooth e n o u g h ) a n d for any subset 70 of 7 with length 70 > 0. In other words, no restriction is imposed on the "geometry" of S nor on the set 7o (by contrast, specific restrictions must be imposed on 0 and 70 in order that the inequality of Korn's type established in the next section hold).
Sect. 2.6]
Inequalities o~ Korn's type on a general Bur/ace
103
The infinitesimal rigid displacement lemma (Thm. 2.6-3) and the inequality of Korn's type "with boundary conditions" (Thm. 2.6-4) were first established by Bernadou & Ciarlet [1976]. A simpler presentation, which we follow here, was then proposed by Ciarlet & Miara [19925] (see also Bernadou, Ciarlet & Miara [1994]). Its first stage consists (as in dimension three; cf. Thm. 1.7-2) in establishing an inequality of Korn's type "without boundary condition", again as a consequence of the lemma o / J . L . Lions: T h e o r e m 2.6-1 ( i n e q u a l i t y of Korn~s t y p e " w i t h o u t b o u n d ary c o n d i t i o n s " on a general surface). L e t w be a d o m a i n in I~2 and let 0 E C3(~;t; 3) be an injective mapping such that the two vectors aa = OaO are linearly independent at all points o/-~. Given
- ( ~ ) e H~(~) • H~(~) • H~(~), ~t 1 p.~ (,1) "- { o.~ v3 - r ~.~o~ ~3 - b~ b ~ ~3 + b~(0~
- r~~)
+ b; ( 0 o ~ - r ~ , ~ )
+ (o~b; + r ~ b ; - r ~o~j~,-' } e L ~(~) denote the covariant components o/ the linearized change o/ metric and change of curvature tensors associated with the displacement field ~ia i o] the sur/ace S = 0(-~). Then there exists a constant co - co(w, O) such that (the notations I" I0,~ and I1" Ilm,~ respectively designate the norms in L2(w) and Hm(w), m > 1; cf. Sect. 1.7)-
{Err,oiL
,,~ 9 + I1~11~,~
a
a
} ~/2 a, f3 a,f3 for all ~ - (77i) e H t ( w ) • H i ( w ) • H2(w).
Proo/. (i) Define the space
w K ( ~ ) "- { , -
(,~) c L~(~) • L~(~) • H~(~); ~,,(U) e L~(~), p,,(U) e L~(~)}.
Inequalities of Korn's type on surfaces
104
[Ch. 2
Then, equipped with the norm il. IIK defined by
I1,~11~ "-
~
2 }i/~
~ I,~oIo, ~ +11,~11~,~+ ~ I'~,~(~)1],~ +~-~ ..... I~o~(,~)1o,~ a
a,t~
a,t~
the space WK(w ) is a Hilbert space. The relations "Ta~(~/) E L2(w) '' and "pat~(~/) E L2(w) '' appearing in the definition of the space WE(W) are to be understood in the sense of distributions. They mean that ~ E L2(w) x L2(w) x Hi(w) belongs to WE(W) if there exist functions in L2(w), denoted 7at~(W) and pa/3(lr/), such that for all ~o E T~(w), L
L {1
}
L Pa~(~I)~ dY - - L {oa~130~o + r:~o~13~o + b:b~I3qo
+,~o~(~g~o) + ~ r ; ~ , ~ o + ~0o(b~) + b~r~~ - (0ob~ + r ~ b ~ - r ~ , ~ ) ~ } ~ . Let there be given a Cauchy sequence (~/k)~~ with elements ~/k _ (r//k) E WK(w). The definition of the norm !1" I]g shows that there exist ~/a E L2(w), r/3 E Ht(w), 3'at3 E L2(w), and Pat3 E L2(w) such that ~/~ --+ 7/~ in
L2(w),
3'at3(~/k) --+ 7at3 in L2(w),
7/3k --+ r/3 in H i ( w ) ,
P~t3(~/k) -~ P~13 in L2(w)
as k -+ cr Given a function qv E T~(w), letting k --+ cr in the relations f~ 3,al3(~/k)qodw - . . . and f~ pa~(Wk)qodw - - . . . then shows that 7a~ -- 7at3(~) and Pal9 -- Pa19(~)" (ii) The spaces WE(W) and Hi(w) x Hi(w) x H2(w) coincide.
Clearly, Hi(w) x Hi(w) x H2(w) C WK(w). To prove the other inclusion, let ~/= (~/i)6 WK(W). The relations 1
O"
~ ( ~ ) .- ~ ( o ~ + o ~ ) - ~ ( ~ ) + r ~ ~ + b~,~3
Sect. 2.6]
105
Inequalities of Korn's type on a general surface
then imply that ea/3(~/) E L2(w) since the functions D ~ and ba/3 are continuous on ~ (even continuously differentiable, since we assume 0 E C3(~; Es)). Therefore, O@~Ta E H - I ( w ) ,
0~(0~) - {0~~(~) + 0~~(~) - 0o~.(~)} e H-~(~), since X E L2(w) implies 0~X E H - i ( w ) . Hence 0~ya E L2(w) by the lemma of J. L. Lions (Whm. 1.7-1) and thus ~a E Hi(w). The definition of the functions Pa/3(~/), the continuity over ~ of the functions r=a/3, b~/3, b~, and Oab~, and the relations pa/~(~/) E L2(w) then imply that 0~/3~/3 E L2(w), hence that ~/3 E H2(w). (iii) Inequality of Korn's type without boundary conditions. The identity mapping t from the space Hi(w) • Hi(w) • H2(w) equipped with its product norm T / - (~7i) --~ { ~ a lllTali2,,,,+iil?si[~,<,,} i/2 into the space W g ( w ) equipped with I]" [Ig is injective, continuous, and surjective by (ii). Since both spaces are complete (cf. (i)), the closed graph theorem then shows that the inverse mapping t -1 is also continuous or equivalently, that the inequality of Korn's type without boundary conditions holds, m
Remark. An earlier "inequality of Korn's type on a surface" is due to Roug6e [1969, Lemma 2.4], who showed the existence of a constant c such that
{ for a11 (W~) e HI(w)
x
2+~[1
<= [2~>
Hi(w),
I
In order to establish an inequality of Korn's type "with boundary conditions", we have to identify classes of boundary conditions to be imposed on the fields ~ / - (Yi) E H i ( w ) x H i ( w ) x H2(w) in order that we can "get rid" of the norms I~la[o,~ and ]lY311i,~ in the right-hand side of the above inequality, i.e., situations where the semi-norm 2 }i/2
,7- (~,)-~ ~ l~<,~(,i)lo~,<,,+ ~ Ip<,~(,i)lo,<,, a,/3
a,/3
becomes a norm, which should be in addition equivalent to the product norm.
106
[Ch. 2
Inequalities of Korn's type on surfaces
To this end, we begin by establishing in Thm. 2.6-3 (as in dimension three; cf. Thm. 1.7-3) an infinitesimal rigid displacement lemma, which provides in particular one instance of boundary conditions implying that this semi-norm becomes a norm. The elegant proof given here was suggested to me by Chapelle [1994]. It relies on the preliminary observation, quite worthwhile per se, that a vector field yia ~ on a surface may be "canonically" extended to a three-dimensional vector field rig i in such a way that all the components eillj(v ) of the associated "three-dimensional" linearized change of metric tensor have remarkable expressions in terms of the components ~/af3(~7) and Paf3(~7) of the linearized change of metric and curvature tensors of the surface vector field: T h e o r e m 2.6-2. Let the assumptions on the mapping 8 9~ -+ E 3 be as in Thm. 2.6-1. By Thm. 3.1-1, there exists eo > 0 such that the canonical extension 0 of the mapping 8 defined by
O(y, X3):: O(y) "~ x3a3(y ) for all (y, x3) e ~0,
where ~o := w•
al/~a2 e0[ and as :-- ]al A a21' is a Cl-diffeomorphism
from f~o onto O(f~0) and the three vectors gi "- OiO are linearly independent at all points of f~o. With any vector field ~liai with covariant components Ya in H 1(w) and ~3 in H2(w), let there be associated the vector field vig i defined on f~o by
for all (y, x3) E -~o, where the vectors gi form the contravariant basis associated with the vectors gi (i.e., g i . g j _ (f~; cf. Sect. 1.2) and
x..-
-(o.,73 + b2o ). .
Then the covariant components vi of the vector field rig' are in H l ( ~ o ) and the covariant components eilli(v ) C L2(~0) of the asso-
Sect. 2.6]
107
Inequalities of Korn's type on a general surface
ciated linearized change of metric tensor (Sect. 1.5) are given by
+ ~ {b~pf3~(r/)+ b;pa~(17) - 2b~b;7ar(17) }, eill3(v) -- O.
Proof. As in the above expressions of the functions eallf3(v), the
dependence on x3 is explicit, but the dependence with respect to y E ~ is omitted, throughout the proof. (i) Given functions ~a, Xa E H i ( w ) vector field v ig i be defined on fro by ~
and r/3 e H2(w), let the
.
rig ~ -- ~Tia* + x 3 X a a a
(in other words, we momentarily ignore that the functions Xa have the specific forms indicated in the theorem). Then the functions vi are in H l ( f l o ) and the covariant components eiljj(v ) of the linearized ~
change of metric tensor associated with the field rig z are given by 1 w3 f
I
1 e3ll3(v ) -- O, a~X~ designate the where nal~ - O/3rla - raf3rl~ and XoI ~ - O~2da - F ~ covariant derivatives of the fields ~lia i and X i a i with X3 - 0 (Thm.
2.3-1). Since Oaa3 -- - b ~ a ,
by the second formula of Weingarten (Thin. 2.3-1), the vectors of the covariant basis associated with the mapping 19 - 0 + x3a3 are given by ga : aa - x3b~a~ and g3 : a3.
108
Inequalities of Korn's type on surfaces
[Ch. 2
The assumed regularities of the functions r//and 2'a imply that vi - ( v j g J ) ' g i - 07ja i + x 3 X a a a ) ' g i
C Hl(f~o)
since gi E C1(~0). The announced expressions for the functions eiiij(v) are obtained by simple computations, based on the relations (Thms. 1.4-1 and 1.5-1): viii/- {Oj(vkgk)}.gi
1 and e/il/(v) - ~(v/ll/+ viii/),
the latter advantageously avoiding the computation of the Christoffel symbols I"/Pj. (ii) W h e n & - -(Oar/s + b~7~), the N n c t i o n s eillj(v) in (i) take the expressions announced in the s t a t e m e n t of the theorem. We first note that Xa E HZ(w) (since b~ e CZ(~)) and that %113(v) - 0 in this case. It thus remains to find the explicit forms of the functions ec~ll~3(v). Replacing the functions Xa by their expressions and using the symmetry relations b~l~ - b~la (Thm. 2.5-1), we find that 1 -- --rl3]af3 -- baT?all3 - b;rlrla - b~laT?r + b~ba/3rl3,
i.e., the factor of :c3 in eall/3(v) is equal to -pa~(r/). Finally,
= b~.(,731~ + b ; l ~
+ b;,7.1~)+ b ; ( ~ l ~ . + b~l~'7~ + b~,7~l~)
= bX(O~(,7) - b;n.l~ + b;b.~,Ts) + b;(O~.(,7) - b7~,7~1. + b~b~.,Ts) = b~O~(,1) + b;O...(,7) - 262b;-y~.(,7). i.e., the factor of ~ in %ll~(V) is that announced in the theorem. II R e m a r k . The otherwise mysterious definition of the functions 2'a
is thus motivated by the requirement that eatlz(v) - 0; cf. part (ii). i
We now establish an infinitesimal rigid displacement l e m m a "on a surface"; "infinitesimal" reminds that only the linearized parts of
Sect. 2.6]
Inequalities of Korn's type on a general surface
109
the change of metric and curvature tensors are required to vanish. Thanks to Thm. 2.6-2, this lemma becomes a simple consequence of the "three-dimensional" infinitesimal rigid displacement lemma in curvilinear coordinates (Thm. 1.7-3), to which it should be profitably compared. This lemma is due to Bernadou & Ciarlet [1976, Thms. 5.1-1 and 5.2-1], who gave a more direct, but less "transparent", proof (see also Bernadou [1994, Part 1, Lemma 5.1.4]). As shown by Blouza & Le Dret [1999], it still holds under weaker smoothness assumptions on the mapping 0 (see Thm. 2.6-5). Part (a) in the next theorem is an infinitesimal rigid displacement l e m m a "without boundary conditions", while part (b) is an infinitesimal rigid displacement lemma "with boundary conditions". T h e o r e m 2.6-3 ( i n f i n i t e s i m a l r i g i d d i s p l a c e m e n t l e m m a o n a g e n e r a l s u r f a c e ) . Let there be given a domain w in IR2 and an injective mapping 0 E C3(~; E 3) such that the two vectors aa - OaO are linearly independent at all points of-~. (a) Let 17 (~?i) e Hi(w) x Hi(w) x H2(w) be such that -
-
7a~(r/) = pad(r/) = 0 in w. Then the vector field yia i is an i n f i n i t e s i m a l r i g i d d i s p l a c e m e n t of the surface S -- 0(-~), in the sense that there exist two vectors @, et E ~3 such that ~ i ( y ) a i ( y ) = ~ + cl A O(y) for all y E ~.
(b) Let 7o be a dT-measurable subset of 7 = Ow that satisfies length 70 > O. Then (Ov denotes the outer normal derivative operator along 7): .-
e
•
•
r/i - O~r/3 - 0 on 70, 7 ~ ( r / ) - pa~(~/) - 0 in w
=:~ r / - - 0 in w.
110
Inequalities of Korn's type on surfaces
[Ch. 2
Proof. Let the set 12o = w • s0, s0[ and the vector field v - (vi) E Ht(gl0) be defined as in T h m . 2.6-2. By the same theorem, Vaf3(Y) = paf3(y) - 0 in w ~ eillj(V) - 0 in 120, and thus, by T h m . 1.7-3 (a), there exist two vectors fi, d C I~3 such that
vi(y, xa)gi(y, xa) - e + h A {0(y) + xaaa(y)} for all (y, xa) E ~0. Hence
~Ti(y)ai(y) - vi(y, xa)gi(y, xa)[zs=o - & + 3 A O(y) for a11 y E ~, and part (a) is established. If ~/i = 0~/3 = 0 on 70, the functions Xa = - ( 0 a y 3 + b~7/~) vanish on 70, since ya = 0~r/3 = 0 on 70 implies 0ar/3 = 0 on V0; consequently ( T a m . 2.6-2),
vi = ( v j g J ) ' g i = (yja j + x 3 X a a a ) ' g i
= 0 on r0 "-- V0 • [-e0, e0].
Since area r0 > o, T h m . 1.7-3 (b) implies t h a t v - 0 in gl0, hence that~-Oon~, m
Remarks. (1) If a field y - (rIi) e HI(w) x HI(w) x H2(w) satisfies Vail(v/) = Pail(Y) = 0 in w, its three components r/i are automatically in g2(~) since ~?i = (TljaJ)'ai and the fields ai are of class r on ~. Remarkably, the field ~lia i - fi + (t A 0 inherits in this case even more regularity, as it is of class Ca on ~! (2) The constant vector d can be expressed in terms of the functions ~/i; cf. Ex. 2.8. (3) If a field y = (Yi) satisfies the assumptions of part (a), the b o u n d a r y conditions 7?i = 0vy3 = 0 on 70 of part (b) are not the only ones t h a t lead to 71 = 0 in w; cf. Ex. 2.9 for another instance. (4) Crucial information can still be gathered about the displacement field yia i if only the components 7aft(Y) vanish in w; cf. Ex. 2.8. m We are now in a position to prove an inequality of Korn's type "with boundary conditions" t h a t plays a fundamental rSle in the analysis of linearly elastic shells, in particular for establishing the
Sect. 2.6]
Inequalities of Korn's type on a general surface
111
existence and uniqueness of the solution to the two-dimensional Koiter equations for a linearly elastic "clamped" shell (another class of boundary conditions is considered in Ex. 2.9) and of the solution to the two-dimensional equations of a linearly elastic "flexural shell", as explained at the beginning of this section. This inequality was first proved by Bernadou & Ciarlet [1976]. It was later given other proofs by Bernadou, Ciarlet & Miara [1994], then by Blouza & Le Dret [1999], who showed that it still holds under a less stringent smoothness assumption on the mapping O (see Thm. 2.6-6). T h e o r e m 2.6-4 ( i n e q u a l i t y of K o r n ' s t y p e on a g e n e r a l surface). Let w be a domain in IR2, let 0 E Ca(~; E 3) be an injective mapping such that the two vectors aa = OaO are linearly independent at all points of-~, let "Yo be a dT-measurable subset of "y = Ow that satisfies
length 70 > 0, and let the space Vg(w) be defined as (0~ denotes the outer normal derivative operator along 7): VK(W) "-- { ~ -
(r]i) e H i ( w ) x H i ( w ) x H2(w); ~7i -- Ou~3 -- 0 on 70}"
~ive?z 71 " - ( ~ i ) E H i ( w ) x H i ( w ) • H2(w), let
1
O"
+ (0ob + r; b$ - r
b
} e
denote the covariant components of the linearized change of metric and linearized change of curvature tensors associated with the dis-
112
[Ch. 2
Inequalities of Korn's type on surfaces
placement field ~lia i of the surface S - 0(~). constant c - c(w, V0, 0) such that
{~
Then there exists a
II~7,~II~,,,,+ II~7311~,,,,}~/2
<_~ ~ 17a~(n)[~,,~+ ~ Io~,~(n)lo,~ a,~
a,r
fo~ ~n n - (,7~) e v~(~). Proof. Let
a
and a,t3
a,t3
If the announced inequality is false, there exists a sequence (r/k) k--1 oo of functions rl k E Vg(w) such that ,,..II~klIHR(,,,)xHI(,,,)xH~(~) -- i for all k and
lim lyh] K - 0.
Since the sequence (r/k)~=i is bounded in Hi(w) x Hi(w) x H2(w), a subsequence (r/Z)~i converges in L2(w) x L2(w) x Hi(w) by the Rellich-Kondra~ov theorem (see, e.g., Vol. I, Thin. 6.1-5); furthermore, each sequence ('yat3(r/t))~i and (pa13(r/Z))~=i also converges in L2(w) (to 0, but this information is not used at this stage) since
lim~_~ IWI5 - 0. The ~ub~eque~e (~)~=~ is thu~ ~ C~uchy ~equence with respect to the norm
,I ->
{
~
a
2
2
2
2 } 1/2
1,7~Io,,,,+ II,7~,II~,,,,+ ~ I~,(~)Io,,,, + ~ Ip,,,(,7)lo,,,, a,~
a,/3
,
hence with respect to the norm I1" IIHi(w)xHi(w)xH2(w) by Korn's inequality without boundary conditions (Thm. 2.6-1). The space Vg(w) being complete as a closed subspace of the space Hi(w) x Hi(w) x H2(w), there exists r/E VK(W) such that
Inequalities of Korn's type on a general surface
Sect. 2.6]
113
and the limit ~/satisfies
i~a~(n)lo,~,
-
]Pc,~(~)]o,~ -
lim
l-+oo
I~,~(nZ)lo,,~ - o,
lim ]pa~(~)]o,,,, - O.
l--+oo
Hence ~/ = 0 by Thm. 2.6-3. But this contradicts the relations [[~lllHl(w)•215 ) -- 1 for all I > 1, and the proof is complete, m If the mapping 8 is of the form 8(yl, y2) = (Yl, Y2, 0) for all
(Yl, Y2) E w, the inequality of Thm. 2.6-4 reduces to two distinct inequalities (obtained by letting first r/a = 0, then ~/3 = 0):
for all 7/3 E H2(w) satisfying ~3 -- 0 ~ 3 -- 0 on 70, and
for all r/a E HI(w) satisfying 7/a - 0 on 70. The first inequality is a well-known property of Sobolev spaces (see, e.g., Vol II, Thm. 1.5-1 and its proof); the second is the two-dimensional Korn inequality in Cartesian coordinates. Both play a central rSle in the existence theory for linear two-dimensional plate equations (Vol. II, Thms. 1.5-1 and 1.5-2). As shown by Blouza & Le Dret [1994a, 1994b, 1999], the regularity
assumptions made on the mapping 0 and on the field ~ = (71i) in both the infinitesimal rigid displacement lemma and the inequality of Korn's type (Thins. 2.6-3 and 2.6-4) can be substantially weakened. This improvement relies on the observation that the covariant components of the linearized change of metric and change of curvature tensors, viz.,
~(~)
1 - ~(0~
Pa~(~)
--0a/~3
+ 0~)
- r~~
- b~3
and --
r~t30~3 b~b~[3~73 -
+ b~(0,,~ - r ~ , ~ ) + b~ (0.,~ - r ~ , ~ ) + (o~b; + r L b 5 - r ~ b ; ) ~ ,
114
Inequalities of Korn's type on surfaces
[Ch. 2
are also given by 1 7~a(n) - ~ ( o ~ 0 9a~ + 0 ~ 0 . ~a) =: ~ ( 0 ) and p~,(n)
= (o~aO - r~,o~O)
9~3 - . P ~ a ( O ) ,
in terms of the field o
0 := ~i a*-
The interest of the new expressions 7afJ(0) and jhaf~(0) is that they still define bona fide distributions under significantly weaker smoothness assumptions than those of Thm. 2.6-4, viz., 8 E Ca(~; E a) and - (yi) E g 1(w) • H 1(w) • H2(w). More specifically, it is easily verified that 7af3(~)) E L2(w) and Fhaf3(@) E H - l ( w ) if 0 E W2'C~(w; E 3) and ~ E I-II(w). Note that, to.avoid any confusion, we intentionally employ the new notations "Yaf3(O) and ~haf3(O). Using this observation, Blouza & Le Dret [1999, Thm. 6] first establish the following extension of Thm. 2.6-3: T h e o r e m 2.6-5 (infinitesimal rigid d i s p l a c e m e n t l e m m a on a g e n e r a l surface w i t h little r e g u l a r i t y ) . Let w be a domain in R 2 and let 8 E W2'~176 E 3) be an injective mapping such that the two vectors aa - Oh8 are linearly independent at all points of-~. Given ~ E Hi(w), let the distributions "~af3(~) E L2(w) and Pa~(O) E H - l ( w ) be defined by 1 ~/o~(0) := -~(o~O . a o + 0 o 0 . a~),
Let 0 E I t l ( w ) be such that ~/~z(O) - #~z(O) - o i= w.
Sect. 2.6]
Inequalities of Korn's type on a general surface
115
Then Q is an infinitesimal rigid d i s p l a c e m e n t of the surface S - 0(-~), in the sense that there exist two vectors i:, d E IRa such that
Q(y) - / : + ~/A O(y) for all y C ~. II
Blouza & Le Dret [1999, Lemma 11] then proceed to establish the following variant of Thm. 2.6-4, which is stated here with the boundary conditions ~?/ - 0 on 7 corresponding to a shell that is simply supported along its entire boundary. T h e o r e m 2.6-6 ( i n e q u a l i t y of Korn~s t y p e on a g e n e r a l surface w i t h little r e g u l a r i t y ) . Let the assumptions on the mapping 0 be as in Thm. 2.6-5 and let the space VK(W ) be defined as
~r~:(~)- {0 c I~(~); o~,0. ~ c n~(~)}. Then there exists a constant c such that
{110112 + ~ ,,,,,
Ioo~-a~l~,,,,,} ,/2 for all ~ e r E ( w ) ,
where the distributions zya~(fT) and Pa# (~7) are defined as in Thm.
2.6-5 (note that/Safe(Q) e L2(w)if Q e ~rg(w)).
II
Remark. As expected, the space ~'g(W) becomes isomorphic to the space H t ( w ) • H~(w) • {H2(w) • H0t(w)} when 0 is smooth enough; cs Ex. 2.10. II
This theorem thus establishes as a corollary the ezistence and uniqueness of the solution to the two-dimensional Koiter equations for a simply supported shell whose middle surface has little regularity,
Inequalities of Korn's type on surfaces
116
once these equations are rewritten as follows: Find
[Ch. 2
~K such
that
E ~e
~
~3
{eaaf3ar'eSo.r(~K)3'af3(fT) + -~aaf3ar'e~ar(~K)~af~(fT) }v/ady _ f f~e. f T ~ d Y for all ~ E V~(w), J~
where the given field/5 e E L2(w) accounts for the applied forces. A w o r d of "ce a u t i o n . In this approach, the unknown is the displacement field ~g of the middle surface S - 8(w), viewed as a vector field with Cartesian components in Hi(w). The displacement field is thus no longer recovered by means of its covariant components on the contravariant bases. I
Remark. Boundary conditions of clamping, viz., Yi = 0 ~ 3 = 0 on V0 C V, as considered in Thm. 2.6-4, can be also handled via the present approach, provided they are first re-interpreted so as to make sense for vector fields ~ that only satisfy "~ E Hi(w) and Oa~'a3 E L2(w)"; cf. Blouza & Le Dret [1999, Sect. 6]. I On the one hand, this approach clearly widens the class of shells that can be modeled by Koiter's equations, since discontinuities in the second derivatives of the mapping 8 are allowed, provided these derivatives stay in L~(w). For instance, it affords the consideration of a shell whose middle surface is composed of a portion of a plane and a portion of a circular cylinder meeting along a segment and having a common tangent plane along this segment. On the other, the asymptotic analysis that will ultimately justify the two-dimensional Koiter equations in the form described at the beginning of this section (i.e., expressed in terms of the functions Va~ (Y) and Pa~(~7)) does seem, however, to require that 0 E C3(~; E3); cf. Sect. 7.2.
S e c t . 2.7]
2.7.
I n e q u a l i t y o f K o r n ' s type o n a n e l l i p t i c s u r f a c e
117
INEQUALITY OF KORN'S TYPE AND INFINITESIMAL RIGID DISPLACEMENT LEMMA ON AN ELLIPTIC SURFACE
The two-dimensional equations of a linearly elastic elliptic membrane shell, which will be fully justified in Chap. 4 through an asymptotic analysis of the three-dimensional solutions under specific restrictions on the boundary conditions and on the "geometry" of the middle surface, take the following form: The unknowns are the covariant components ~ 9~ --+ IR of the displacement ~ a i of the middle surface S = 8(~) of the shell and ~ := ( ~ ) satisfies:
r e v ~ ( ~ ) : = H0~(~) • H~(~) • L~(~), f
eaa~ar'~7~r(~)Taf3(~l)v~dy = ~ pi'e71ix~dy for all r / - (r/i) C VM(~9),
where 2e > 0 is the thickness of the shell, a a~OT~ $ ,_- -
As + 2# ~
aa~ aar + 2tze (a a~rafar + a ar afhT)
denote the contravariant components of the two-dimensional elasticity tensor of the shell (the same as in Sect. 2.6), 7a~3(v/) denote the covariant components of the linearized change of metric tensor associated with a displacement field v/iaz of S, and the given functions pi'e E L2(w) account for the applied forces. As already noted in Sect. 2.6, there exists a constant Ce - Ce(W, O, #~) > 0 such that
]~.~12 _< ~ ~ , ~ ( y ) ~ ~ a,~
for all y C ~ and all symmetric matrices (ta~) (Thm. 3.3-2). Establishing the existence and uniqueness of a solution to the above variational problem by means of the Laz-Milgram lemma thus amounts to proving the existence of a constant CM such that
{Ell oLi
w
--
{
O~w
fo~ ~n ,7 = ( ~ ) e v ~ ( ~ ) .
118
[Ch. 2
Inequalities of Korn's type on surfaces
The objective of this section, based on Ciarlet & Lods [1996a] and Ciarlet & Sanchez-Palencia [1996], is to find sufficient conditions (which, remarkably, turn out to be also necessary; cf. Thm. 2.7-4), essentially bearing on the "geometry" of the surface S, guaranteeing that such an "inequality of Korn's type" holds. It is also worth noticing that the justification alluded to above of these two-dimensional membrane shell equations from three-dimensional elasticity will be carried out under precisely the same assumptions on the geometry of S (Chap. 4). We follow the usual pattern, i.e., we begin by proving an inequality of Korn's type "without boundary condition", which in fact holds for "arbitrary" geometries, even though it only involves the linearized change of metric tensor. Everything has its price, however: The norm !1~73112,~appearing in the left-hand side of the "first" inequality of Korn's type on a general surface (Whm. 2.6-1) is now replaced by the norm lY310,~. T h e o r e m 2.7-1 (second i n e q u a l i t y of K o r n ' s t y p e " w i t h o u t b o u n d a r y c o n d i t i o n s " on a g e n e r a l surface). Let w be a domain in I~2 and let 0 E C2(~; s be an injective mapping such that the two vectors aa - OaO are linearly independent at all points of-~.
1
"1
IT
denote the covariant components of the linearized change of metric tensor associated with the displacement field rlia~ of the surface S - 0(-~). Then there ezists a constant c o - co(w, O) such that
a
i
a,t3
Proof. The proof is analogous to that of Thm. 2.6-1 and, for this reason, is only sketched; it relies on the following steps" First, the space wM( ) . - { . c c
Sect. 2.7]
Inequality of Korn's type on an elliptic surface
119
becomes a Hilbert space when it is equipped with the norm !1" I]M defined by
i
a,f3
Next, the two spaces WM(w) and H i ( w ) x H i ( w ) x L 2 ( w ) coincide, thanks again to the identities -
+
-
and to the lemma of J.L. Lions (Thm. 1.7-1). Finally, the closed graph theorem shows that the identity mapping from the space Hi(w) x Hi(w) x L2(w) equipped with the product norm y - (~?i) ~ { ~ a l]Yall2,~ + 177312,~}i/2 onto the space WM(w) equipped with I1" IIMwhas a continuous inverse. Hence the announced inequality holds, m The next step consists in identifying sufficient conditions affording the "elimination" of the norms ]Yil0,~ in the right-hand side of the above inequality of Korn's type. Whether it be for the threedimensional Korn inequality in curvilinear coordinates (Thm. 1.7-4) or for the inequality of Korn's type on a general surface (Thm. 2.6-4), the corresponding eliminations simply resulted from imposing ad hoc boundary conditions on the displacement fields. This idyllic scheme needs to be drastically amended in the present case! If we were following the same pattern, we would first prove an "infinitesimal rigid displacement lemma without boundary conditions", characterizing those displacement fields y~a z with covariant components Ya E Hi(w) and y3 E L2(w) that satisfy 7af3(Y)- 0 in w. The most powerful result in this direction is due to Blouza & Le Dret [1999, Thin. 6]: /f the three covariant components ~7~ of the displacement field are in L2(w) and V~f3(Y) - 0 in w, then there exists r E H - i (w) such that - r A 0
0.
However, this characterization does not provide an explicit form of the corresponding displacement field (compare with Thm. 2.6-3, or with Thm. 1.7-3 in the three-dimensional case); in fact it does
120
Inequalities of Korn's type on surfaces
[Ch. 2
not even allow to conclude that the space formed by such fields is finite-dimensional! We are thus left to directly establishing an "infinitesimal rigid displacement lemma with boundary conditions", i.e., to finding in particular boundary conditions guaranteeing that the semi-norm
.
(,,I
I o l.lr } a,f~
becomes a norm for the displacement fields ~7ia i that satisfy them. Since 773 is only in L2(w), the only choice consists in "trying" the boundary conditions ~?a = 0 on 70 C 7, with length 70 > 0. It then turns out that such an infinitesimal rigid displacement lemma does hold, but only for special geometries of the surface S and special subsets 0(70) of the "boundary" of S. We refer to Lods g~ Mardare [1998a], Mardare [1998c], and SUcaru [1998] who have identified various situations of interest where this lemma holds; such situations are described in Chaps. 4 and 5. Note in passing that such geometrical restrictions occur "for the first time" in our derivations of inequalities of Korn's type.
But the worst is yet to come! For, even though this infinitesimal rigid displacement lemma "often" holds, it "seldom" implies that the norm rl --+ {~a,f~ I')'afl(rl)12,o~}1/2 is equivalent to the product norm
n -
2
Ii, ll , + Jvsl0, ) More precisely, we shall see (Thm. 2.7-3) that, under ad hoc regularity assumptions on the mapping 0, these two norms are equivalent /f 7 = 70 and the surface S is "elliptic" according to the definition given below. In addition, Slicaru [1997] has shown the remarkable result that, even under "minimar' regularity assumptions, the same sufficient conditions are also necessary for the equivalence of these norms (see Thin. 2.7-4 for a more precise statement), which thus "seldom" occurs indeed! We begin by a fundamental definition: Let a surface S = 0(~) be given, where 0 E C2(~; E 3) is an injective mapping such that the two vectors a a -- OaO are linearly independent at all points of ~. Then S is elliptic if the symmetric matrix (baf3(y)) formed by the -+
Inequality of Korn's type on an elliptic surface
Sect. 2.7]
121
covariant components of the curvature tensor of S is positive, or negative, definite at all points y E ~; or equivalently if there exists a constant c such t h a t
I~a[ 2 < clba#(y)~a~#l for all y E ~ and all (~r
e R2
or equivalently if the Gaussian curvature of S (Sect. 2.2) is everywhere
strictly positive, i.e., if 1
R1 (y)R2(y) > 0 for all y E ~, where R~(y) and R2(y) are the principal radii of curvature of S at 8(y). A portion of an ellipsoid provides an instance of elliptic surface. In the next theorem, analytic functions of two real variables in an open subset of ]~2 a r e considered; we simply recall here their deftnition~ referring to Dieudonn~ [1968] for a particularly elegant treatment of analytic functions of any finite number of real or complex variables: Let w be an open subset of I~2; a function f : w -+ R is analytic if, given any y = (Yl, Y2) C w, there exists r > 0 and ainu E ~, m > O, n > O, such that the open ball of radius r centered at y is in w and OO
f(yl, m~ n - - 0
for all y' - (y~, y~) such that lY'-Y[ < r, these series being absolutely convergent. We also recaU t h a t a function belongs to the space C2'1(~) if it is in C2(~) and its second-order partial derivatives are Lipschit~continuous on ~. We now prove the announced "infinitesimal rigid displacement lemma", directly under the assumptions (70 = 7 and S is elliptic) t h a t will eventually lead to the equivalence of norms. For pedagogical purposes~ we assume in our proof more smoothness on the b o u n d a r y 7 and on the mapping 8 t h a n is necessary; in so doing, we essentially combine the proofs given by Ciarlet & Lods [1996a] and Ciarlet & Sanchez-Palencia [1996]. We refer to
122
[Ch. 2
Inequalities of Korn's type on surfaces
Lods & Mardare [1998a] for a proof under the more general assumptions stated in the theorem below. An earlier version of this lemma is due to Vekua [1962], who proved it under the assumptions that 7 is of class C3 and 0 E W3'P(w; Ea), p > 1, using the theory of "generalized analytic functions". T h e o r e m 2.7-2 ( i n f i n i t e s i m a l rigid d i s p l a c e m e n t l e m m a o n a n elliptic s u r f a c e ) . Let there be given a domain 03 in IR2 and an injective mapping 8 C C2'1(~; C 3) such that the two vectors aa - OaO are linearly independent at all points of-~ and such that the surface S : 0(-~) is elliptic. Then
n -- (r/i) E H i ( w ) • H i ( w ) • L2(w), ~ :ff ~ / _ 0 in 7af3(r/) - 0 in w
03.
J
Proof. We give the proof under the additional assumptions that the boundary 7 is of class C3 and that the components of the mapping 0 are restrictions to -~ of analytic ]unctions in an open set w' C R 2 containing -~.
(i) We first note that establishing this implication is equivalent to proving a uniqueness theorem, viz., ~7 = (vii) = 0 is the only solution in the space H i ( w ) x H i ( w ) • L2(w) of the linear system formed by the three partial differential equations 7af3(r/) = 0 in w together with the two boundary conditions (understood in the sense of traces) r/a - 0 on 7, or in eztenso, 01771 la2~i
--
+
i
r~l
T]a
0i~/2 -- r i ~ a 02,72 -
-
bll r/3 -- 0 in w,
-
bi2~a -
Oinw,
b22,/3 -
0 in w,
~r r22,/~ -
r]l
= 0ong,,
Y2
= 0onT.
(ii) A n y field n -- (77i) e Hlo(W) • H i ( w ) • L2(w) satisfying 7aft(Y) - 0 l a w and ya - 0 on 7 is in the spaceCt(-~)•215176 This regularity result relies on a crucial observation, due to Geymonat & Sanchez-Palencia [1991, 1995]: The partial differential equations 7af3(W) = 0 in w constitute a system, of the first order with
Inequality of Korn's type on an elliptic surface
Sect. 2.7]
123
respect to the functions ~?a and of order zero with respect to the function ~/3, that is "uniformly elliptic" in the sense of Agmon, Douglis & Nirenberg [1964]. This means that there exists a constant A > 0 such that (here and subsequently in this part of the proof, we use the notations of Agmon, Douglis & Nirenberg [1964]):
I ol < IL(y,
< A
(It
for all y E ~ and ~ = (~a) E R 2, where
~1 L(y, ~) := det
1
0
-bll(y)
1 9
0
~2
-b22(Y)
The way the above matrix of order three is constructed from the equations 7af3(r/) = 0 should be clear; suffice it to specify that only the coefficients of the partial derivatives of the highest order for each unknown (one for r/a and zero for ~73) are taken into account. The uniform ellipticity of the system "Taf3(v/) = 0 in w" thus holds since
L(y,
l~) - -
1
-~(~2 - ~I)
(
bii (y) b21(y) b22(Y)
~2
in the present case, and the symmetric matrix (baj(y)) is either positive, or negative, definite at all points y E ~ by the assumed ellipticity of S. In addition, the "supplementary condition on L " (which needs to be verified only in two dimensions, as here) is also satisfied: The degree m of the polynomial L with respect to ~1 and ~2 being two, the polynomial r E C ~ L(y, ~ § EC has exactly ~m _ one root T+ with Imv + > 0 for all y E ~ and all linearly independent vectors ~ - (~a) and v / - (r/a) in I~2. Finally, when ~one boundary condition, e.g., rh - 0 on 7, is appended to the equations "Taf3(v/) -- 0 in w', the "complementing boundary condition" is also satisfied: This means in this case that the polynomial r E C --+ ( T - T+) divides the polynomials T -+ c(~1 + T7/1) and r -+ c(~2 -4- r7/2) only if the constant c vanishes. ~Tt
__
124
[Ch. 2
Inequalities of Korn's type on surfaces
It then follows from Agmon, Douglis & Nirenberg [1964, Thm. 10.5] that, if-y is of class C3 and the coefficients of the uniformly elliptic system 3'a~(~/) - 0 are in the space C2(~), any solution r / E Hi(w) x Hi(w) • L2(w) of 3'a;3(r/) - 0 in w together with, e.g., 7/i = 0 on 3' is in the space HJ(w) x H J ( w ) xH2(w). The assertion then follows from the continuous imbeddings Hm(w) r cm-2(~), m > 2. (iii) Local uniqueness of the solution of "7a/3(~/) = 0 in w and ~la = 0 on 7 ". The assumed eUipticity of the surface S shows that there exists a constant c > 0 such that [btx(y)l _> c for all y E ~. Hence the unknown ~/3 may be eliminated by means of the equation 7ii (~/) - 0 (naturally, ~73 could be likewise eliminated by means of the equation 722(~/) = 0). This elimination shows that 1 Oll and that yi and 7/2 are solutions of the "reduced" system
b12 ( tr - 2 ~lt 0i 7/i + 027/i + 0i ~/2 - 2 ri2 - bi2 r~ri )7/~ -- O i n w , b22 01~1 + 02?72- (kr ~ 2 - b22r~ i ) ~ bii
-
0
in w,
~i -- O o n T , r/2 - 0 0 n T . Since the coefficients of this reduced system are analytic in w' and since the boundary 7 is of class C3 and is not a characteristic curve for this system, as is easily verified by using again the assumed ellipticity of the surface S, Holmgren's uniqueness theorem (see, e.g., Courant & Hilbert [1962, p. 238], Bers, John & Schechter [1964, p. 47], or Dautray & Lions [1984, Chap. 5, Sect. 1]) shows that "locally', i.e., in a small enough open neighborhood ~v C w' of any point of 7, (~/i, 7/2) = (0, 0) is the unique solution to this reduced system in ci(5~) x ci(&). Recalling that any solution T / - (7/i) of the "full" system is such that ~/a E c i ( ~ ) by (ii), we have thus shown: Any point of 7 possesses an open neighborhood ~v C w ~ such that the only solution 9/ - (7/i) E Hi(w) • Hi(w) x L2(w) of the "full" system "7a;3(~/) - 0 in w and Tla = O on T" is ~l = O in (v N-~.
Sect. 2.7]
Inequality of Korn's type on an elliptic surface
125
(iv) Global uniqueness of the solution of "Tab(r/) = 0 in w and 71a = 0 on 7 ' " By a theorem of Morrey & Nirenberg [1957], any solution of a u n i f o r m l y elliptic system whose coefficients are analytic in w is analytic in w. Since any solution ~/ = (7/i) of "Ta~(~/) = 0 in w and 7/a = 0 on 7" vanishes in a neighborhood of 7 by (iii) and ~ / = 0 is an analytic solution, the analytic continuation theorem for analytic functions of several variables (see, e.g., Dieudonn6 [1968, Thm. 9.4.2]) thus shows that r / = 0 is the only solution, m Several comments are in order about this theorem and its proof." As expected, the infinitesimal rigid displacement lemma is an "intrinsic" property of an elliptic surface" It also holds if the same surface is equipped with another system of curvilinear coordinates; cf. Ex. 2.11. The regularity theorem of Agmon, Douglis & Nirenberg [1964] used in part (ii) requires that only one boundary condition be added to the partial differential equations q'a~(~/) - 0 in w, but this boundary condition must be imposed on the entire boundary q,. Hence this analysis precludes the consideration of boundary conditions such as ~Ta - 0 on q'0 with length ~o < length V. A particularly interesting discussion about Holmgren's theorem and uniformly elliptic systems, especially adapted to linear shell theory, is given in Sanchez-Hubert & Sanchez-Palencia [1997, Chaps. 2 and 3]. We already mentioned that the regularity assumptions "8 analytic in w' D ~ and 7 of class C3'' can be substantially weakened. For instance, Ciarlet & Lods [1996a] assumed instead "0 E gs(~; E3) and 0' of class C4''. The analog of part (ii) in this case shows that the functions 7/1 and r/2 are in C2(~), again by resorting to the regularity theorem of Agmon, Douglis & Nirenberg [1964]; the "local uniqueness" of the solution 7/~ and r/2 of the reduced system (part (iii)) is then obtained by combining results of Carleman [1938] and Calder6n [1958] (by contrast with Holmgren's theorem, these results do not require the assumption of analyticity of the coefficients); finally, the "global uniqueness" (part (iv)) is obtained through the unique continuation theorem of Aronszajn [1956] applied to an elliptic equation satisfied by a single unknown, the auxiliary function (01~/2 - 0 2 r / l ) (also introduced in a different context by Bernadou, Ciarlet & Miara [1994]).
126
Inequalities of Korn's
type on
[Ch. 2
surfaces
The final word in this direction seems to have been achieved by Lods & Mardare [1998a] who were able to weaken the assumptions to those stated in Thm. 2.7-2, i.e., to "0 E C2'1(~; E 3) and V is Lipschitz-continuous". To this end, they notably made use of the unique continuation theorem of Hhrmander [1983]. In some special cases, Thm. 2.7-2 is easier to establish. For instance, an elegant and short proof, due to Brezzi [1994], applies to the special case where O(yl, y2) (Yl, Y2, f ( Y l , Y2)) (Ex. 2.12); as noted by Ciarlet & Lods [1996a], a portion of a sphere is likewise amenable to a simple proof (Ex. 2.13). It seems, however, that under the minimal regularity assumption "0 E C2(~; s all that can be proved is the finite-dimensionality of the space -
{,
=
e
•
-
•
0
This finite-dimensionality was established by Ciarlet & Lods [1996a], who reduced this issue to showing that the pair (YI, ~/2) belongs to the kernel of an operator of the form ( I - T), with T compact. As noted by Geymonat & Sanchez-Palencia [1991], this issue can also be resolved by resorting to Agmon, Douglis & Nirenberg [1964] or Geymonat [1965], under the assumptions "0 E C2(~; E 3) and ~' of class C 1". We are now in a position to prove the main result of this section, due to Ciarlet & Lods [1996a] and Ciarlet & Sanchez-Palencia [1996]; special mention must also be made of the early existence and uniqueness theorem for elliptic surfaces of Destuynder [1985, Thms. 6.1 and 6.5], obtained under the additional assumptions that the surface S can be covered by a single system of lines of curvature (Sect. 2.2) and that the C~ of the corresponding Christoffel symbols of S are small enough. In particular, Thm. 2.7-3 yields the ezistence and uniqueness of the solution to the two-dimensional equations of a linearly elastic membrane shell, as explained at the beginning of this section. The subscript " M " is appended to the constant as a reminder of this property. T h e o r e m 2.7-3 ( i n e q u a l i t y of K o r n ' s t y p e o n an elliptic surface). Let w be a domain in R 2 and let 0 E ~2,1(~; ~3) be an injective mapping such that the two vectors aa = OaO are linearly
Sect.2.7]
127
Inequality of Korn's type on an elliptic surface
independent at all points of-~ and such that the surface S elliptic. Given 0 - (m) ~ Hi(W) • Hi(W) • L2(w), let
0(-~) is
1
denote the covariant components of the linearized change of metric tensor associated with the displacement field rliai of the surface S. Then there ezists a constant CM = CM(W, O) such that
{
} 1/2 _< ,:M
I1oll Ol
for a11 ~ -
2 }1/2
~ I~(,)1o,,.,, a,fl
(~7i) C V M ( w ) " - H~(w) x H~(w)
x
L2(w).
Proof. (i) By the second inequality of Korn's type "without boundary conditions" on a general surface (Thm. 2.7-1), there exists a constant co such that
II,II~"c,,,)•215
:-
2 }I/2
I1~,~11~,,.,,+ I~lo,,,,
~
i
a,fl
for a11 U E Hi(w) • Hi(w) • L2(w) ~ VM(W). Hence it suffices to show that there exists a constant c such that
{
2}1/2
I~,1o,,., i
{
-< ,: ~ I~,~(.)lg,,,.,
}1/2 fo~ all n e VM(W).
a,/3
O0 (ii) If the last inequality is false, there exists a sequence (r/k)k_l of functions r/k --(y/k) e VM(w) such that
I Y~ 1~/~]0,~ 2) 1/2-
i
1 for all k and lim k ~ o o
IZ
211/2
lTat3(r/k)[0, ~
-0.
a,/3
In particular then, the sequence (r/k)~~ is bounded with respect to the norm [[ 9[[H~(~)•215 thanks again to the second inequality of Korn's type of Thm. 2.7-1. Since any bounded sequence in
128
[Ch. 2
Inequalities of Korn's type on surfaces
a Hilbert space contains a weakly convergent subsequence (see, e.g., Vol. I, Thm. 7.1-4), there exists a subsequence (~ll)~t and an element =
e
that
r/la ~ r/a in Hi(w) and rlla -+ Ya in L2(w), Y~ ~ Y3 in L2(w), where --~ and --+ denote weak and strong convergences (the compact imbedding H i ( w ) 9 L2(w) is also used here; see, e.g., Vol. I, Whm. 6.1-5).
(iii) Naturally, the difficulty rests with the subsequence (~/~)oo I:I which converges only weakly in L2(w). Our recourse for showing that it in fact strongly converges in L2(w) will be the assumed ellipticity of the surface S (see (iv)), but first, we prove that r / = (r/i) = 0. To this end, we simply note that yta ~ r/a in Hi(w) and yta ~ r/3 in L2(w) imply that 7~(r/l) ~ "ya;s(r/) in L2(w) on the one hand; since 7a~(n l) -+ 0 in L2(w)
on the other, we conclude that ~'afJ(~) : 0. Hence ~7 : 0 by Thm. 2.7-2. (iv) We next show that r]~ -+ 0 in L2(w). The strong convergences 7afj(~ l) -+ 0 in L2(w) and ,la --+ 0 in L2(w) combined with the definition of the functions 7af3(Y) imply the following strong convergences:
017"}~ -- b11,/3 -- {')'11(, I) d- F~I,~}
--+ 0 in L2(w), t --+ 0 in L 2 (w), 02~}I -Jr- 01./2 -- 2b12~7/3 -- {2")'12(r//) -4- 2F12r/,)
02?}/2 --b22,I -- {Q'22(, l) +
-+
0 i.
Since the function bll C C~ does not vanish in ~ by the assumed ellipticity of the surface S, we can eliminate 7/13between the first and second, and between the first and third, relations; this elimination yields"
{ 02 ?']i -~- 017}/2 -- 2
b12
522 1,02._ -- ~
0tr/~ } -+ 0 in L2(w)
Sect. 2.7]
129
Inequality of Korn's type on an elliptic surface
Multiplying the first relation by 02~?~and the second by 0z y~, then integrating over w, we get b12
{
I
l}
77102771
-
--~ 0,
~
since each sequence (Oa~7[)~z is bounded in L2(w) (each sequence even weakly converges to 0 in L2(w)). Subtracting the last two relations and using the relation f~ 02~?z01 ~72dy - f ~ 01~7102~72dy satisfied by all (~i,~72) E H~(w) x H~(w), we thus obtain + (511)2 (511522 - (512)
2
--~ 0.
Consequently, 0 1 ~ ~ 0 in L2(w),
since b11b22 -- (b12) 2 -- det(ba~) E C~ assumed ellipticity of S. Hence
does not vanish in ~ by the
(v) The relations ~ --~ 0 in L2(w) established in parts (iii) and (iv) thus contradict the relations { ~ , 1~?~12,~}1/2 - 1 for all I, and the proof is complete, m Finally, we state without proof a noteworthy result of Slicaru [1997, 1998], who showed that the sufficient conditions of Thm. 2.7-3 for obtaining an inequality of Korn's type (of the form given in this theorem) are also necessary! T h e o r e m 2.7-4. Let w be a domain in I~2, let 70 be a d 7measurable subset of V - Ow, and let 8 E C2(~; C3) be an injective mapping such that the two vectors a a -- Oa~ are linearly independent at all points of-~. Assume that there exists a constant c such that
z/2
}1/2 a,/3
130
Inequalities of Korn's type on surfaces
[Ch. 2
for all r / = (r/i) e V(w; 3'0), where
V(w; 70)"-- {T/-- (~7i) E H i ( w ) • HI(w) • L2(w); r/a -- 0 on ")'0} 9 Then "To = 7 and the surface S = 0(-~) is elliptic.
2.8 b.
m
COMPLEMENT: RECOVERY OF A SURFACE FROM ITS METRIC AND CURVATURE TENSOR FIELDS
The content of this section should be advantageously compared to that of Sect. 1.9. Let w be an open subset in ]R2 and let 8 : w -+ t; 3 be a thrice continuously differentiable mapping such that the two vectors aa = Oa8 are linearly independent everywhere in w. As before, let al Aa2 a3 = ]al A a21' let the three vectors a i be defined by the relations a J . a i - 5~, let aa[3 - aa "a/3 and ba[3 - a3" O/3aa, and let the Christoffel symbols be defined by r~a/3 = a ~ . O/3aa. In addition, let the Christoffel symbols of the first kind F a i r be defined by
r o , . := aarr~. In this context, the functions r ~ at3 are also called the Christoffel symbols of the second kind. Then it can be shown that the Christoffel symbols and the covariant components of the curvature tensor satisfy the following compatibility conditions (Ex. 2.14): 0~r.~
- 0~ro~ + r"~.r~, - r~.r~.~ O ~ b ~ - O~bo. + r ~ b ~
- r~.b~
- b . . b , ~ - b . ~ b , ~ i~ ~ , -
0 ~ ~.
These conditions are in effect relations between partial derivatives of the first, second, and third order of the mapping O. They are also relations between the covariant components aa~ and ba/3 of the metric and curvature tensors and their derivatives, since the Christoffel symbols r~t3r may be directly defined in terms of the functions aat3 as (Ex. 2.14): r~
1 - ~(~~
+ o~~
- o~~).
Recovery of a surface
Sect. 2.8 b]
131
R e m a r k a b l y , these necessary conditions are also sufficient for the existence of a m a p p i n g 0 :w C I~2 -+ E 3 whose metric a n d c u r v a t u r e tensor fields are given on w:
T h e o r e m 2.8-1. Let w be a simply connected open subset of R 2, and let there be given a twice continuously differentiable, symmetric, and positive definite matrix field (aa/3) on w and a continuously differentiable symmetric matrix field (ba[3) on w that together satisfy
-
O~ba# - Of3baa + r ~ b ~ :=
:=
- b,~b~r - b,~rb~ in w,
-
1
+
- r~b~
-
0 in w, where
-
:=
Then there exists a mapping O E C3(w; IR3) such that 0 a 0 " 0f30 = aaf3 in w,
0~o A 020 90af3O = bar3 in w. 10~o A 0201 Furthermore, this mapping is unique "up to rigid deformations in I~3 ": This means that any other solution is necessarily of the form y E w --+ c + Q 0 ( y ) , where e is a vector in IR3 and Q is an orthogonal matrix of order three. II For a direct p r o o f of this delicate result, see Klingenberg [1973, Thin. 3.8.8]. Otherwise, Thin. 2.8-1 can be also obtained as a corollary to Thin. 1.9-1; see Ciarlet & L a r s o n n e u r [2000]. T h e first relations constitute one version of the "Theorema egreg i u m " of Gaufl [1828] (Ex. 2.14), while the second ones are equivalent to the "Codazzi-Mainardi identities" (Ex. 2.7).
Inequalities of Korn's type on surfaces
132
[Ch. 2
EXERCISES
2.1. (1) Compute the vectors of the covariant and contravariant bases, the area and length elements, the Christoffel symbols, the covariant and contravariant components of the metric tensor, and the covariant and mixed components of the curvature tensor corresponding to a portion of a circular cylinder and to a portion of a torus equipped with the curvilinear coordinates shown in Fig. 2.1-2. (2) Carry out the same computations for a portion of a sphere equipped with the three kinds of curvilinear coordinates shown in Fig. 2.1-3. (3) In each case considered in (2), verify that the radius of the sphere indeed satisfies the relation established in Thm. 2.2-1. (4) Carry out the same computations as in (1) for a portion of a hyperbolic paraboloid represented by a mapping of the form
(~, y)-~ (~, y,
h
y),
where a, b, and h are three given lengths. 2.2. Show that the relation dS(~l) = v/a(Y)dy providing the area element at ~) = 8(y) e S = 8(~) (Thm. 2.1-1) can be recovered from the relation (Thm. 1.2-1)
aT(~)- ICofVO(x)n(x)ldr(x)
at ~ - O(x) e ~ - o(r),
by specializingthe mapping | to be the canonical extension of the mapping 8 (Thin. 3.1-1). 2.3. The notations and assumptions are as in Sect. 2.1. Let y E ~ be given. Show that the angle a(y) between the coordinate lines passing through the point O(y) satisfies cos a(y)
=
{~ (y)~2~(y))~/~"
2.4. Let the assumptions and notations be as in Thm. 2.2-1. A 1 1 point O(y) E S is called a planar point if R1 (y) - R2(y) = 0. Show that, if all the points of S are planar, S is a portion of a plane. Hint: See, e.g., Stoker [1969, p. 87].
Ezercises
133
2.5. Let the assumptions and notations be as in Thin. 2.2-1 and assume in addition that 0 E C3(~; E3). A point O(y) e S is called an 1 1 umbilical point if Rt (y) -- R2(y) ~ 0. Show that, if all the points of S are umbilical, S is a portion of a sphere. Hint: See, e.g., Stoker [1969, p. 99]. 2.6. Let K : S -+ IR denote the Gaussian curvature of a (smooth enough) surface S. (1) Let S be a sphere; show that fs g(~/)dS(~/) = 47r. (2) Let S be a torus; show that fs g(~/)dS(~/) = O. Remark. These are instances of applications of the Gaufl-Bonnet theorem (Sect. 2.2). 2.7. Let ba~l~ :-- O~ba~ - r [ ~ b ~ - r ~ b ~ denote the first-order covariant derivatives of the curvature tensor, defined here by means of its covariant components. Show that these covariant derivatives satisfy the Codazzi-Mainardi identities
which are themselves equivalent to the relations (Thm. 2.8-1) 0.bo,
- 0,bo~ + r;,b~
- r;~b~,
- 0.
Hint: The proof is analogous to that given in Thm. 2.5-1 for establishing the relations b~la - b~l#. 2.8. Let the assumptions on the mapping O be as in Thm. 2.6-3. (1) Let there be given ~/ - (7}i) e Hi(w) such that 3'a#(~/) - 0 in w. Show that
o~(~ia i)
-
r A o~o,
where
10~
with~tl-s22-O s t 2-_~ ,1 ,
ands2t-
1 ~.
Remark. The vector field ~ is the infinitesimal rotation field, introduced by Vekua [1962], then put to various uses by Bernadou & Ciarlet [1976, Lemma 2.5], Choi [1993], Choi & Sanchez-Palencia [1993]. Its existence under minimal regularity assumptions is established by Blouza & Le Dret [1999, Thm. 6].
134
[Ch. 2
Inequalities of Korn's type on surface8
(2) Let there be given rl - (r/i) E Hi(w) x Hi(w) x H2(w) such that 7aft(r/) -- Pat3(r/) - 0 in w. Show that the vector field r found in (1) is constant in this case and equal to the vector t/found in Thm. 2.6-3 (a). 2.9. Let the assumptions on the mapping 0 be as in Thm. 2.6-3 and let 70 be a relatively open subset of 7 that satisfies length "ro > O. (1) Show that the implication (the analog of Thm. 2.6-3 (b) for another class of boundary conditions):
~i - - 0 on 70,
=~ r / - 0 i n w
"Ya/J(r/) - Pa/3(~/) - 0 in w holds if and only if 0(70) is not a subset of a straight line. Hint: One way to solve this problem is to combine Thm. 2.6-3 (a) with Ex. 2.8. (2) Assuming that the implication of (1) holds, show that an inequality of Korn's type on a general surface analogous to that of Thm. 2.6-4 holds, where the space
v~c(~) := {. - (~) e H ~ ( ~ ) x H~(~) x H~(~); ~ = 0 on 70) replaces the space V g ( w ) found in ibid. The exponent "s" reminds that the space V}c(w ) corresponds to a shell that is simply supported along the curve 0(70), i.e., whose displacement field ~ia i satisfies the two-dimensional boundary conditions of simple support r - 0 on "y0. 2.10. (1) The notations and assumptions are the same as in Thm. 2.6-6. Show that the space V g ( w ) becomes a Hilbert space when it is equipped with the norm --+
]lOii2,~, + ~
10at3~/" a3]0,o,
9
a,/3
(2) Assume in addition that 0 E W3'~176 E3). Show that the mapping r / - (7/i) --+ ~ - 71iaz is an isomorphism between the space
v~(~)
.
-
H~ (~) • g~ (~) • {H ~(~) n H~(~)),
equipped with the product norm r/ --4 { ~ a IIr/a[[2,~ + IIr/sl]2,oJ}1/2, and the space ~'g(W) equipped with the norm of (1). Remark. This observation is due to Blouza & Le Dret [1999, Lemma 4].
Ezercises
135
2.11. Let an elliptic surface be equipped with two different systems of curvilinear coordinates. Show that, if the infinitesimal rigid displacement lemma on an elliptic surface (Thm. 2.7-2) holds for one system, it holds for the other. 2.12. Let w C I~2 be a domain with a boundary of class C2. (1) Let there be given functions Aaf3 E Ct(~) satisfying the following "ellipticity condition": There exists a constant c > 0 such that
Aa~(y)~a~ ~ ~ c ~
I~a[2 for aU y e ~ and all (~a) e R 2.
a
Show that, if X e H~(w) satisfies Aa~Oa~X e L2(w), then X e H2(w). Hint: Indications may be found, e.g., in Brezis [1983, Sect. 9.6]. (2) Let S = 8(~) be an elliptic surface with a mapping 8 of the
fo~m e - (y~, y~) e ~ ~ (y~, y~, f(y~, y~)), where f e C3(~). Show that a field v / - (v/i) e Hi(w) x Ht(w) x L2(w) satisfies 7af~(v/) - 0 in w if and only if there exists g C L2(w) such that 1
~ . z ( n ) . - ~(oz,~. + o.,~z) - g o . z f . (3) Show that, for all 7/a e H~(w) and all X E H~(w)N H2(w),
L
{ell (v/)022X - 2e12(~l)O12X + e22(~l)OllX} dy = O.
(4) Show that, if v/C H~(w) • H~(w) • L2(w) satisfies 7a/3(v/) - 0 in w, then v / - 0 in w. Remark. This result, which is due to Brezzi [1994], thus provides a considerable simplification of the proof of Thm. 2.7-2 for the special class of elliptic surfaces considered in (2). 2.13. Let S be a portion of a sphere with radius R, equipped with stereographic coordinates u and v (Fig. 2.1-3). Given any element
- (~) e H~(~) • H~(~) • L2(~), l~t Xa :-- rla(u2 -4- v 2 -4- R2) 2. Show that the relations 3'a/3(v/) = 0 in w imply that 02X1 -~-
O1X2 =
0 and
02X2 -
01X1 = 0 i n
w,
Inequalities of Korn's type on surfaces
136
[Ch. 2
and conclude in this fashion that 7/a = 0 in w, thus providing in this case a quick proof of Thm. 2.7-2 (this observation is found in Ciarlet & Lods [1996a, Sect. 3]). 2 914 9 (1) The Christoffel symbols are defined as r ctl3 ~ - a ~ . O~aa in the text (Thm. 2.3-1), i.e., by means of the vectors of the covariant and contravariant bases of the tangent plane. It is remarkable that they can be also defined solely in terms of the metric tensor of the surface. More precisely, show that r~
- ~ro~,
1 where r o ~ . - ~ ( 0 ~ ~
+ 0~~
- 0~a~).
(2) Show that the Christoffel symbols satisfy the compatibility conditions o~ro~ - o~ro~ + r"~r~.
- r~r~
0~boz - 0 z b ~
- bo~b~ - bo~b~ i~ ~,
+ rxzb,~ - r~b~z
- 0 in
w.
(3) ~ t R~
:= 0 ~ r ~
- o~ro~ + r.~r~.
- r~r~.
Show that R~
= -R~~
Raj311 = RaD22
= -R~~ -- Rlltrr
-
= R~, R22~rr --0,
R1212 = R2121 =-R1221 =-R2112, and that, consequently, the compatibility conditions found in (2) in fact reduce to R 1 2 1 2 - - bllb22 - - ( b 1 2 ) 2 . (4) Deduce from (3) that the Gaussian curvature K (Sect. 2.2) satisfies K - (alia22 - ( a 1 2 ) 2 ) -1 (02rl12 - 01r122 + r ~"r ~ .
- r ~ l r 22~).
This is the astonishing T h e o r e m a egregium of Gaul] [1828]: The Gaussian curvature depends only on the knowledge of the f i r s t fundamental form of a surface, i.e., only on its metric! R e m a r k . The functions Radar are the covariant components of the R i e m a n n c u r v a t u r e t e n s o r of the surface.
CHAPTER 3 ASYMPTOTIC ANALYSIS SHELLS: PRELIMINARIES
OF LINEARLY ELASTIC AND OUTLINE
INTRODUCTION
The purpose of this chapter is twofold: First and foremost, it gathers the fundamental preliminaries needed in the remainder of this volume for carrying out the asymptotic analysis of all the kinds of linearly elastic shells that we shall encounter: After ad hoc "scalings" of the unknowns (the covariant components of the displacement field) and ad hoc "asymptotic" assumptions on the data (the Lam6 constants and applied force densities) have been made, we transform the problem of a linearly elastic clamped shell with thickness 2e > 0 into a "scaled problem", defined over a domain that is independent ore. Second, it justifies the two fundamentally distinct classes of twodimensional equations that mathematically model linearly elastic shells~ through a formal asymptotic analysis of the solution of the scaled problems, where e is considered as the "small" parameter. Since this formal analysis will be later replaced by the more satisfactory (but also considerably more delicate) convergence theorems of the next three chapters~ this chapter thus essentially serves a pedagogical purpose. More specifically, let w be a domain in I~2 with boundary ~/and let ~ 9~ --~ 1~3 be a smooth enough injective mapping such that the two vectors aa : 0aS are linearly independent at all points in ~. Consider a shell with middle surface S = 8(~) and thickness 2e > 0, i.e., a body whose reference configuration is the set O ( ~ ) , where :,.,.,x]
-
O(x ~) - e ( y ) + xga3(y) for all x e - (y, xg) - (x~) e ~ , and a3(y) is the unit outer vector normal to S at #(y) constructed as in Sect. 2.2.
Asymptotic analysis of linearly elastic shells: Outline
138
[Ch. 3
The shell is subjected to applied body forces with contravariant components f/' ~ 9f~e -+ I~ and is subjected to a boundary condition of place along a portion (9(70 • I - e , el) of its lateral face (9(7 • I - e , e]), where 70 C 7 and length 70 > 0. Applied surface forces acting on the upper and lower faces O(w • {e}) and O(w x { - e } ) may be also considered; for simplicity, we assume in this introduction t h a t they vanish. Let )~e a n d / z e denote the Lam6 constants of the elastic material constituting the shell. T h e n in linearized elasticity, the unknown u ~ - (u~), where ui -+ R are the covariant components of the displacement field of the points of the shell, satisfies the following three-dimensional equations in terms of the "natural" curvilinear coordinates x i of the shell (we quickly review in Sect. 3.1 the main features of this variational problem, otherwise extensively studied in Chap. 1):
~," e v ( n ' ) -
{,," - ( W ) e I-I~(n'); ,, e = 0 on 3'0 x [ - e , e]},
fft AiJkl'ee~lll(ue)e~llj(Ve)~ dx e c
--
f n fi, eV ie v / ~ dz ~ for all v e E V(f~6), g
where
Aijkl, e = Ae gij, egkl, e + #~ (gik, egjl, e + git, egjk, e) , e~ (v ~
1
Ve
p, e
illJ -- O~v~ - r i j
V~.
In Sect. 3.2, we transform this problem into an equivalent problem, but now posed over the set ft = w x ] - 1, 1[, which is independent of e. This transformation relies in a crucial way on appropriate scalo ings of the unknowns u ie and assumptions on the data )t e, #~ and fz, ~. More specifically, we define the scaled unknown u(e) = (ui(e)) by letting
=~(~)
-
= ~ ( ~ ) ( ~ ) fo~ a n ~
-
~
e
~,
where lre(Zl, z2, x3) = (Zl, x2, ez3). We then assume t h a t there exist constants A > O, # > 0 and functions fi independent of e such that ~e = % and #e = #, i~,~(~~) - ~y~(~) fo~ an ~ - ~
E n~,
Introduction
139
where the exponent p is unspecified at this stage. It is found in this fashion that the scaled unknown u(e) satisfies a variational problem of the form (Thm. 3.2-1):
u(e) E V(ft) -- {v -- (vi) e H l ( f t ) ; v - 0 on 70 • [ - 1 , 1]}, f
AiJkl(e)eklll(e; u(e))eillj(e; v)v/g(e)dz
-- ~P ffl fivi v/g(e) dx for all v e V(ft), where, for any vector field v = (vi) E H l ( f t ) , the scaled linearized strains eillJ(S; v) -- ejlli(e; v) E L2(f~) are defined by:
1
-
eatlz(e;v) e3113(e; v )
p
-
l(103va+Oav3)
~ e u
--
+
-
e103v 3.
The specific form of this variational problem suggests that we use the method of formal asymptotic ezpansions, i.e., we let U ( g ) -- U 0 -~- g U 1 -~- S2U 2 -+- g3U3 q- g4U4 -+- " ' " , w i t h
u ~ ~ 0,
in the variational equations, and then we equate to zero the factors of the successive powers of e found in the resulting equations until the leading t e r m u ~ can be fully identified, without imposing any restriction on the applied force densities. In so doing, we are led after a series of delicate computations (which themselves rely on various "geometrical" and "mechanical" preliminaries established in Sect. 3.3) to the conclusion that the leading t e r m u ~ satisfies a two-dimensional problem that falls in one of the following two categories: Assume first that the space Vo(w) "- {W E Hi(w); r/- 0 on 7o, 7a/3(W) - 0 in w}
contains only the function 17 = O. Then the functions fi, e m u s t be of the form
fi'e(xe)
-- fi'O(x) for all x ~ - 7 r ~ x C ~Y,
140
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
where the functions f i , 0 a r e independent of e (i.e., the exponent p must be set equal to 0 in this case). Furthermore, the leading
term u ~ 9 -+ I~3 is independent of the transverse variable x3 and ~o _- 12f1-1 u~ dx3 should satisfy the following (scaled) two-dimensional variational problem of a linearly elastic "membrane" shell (see Thm. 3.4-2; this problem is in fact provisional, since the function spaces and boundary conditions appearing in the definition of the space V(w) definitely need to be modified; see Chaps. 4 and 5, or Sect. 3.5)" r
E V(w)-
{ r / - (r/i)E Hi(w); r / - 0 on 70},
~ aa~'TrT~r(C~~ ~o~ ~n n -
f~ { f : .fi'~ dx3} r l i ~ d y
(,i) e v ( ~ ) , where 1
aa[3~r _
4)~tt aa[3a~r + 2#(aa~aS3r + aar a~), A+2/z
a - det(a~s), and a a~ - a ~. a Is are the contravariant components of the metric tensor of S. Assume next that the space V0(w) contains nonzero functions *7. Then the functions fz, e must be of the form ~
i~.'(~ ") - ~2f~.~(~) fo~ an ~ - ~'~ E n ' , where the functions fi,2 are independent of e (i.e., the exponent p must be set equal to 2 in this case). Furthermore, the leading term
u ~ 9-~ -+ I~3 is independent of the transverse variable x3 and the field r 1 f ~l u~ dx3 satisfies the iollowing (scaled) two-dimensional variational problem o] a linearly elastic "flexural" shell (Thin. 3.4-3)" r e VF(6O):-- { n - (TIi) E Hi(w) x Hi(w) x H2(60); r/i -- Ou?13 -- 0 o n "70, "~a[3(O) -- 0 ill 60},
1 r 3 L aa~vTp'7~'( ) p a / 3 ( r l ) ~ d y - f ~ { f _ : for all n -
]i'2dxa}~Tiv~dY
(ha) E V v ( ~ ) , where
P . , (n) -
0.~ 03 - r2~ O. ~3 - b~ b~, 03 + b~ (0, ~ - r ~ ~. ) + ~ (0o0. - r ~ , ~ )
+ (0o~ + r~~
- r~~),~,
Sect. 3.1]
The three-dimensional equations
141
and the functions Van(r/) and a a ~ r are the same as above. Although the method of formal asymptotic expansions thus admirably serves the purpose of clearly identifying two fundamental classes of possible limit equations, it needs to be substantiated by a convergence analysis. This is why we conclude this chapter by a quick review (Sect. 3.5) of the convergence theorems that will be established in the next three chapters, showing that the scaled unknown u(s) has indeed a limit as --> 0 in an ad hoc functional space, which in each case depends on the geometry of the middle surface S and on the subset V0 of-y. This "refined" analysis also illustrates the limits of the otherwise quite efficient formal approach, as it shows that the "membrane" shells found above need themselves to be subdivided into two subcategories. 3.1.
THE THREE-DIMENSIONAL LINEARLY ELASTIC SHELL
EQUATIONS
OF A
To begin with, we briefly review the main notations, definitions, and results, mostly from three-dimensional linearized elasticity in curvilinear coordinates and from the differential geometry of surfaces, that will be needed in the sequel, and we provide ad hoc cross-references to Chaps. 1 and 2, where these notions have been expounded in detail. Note that the real three-dimensional affine Euclidean space formerly denoted E 3 in Chaps. i and 2 will be henceforth denoted I~3. Greek indices and exponents (except s) belong to the set {1, 2}, Latin indices and exponents (except when otherwise indicated, as when they are used to index sequences) belong to the set {1, 2, 3}, and the summation convention with respect to repeated indices and exponents is systematically used. Symbols such as 6~ or 6~ designate the Kronecker's symbol. The Euclidean scalar product and the exterior product of a, 5 E IR3 are noted 6 . 5 and a A 5; the Euclidean norm of a E IR8 is noted lal. A domain ~ in R n is a bounded, open, connected subset of I~n with a Lipschitz-continuous boundary 0~, the set ~ being locally on one side of 0~. For each integer m > 1, H m ( ~ ) and H ~ ( ~ ) denote the usual Sobolev spaces of real-valued functions. Boldface letters denote vector-valued functions and their associated function spaces. The norm in L2(~) or L2(~) is noted I 9]0,~ and the norm in Hm(f~)
142
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
or ttm(~2), m > 1, is noted II. IIm,~ (s~ct. 1.7). Let w be a domain in I~2 with boundary -),. Let y - (ya) denote a generic point in the set ~ and let 0a := O/Oya. Let 0 E C2(~; I~3) be an injective mapping such that the two vectors
a.(y) := o.o(y) are linearly independent at all points y E -~. These two vectors form the covariant basis of the tangent plane to the surface
s := o(~) at the point 0(y) and the two vectors aS(y) of the tangent plane at 0(y) defined by the relations
form its contravariant basis. Also let
a~(y) A a~(y) a s ( y ) - a3(y):= la~(y)A a2(y)l" Then I~(y)l- z, the vector as(y) is normal to S at the point O(y), and the three vectors ai(y) form the contravariant basis at 0(y); cf. Fig. 2.3-1. Note that (Yl, Y2) constitutes a system of curvilinear coordinates (Sect. 2.1) for describing the surface S. The covariant and contravariant components aaf3 and a afJ of the metric tensor, also called the first fundamental form (Sect. 2.1), the covariant and mixed components ba~ and ~ of the curvature tensor, also called the second fundamental form (Sect. 2.2), and the Christoffel symbols r ~ (Sect. 2.3), of the surface S are then defined by letting (whenever no confusion should arise, we henceforth drop the explicit dependence on the variable y E ~): aao := a a 9af3,
a af3 "= a a 9a f3,
bafj := a 3" O~aa ,
r ~ :=
bfla "- a[3~baa ,
a ~ 9O ~ a ~ .
Note the symmetries: ao, = ~,o,
~~
-- ~ ' ~
bo, -- b,o,
r~, - r~o.
Sect. 3.1]
The three-dimensional equations
143
The area element along S is x/rd dy, where a := det (aa~l). All the functions aa~l, a a~, ba~, bfla,r -a~, and a are thus at least continuous over the set ~. Let V0 denote a dT-measurable subset of the b o u n d a r y 3, of w satisfying
length Vo > O. For each e > 0, we define the sets
a~.- ~• r t "= ~ • {~},
r'
~, ~[,
:= ~ x {-~},
r~ := ~o • [-~, ~].
Let x ~ - (x[) denote a generic point in the set ~ and let 0~ := O/Ox~; hence x a - y a and O~ - Oa. Consider an e l a s t i c s h e l l with m i d d l e s u r f a c e S - 0 ( ~ ) and t h i c k n e s s 2e > O, i.e., an elastic b o d y whose reference configuration consists of all points within a distance <_ e from S (Fig. 1.2-1). In other words, the reference configuration of the shell is the image
0(-~ ~) C I~3 of the set-~6 C R 3 through a mapping O given by
9
--~ ~
e ( ~ ~) := e(y) + ~ i ~ s ( y ) fo~ an ~ - (y, ~i) - (y~, y2, ~i) e ~ ' .
Remark. Shells whose middle surface S has "no b o u n d a r y " , such as an ellipsoid or a torus, are thus not covered by the present analysis, which assumes t h a t S - 8(~) can be described by a single injective m a p p i n g 8 9~ ~ IR3. References for such shells are provided in Sects. 4.6 and 5.8. II The next theorem is essential for describing the geometry of such a shell: It shows t h a t , if the injective mapping 8 9 -+ I~a is smooth
enough, the mapping 0 enough.
9 e --3 I~a is also injective for e > 0 small
144
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
T h e o r e m 3.1-1. Let w be a domain in I~2, let 8 E C2(~; R 3) be an injective mapping such that the two vectors aa - OaO are linearly al A a2 independent at all points o]-~, and let a 3 -- l a 1 A a21" Then there exists eo > 0 such that the mapping 0 defined by
9-~o -4 R 3
m
O(y, x3) "- O(y) Jr xaa3(y) for a11 (y, x3) E l~0, where ~o " - , , , x ] - ~o, ~o[, is a Cl-diffeomorphism from ~0 onto O(~0) and det(gi, g2, g3) > 0 in -~0, where gi := 0 i 0 . The mapping 0 9-~0 -4 I~3 is called the c a n o n i c a l e x t e n s i o n of the mapping O"-~ -4 ~3. Pro@ The assumed regularity on the mapping 0 implies that O E c i ( ~ x [-e, el; I~3) for any e > 0. The relations ~la -- Oa ~) -- a a ~- x3cgaa3 a n d g3 -" 0 3 0
- a3
imply that det(gl, g2, g3)lx3=o -- det(al, a2, a3) (the last equality is proved in part (i) of the proof of Thm. 2.5-1). Hence it is clear that det(gi, g2, g3) :> 0 on ~ • [-e, e] if e > 0 is small enough. Hence the implicit function theorem can be applied: It shows that, locally, the mapping O is a Ci-diffeomorphism: Given any y E ~, there exist a neighborhood U(y) of y in ~ and e(y) > 0 such that O is a Ci-diffeomorphism from the set U(y) x [-e(y), e(y)] onto O ( U ( y ) • [-e(y), s(y)]). See, e.g., Schwartz [1992, Chap. 3, Sect. 8]; the proof of the implicit function theorem, which is almost invariably given for functions defined over open sets, can be easily extended to functions defined over closures of domains, such as the sets ~ x [-e, e] (in particular, the definition of differentiable functions over a domain poses no difficulty; see Stein [1970]). To establish that the mapping O 9~ • I-e, e] -+ I~3 is injective provided e > 0 is small enough, we proceed by contradiction: If this property is false, there exist en > 0, (yn, x ~ ) , and (9 n, ~ ) such that
~-~0, y~~, 9"~, I~l<~,, I~l<_~n, (yn, x~) ~ (9n, $~) and O(y n, x~) - | n, $~).
Sect. 3.1]
The three-dimensional equations
145
Since the set ~ is compact, there exist y E ~, 9 E ~, and a subsequence, still indexed by n for convenience, such that yn__+y, as n --+ cr
9n - + ~ ,
x~O,
&~--+0
Hence O(y)-
lira O(y n, x ~ ) n---4 o o
lira |
& ~ ) - 0(9),
n--4 oo
by the continuity of the mapping @ and thus y - ~ since the mapping 8 is injective by assumption. But these properties contradict the local injectivity (established supra) of the mapping @. Hence there exists e0 > 0 such that O is injective on the set ~0 - w • [-s0, e0]. Showing that @-1 . @(~0) --+ ~0 is also of class C1 is akin to proving the differentiability of an implicit function (seer e.g., Schwartz [1992, Chap. 3, Sect. 8]). Again, such a proof is usually given for functions defined over open sets, but it can be easily extended to functions defined over closures of domains. II Thm. 3.1-1 thus shows that, if so > 0 is small enough, (yl, y2, x~) constitutes a bona fide system of "natural" curvilinear coordinates (Sect. 1.2) for describing the reference configuration @(~e) of the shell and the physical problem is meaningful as the set @(~e) does not "interpenetrate itself". The curvilinear coordinate x~ E [-~, ~] is called the t r a n s v e r s e variable. The same theorem also shows that, again for s > 0 small enough, the three vectors .
-
form the covariant basis (of the tangent space, here R 3, to the manifold O ( ~ ) ) at the point @(xe), and the three vectors gi'e(xe) defined by form the contravariant We also define the and gij, ~ of the metric riP~~ (Sect. 1.4) of the dependence on x~) 9 gij
basis at @(x ~) (Sect. 1.2). g covariant and contravariant components gij tensor (Sect. 1.2) and the Christoffel symbols set O ( ~ ~) by letting (we omit the explicit
gi " gj , .
-
gt' .
, gj, e
146
A s y m p t o t i c analysis of linearly elastic shells: Outline
[Ch. 3
Note the symmetries: e
gij
__ g ;
i,
g
ij, e
=
gji, e
r~," r;,"i " ij --
,
The volume element in the set e ( ~ e) is V ~ dxe, where
g' := det(g,~). We assume that the material constituting the shell is homogeneous and isotropic and that O ( ~ e) is a natural state, so that the material is characterized by its two Lamd constants Ae > 0 and #e > 0. The u n k n o w n of the problem is the vector field u e - (u~) 9~e --+ I~3 where the three functions u ie . ~ e -+ IR are the covariant components of the displacement field Ue.o ,o i'e 9-~e -+ i~ 3 that the shell undergoes under the influence of applied forces. This means that u[ ( x e )9i, e ( xe ) is the displacement of the point O(xe); see Fig. 3.1-1. Finally, we assume that the shell is subjected to the boundary condition of place u e - 0 on r~, i.e., that the displacement field vanishes along the portion o ( r ~ ) of its l a t e r a l face 0 ( 7 x [-~, ~]). A w o r d of c a u t i o n . We refrain from calling "boundary conditions of clamping" the boundary conditions u~ - 0 on r~, in order to avoid any confusion with the "two-dimensional boundary conditions of clamping", which apply only to some specific classes of twodimensional shell equations (Chaps. 6 and 10). I In linearized elasticity, the unknown u* = (u~) then satisfies the following t h r e e - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 79(f~*) of a line a r l y elastic shell in c u r v i l i n e a r c o o r d i n a t e s , i.e., expressed in terms of the "natural" curvilinear coordinates x i of the shell (Thm.
1.3-1): ~
e v ( n ~) . - { ~ ' - (~f) e I ~ ( n ' ) ;
. ' - o o . r~},
ta AiJ kl, * , , e~llt(ue)eillj(ve)v/~ dx e -__
/o
f$,e
'
v ie v ~ d x ,
+
Z
~ur'_ for
hZ, ev~e ~ / ~ d r ~
all v ~ E V(f~e),
Sect. 3.1]
The three-dimensional equations
147
g
2,s
e(r:)--
Fig. 3.1-1: The three-dimensional shell problem. Let fl" = w x ] - e, e[. The set O(~e), where lD(y, z~) = O ( y ) + z~as(y) for all z r = (y, z~) E ~e, is the reference configuration of a shell, with thickness 2e and middle surface S = 0(~), which is subjected to a boundary condition of place along the portion (D(I~) of its lateral face (i.e., the displacement vanishes on O(P~)), where P~ = V0 • [-e, s] and Vo C 7 = Ow. The shell is subjected to applied body forces in its interior lD(fl ~) and to applied surface forces on its upper and lower faces lD(I~_) and ID(PL) where F~: = ~ • {• Under the influence of these forces, a point lD(z ") undergoes a displacement u~(z')gi"(~.'), where the three vectors gi'c(z~) form the contravariant basis at lD(z:). The unknowns of the problem are the three covariant components ui : -+ R of the displacement field ui " --~ of the points of O ( ~ ' ) , which thus satisfy the boundary conditions u~ = 0 on I~. The objective consists in finding ad hoc conditions affording the replacement of this three-dimensional problem by a "two-dimensional problem posed over the middle surface S" if e is "small enough"; see Fig. 3.1-2. Note that, for the sake of visual clarity, the thickness is overly exaggerated.
where
AiJkt,~ .= )dgij,~ gkl, e + #e(gik, e git, e + git, e gjk, e) designate the contravariant components of the three-dimensional elas-
148
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
ticity tensor,
.:
1
~
~ e
+ o,,j)
p,e
- r,j
e
designate the linearized strains in curvilinear coordinates associated with an arbitrary displacement field-e_i,e of the set O ( ~ ~) (Thm. 1.5-1), fi,~ E L2(f~ e) and h i'e E denote the contravariant components of the applied body and surface force densities, respectively applied to the interior @(f~e) of the shell and to its " u p p e r " and "lower" faces o(r~_) and lD(re__), and d r e designates the area element along 0 ~ e. We thus assume that there are no surface forces applied to the portion O ( ( ~ - 70) • I-e, el) of the lateral face of the shell. We record in passing the symmetries AiJkl,
e _
Ajikl,
e _
Aklij,
e
and the relations (satisfied because the mapping O is of the special form given earlier) r3,e a3 - F ~ e - 0 and A a/3~3, ~ = Aa333 ' e = 0 in ~e . The above three-dimensional equations of a linearly elastic shell have one and only one solution for e > 0 small enough. As shown in Thm. 1.8-2, this existence and uniqueness result relies on Korn's inequality in curvilinear coordinates (Thm. 1.7-4), combined with the uniform positive definiteness of the elasticity tensor (Thm. 1.8-1). Our objective in the rest of this chapter consists in showing that, if e > 0 is small enough and the data are of an appropriate order with respect to e, the above three-dimensional problems are "asymptotically equivalent" to a "two-dimensional problem posed over the middle surface of the shell". This means that the new unknown should be ~ = ( ~ ), where ~ are the covariant components of the displacement ~ a i " -~ -+ I~3 of the points of the middle surface S - 0(~). In other words, ~ ( y ) a i ( y ) is the displacement of the point O(y) e S; see Fig. 3.1-2.
Sect. 3.2] The three.dimensional equations over a domain independent of e 149
3~ 13t3
Fig. 3.1-2: A two-dimensional shell problem. For e > 0 "small enough" and data of ad hoe orders of magnitude, the three-dimensional shell problem (Fig. 3.1-1) is replaced by a "two-dimensional shell problem". This means that the new unknowns are the three covariant components ~ : ~ --+ IR of the displacement field ~ a i : ~ --+ R 3 of the points of the middle surface S = 0(~). In this process, the "three-dimensional" boundary conditions on F~ are replaced by ad hoe "twodimensional" boundary conditions on 70.
3.2.
THE TttREE-DIMENSIONAI, A D OMAIN INDEPENDENT
EQUATIONS OF
OVER
We describe in this section the basic preliminaries of the asymptotic analysis of a linearly elastic shell, as set forth by SanchezPalencia [1990~ 1992] in the slightly different, b u t in fact equivalent, framework of a "multi-scale" a s y m p t o t i c analysis, t h e n by M i a r a & Sanchez-Palencia [1996], Ciarlet & Lods [1996b, 1996c, 1996d], a n d Ciarlet, Lods & M i a r a [1996]. " A s y m p t o t i c analysis" means t h a t our objective is to study the -+ I~3 as ~ -+ O~ an behavior of the displacement field u~g i'e 9 endeavor t h a t will be achieved by s t u d y i n g the behavior as ~ ~ 0 of the covariant components u i --~ I~ of the displacement field, i.e, the behavior of the u n k n o w n u e = (u~) " - ~ ~ I~3 of the threed i m e n s i o n a l variational p r o b l e m ~ ( ~ e ) described in Sect. 3.1.
150
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
Since these fields are defined on sets ~e that themselves vary with ~ our first task naturally consists in trans]orming the threedimensional problems 7~(~ e) into problems posed over a set that does not depend on e. The underlying principle is thus identical to that followed for plates (Vol. II, Sect. 1.3). As we shall see, there are, however, striking differences! Let
f~ := r0 . - ~ 0 • [ - 1 , 1],
w•
F+ "-- w
1, 1[, •
{1},
r_ .- w • {-1}.
Let x -- (xl, x2, xa) denote a generic point in the set [2 and let (9/ := 0/0xi; hence xa - Ya (a generic point in the set ~ is denoted y - (Yl, y2); cf. Sect. 3.1). With each point x C ~, we associate the point x ~ E ~e through the bijection (Fig. 3.2-1)
7re "X -- (Xl, X2, X3) E ~ ---Y Xe -- (X~) -- (Xl, Lg2, ~g3) E ~ e .
Consequently,
O~ -
Oa and 0~ _ _103.
The coordinate x3 C [-1, 1] will be also called t r a n s v e r s e variable, like x~ E [-e, e] in Sect. 3.1 ("scaled" transverse variable could be preferred; however, no confusion should arise). In order to carry out our asymptotic treatment of the solutions u e = (u~) of problems ~(f~e) by considering e as a small parameter, we must: (i) ~ p ~ i ~ the ~ y th~ ~ k ~ o w ~ ~ = ( ~ ) ~ d m o ~ g e ~ ~ n y th~ ,~to~ ~Id~ ~ = ( ~ ) ~ p p ~ ~ g i~ the fo~m~t~tio~ of p~obt~.~ ~'(~) are mapped into vector fields over the set ~; (ii) control the way the Lamd constants and the applied forces depend on the parameter e. W i t h the unknown u e - (u~) 9~e ~ R3 and the vector fields --+ appearing in the three-dimensional problem 7~(fle), we associate the s c a l e d u n k n o w n u(e) - (ui(e)) 9 -+ ~3
Sect. 3.2]
The three-dimensional equations over a domain independent of e 151
(
0
)
~=~ Z~ I
4
i
I
2
Fig. 3.2-1: Transformation of the shell problem into a scaled problem posed over a set f~ that is independent of e. Each point of the reference configuration of the shell is the image | ") of a point z" = (z~) of the set ~" = ~ • [ - e , e], over which the shell problem is posed in terms of the "natuxal" cuxvilinear coordinates z~. Each point z" = (z~) E ~c is itself the image 7r~z of the point z = (zi) of the set ~ = ~ • [-1, 1] with za = z~ and z"3 = ~zs . Thanks to these changes of variables, the shell problem is transformed into a problem posed over the set 1~.
152
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
and the s c a l e d v e c t o r fields v = (vi) 9~ -+ I~3 defined by the scalings: u~(z e) =: ui(e)(z) and v~(z e) -" vi(z) for all z e - 7fez e ~e.
The three components ui(s) of the scaled unknown u(s) are called the s e a l e d d i s p l a c e m e n t s . We next make the following a s s u m p t i o n s on t h e d a t a , i.e., on the Lamg constants and on the applied body and surface forces: There exist constants A > 0 and/z > 0 independent of e and there exist functions fi E L2(~) and h i C L2(r+ t0 r_) independent of such that A e = A and/~e =/~,
hi, e(z e) = ep+lhi(z)
for all
z e = 7r~zEf~,
for all
z e - re z E F e u r ~
where the exponent p is for the time being left unspecified; needless to say, p is not subjected to the usual rule governing Latin exponents! Since the problem is linear, we assume without loss of generality that the scaled unknown u(e) is "of order 0 with respect to e". This means that the leading term of a formal asymptotic expansion of u(e) is a priori assumed to be of order 0 (Sect. 3.4), an assumption later justified by the proof that the scaled unknown u(s) converges in ad hoc space to a nonzero limit (Chaps. 4, 5, and 6), when the applied forces are of the right orders. A complete justification of the scalings of the unknowns and the assumptions on the data, including the determination of the exponent p, will be given in Sect. 3.4. At this preliminary stage, we simply record three observations: (i) For plates, the "passage from f~ to f~" is identical, but different scalings can be made on the "horizontal" components u a'* and "vertical" components u 3,* of the unknown, and different assumptions can be made on the "horizontal" components fa,~ or h a,6 and "vertical" components f3,e or h 3'~ of the applied forces. This was the basis of the asymptotic analysis set forth for plates by Ciarlet & Destuynder [1979a], also described at length in Vol. II, Sect. 1.3.
Sect. 3.2]
The three-dimensional equations over a domain independent of e 153
(ii) As the problem is linear, there is no loss of generality in choosing ~0 = 1 as the (same) scaling factor for all three covariant components of the displacement field. (iii) For definiteness, we assume that the Lamd constants are independent of s. However, this assumption is merely a special case among a whole class of assumptions, which permit in particular the Lam~ constants to vary with e as s --+ 0 if one so wishes. More precisely, a multiplication of both Lam~ constants by a factor s t, t E IR, is always possible, as we shall explain with more details in the next chapters. The choice t = 0 is merely made here for definiteness. A simple computation then yields the variational problem that the scaled unknown u(s) satisfies over the set ft, thus over a domain that is independent of ~ (as noted in Sect. 3.1 the Christoffel symbols r3'e and FP~e vanish in ~1e for the special class of mappings O considered here; consequently, the functions r3a3(s) and r~3(e ) defined below likewise vanish in ft): T h e o r e m 3.2-1. Let w be a domain in I~2, let 0 E C2(~; I~3) be an injective mapping such that the two vectors aa - OaO are linearly independent at all points of-~, and let so > 0 be as in Thm. 3.1-1. With the functions r i~ , g~ , AiJkl,~ . - ~ -+ R appearing in problem p ( ~ e ) (Sect. 3.1) we associate for each 0 < e < so the "scaled" ]unctions FiP(s), g(e), AiJkl(e) 9-~ --+ I~ defined by
.-
.-
")
g'(,')
for all
ze - ~zE~,
for all
z e - lre z E ~ e ,
for all
z e - ~ezE~.
With any vector field v - (vi) C H l ( f t ) , we associate the s c a l e d l i n e a r i z e d s t r a i n s eillj(s;v ) - ejlli(s; v) E L2(ft) defined by:
.=
1
+
-
) -- _103v3.
p
154
Asymptoticanalysis of linearly elastic shells: Outline
[Ch. 3
Let the assumptions on the data be as above. Then, for each 0 < e <_ eo, the scaled unknown u(e) satisfies the following scaled t h r e e - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 7~(e; f~) of a l i n e a r l y elastic shell:
u(e) E V(f~):= (v - (vi) E Hl(fl); v - 0 on fa AiJkt(e)eklll(6; u(e))eillJ(e; v ) x / ~
r0},
dx
for all v E V(f~). m
The functions AiJkl(e) are called the c o n t r a v a r i a n t c o m p o n e n t s of the scaled t h r e e - d i m e n s i o n a l e l a s t i c i t y t e n s o r of t h e shell. The functions eillj(e; v) are called "scaled" linearized strains because they satisfy eillj(~; v)(x) -- e~llj(v~)(xe ) for all x e - ~r~x e ~ . Note that the scaled strains eilla(s; v) are not defined for e - 0. Hence problems T'(g; f~) provide instances of singular perturbation problems in variational form, as considered and extensively studied by Lions [1973].
Remark. By contrast, the functions riP(e), g(e), AiJkl(e) converge in the space C~ 3.3.
as e --+ 0 (Whm. 3.3-1).
m
GEOMETRICAL AND MECHANICAL PRELIMINARIES
Our first result, based on Ciarlet & Lods [1996b, Lemma 3.1], Ciarlet & Lods [1996d, Lemma 3.1], and Ciarlet, Lods & Miara [1996, Lemma 3.1], gathers all the "geometrical" preliminaries needed in the sequel for the asymptotic analysis of linearly elastic shells, regarding the behavior of the functions ri~(e ) and g(e) (defined in Thm. 3.2-1)
Sect. 3.3]
Geometrical and mechanical preliminaries
155
as e -+ 0. A n o t e w o r t h y conclusion is that, while these are functions of x - (y, x3) E f~ - ~ • [ - 1 , 1], their limits for e - 0 are functions of y E-~ only, i.e., the limits are independent of the transverse variable x3; note in this respect t h a t the functions r~fl, ba~, b~, a n d a are identified in the next t h e o r e m with functions defined over the set f~ by letting these be constant with respect to x3. Observe also t h a t the n o t a t i o n a l distinction between the "three-dimensional" and "two-dimensional" Christoffel symbols r~f3(e ) and r~, is a u t o m a t i c since the symbol e appears only in the former. T h e o r e m 3.3-1. Let w be a domain in R 2, let 0 E C3(~; ]~3) be an injective mapping such that the two vectors aa - OaO are linearly independent at all points of-~ and let eo > 0 be as in T h m . 3.1-1. The functions riP(e) = rPi(e) and g(e) are defined in Thin. 3.2-1, the functions ba~ , b~ , FIT af3 , a are defined in Sect. 3.1, the covariant d ~ , a t ~ , ~ , b~l o ~ d ~ ~ d by ( T a ~ . 2.5-~)
and the functions bar3, b~, tITan, b~la,r tions inC~
and a are identified with func-
Then
~(~) -
tiT
r .ITz - ~ 3 b ~ l . + o(~ ~)
r~z(~) - b.z - ~ b ~ b ~ , , 03 F p~ ( ~ ) -
0(~),
r;3(~) = - b ~ - ~=3b~b; + O(~2), r i 3 ( ~ ) - r;~(~) - o, g(~) - ~ + o ( ~ ) , /or all 0 < e < co, where the order symbols O(e) and O(e 2) are meant with respect to the norm ]1 " II0,oo,~ defined by IIw]10,c~,~ - sup {]w(x)l; x E ~ } .
Finally, there exist constants a0, go, and gt such that O < ao <_ a(y) for a l l y E ~ , 0 < go <_ g(6)(x) < gt for all x C f~ and all 0 < e < e0.
156
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
e gij, e The vectors ai, a ' ,9 g~, gS," e and the scalars aaf~, gij, are defined in Sect. 3.1. Let Proof.
gi(~)(x)
:= g ~ ( x e) and g i ( ~ ) ( x )
gij(e)(x)
:= gi](x e) and g i i ( e ) ( x ) := gij'e(xe) for all x ~ - 7rex e - ~ .
:= g i ' e ( x e )
for all x e - 7rex e ~ ,
By definition, g a ( e ) : a a + exaOaa3 : a a - ex3bao-a ~
since Oaa3 = - b a , r a a by the formula of Weingarten (Thm. 2.3-1). Hence ga/3(e) - aa/3 - 2ex3ba/3 + O(e 2) and gia(e) - (fi3,
since ga(e) : a3. The matrix (gij(e)) being thus of the form
(g~j(~)) = G + eH + O(~2), its inverse satisfies (g~j(~))-~ = (g~J(~)) - G - ~ _ ~ G - ~ G - ~
+ 0(~).
Combined with the expressions found supra for the elements of the matrices (~ and H and the relations b~ - aaabar~ the last equality gives ga~(e) = a af3 + 2 e x 3 a a ~ + O(e 2) and gi3(~) _ j i 3
so that ga(~) : g a i ( ~ ) g i ( e ) _ a a + e x a b ~ a ~ +
O(~2).
By definition, r ~ j ( ~ ) - g ~ ( e ) . Oagj(e); consequently,
r:,(~) -
(." + ~ 3 b : . ~ + 0(~2)) 9( 0 . . ,
r:~(~) -
(." + ~b:.
~ + O(~))
+ ~0.~.~),
90 o . ~ .
Combining these expressions with the formulas of Gaus and Weingarten (Thin. 2.3-1), viz.,
O.a~ = r ~O" a . + baf3a3 and Oaa3 - - b ~ a ~ , together with the definition of the covariant derivatives b~la, we find that r:,(~)
- r:, - ~b~l.
+ 0(~2),
r~3(~) - -b~ - ~ b ~ b ~ + O(~2).
Sect. 3.3]
157
Geometrical and mechanical preliminaries
Since ga(s) -- gai(e)gi(c ) -- a a, we likewise have, thanks again to the formulas of Gau~ and Weingarten,
r~,(~)
a~(~), o.g,(~) = ~ . (o.~, + ~ o . , ~ )
-
-
~,~,~
-
~~2~,.
The relations
r~(~)
-
~(~). o~g.(~)
-
g~(~). ( o ~ . + ~ s o . ~ 3 )
show that
o3r~(~)
-
o ~ ( ~ ) . 0 ~ . + o(~).
Let i
Since only first-order and second-order derivatives of 8 occur in the functions
and since O C C3(~; R3), we conclude that there exists a constant cl such that g ~ ( e ) - a ~ + c g ~ ( c ) with [[g~ (c) [ll, oo,~ -< ct. Consequently~ g(e) -- a -- O(~), and there likewise exists a constant c2 such that
g"~(~) - ~" + ~g"~,~(~) with Ilg"~,~(~)ll~,~,~ <_ ~2. The relations 03rPf3(s) - O(e) then follow by noting that ga(c ) _ gaf3(s)gf3(c ) a n d g3 (~) _ a 3. Since the vectors a a are linearly independent at all points of ~ there exist constants a0 and al such that
O
for a l l y E ~ .
Hence the above asymptotic behavior of the function g(s) shows that, if s0 > 0 is chosen as in Thm. 3.1-1, there exist constants go and gl such that O < g o _ < g ( c ) ( x ) _ < g t for a l l x E ~ a n d a 1 1 0 < s _ < s o .
158
Asymptotic analysis of linearly elastic shells: Outline
Finally, it is clear that the relations 1~3, e immediate consequences of the relations -a3 -
-
-
0
[Ch. 3
- 0 (Sect. 3.1). II
Our second result, based on Bernadou, Ciarlet & Miara [1994, Lemma 2.1] and Ciarlet & Lods [1996b, Lemma 3.1], gathers all the "mechanical" preliminaries needed in the sequel. It comprises two parts" In part (a), the uniform positive definiteness of the scaled two-dimensional elasticity tensor of the shell, given here in terms of its contravariant components aat~crr~ is established; this tensor will appear naturally in the formulation of all the "limit" scaled two-dimensional shell equations found at the outcome of the asymptotic analysis. In part (b), the behavior of the scaled contravariant components AiJkt(e) of the three-dimensional elasticity tensor (defined in Thm. 3.2-1) as e ~ 0 is analyzed; it is shown that their limits are functions of y E ~ only and that the positive definiteness of the associated scaled three-dimensional elasticity tensor is uniform, not only with respect to x E f~, but also with respect to 0 < e < e0. T h e o r e m 3.3-2. (a) Let w be a domain in I~2, let 0 E C1(~; R s) be an injective mapping such that the two vectors aa = OaO are linearly independent at all points of-~, let a ao denote the contravariant components of the metric tensor of S = 0(~), let the e o n t r a v a r i a n t c o m p o n e n t s of the sealed t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of t h e shell be defined by aO~/~Or'r ,-'-
+
.
+
)~ + 2/~ and assume that
A>_ 0 a n d # > O . Then there ezists a constant ce - ce(w, O, Iz) > 0 such that
for all y E -~ and all symmetric matrices (tao).
Sect. 3.3]
159
Geometrical and mechanical preliminaries
(b) Let 0 E C2(@ R 3) and let the other assumptions on the mapping 0 and the definition of ~o be as in Thin. 3.1-1. The contravariant components AiJkt(e) - AJikt(e) - AkUj(e) of the scaled threedimensional elasticity tensor (Thin. 3.2-1) satisfy AOkt(e) -- AiJkl(0) + O(~) and Aafkr3(c) - Aa333(c) -- 0 for all 0 < ~ <_ so, where the order symbol O(e) is meant with respect to the norm [t 9 !10,oo, (defined in Thm. 3.3-1) and
Aaf3~rr(o) := Aaa~a 'rr + ~(aa'ra ~ + aa~aO'~), Aa/333(0) := /~aa~, Aa3tr3(0) :-- pa act, A3333(0) "-- ,,~ -4- 2p, A a13~r3(O) -- A a333(0) "-- O. Finally, there exists a constant Ce > 0 such that
E Itij12 -< CeAiJkt(e)(x)tkttiJ i,j ]or all 0 < e < co, all x E f~, and all symmetric matrices (tii). Proof. We note, first that a a~ (y)a ~T (y)tzrta~ -- (a a~ (y)ta~)2 > 0
for all y E ~ and all matrices (tar3), next that = 2tTA(y)t
for all symmetric matrices (ta13), where A(y) " -
t :=
t
2alla12 alia11 2alia 12 2(a12a 12 -4- a l i a 22) a12a12 2a12a22 t12 t22
9
a12a12 \ 2a12a 22 a22a22
)
(y),
Asymptotic analysis of linearly elastic shells: Outline
160
Since
det
[Ch. 3
a l i a 11 > 0 in ~,
alIa11 2alla12 \ a 11a11 2allal2 2(a12a12+ a11a22) ) - 2 ~ a ~ 0 in ~, 2 det A - ~-~ > 0 in ~,
where a - det(aa#), we infer from a well-known characterization that the symmetric matrix A(y) is positive definite at all y E ~, and the existence of the constant Ce > 0 follows from the compactness of the set ~; this proves (a). An argument similar to that used in the proof of Thin. 3.3-1 shows that ga~ (e) -- a a~ + O(e) and gi3 (e) - t~i3, where gii(e)(x):= gij'e(xe) for all x e - 7rex e ~ . combined with the definitions (Thm. 3.2-1)
These relations,
imply that
AiJkl(e) -- Aijkl(O) + O(e) and A a ~ a ( e ) - Aa333(e) - 0. For each 0 < e <_ eo, the tbxee-dimensional elasticity tensor defined by means of its contravariant components A ijkl'e is positive definite, uniformly with respect to x e E ~ (Thin. 1.8-1). This implies that there exists c(e) > 0 such that
It jl < i,] for all x E f~ and all (tij) E S, where S denotes the set of all symmetric matrices of order three, on the one hand. The definitions of the functions Aifla(0) imply that
Aqkl(O)tkltii -- A(aa~ta~ + t33) 2 + #(aa~a ~r + aara[3~)t~rta[3 +41zaa#tazt~3 + (A + 2#)tzztza, on the other. Since there exists Cl such that (see the first part of this proof)"
a,/9
Sect. 3.4]
Linearly elastic "membrane" and "flezural" shells
161
for all y E ~ and all symmetric matrices (ta~), it follows that there exists c2 such that
It j[ 2 _< c2A Jk (O)(x)tk t j i,j
for all z E f~ and all (tij) C S. The continuity of the mapping
(~, z, (tij)) E [0, eo] • ~ • $1 --+ Ai3kl(e)(x)tkltij, where ~1 {(tij) C ~; Ei, j It jl 2 - 1}, then yields the existence of a constant Ce > 0 such that the inequality "
-
-
E ItiJ[2~ CeAiJkl(~)(z)tkltiJ i,j holds for all (tu) E S, x E f~, and 0 _ e ~ e0; this proves (b).
II
Remarks. (1) The uniform positive definiteness of the scaled twodimensional elasticity tensor of the shell (part (a)) can be established in a different manner; cf. Ex. 3.1. (2) Although A < 0 is not realistic from a mechanical standpoint (the Lam~ constants of actual elastic materials are strictly positive), the elasticity tensor remains positive definite even if A is "slightly negative"; cf. Ex. 3.2. II Other "technical", but of a more specialized nature, preliminaries will be also proved in the next chapters; see notably Sects. 4.2, 4.3, 5.2, and 5.3. 3.4.
THE TWO-DIMENSIONAL EQUATIONS OF LINEARLY ELASTIC "MEMBRANE" AND "FLEXURAL" SHELLS DERIVED BY MEANS A FORMAL ASYMPTOTIC ANALYSIS
OF
The formal asymptotic analysis of linearly elastic "thin" shells is a relatively recent subject. After the landmark contributions of Goldenveizer [1963, 1964], a major step was achieved by Destuynder [1980] in his Doctoral Dissertation, where a convergence theorem for elliptic membrane shells (as they will be defined in the next chapter) was in particular "almost proved". Another major step was achieved in the
162
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
pioneering Note aux Comptes Rendus of Sanchez-Palencia [1990] (see also Sanchez-Palencia [1992]), who for the first time clearly demonstrated that the asymptotic behavior of a shell as its thickness goes to zero is governed, either by "membrane", or by "flexural", twodimensional equations and that the distinction rests on whether a certain space of linearized ineztensional displacements reduces to {0} or not. His analysis was further refined by Miara & Sanchez-Palencia [1996], whom we faithfully follow in this section. Finally, Caillerie & Sanchez-Palencia [1995b] have shown that, by contrast, the "membrane" and "flexurar' terms remain coupled in the limit equations (again obtained via a formal asymptotic analysis) when the linearly elastic material is anisotropic and nonhomogeneous.
Within the linear theory, this section has an optional character: Stricto sensu, only the content of Sects. 3.1, 3.2, and 3.3 is needed for the asymptotic analysis of linearly elastic shells carried out in the next three chapters; besides, its conclusions, which are based on the method of formal asymptotic expansions, might not have the same appeal as those derived from the convergence theorems proved in the next chapters. Nevertheless, its content admirably serves ]our major purposes: First and foremost, it clearly leads through a perhaps formal, but certainly subtle, analysis to the somewhat unexpected conclusion that there are two fundamentally distinct kinds of linearly elastic shells, the "membrane" shells and the "flezural" shells (the "membrane" shells being themselves later subdivided into two subclasses; cf. Chaps. 4 and 5). This conclusion is of course in sharp contrast with that drawn for linearly elastic plates in Vol. II, Chap. 1. Secondly, it justifies the scalings on the unknowns u~ and the assumptions on the data ]i,e and h i'e made in Sect. 3.2, notably by specifying how the exponent p must be chosen in each case. Thirdly, it explains why the logical order of presentation consists in treating the "membrane" shells before the "flexural" ones; otherwise it makes as much sense, from a mathematical standpoint at least, to first consider the "flexural" shells. Fourthly, it plays an indispensable rSle in the justification of the two-dimensional equations of the nonlinear theory (Chaps. 8 to 10) by the method of formal asymptotic expansions. Consider the three-dimensional shell problems :P(~te), as they are defined in Sect. 3.1 for each e > 0. We assume that the mapping 0
Sect. 3.4]
Linearly elastic "membrane" and "flezural" shells
163
that defines the middle surface of the shell satisfies
0 ~ c 3 (~; R 3). We also assume ab initio that the Lamd constants )~e and #e are independent of e, i.e., that there exist constants A > 0 a n d / ~ > 0 such that Ae = A and #e = # for all e > 0.
First, vector fields u(e) - (ui(~)) e H I ( ~ ) and v - (vi) e HI(~) and functions fi(e) e L2(f~) and hi(e) e L2(F+ U F_) are defined by letting u~(x ~) - ui(e)(x) and v~(x ~) - vi(x) for all x e = r e x e ~e, /i'~(z~)
-
/ i ( e ) ( z ) for all z ~ - 7r~z E f~,
h i ' e ( x e)
--
hi(e)(x) for all z e - 7fez E r~ u r~.
In this fashion, each variational problem 7~(I2~), e > 0, is transformed into the following variational problem 7~*(~; I2): e v(a)
-
=
e rI
(a); . = 0
r0},
ff Aijkl(c)eklll(~; u(E))ei[[j(~; V)x/ g(~) dx C
+ur_
for all v c V ( ~ ) ,
the "scaled" functions A~Jkt(e) and g(e) and the scaled linearized strains eil[j(e; v) being defined as in Thm. 3.2-1, where they appeared in the problems then denoted 79(e; I2). As expected, the left-hand sides of the variational equations of problems ~o*(~; f~) and 79(6; [2) coincide since the same transformation was already performed in Thm. 3.2-1 on the covariant components of the unknown displacement. The difference lies in the right-hand sides, where the "right" scalings are yet to be found on the contravariant components of the applied forces. The notation 7~* (c; f~) is precisely intended to remind of this difference.
Asymptotic analysis of linearly elastic shells: Outline
164
[Ch. 3
Second, it is assumed that there ezists a formal asymptotic ezpansion of the scaled unknown of the form: u ~ ~ O.
U ( S ) -- U 0 -~- SU 1 "~- 82U2 -~- S3U 3 -[- S4U 4 -[- " ' " , w i t h
As already observed in Sect. 3.2, there is no loss of generality in assuming that the leading term u ~ is of order 0 with respect to e since the problem is linear. Note that, by contrast, this freedom is lost in the nonlinear case; eL Sect. 8.6. According to the basic Ansatz of the method of formal asymptotic expansions (more details and references about this method are found in Sect. 8.6, where it is put to use in an even more general setting), we wish to identify the leading term u ~ by equating to 0 the factors of the successive powers of e found in problem 79*(e; f~) when u(e) is replaced by its formal expansion, then by solving the resulting variational equations. In this process, each needed term u q, q >__0, is required to be in the space Hl(f~), but the boundary condition on F0 is imposed on it only if it is needed for determining u ~ Finally, it is required that no restriction should be imposed on the
applied force densities entering the right-hand side of the equations used for determining the leading term eventually found, since no such restriction is imposed on the original problems 79(f~). It is this natural requirement that allows to successively "eliminate" unwanted models. The above formal expansion of the scaled unknown combined with the asymptotic behavior of the functions riPj(e) (Thin. 3.3-1) induces formal asymptotic expansions of the scaled linearized strains of the form: 1 -i
0
1
eillj(e; u(e)) - ~eillJ +eillJ +eeillj +
~2 2
eillJ +
~3 3
eillJ + . . . ,
where
e-1 = O, allt3 -1
%113
Z --
03u O,
o
1 -
-
eOll3 - 1 o
e3113
-
,7 o + OauO ) - Fa~U~r
+
~ +
~
-
-
Sect. 3.4]
Linearly elastic "membrane" and "flezural" shells
165
eta I - -1 (O~u~ + Oau~) - r .,r, . ~ t - b.~.~3 + ~ { b , l,ro - . 0 + bo,rb,r~u033},
1 elalt3 --- ~(03Z$ 2 + OotU~) 2r" b~'u 1 + x3b~b~u 0,
and each function e illJ' q q >- 2, is a linear combination of u q- 1, U q , U q+l and of their first-order partial derivatives (only the values q = 2 and q - 3 are needed in the sequel). The functions eillJ(S; v) likewise admit formal asymptotic expansions (again resulting from the asymptotic behavior of the functions r,~ (~)) of the form:
le-1 0 (v)+ i (v)+ 2 2 (v)+'", eillJ(e; v ) - - ~ illj(V)+ eillJ seillJ s eillJ where e-1 ~ l ~ ( . ) - 0, e- 1 .ll3(~)-
1 ~o3~,
~(~)-
o~.
I~
~o I[fil( ~ ) _
1 2(0/3va + OqaVfl) -- raflVcr -- ba/3v3' 1
0 e3113(v) -- 0,
3b~l~,~ + 9
,
~ll~(~) - ~ b ; b ~ . . , ~ll~ (v) - o. The asymptotic behavior of the functions g(s) and AiJkt(s) (Thms. 3.3-1 and 3.3-2) also implies that (recall that O E C3(W; I~3) by assumption) g(s) = a + O(e), A i j k t ( s ) V / g ( s ) = AiJk'(O)v/~ + eBijkt, 1 + e2BiJht, 2 + o(s2),
where the functions AiJkl(O) are defined in Thm. 3.3-2.
Asymptotic analysis of linearly elastic shells: Outline
166
[Ch. 3
Note that the above expansions comprise all the terms that will actually appear in an equation at some stage of the subsequent analysis. It does not mean, however, that each one of their specific expressions will be actually needed. As a last preliminary, we also record a simple result frequently used in this volume (as in Vol. II):
T h e o r e m 3.4-1. Let w be a domain in I~2 with boundary 7, let = w • - 1, 1[, and let w E IF(~), p > 1, be a function such that
wO3v d= = 0 for all v E Cr162 Then w
-
-
that satisfy v - 0 on 7 x [ - 1 , 1].
O.
Proof. Let 7~ be an arbitrary function in the space T~(~) and let the function v" ~ -+ R be defined by ~V(zl, z2, t ) d t for all (Zl, z2, z3) E ~2. T h e n v e Ccr
and v = 0 on 3' • [-1, 1]; hence
f n wcp dx - f a wOzv dx = O, and thus w = 0 by a classical property of the space LP(~), p > 1. II The above implication a fortiori holds if fa wO3v dx - 0 for all v E HI(F~) that vanish on r0; this is how it is used in the remainder of this section. We are now in a position to start the cancellation of the factors
of the successive powers of e in the variational equations of problem 7~*(e; ~), until the leading term u ~ of the formal asymptotic expansion can be fully identified. It will be found in this fashion t h a t u ~ satisfies a two-dimensional variational problem, which may be either that of a "membrane" shell or that of a "flexural" shell and that in each case the "right" orders on the components of the applied forces (which we are seeking) appearing in the right-hand sides are simulta-
neously determined. Note that, by contrast with our usual practice, these major conclusions will be formally stated (Thms. 3.4-2 and 3.4-3) only after
Sect. 3.4]
Linearly elastic "membrane" and "flezural" shells
167
they have been established, in a series of eight steps n u m b e r e d (i) to (viii), as this procedure seems more n a t u r a l in the present case. (i) Since the lowest power of e appearing in 7~*(e; fl) is e -2, we are n a t u r a l l y led to first try
1 9 and hi ( e ) _fir,,-2
fi(r
lhi,_ 1
where, here and subsequently, any ]unction fi, r
C
L2(~) and any
]unction h i'r+l E L2(r+ u r_), r > - 2 , is meant to be independent of ~. T h e cancellation of the coefficient of e -2 in the variational equations of p r o b l e m :P* (e; ~) then leads to the equations
ff~ AiJkt(O)e~ltl e -illj l ( v ) v / - a d x - - ff~fi'-2viv/-ddx+ fs t + o r _ hi,-lviv/-adr for all v E V(fl). These equations imply that the ]unctions fi,-2 and h i , -1 cannot be chosen arbitrarily, since we should have in particular
ff fi,-2viv/-a dx + fr+ur_ hi' -lvi ~/-a dr - 0 for all v E V ( f l ) t h a t are independent of x3; to see this, recall t h a t e -1 (v) -- 0, e-1 (v) -- 89 03va, and Consequently,
must let f i , - 2 _ 0 and h i'-1 - O. The expressions of the functions Aijkl(o) and e~-i~l t h e n show t h a t we must have
illJ
-- f~2 (4Aa3Cr3(O)e~[~3e-1 (V) + A3333 _
=
fa
o
03u~O3v~ + ()~ + 2,)03u~
} vZadx - 0
for all v E V ( ~ ) . C o m b i n e d with T h m . 3.4-1, these equations in t u r n imply t h a t
03u ~ - -
0
in f~,
since _ a ~u. 3 %0 - 0 in ~ implies t h a t 03u a0 _ 0 in ~ (the m a t r i x (a af3) is positive definite); hence the leading term u ~ C V ( ~ ) is independent of the transverse variable x3 and it can therefore be identified with
Asymptotic analysis of linearly elastic shells: Outline
168
[Ch. 3
a vector field ~o E Hi(w) satisfying ~o _ 0 on 70. Canceling the coefficient of ~-2 thus yields the relations" r
{~ -
e v(~).-
( ~ ) e a x ( ~ ) ; ~ - 0 on ~o},
e -1 = 0 i n f , . illj (ii) Our second try is thus fi(e) -- I f i ' - I and hi(s) - h i'O. Combined with the relations e -1 - 0 obtained in step (i), the cancelillJ lation of the coefficient of e - t in the variational equations of problem 79"(e; f~) then leads to the equations
fo
AiJkt(O)e~
illJ
+ur_
hi'~
Letting v E V(f~) be independent of xs then implies as in step (i) that the functions fi,-1 and h i'O cannot be chosen arbitrarily, since for such functions v,
for an v ~ v ( ~ ) .
f f fi,-lviv/-ddx-l- fr
+ur_
hi'~
- O.
Consequently, we must let fi,-1 _ 0 and h i'O - 0 and, accordingly, we must then have
faAiJkt(O) ekllt o e-l(v)v/-ddx illJ
= fo 4Aa3~3(O)e~
=0 for aU v E V(f~). Combined with Thm. 3.4-1, these equations in turn imply that e~
-0
and e~l]3 =
aaD ~0
A+2p
Linearly elastic "membrane" and "flezural" shells
Sect. 3.4]
(iii) Our
third try
169
is thus
fi(8 ) __ fi,
O and hi(e) - eh i'l.
The cancellation of the coefficient of s o in the variational equations of problem 79"(~; ft) then leads to the equations (recall that e -1 - 0 ; cf. step (i))" illJ ,A/jkl(0){
o o kll ,llJ(
+
e-~
il,j(")}
+ fo BiJhl'le~ lit e-1illJ (v) d~, = ~ fi'~
fr
+ur_
hi, lviv/-ddr
for all v E V(ft). Observe in passing that the occurrence of the functions e~lll in these equations implies that the formal asymptotic expansion of u(e) should be "at least" of the form u ( e ) - u ~ + ~ u 1 + e2u2 + . . -
.
Let v = r/ E V(w), i.e., v C V(ft) and v is independent of z3. Noting that e illJ - t (v) - 0 for such functions v and using the expressions 0 of the functions AiJkt(O), the relations satisfied by the functions ei[[3 (step (ii)), and the definitions of the functions e~ r and e~ find that
%ll~.eall~(rl)~/~ dx
-
/o 1 /~ = -~
o
aa~O.r 0
0 %ll,eallB(rl)v/-ddx
+UI"_
we
170
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
for all v/E V(w), where the functions
4A~ aC~f3crT .__ ~ a a~
aZ'r + 2p(a a'7 a ~r + a ar a ~(r)
A+2#
are the contravariant components of the s c a l e d t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of t h e shell (already introduced, but without any justification, in Thm. 3.3-2). Let
~,(~)
O"
:= ~(o,~o + oo~,) - r o , ~
- bo,~3
denote the covariant components of the l i n e a r i z e d c h a n g e of m e t ric t e n s o r associated with a displacement field riia' of surface S (Sect. 2.4). Note that Vaf3(v/) E L2(w) if v / - (rIi) E H i ( w ) x H i ( w ) x L2(w). Then remarkably, we precisely have
~o IIf3 - Vaf3(r ) and eallf o 3(r/) - ~'af3(~?) for all ~7 e V(w)
(iv) Define the space
v 0 ( ~ ) := {n e v ( ~ ) ; ~.z(n) - 0 i= ~}, and assume first that
v 0 ( ~ ) = {0}, in which case the shell will be called a linearly elastic "membrane shell. A w o r d of c a u t i o n . This definition is essentially provisional, as it will have to be subsequently amended; cf. Sects. 4.1 and 5.1. The quotation marks emphasize this temporary character, i
Linearly elastic "membrane" and "flezural" shells
Sect. 3.4]
171
Then by step (iii), the vector field ~o should satisfy the following two-dimensional variational problem '~PM(W) ":
r
e v ( ~ ) - {n - ( ~ ) e H~(~); n - o on 70},
fwaa#~rV~r(,~
f pi'~
9
where pz, O :=
for all rl E V(w),
:f2,o dx~ + hi'l( ., 1) + hi't( ., - 1 ) . 1
Besides, the induction stops here since
"~E:~M(W)"is
a bona fide variational problem, which can be studied for its own sake: In this direction, a first observation is that the issue of uniqueness of a solution to problem "7~M(W)'' is immediately resolved when V0(w) - {0}. For, if pi,0 _ 0, we must have in particular
~w aa~ar'y~r(~~176
dy - 0,
and thus 7 ~ ( r 1 7 6 _ 0 in ~o (the scaled two-dimensional elasticity tensor is uniformly positive definite; cf. Thm. 3.3-2 (a)). Therefore, ~0 E V0(w) and thus ~0 _ O. By contrast, the issue of existence of a solution is a delicate one. If Vo -- "Y and the surface S = 0(~) is elliptic, we have already shown through a delicate analysis that problem "~M(W)" has a solution, albeit in the l ~ g ~ space V ~ ( ~ ) - H~(~) • H~ (~ ) • L~(~) (Thins. 2.7-3 and 3.3-2). In the other cases where V0(w) = { 0 } , the c o n c l u s i o n s are even more subtle since the "right" spaces where to seek
a solution turn out to be "abstract" completions (see in particular Thm. 5.6-1)! Such observations already indicate why the definition of a "membrane" shell and that of problem "7)M(W) '' (note the quotation marks) will have both to be modified later. (v) Assume next that the space V0(w) contains nonzero functions, i.e., that
V o ( ~ ) ~ {o}, in which case the shell will be called a linearly elastic "flezural" shell.
172
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
A w o r d of c a u t i o n . Like that of a "membrane" shell, this definition is essentially provisional, as it will have to be subsequently amended; cf. Sect. 6.1. The quotation marks again emphasize this temporary character. II If Vo(w) # {0}, the functions fi, o and h i,1 cannot be chosen arbitrarily since we must have
f pi,~176
+ur_
hi'l~Ti~dF-O
for all r / E ( V o ( w ) - { 0 } ) by definition of problem "~M(W)". Hence we must let fi, o _ 0 and h i' 1 _ 0 and accordingly pursue the induction. Before continuing the induction proper (step (vi)), let us have a second look at the cancellation of the coefficients of e - i and e ~ (steps (ii) and (iii)). Since pi, O _ 0, letting r / - r in the variational equations of problem "7~M(W)'' gives
f a ai3crrTcrr(~0)7a/3 (~0)v/a dy -- O. Hence 7a#(~ ~ - 0 and thus ~o E Vo(w). Since then Call# o - 7a~(~ ~ - 0, the relations e~ 3 - 0 and (A + 2p)e~l13 + Aaa/3e~
-- 0 ill fl
established in step (ii) imply that
0 3 u ~ - e3113 - 0 o
and
e~ll 30 -
cr 0 - 0. ~i ( o ~ ~ + 0 3 ~~) + b~~ 0. - ~i(~162o + o3=~) + b~r
Hence
03u~ -
-
cr 0 (oo( ~ + 2b~i~).
Assume that u i E V(f~). Since each function (0a~ ~ + 2 bcra ~0) is independent of z3, there exists ~i _ (~/1) e V(w) such that
uai _ ~a _ x (Oh ~o + 2baa~o) and u~ - ~J, the first relations forcing in addition the function ~o to be in the space H2(w) and to satisfy the boundary condition 0v~ ~ - 0 on 70
Sect. 3.4]
Linearly elastic "membrane" and "flezural" shells
173
(0v denotes the outer normal derivative along 7) since ~o = 0 on 70 by step (i) (Ex. 1.1 in Vol. II can be handy here). To sum up, we have shown that e~Ij
-- Oinf~,
~O E VF(60)"--
{r/- (~7i) e HI(w) • Hi(w) • H2(w); ~i = 0 ~ 3
Ual __
~al
= 0 011 ")'0, ")'a/3(~'/) = 0 i n w } ,
_ x3 (0a ~g + 2b~ r
and u~ - ~] where r 1 _ (~/1) e V(w).
Since e/~ = 0, the cancellation of the coefficient of e ~ in 7~*(e; ~t) reduces to
ff~ AiJkt(O)e~llt e -illg l(v)v~dx
for all v e V(ft)
-0
By a computation similar to that in step (ii), we then conclude that A a at3etall~ in ft. A+2#
clair3 --0 and elll 3 --
It remains to compute the functions eallt3, 1 as their expressions play a key r61e later. Combining their definitions with the expressions found supra for the components u i1 in terms of ~o and ~ and using the symmetry relations (Thm. 2.5-1) o"
7"
o"
T
we obtain, after some manipulations, 1 1 (0.,11,1 -[- O a U l ) - r ~r ~ 1 - b ~ ~ - -2 ec~ll~--
1 ('"~ l'-'.a
-
+ Oa~)
2
_~{o~o +b~(o~o
Let
_ -
~ 1 1 -- r al~r -- ba~i3
r~ a~ o~r ~
+ ~ 3 { b ~ l ~ 0 + ba" ~b~
o
_ b~a b ~ 3 o o
r ~~~ ) o +
~
174
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
denote the covariant components of the l i n e a r i z e d c h a n g e of curv a t u r e t e n s o r associated with a displacement field yia i of the surface S (Sect. 2.5). Note that
pa~(rl) e L2(w) if r / - (r/i) e HI(u)) • HI(w) • H2(w). Then remarkably, the functions e~ll~ have the following simple expressions in terms of the functions Va~(~ t) (the functions Van(r/) are defined in step (iii) for any r / e V(w)) and pal~(~~ 1
%11~-
7~(r
1
) - ~3p~(r176
9
(vi) We are now in a position to continue the induction V0(w) ~ {0}, with
when
fi(e) = ef i'l and hi(e) = e2h i'2 as our fourth try. The cancellation of the coefficient of s in ~* (e; f~) then shows that (recall that e-tilli = eilli~_ 0; cf. steps (i) and (v))"
fflAijkl(o){ ekllteilli 1 0 (?3) + ekllt 2 e-l(?3))V/-~dx illi illJ + f BiJk"~ lit~-~(.)a~f , i'Xvi ~
dx +
f~+or_ h""2vi v/-d dr
for a U v E V(f~) 9 Let v -- r/ E V(w). Noting that e-t(v) - 0 for illi such functions v and using the expressions of the functions Aiikt(O) and the relations (found in step (v)) satisfied by the functions e illJ' 1 we are led to the equations
fn AiJk~(O)%llteilli ~ o (rl ) v~ dr, 1 [ _a~~o
o
= ~ aa~rT~'(f'l)Va~ (n)V~dy
= f fi,~niV~ddx+fr+ur_ hi,2~Tiv/-ddrforallrleV(w).
Sect. 3.4]
Linearly elastic "membrane" and "flezural" shells
175
Hence the functions fi, t and h i'2 cannot be chosen arbitrarily, since we must have
fwfi'lrliV/-ddy + f r
+ur_
hi'2rli~dr - O
for all r/ E ( V 0 ( w ) - {0)); consequently, we must let fi,1 _ 0 and h i'2 - O. But then, letting r / - ~t in the above variational problem shows that
Hence 7a~ (~ t) = 0 and consequently,
c v0@), since on the other hand, ~l e V(w) by step (v). (vii) As a preparation to step (viii), let us further exploit the cancellation of the coefficient of e in 7~*(~; 12) when V0(w) # {0}, which now leads to the equations
fftAiJkl(O){ ek[[leillj(v) 1 0 + e~llle-l(v))v/-adx i[IJ -4- f~ BiJkt'ie~lll e-1 illj(V)dx - 0 for all v C g(f~), since we are now assuming that fi, x = 0 and h i'2 --O. Given an arbitrary element ~/ in the space Vy(w) defined in step (v), let v(~/)= (vi(~l)) be defined by va(r/) := zsI2b~,r/~ + 0at/s} and v3(r/):= 0. Since v(r/) e V(i2), we may let v = v(r/) in the above equations; this leads to the equations:
ft AiJkl(O) ekllteillj 0 (v(r/))~dx +4
/a
/o
+
r
l oa ~13) v/-d dz
176
[Ch. 3
Asymptotic analysis of linearly elastic shells: Outline
Remark. The relations -),a~(rl) = 0 in w satisfied by the functions rl E VF(w) are not needed at this stage; in other words, the above equations are in fact valid for all rl - (r/i) e Hi(w) • Hi(w) • H2(w) that satisfy the boundary conditions r/i = Ourl3 = 0 on 70. i (viii) We conclude the induction when V0(w) r {0}, with
fi(e) __ e2 fi, 2 a n d hi(e) - ~3hi'3 as our fifth, and final, try. The cancellation of the coefficient of e 2 in :P*(e; 12) then shows that (recall that e-lilli - eillJ~_ 0)" fa
3
-1 (v)}v/-~ d x
+ foBiim, l~l ehllleillj 1 o (v)+ ekllZ 2 e-1 ill./(v)} dx
...Bijkl,2elklit illJ(v) dx - f "f"2viv/-ddx+ e -1
+ur_
" h"3v~v/Sdr
for all v C V ( ~ ) . Observe in passing that the occurrence of the functions e3]lz in these equations implies that the formal asymptotic expansion of u(e) be "at least" of the form U(e) -- U 0 -~- eU 1 -~ e2U 2 -~- e3U 3 -~- e4U 4 ~ - ' ' " ,
as e~llt is a function of u 2 u 3 and u 4 Let v = rl E VF(w). Then for such functions v, ~
e-l(v)
illJ
0
:
-
0,
9
= 0
1
=
+
a
so that we are left with the equations
faA~jh~(0) ekllteill 1 1 I ( rl ) v/adx + 4 in Aanaa(O)e21jn{bTarl~ + ~Oay3}v/adx 1 -+-4 .fc~ Ba3a3' 1 e~li3 1 {brarlr + -~Oarl3 1 }~
=
fp~'2mv~dy d~
dx
for all rl E VF(w),
0 (v)-0
Sect. 3.4]
Linearly elastic "membrane" and "flezural" shells
177
where p,,29 := f ~
1
It," 2 dx3 -4- h "s (. ,
1) + h""s (., - 1 ) .
Subtracting the equations found in step (vii) from these equations, we find that
f~ AiJkl (O)e~lll{eillj(~l) t O ( v ( ~ l ) ) } ~ d x _ ~ p"" 271iV/-~dy - eillJ for all v/e VF(w). First, the relations (established in step (v)) t -0and eall3
t A aaf3"l inf~ e3113 = - A + 2------~ ~all~ '
and the relations esllst(~/) _ esll 3o (v(r/)) - 0 together imply that
AiJk'(O)e~llt{e~llJ(~l) - ei~ _
_
AOt~a'v[Nh,~l
- "-"
el
0
<"J~llr{ allf3(n) - eall~
+ A~(o)~ltl~{~ll~(n
(V (n)))
) _ ~oallf~(,,(n))}
laa~,~- ~11~{~11~ ~ ~ (n)
= ~
o (v(y))}. ~,~11~
-
Second, by steps (v) and (vi). Finally, eall~(~7) - Call~
3
-x3{Oa(b~Tl~) + 0/3(b~71~) + Oaf371s r~O~71s - 2r ~af~b~71~ , -
and a computation analogous to that performed in step (v) then shows that %llf3(v/) _ 1
e~
( v ( ~ / ) ) - -zzp~/3(~l) for all ~/e VF(w)
We have thus derived the equations
f AiJk'(O)e~lll{e~llJ(ll) - ei~
1 fa z2a~" = ~1 f~
aal3crr p ~ ( r 1 7 6
for all n e Ve(w).
178
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
To sum up, we have established that, when V0(w) ~ {0}, the vector field ~o must satisfy the/ollowing variational problem 79F(W) 9
r
e VF(W) -- { ~ -
(r/i) e Hi(w) • Hi(w) • H2(w); Yi = Ov~?a = 0 on 70, ")'af3(r/) = 0 in w},
g
aa/3arpar(r176
where p~,2 .=
=
p " Z ~ T i ~ d y for all n e VF(W),
.fi,2 dxs + hi'3( ., 1) + hi'3( ., - 1 ) .
1 The issues of existence and uniqueness of a solution to problem 7)F(W) are already resolved, as corollaries to Thms. 2.6-4 and 3.3-2. Let us now formally recapitulate the main conclusions of the above formal asymptotic analysis. We recall that the following theorems, which are found in Miara & Sanchez-Palencia [1996], were themselves based on a prior work of Sanchez-Palencia [1990]. Another noteworthy reference is Caillerie & Sanchez-Palencia [1995b], who analogously carried out the formal asymptotic analysis of linearly elastic shells made of nonhomogeneous and anisotropic materials. We logically begin with "membrane" shells (an example is proposed in Ex. 3.3): T h e o r e m 3.4-2. Assume that the space
Vo(w) : : {r/C Hi(w); r / - 0 on 70, 7a~(r/) - 0 in w} contains only rI = 0 and that the scaled unknown u(e) = (ui(s)), where ui(s)(x) := ui(x " e ) for all x e = 7rex E - ~ , admits a formal asymptotic expansion of the form u(s) = u ~ + eu 1 + s2u 2 + . . .
, with
u ~ e V(f~), u ~ r 0, and u q e Hl(f~), q -
1, 2.
Assume for definiteness that the Lamd constants are independent of e. Then in order that no restriction be put on the applied forces
Sect. 3.4]
Linearly elastic "membrane" and "flezural" shells
179
entering the equations satisfied by the leading term u ~ their components must be of the form fi'e(Xe) -- fi'O(x) for all x ~ --7r~x E f~e, hi,~(x e) - r
for all x e - 7r~x E r~_ U I~_,
where the functions fi,0 6 L2(f2) and h i'1 E L 2 ( r + U r _ ) are independent of ~. This being the case, the leading term u ~ 9 -+ R 3 is independent of the transverse variable and, once identified with a vector field ~o . -~ __+ ire, it should satisfy the following scaled two-dimensional variational problem "7~M(W)" of a linearly elastic "membrane" shell: r
{~ -
e v(~):=
(v,) e ~ I ~ ( ~ ) ; ,
~ aa~r T~r(,~
- o on ~0}.
dy - ~ pi' ~yiv~ dy fo~ ~11 n -
(~) e V(~).
where ~(~)
.-
a a ~ ~ . _-
pz, O .__
9
1
~(o~
+ o~)
o"
- r.,~
- b.,~s.
4A# aa~aCrr + 2#(aaCra~r + aara~Cr) ' A+2# fi, O dx 3 ._~ h ~ l _[_ hZ, 1 a n d h ~ t - h " t ( ., i l ) .
f.._..~ 1
.
.
.
. II
A w o r d of c a u t i o n . The variational problem "~PM(W)" is not well posed in general. More precisely, the space V(w) has to be replaced, either with the space
v M ( ~ ) "- HJ(~) • H~(~) • L~(~) if the shell is an "elliptic membrane" one (Thm. 4.4-1), or with the space VM(W) := completion of V(w) with respect to I" ]M
Asymptotic analysis of linearly elastic shells: Outline
180
[Ch. 3
where
a,/3
if the shell is a "generalized membrane" one (Thm. 5.6-1); see also Sect. 3.5. II We likewise summarize the analysis of "flexurar' shells (an example is proposed in Ex. 3.4)" T h e o r e m 3.4-3. Assume that the space
V0(w) : : { r / e Hi(w); r / -
0 on 70, Van(r/) : 0 in w}
contains nonzero fields ~ and that the scaled unknown u(e) - (~i(~)), where ui(s)(x) "- u~(x e) for all x" = r e x C -~e, admits a formal asymptotic expansion of the form
U ( ~ ) -- U 0 2r- ~U 1 -~- e 2 U 2 + C3U 3 -q--~4U4 -~- ' ' -
u~
u 1
E
u ~
, with
r 0, and u q E HI(f~), q -
2, 3, 4.
A s s u m e for definiteness that the Lamd constants are independent of s. Then in order that no restriction be put on the applied forces entering the equations satisfied by the leading term u ~ their components must be of the form
fi'e(xe) - e2f/'2(x) for all x ~ - ~rex E 12~, hi'e(x e) - sShi'3(x) for all x e - r e x E r ~ t2 r ~
where the/'unctions fi,2 E L2(f~) and h i's C L2(r+ u r_) are independent of e. This being the case, the leading term u ~ 9 --~ I~3 is independent of the transverse variable and, once identified with a vector field ~0 . -~ ~ i~3, it should satisfy the following scaled twodimensional variational problem 7)F(W) of a linearly elastic "flesural"
Linearly elastic "membrane" and "flezural" shells
Sect. 3.4]
181
shell: r e v~(~).-
{ n - (,,) e H~(~) • H~(~) • H~(~); -- 0~r/3 -- 0 o n 3'0, 7 a ~ ( ~ / )
1
~ f a~/3~rp~(~~
= 0 in w},
f p"2~lix/rady for all v / - (~/i) e VF(W),
where
~.z(n)
:=
1 ~(oo~ + o~.)
O"
-
p.z (n) := o.~ ~3 - r~, o~~3 -
r~~ b~ b~
-
.3
b.z~3. + b~ (o~~
-
r~ ~ )
aa/3ar .-'- 4Ap aaf3a~r + 2#(aaZ a~T + a,~.a~) ' A+2# pi, 2 : =
l_ f i , 2 dx3 + ,~+ t,i, 3 + hi,3 i~i,3 -- h ~'"3 (., +1). _ a n d ,~+ 1
II Several important comments are in order about these two theorems: First, it is remarkable that the linearized change of metric and change of curvature tensors have been exactly recovered (by means of their covariant components 3'af~(~/) and paf~(~/)) as the asymptotic analysis proceeded, without any explicit reference to their geometrical definitions (Thins. 2.4-1 and 2.5-1)! No less remarkable is the increase of regularity (from Hi(w) to H2(w)) that is automatically induced on the normal component ~3 by the assumption V0(w) # {0} satisfied by a "flexural" shell; see step (v) of the recursion. Another virtue of this procedure is to likewise automatically provide the expression of the scaled two-dimensional elasticity tensor of the shell, by means of its contravariant components a Olf~o'T
._.
~ 4Ap aa/3 A+2#
a~rr + 2p(a az a zr + aar a[3Z).
Asymptotic analysis of linearly elastic shells: Outline
182
[Ch. 3
A w o r d of c a u t i o n . Although reminiscent of those of the "originar' three-dimensional elasticity tensor, viz.,
A ijkl'e-- Agij'eg kl'e + t~(gik'eg jl'e -b gil, egjk, e) (recall that the Lam~ constants are assumed to be independent of ~; hence Ae = A and #e = #), these components differ in that "the coefficient A is replaced by A2)~# § 2---~,, " The asymptotic analysis thus spares
the arduous task of justifying such an intriguing "replacement"!
m
A guiding principle for carrying to its term the asymptotic analysis was that, whenever applied forces of a given order had to satisfy some compatibility conditions, they were replaced by higher order ones. It should be emphasized, however, that "unwanted" models discarded for this reason likewise deserve to be studied for their own sake, as they have been in the case of plates by Dauge, Djurdjevic & Rhssle [1998a]. We conclude this section by linking the present formal asymptotic analysis to the content of the next chapters, where what was defined in this section as the leading term of the formal asymptotic expansion of the scaled unknown u(e) will be rigorously identified as the limit of u(e) when e approaches 0 in ad hoc functional spaces; see also Sect. 3.5 for a brief review of the corresponding convergence theorems. The present distinction between "membrane" shells and "flexurar' shells depends on whether V0(w) = {0} or V0(w) ~ {0}, where the space V0(w) is defined by V0(w) - { r / - (~?i) e Hi(w); r / - 0 on 70, 7a~(r/) - 0 in w}. The convergence analyses of the next chapters will, however, entail a crucial amendment of the present classification: The decisive criterion will be instead whether Ve(w) = {0} or VF(w) ~ {0}, where the space VF(w) (already encountered in Thm. 3.4-3) is de-
fined by vF(
) - {v -(w)
e
•
•
rli = 0v~/a = 0 on 3'0, "ya~(r/) = 0 in w}, Besides, the linearly elastic membrane shells~ i.e., those corresponding to VF(w) = {{}}, will have to be themselves separated into two distinct categories, formed either by the elliptic membrane shells
Sect. 3.5]
Summary of the convergence theorems
183
(Chap. 4) or by the generalized membrane shells (Chap. 5). Those corresponding to VF(w) # {0} are the flezural shells (Chap. 6). See also Sect. 3.5. Note that we are, and will henceforth be, no longer using quotation marks in these "final" definitions. By contrast~ the quotation marks appended in this section to the definitions of "membrane" and "flexurar' shells served the purpose of emphasizing their provisional character. As already noted, problem "~M(W)" has likewise a temporary character as it will be later replaced by one among two closely related, but not identical, variational problems, respectively denoted ~:~M(W)and 7~M(W)in Chaps. 4 and 5. By chance, problem :PF(w) is already in its "final" form; it will thus be re-stated verbatim in Chap. 6. See also Sect. 3.5. 3.5.
S U M M A R Y OF T H E C O N V E R G E N C E THEOREMS
As Sect. 3.4, this section has an optional character: Its purpose is simply to briefly review the way the formal asymptotic analysis of Sect. 3.4 will be superseded in Chaps. 4 to 6 by rigorous, but substantially more delicate, convergence theorems, whose lengthy proofs may thus be spared to the reader in haste... This section also provides references to the main results. Inspired by the formal analysis of Sanchez-Palencia [1990] and by its later refinement by Miara & Sanchez-Palencia [1996] (the content of Sect. 3.4), Ciarlet & Lods [1996b, 1996d] and Ciarlet, Lods & Miara [1996] have carried out a no longer formal asymptotic analysis of linearly elastic shells. Under three distinct and carefully circumscribed sets of assumptions that cover all possible cases, they established the convergence of the scaled unknown in ad hoc functional spaces as ~ 0, justifying in this fashion the linear two-dimensional equations of linearly elastic elliptic membrane, or generalized membrane, or flexural, shells. The main conclusions of their analysis, which should be advantageously compared to those drawn in Thms. 3.4-2 and 3.4-3, are summarized below. Note that the definition of each category of shell is governed by the geometry of the middle surface, the boundary conditions, and the order of magnitude of the applied forces. Consider a family of linearly elastic shells with thickness 2~ approaching zero, with each formed by the same homogeneous and iso-
184
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
tropic material, with each having the same middle sur]ace S = 0(~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(70 ) as its middle line, where 70 C ")' = Ow and length 70 > 0. For each ~ > 0, the shells are subjected to applied body and surface forces. For each e > 0, let ui(e) denote as in Sect. 3.2 the three scaled displacements, i.e.~ the scaled covariant components of the displacement field of the points of the shell; each scaled displacement ui(e) is thus defined over the fixed domain f~ = w • 1, 1[. Assume first that 70 = 7 and that the surface S is elliptic, in the sense that its Gaussian curvature is > 0 everywhere. Then Ciarlet & Lods [1996b] have shown that, if the applied body force density is O(1) and the applied surface force density is O(e) with respect to 6 in the sense of Tam. 3.4-2, the scaled unknown u(e) = (ui(e)) converges in Hl(f~) • Hl(f~) • L2(f~) as ~ --+ 0 to a limit u, which is independent of the transverse variable (this convergence result has been recently improved by Mardare [199Sa], who obtained art O(~ 1/6) error estimate). Furthermore, the average ~ := 89f~l u dx3 satisfies the s e a l e d t w o - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 7~M(W) of a l i n e a r l y elastic elliptic m e m b r a n e shell:
r e
•
•
f aaf3~r 7at (~) 7af3 (r/) V/-ddy
= ~pi'~
for all rl - (rli) C VM(w),
where the functions a afl~r, ")'aft(r/), and pi, O are defined as in Thin. 3.4-2. Notice, however, that the space VM(W) does not coincide with the space V(w) found in Thin. 3.4-2! The detailed proof of this convergence result, which occupies most of Chap. 4, is concluded with Thin. 4.4-1. Naturally, these equations have to be "de-scaled'~ so as to be expressed in terms of "physical" unknowns and data. To this end, we let ~e(y) = ~(y) for y E ~ (in view of the scalings ue(x `) = u(e)(x) for x e - rex C f~); it is found in this fashion that the de-scaled
Summary of the convergencetheorems
Sect. 3.5]
185
unknown ~e satisfies the two-dimensional variational problem:
Ce E VM(W),
L aa~~176 = f p~'~wV~dy fo~ an ,1 = (w) Jw
VM(W),
where 4)~ePe aa~a~r + 2pe(aa~a~r + a a r a ~ ) , ,Xe + 2# e o
/
~
~
9
~
o
f',~ d ~ + h ; ~ + h ',~ ~na h~ ~ : : h ',~(.
+~)
e
Note that these equations are of the form announced in Sect. 2.7. As shown by Ciarlet & Lods [1996a] and Ciarlet & SanchezPalencia [1996] (under rather stringent regularity assumptions, later substantially relaxed by Lods & Mardare [1998a]; cf. Thm. 2.7-2), the conjunction of the two assumptions "V0 - V" and "S is elliptic" provides a first instance where the space VF(W) "-- {r/--(r/i) e Hi(w) • HI(w) • H2(w); ~?i - 0v~73 - 0 on 70, 7a~(~/) = 0 in ~} contains only the function r / = 0. The space VF(W) was introduced by Sanchez-Palencia [1989a]. Then Ciarlet and Lods [1996d] studied all the "remaining" cases where VF(w) = {0}, for instance, when the surface S is elliptic but 0 < length 70 < length V (as shown by Vekua [1962] and Lods & Mardare [1998a]), or when S is a portion of a hyperboloid of revolution and V0 is "large enough" (as shown by Sanchez-Palencia [1993] and Mardare [1998c]), etc. To give a flavor of their results, consider the most common case, where the space Vo(c,,) := {n ~ Hi(,,,); ,7 - o on 7o, ~',l~(n) = 0 in ~,},
186
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
which contains VF(w), "already" reduces to {0}, or equivalently, when the semi-norm
l" [M'~--(~i)~ -+
1271M"- { ~ lT,~(n)i~),,,,}
1/2
a,/3
becomes a norm over the space := { n e
n - 0 on 70).
In this case, if the applied forces are "admissible" in a specific sense (too technical to be reproduced here; cf. Sect. 5.5), the average !2 fl_ 1 u ( e ) d x 3 converges as e -4 0 in the space
V~M(W) '-- completion of V(w) with respect to I" IM. A convergence result also holds for the scaled unknown u(e) itself (again too technical to be reproduced here). Furthermore, the limit satisfies the scaled t w o - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 79~M(W) of a l i n e a r l y elastic g e n e r a l i z e d m e m b r a n e shell:
r e V~M(~O)and B~M(r O) -- L~M(n) for all r/E
V~M(w),
where B~M is the unique extension to V~M(W) of the bilinear form (r
already found in the variational equations of an elliptic membrane shell (see supra) and L~M 9V~M(W) -4 IR is an ad hoc linear form, determined by the behavior as e -4 0 of the admissible forces. Notice that the space V~M(W) neither coincides with the space V(w) found in Thm. 3.4-2! The detailed proof of this convergence result, which occupies most of Chap. 5, is concluded with Thm. 5.6-1. A particularly noteworthy feature of such variational problems is that they provide instances of "sensitive problems", introduced and studied by Lions & Sanchez-Palencia [1994, 1996, 1997].
Summary of the convergence theorems
Sect. 3.5]
187
Remark. In the "last", but seemingly uncommon, case where VF(w) -- {0} but I 9IM is a "genuine" semi-norm over the space V(w), a similar convergence result can be established, but instead in the completion ~r~M(W) with respect to I" IM of the quotient space .
-
The limit found in this fashion in the space ~r~M(W) then satisfies the scaled two-dimensional variational problem of a linearly elastic generalized membrane shell "of the second kind"; see Thm. 5.6-2. II Finally, Ciarlet, Lods & Miara [1996] have considered the case where the space VF(w) contains non-zero functions. This assumption is satisfied in particular if S is a portion of a cylinder and 0(70) is contained in one or two generatrices of S (as shown by Lods & Mardare [1998a]; see also Ex. 3.4) or if S is contained in a plane, i.e., the shell is a plate. They then showed that, if the applied body force density is O(e 2) and the applied surface force density is O(e 3) with respect to e in the sense of Thm. 3.4-3, the scaled unknown u(e) = (ui(e)) converges in Hi(12) to a limit u, which is independent of the transverse variable. Furthermore, the average ~ "- 1 f ! l u din3 satisfies the s c a l e d twod i m e n s i o n a l v a r i a t i o n a l p r o b l e m T'F(w) of a l i n e a r l y elastic f l e x u r a l shell: r E V f ( w ) : = {r/--(r/i) E Hi(w) • HI(w) x H2(w); 7/i - 0v7/3 - 0 on 3'0,7a#(r/) - 0 in w},
1Z for all r / = (r/i) E VF(w),
where the functions a a#~rr, pa#(r/), and pi, 2 a r e defined as in Thm. 3.4-3. Notice that, for once, the "right" space Vv(w) was already found in this theorem! The detailed proof of this convergence result is the object of Thin. 6.2-1. In order to acquire physical significance, these equations must be de-scaled, like those of a membrane shell. The de-scaling then shows that the de-scaled unknown ~e satisfies the two-dimensional
188
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
variational problem:
r ~ vp(~), C3 f aaf~ar, e
Y J~
P
par(~e)pa#(17)~/-a dy - J~ Pi'e~Ti~/a dy for all r / = (~/i) e VF(W),
where
~~,~
._.- 4 ~ , ~ ~ ~
+ 2,~(~~ ~ + ~~'~),
Ae + 2# e 9
f ~
9
o
.
.
;',~ : : j _ I ',~ d ~ + h ; ~ + h ''~_ ~ a h~ ~ : : h ',~(., +~). $
Note that these equations are of the form announced in Sect. 2.6.
All these convergence results heavily rely on the inequalities of Korn's type on a surface established in Chap. 2, first for a general surface (Thm. 2.6-4), then for an elliptic surface (Thm. 2.7-3). Not only do these inequalities play a crucial r61e for establishing the existence and uniqueness of the solution to the "limit" two-dimensional equations (as already shown in Sects. 2.6 and 2.7), but they also constitute the basis for proving "three-dimensional" inequalities of Korn's type, either for a family of general shells or for a family of elliptic membrane shells, that are themselves the keystones of the convergence proofs. These convergences for the displacements have been recently complemented by Collard & Miara [1999] who have established the convergence of the corresponding linearized stresses, by Giroud [1998] who has considered linearly elastic shells made of nonhomogeneous and anisotropic materials, and by Xiao Li-ming [1998, 1999a] who has considered time-dependent linearly elastic membrane and flexural shells. Ciarlet & Lods [1996c, 1996d] have also fully justified the twodimensional Koiter equations for a linearly elastic shell, described at the beginning of Sect. 2.6, by combining the above convergences with
Sect. 3.5]
Summary of the convergence theorems
189
former asymptotic analyses of Koiter's equations due to Destuynder [1985] and Sanchez-Palencia [1989a, 1989b]. More specifically, Ciarlet Lods [1996c, 1996d] have shown that the average 1 fe_e ue dx~ and the solution ~ g of Koiter's equation have for each kind of shell the same asymptotic behavior as ~ -~ O.
The detailed proof of this result is the object of Thms. 7.2-1 to 7.2-3. A somewhat unexpected phenomenon, identified by Ciarlet [1992a, 1992b], arises when "a shell becomes a plate": Suppose first that the middle surface S "converges" in a natural sense to a planar domain, while the thickness 2~ is held fixed: Then the solution of the three-dimensional shell equations converges toward that of the threedimensional plate equations (Ex. 3.5). If then e approaches zero, the solution of the three-dimensional plate equations converges toward the solution of the two-dimensional plate equations, i.e., those of the linear Kirchhoff-Love theory, which simultaneously incorporate membrane and flexural terms (Vol. II, Thm. 1.4-1), regardless of the geometry of S and of the boundary conditions. By contrast, suppose first that e approaches zero while the middle surface is held fixed. Then the above ~onvergence theorems show that the three-dimensional solutions converges towards a two-dimensional limit that satisfies either the elliptic membrane, or the generalized membrane, or the flexural equations, according to the geometry of S and to the boundary conditions. If then the surface S becomes planar, the limit equations obtained in this process are thus either the membrane, or the flexural, plate equations. In other words, "the two passages to the limits do not commute"! Additional considerations about this puzzling question are found in Sanchez-Palencia [1994] and
Loa [1996]. As shown by Busse, Ciarlet & Miara [1997], the two-dimensional equations of a linearly elastic "shallow" shell in curvilinear coordinates (these equations are given in Sect. 7.6) can be also justified by means of a convergence theorem, which is reminiscent of that used for a linearly elastic shallow shell in Cartesian coordinates (Vol. II, Chap. 3). We recall that, according to the definition justified by Ciarlet & Paumier [1986] in the nonlinear case and by Ciarlet & Miara [1992a] in the linear case, a shell is "shallow" if, in its reference configuration, the deviation of its middle surface from a plane is of the order of its thickness (see again Vol. II, Chap. 3 for details).
190
Asymptotic analysis of linearly elastic shells: Outline
[Ch. 3
EXERCISES
3.1. Let the assumptions be as in Thin. 3.3-2. (1) Show that the uniform positive definiteness of the two-dimensional elasticity tensor on S can be also established in a manner reminiscent of that used in Thm. 1.8-1 for establishing the uniform positive definiteness of the three-dimensional elasticity tensor. (2) Show that yet another proof consists in adapting that suggested in Ex. 1.8 in the three-dimensional case. 3.2. Let a a ~ r denote the contravariant components of the twodimensional elasticity tensor of a shell (Thm. 3.3-2), and let # > 0 be given. Show that there exists a constant A0(/z) < 0 such that, if X > )~0(#), there exists a constant c > 0 such that E
Iraqi2 -< caa/3~r(y)tarta[3
a,/3
for all y C ~ and all symmetric matrices (tat3). Remark. Such a uniform positive definiteness of the two-dimensional elasticity tensor is established in Thm. 3.3-2 under the assumptions )~ >_ 0 and # > 0. 3.3. Consider a linearly elastic shell whose middle surface is a portion S - 0(~) of a sphere, with (Fig. 2.1-3) := {(~, , ) ~ R 2 ; o < u 2 + v 2 < r2},
( O(u, v) :=
2R2u 2R2v u 2 + v 2 - R 2) u 2 + v ~ + R 2' u~ + v 2 + R 2' Ru2 ~- v 2 + R 2 '
i.e., stereographic coordinates are used for representing the surface S. This spherical shell is subjected to a boundary condition of place along its entire lateral face, i.e., 7o = 7. (1) Show that such a shell is a "membrane" shell, according to the definition given in Thm. 3.4-2. Hint: Use Ex. 2.13. (2) Compute explicitly the functions a a/3~r, 7at3(r/), and a appearing in the formulation of the associated variational problem "79M(W)'' (Thm. 3.4-2).
Ezercise8
191
(3) Carry out the same computations when the surface S is represented by means of spherical coordinates (Fig. 2.1-3). Remark. The expressions sought in (2) and (3) immediately follow from Ex. 2.1 (2). 3.4. Consider a linearly elastic shell whose middle surface is a portion S - 0(~) of a circular cylinder, with (Fig. 2.1-2)
:= (R cos
R sin
z).
This cylindrical shell is subjected to a boundary condition of place along both "vertical" portions of its lateral face, i.e., 70 - {(0, z) C I~2; 0 _< z _ h} U {(Tr, z) E I~2; 0 _ z _< h}. (1) Show that such a shell is a "flexural" shell, according to the definition given in Thm. 3.4-3. (2) Compute explicitly the functions a a ~ r , 7a~(r/), pa~(~/), and a appearing in the formulation of the associated variational problem T'F(w) (Thm. 3.4-3). Remark. The sought expressions follow from Ex. 2.1 (1). (3) Assume now that this shell is subjected to a boundary condition of place along its entire lateral face, i.e., that V0 - V. Is it still a "flexural" shell? 3.5. Let there be given an injective mapping 0 = (~i) E C2(~; I~3) such that the two vectors 0a0 are linearly independent at all points of ~. Following Ciarlet [1992a], consider a one-parameter family of mappings O(t) - (Oi(t)) 9 w ~ I~3 defined for 0 ~ t ~ 1 and (yl, y2) E ~ by e
(t)(yl, y2) : te (y , y2) + (1 - t)y ,
~3(t)(Yl, Y2) -- t~3(Yl, Y2), and assume for simplicity that the contravariant components of the applied forces are independent of t. Let u~(t) E V ( ~ e) denote for 0 < t ~ 1 the solutions of the corresponding variational problems ~(~2~), as described in Sect. 3.1 but now parametrized by t.
192
Asymptoticanalysis of linearly elastic shells: Outline
[Ch. 3
Show that the mapping t E [0, 1] -+ ue(t) E V(I2 e) is continuous. In particular, ue(t) --+ ue(O) in Hl(f~ e) when t -+ 0 +, i.e., when "the shell becomes a plate", where ue(O) E V(f~ e) satisfies the threedimensional equations of a clamped plate:
ffl BiJkt, eei~(ue(O))e~t(ve)dxe- ffl fi'e edxe+ f 9
~ u r ~_
hi'e iecIF
where
Hint: Observing that the Lax-Milgram lemma can be proved by means of the contraction mapping theorem (Lions & Stampacchia [1967]), use the local uniform ellipticity of the associated bilinear forms to show that the contractions are also loeaUy uniformly contraeting.
CHAPTER 4 LINEARLY SHELLS
ELASTIC
ELLIPTIC
MEMBRANE
INTRODUCTION
A linearly elastic elliptic membrane shell is one whose middle surface S = 8(~) is elliptic, i.e., that has a Gaussian curvature that is everywhere > 0, and which is subjected to a boundary condition of place along its entire lateral face (Sect. 4.1). The infinitesimal rigid displacement lemma on an elliptic surface (Thm. 2.7-2) thus shows that an elliptic membrane shell provides one instance where the space VF(6Q) -- {17 --(~7i) e Hl(ag) x H i ( w ) x H2(Qg); i T i - 0v~73 - - 0 oil ")'0, =
0
reduces to {0}. The other instances where Vy(w) = {0} are provided by the "generalized membrane" shells, which will be studied in the next chapter. As we shall see, whether VF(w) = {0} or VF(W) r {0} is the fundamental criterion for classifying linearly elastic shells: Those corresponding to VF(w) = {0} constitute the membrane shells, while those corresponding to Vv(w) # {0} constitute the flezural shells (Chap. 6). The purpose of this chapter is to identify and to mathematically justify the two-dimensional equations of a linearly elastic elliptic membrane shell, by establishing the convergence in ad hoc functional spaces of the three-dimensional displacements as the thickness of such a shell approaches zero. More specifically, consider a family of linearly elastic elliptic membrane shells with thickness 26 approaching zero and with each having the same middle surface S = 8(~). As in Chap. 3, the associated three-dimensional problems, posed in curvilinear coordinates over the sets Fte = w• - e, s[, are transformed for each e > 0 into equivalent problems, but now posed over the set ~ = w • 1~1[, which
194
Linearly elastic elliptic membrane shells
[Ch. 4
is independent of e. This transformation relies in a crucial way on appropriate scalings of the unknowns u~ (the covariant components of the displacement field) and assumptions on the Lam6 constants
)i ~ and Ize and on the contravariant components fi, e of the applied body forces (for simplicity, we assume in this introduction that there are no applied surface forces). More specifically, we define the scaled unknown u(e) = (ui(e)) by letting u~(x e) - ui(e)(x) for all x ~ - ~r~x e ~e, where 71"e(Xl, X2, X3) (Xl, X2, EX3). Guided by the formal asymptotic analysis of Sect. 3.4, we then assume that there exist constants ,k > 0 and # > 0 and functions fi E L2(f~) independent of e such that -
-
%e=)~
and
#e=#,
fi,~(x ~) - f i ( z ) for all x ~ - ~'ex E ~26. It is found in this fashion that the scaled unknown u(e) satisfies a variational problem of the form (Thin. 4.1-1):
u(e) -- (ui(~)) E V(f~) -- {v -- (vi) E Hl(f~); v - 0 on "y • f n AiJkl(e)eklll(e; u(e))eillj(e; v ) ~ ) =
[--I,1]},
d~,
fo
for all v e V(f~),
where, for any vector field v = (vi) E Hl(f~), the scaled linearized strains eillj(e; v) -- ejlli(e; v ) E L2(12) are defined by"
~jl~(~; ~) - ~I ( o ~ + o ~ ) ~ll~ (s;
")
1(i
= ~ ~ o3,~ + o ~ 3
)
r P~
(e)vp ,
- r~3 (e)~,
eall3(e; v) = e103v 3. To begin with, we establish several properties of "averages with respect to the transverse variable" that will be of frequent use (Thm. 4.2-1). T h e n we prove a crucial three-dimensional inequality of Korn's type (Thin. 4.3-1): Given a family of linearly elastic elliptic membrane shells with each having the same (elliptic) middle surface S -- 0(~),
Introduction
195
there exists a constant C such that, for e small enough and for all = (~) E v(~),
~l~llj(~;~)l~,~
}1/2
9
i,j
s
The restriction that the shells be elliptic membrane ones is essential here! Equipped with these preliminaries, we then establish the main result of this chapter (Thm. 4.4-1), by showing that the family (u(e))~>0 strongly converges in the space Ht(fl) • Ht(f~) x L2(~t) as r -+ 0 and that u = lime_,0 u(e) is obtained by solving a two-dimensional problem. More specifically, we show that the limit u is independent of the transverse variable and that ~ - ~1 f_l 1 u dx3 satisfies the following (scaled) two-dimensional variational problem of a linearly elastic membrane shell:
r -(r
e v M ( ~ ) - H ] ( ~ ) • H0~(~) • L~(~),
1
O"
~ o , ( ~ ) - ~ ( 0 , , ~ + oo~,) - r o , , ~ - bo,,~, aa~ r _
4)~p aa~ a,Tr + 2#(aa(r a~r + aar a~,r) ' )~+2#
the functions 7a~(r/) and a a~ar being respectively the covariant components of the linearized change of metric tensor associated with a , displacement field yia ~ of the middle surface S (Sect. 2.4) and the contravariant components of the (scaled) two-dimensional elasticity tensor of the shell. To conclude this chapter, we review the existence, uniqueness, and regularity properties of the solution to the above variational problem and we describe the associated minimization problem and boundary value problem (Thm. 4.5-1). We also rewrite these two-dimensional equations and the fundamental convergence theorem in terms of descaled unknowns and data, thus providing a justification in terms of
196
Linearly elastic elliptic membrane shells
[Ch. 4
"physical" quantities of the two-dimensional equations of a linearly elastic membrane shell (Thins. 4.5-2 and 4.6-1). In particular, the limit displacement field ~ a i of the middle surface is such that ~e _ (i~) satisfies the following minimization problem: ~e C VM(W) and j ~ ( ~ e ) = inf j ~ ( r / ) , r/eVM(tO)
where the two-dimensional energy j ~ " V M ( w ) ~ I~ is defined by
J ~ ( Y ) = -~
e a a~ 'Tr' eVar ( Y ) Va~ ( Y ) v/-a d y _
L{f_
}
fi, edxe3 ~Tiv/-ddy, e
and a a[J(rT, e =
Ae
4.1.
+ 2# e
aal3 aO'r + 2/.te (a a~
+ a ar a13O').
LINEARLY ELASTIC ELLIPTIC MEMBRANE SHELLS: D E F I N I T I O N , E X A M P L E , A N D A S S U M P T I O N S O N T H E DATA; T H E THREE-DIMENSIONAL EQUATIONS OVER A D O M A I N I N D E P E N D E N T OF g
Let w be a domain in ]~2 with boundary 7 and let 0 C C2(~; I~3) be an injective mapping such that the two vectors OaS(y) are linearly independent at all points y C ~. A linearly elastic shell with middle surface S = 8(~) is called a l i n e a r l y e l a s t i c elliptic m e m b r a n e shell if the following two conditions are simultaneously satisfied (the definitions and notations are those of Sect. 3.1): (i) The shell is subjected to a (homogeneous) b o u n d a r y cond i t i o n of place along its entire lateral face 0 ( 7 • I - e , e]), i.e., the displacement field vanishes there; equivalently, 70 -- "Y.
(ii) The middle sur]ace S is elliptic, in the sense that there exists a constant c such that
i~12 ~ alba~(y)~a~[ c~
for a11 y E ~ and a11 (~a) E R 2
Sect. 4.1]
Definition, ezample, and assumptions on the data
197
Fig. 4.1-1: A linearly elastic elliptic membrane shell. A linearly elastic shell whose middle surface S = 8(~) is a portion of an ellipsoid E, and which is subjected to a boundary condition of place (i.e., of vanishing displacement field) along its entire lateral face 19(7 • [-e, e]) (darkened on the figure), provides an instance of an elliptic membrane shell. Note that stereographic coordinates (Fig. 2.1-3) afford the representation of such a surface S. Elliptic membrane shells provide a first instance (70 = 7 and S is elliptic) where the space VF(w) = {~ = ('7,) e Hi(W) • Hi(oJ) • H2(oJ); ~, = 0~Os = 0 on 70, 7o~C,7) = 0 in
~},
which is the key to the classification of linearly elastic shells, reduces to {0} (in fact, a larger space "already" reduces to {0} in this case; cf. Tlun. 2.7-2). The "generalized membrane shells" (Chap. 5) exhaust all the remaining cases where V~(~) = {0}.
where the functions ba~ : ~ --~ I~ are the covariant c o m p o n e n t s of the c u r v a t u r e tensor of S (this definition was a l r e a d y given in Sect. 2.7). This a s s u m p t i o n m e a n s t h a t the Gaussian curvature of S is everywhere > 0; equivalently, the two principal radii of curvature are either both > 0 at all points of S, or both < 0 at all points of S (see Sect. 2.2 for a detailed exposition of these notions). A shell s u b j e c t e d to a b o u n d a r y c o n d i t i o n of place along its entire lateral face a n d whose middle surface is a p o r t i o n of an ellipsoid provides a n e x a m p l e of a linearly elastic m e m b r a n e shell (Fig. 4.1-1).
Let there be given a linearly elastic elliptic membrane shell such that 8 C C2'1(~; IR3). T h e n the following inequality of K o r n ' s t y p e on the elliptic surface S = 8 ( ~ ) holds ( T a m . 2.7-3): There exists a
Linearly elastic elliptic membrane shells
198 c o n s t a n t CM
[Ch. 4
such that
11,7~11~~,~
~ ,~3,o,~
_< CM
~
I~,~(.11o2, ,,,
fo~ ~11. = (vi) e v M ( ~ ) : = H~(~) • H~(~) • L2(~), where the functions 1
7 ~ ( n ) := ~ ( 0 ~ , + 0 ~ )
O"
- r~.
- ba/3r/3
are the covariant components of the linearized change of metric tensor associated with a displacement field ~ia 2 of the surface S; the subscript "M" announces that VM(W) is the functional space over which the limit two-dimensional equations of such a membrane shell are posed (Thin. 4.4-1). This inequality, consequence of the definition of an elliptic membrane shell, will be the key to the ensuing analysis of this chapter. A w o r d of c a u t i o n . The definition of a linearly elastic elliptic membrane shell thus depends only on the subset of the lateral ]ace where the shell is subjected to a boundary condition of place (this subset should be the entire lateral face) and on the "geometry" of its middle surface (its Gaussian curvature should be > 0 everywhere). This definition is thus independent of the particular system of curvilinear coordinates employed for representing the surface S. m If assumptions (i) and (ii) are satisfied and O e C2'1(~; IR3), we thus have
{ . - ( y i ) e H~(w) • H~(w) x L2(w); 7 a ~ ( . ) -
0 in w} - {0}.
Hence linearly elastic elliptic membrane shells provide a first instance (70 = 7 and S is elliptic) where the space (already introduced in Thin. 3.4-3)
VF(~) "-- {n = (~) e HI(~) • H~(~) • H2(~); ~?i = Ou~3 = 0 on 70, 7a~3(~) = 0 in w} a fortiori reduces to {0) (we recaU that Ov denotes the outer normal derivative operator along 7; the subscript "F" reminds that this space is central to the study of flezural shells, undertaken in Chap. 6).
Definition, ezample, and assumptions on ~he da~a
Sect. 4.1]
199
The importance of this observation lies in that the issue of whether VF(w) = {0) or VF(w) ~ {O) is the basis of the classification of linearly elastic shells: A shell is caUed a l i n e a r l y e l a s t i c m e m b r a n e shell if VF(w) = {0) or a l i n e a r l y e l a s t i c f l e x u r a l shell if VF(~)
# {0).
In this direction, note that elliptic membrane shells far from exhaust all the instances where VF(w) = {0), the remaining instances corresponding to the generalized membrane shells studied in the next chapter. The formal analysis of Sect. 3.4 then naturally leads us to make the following s e a l i n g s of t h e u n k n o w n s and a s s u m p t i o n s o n t h e d a t a for a family of linearly elastic elliptic membrane shells with each having the same (elliptic) middle surface S = 0(~) as their thickness 2~ approaches zero. First, we define the s c a l e d u n k n o w n u ( e ) - (ui(e)) by letting
u~(x ~) "-- ui(6)(x) for all x e - 7r~x e ~ .
Next, we require that the Lamd constants and the applied body and surface force densities be such that
)~ - )~ fi,~(x e) - f i ( x )
h',~(~ ~) - ~h'(~)
and for all
#e _ #, x ~ = 7r~x e a ~,
fo~ ~11 ~ - ~
e r ~W u r ~- '
where the constants )~ > 0 and ~ > 0 and the functions f i E L2(f~) and h i E L2(r+ u r_) are independent of e (Fig. 3.2-1 recapitulates the definitions of the sets ~e, f~, r ~ , r + , r ~_, a n d r _ ) . Remark. For notational brevity, the functions f i and h i stand for the functions that were respectively denoted fi, 0 and h i' 1 in Sect. 3.4. II As an immediate corollary to Thm. 3.2-1 (simply corresponding to p -- 0), we obtain the problems satisfied by the scaled unknown over the set fl, thus over a domain that is independent of ~:
200
Linearly elastic elliptic m e m b r a n e shells
[Ch. 4
T h e o r e m 4.1-1. Let w be a domain in I~2, let 0 E C2(~;IR 3) be an injective mapping such that the two vectors a a - - OaO are linearly independent at all points of-~, and let eo > 0 be as in Thm. 3.1-1. Consider a family of linearly elastic elliptic membrane shells with thickness 2e with each having the same elliptic middle surface S - 0(~). Let the assumptions on the data be as above. Then, for each 0 < e <_ co, the scaled unknown u(e) - (ui(e)) satisfies the following sealed t h r e e - d i m e n s i o n a l variational p r o b l e m 79(e; f~) of a linearly elastic elliptic m e m b r a n e shell:
u(e) e V ( ~ ) ' -
{v - (vi) e Ht(~); v - 0 on 7 • [-1, 1]},
- ~ f i v i v ~ ( e ) dX + f r
hivi~
dF for all v e V(~t),
+ur_
where the scaled linearized strains eil2j(e; v) are given by
-
1 (i)~va + Oars)
-
v
l ( 1 0 3 v a + O a v 3 ) - Y:s(e)vz,
ealla(e; v) _ _10sva, g
and the "scaled" ]unctions Aiikt(e), rib(e), g(e) are defined as in Thin. 3.2-1. I
Our main objective in this chapter consists in analyzing the behavior of the solutions u(e) e Ht(ft) of problems ~(e; ~2) as e --+ O. To this end, we begin by proving various "analytical" preliminaries, which complement the "geometrical" and "mechanical" preliminaries proved in Sect. 3.3. Those of Sect. 4.2 are common to all types of linearly elastic shells while the inequality of Sect. 4.3 is special to elliptic membrane shells.
Sect. 4.2]
4.2.
Averages with respect to the transverse variable
AVERAGES WITH RESPECT TRANSVERSE VARIABLE
201
TO THE
Averages with respect to the transverse variable x3 play a fundamental r61e in this chapter as well as in the next chapters. If v, or v, are real-valued, or vector-valued, functions defined a.e. over f~ = w • 1, 1[ (the abbreviation "a.e." stands for "almost everywhere"), these a v e r a g e s ~, or ~, are the real-valued, or vector-valued, functions defined a.e. over w by letting
1/
if
v(y, x3 ) d~,s
for almost all y E w whenever these definitions make sense (Thm. 4.2-1 (a) provides such an instance). The next theorem gathers the properties of averages that are relevant in the ensuing analysis. Note that the functions 7~(~/) introduced in part (d) are the covariant components of the linearized change of metric, or strain, tensor, associated with an arbitrary displacement field ~7iai of the surface S (Sect. 2.4). We follow here Ciarlet & Lods [1996b, Lemma 3.2]. Observe that the convergence established in part (d) holds whether the surface S = 0(-~) is elliptic or not. T h e o r e m 4.2-1. Letw be a domain in IR2 and let ft - w • 1[. (a) Let v E Lg(f~). Then ~(y) as defined supra is finite for almost all y E w, the function ~ defined in this fashion belongs to LZ(w), and 1
I 10, <__
If 03v - 0 in the sense of distributions, i.e., if fn vO3qodx - 0 for all qo C 9(f~), then v does not depend on x3, and v(y, x3) : ~(y) for almost all (y, x3) E C~.
1
Linearly elastic elliptic membrane shells
202
[Ch. 4
Let Vo denote a &y-measurable subset of V. If v - 0 on V0 x [-1, 1], ~
~-
0 o~ ~0; i~ , ~ i ~ e ~ ,
~ e H I ( ~ ) iI ~ - 0 o~ ~ • [ - ~ , 1].
(c) Let (v(s))~>0 be a sequence o f f unctions v(s) e Hl(f~) and let E L2 (w) be such that
03v(s) -+ 0 in L2(f~) and ~(s) --+ ~ in L2(w) as ~ -+ O. Then v(e) --+ ~ in L2(f~) as s ~ 0, where the function ~ E L2(w) is identified with a function in L2(f~)
(d) Let the assumptions on the mapping 0 and the definitions be ~ i~ T a m . 3.3-1. L ~ t h ~
b~ g i , ~
~ ~q~~
(~(~))~>0 oI , ~ o ~
fields v(s) - (vi(s)) e ttl(f~) that is bounded in L2(f~), let the scaled linearized strains eall~(s; v(s)) be defined as in Tam. 3.2-1, and let 1
~o~(~) . - ~ ( o ~ , o + oo,~) - r i , , ~
fo~ ~ y ~ -
- bo,~
(~) c H~(~) • H~(~) • L2(~). T ~
{eall~(s; v(s)) -Va~(v(s))} --+ 0 in L2(w) as s --+ O. Proof. (i) Let v e L2(f~). For almost all y e w, the function v(y, .) belongs to the space L 2 ( ] - 1, 1 D by Fubini's theorem. For
such points y, Cauchy-Schwarz inequality gives
I
f
1
v(y,
x3)dx3
12 f < 2
Iv(y,
X3)[2 dx3 < +00.
1
Hence [~(y)l 2 dy < -~ and thus
1Iv(y, x3)l 2 dxs
dy = -~
dx,
I~lo,~ ~ ~2 [vl~ a" If Oar = 0 in the sense of distributions, there exists 7/ C L2(w) such that v(y, x3) = y(y) for almost all (y, xs) e f~ (Ex 4.1). But = 7/in this case, and thus (a) is proved.
Sect. 4.2]
203
Averages with respect to the transverse variable
(ii) Let v q,: ~ --+ I~ be vanishes on 7 outer normal gives
E H I ( ~ ) . Given an arbitrary function ~o E D(w), let defined by q,(y, x3) = ~o(y) for all (y, x3) E f~. Since ,]i x [-1, 1] and the "horizontal" components of the unit vector vanish on r + lj r _ , Green's formula in H t ( ~ )
/a v Oaff~dz = - /a Oav q~dz. Since v E L2(a), Oar E L2(~), and ,~ and 0a(I' are independent of z3, Fubini's theorem yields
~ ~O,~ody - - ~ Oar ~ody; hence ~ E Hi(w) and 0a~ - Oar. These relations, combined with the inequality proved in (i), imply that II~llt,o, _< ~2211vllt,a. Let 70 be a dT-measurable subset of 7, and assume in addition that v = 0 on F0 "- 70 x [-1, 1]. There exists a sequence (q~k)~=t of functions q,k E C~176 such that q)k _+ v in H t ( ~ ) as k --+ oo. Hence, --h
9 klr ~ --+ 0 - vlr o, in L2(F0) as k -+ oo; consequently, !li 17o --+ 0 in
II~kllL~(~o)<_ ~ll~kllL~(ro),
L2(701 since in L2('y0). This proves (b).
and thus ~kl7o --+
0 -
-v l ~ o
(iii) Let v E L2(~) be such that its derivative 03v in the sense of distributions belongs to L~(~). Then we may write, for almost all (y, ~) e ~• ] - 1, 1[ (E~. 4.1),
.(y. ~) - .(y. -1) + f ' 1 Oar(y,
x3)
dx3,
and thus
17
if(i;
v(y, s)ds - v(y, -1) + ~
1
t
03v(y, x3)dx3) dr.
Hence the following identity holds:
lf)(/~_
~(y. ~) - ~(y) + f_' 1
O~v(y, z~)dz~ - -~
1
1 03v(y, z 3 ) d z s
204
[Ch. 4
Linearly elastic elliptic membrane shells
This identity, combined with the triangular inequality and the relations
I~(y)l 2 ds dy
=
21~10,~,
1
03v(y, x3)dx3 1
/.{/_:l;(/j 1
l
ds dy < 4103vl 20 , ~ -
-
)Idt ds }dy < 8103v10',a,
03v(y, X3) dT,3 1
shows that
The desired convergence v(s) -+ ~ in L2(f~) is then proved by letting v -- v(e) - ~ in the last inequality, and thus (c) is proved. (iv) From (ii), we infer that 1 ~(o~,,(~) + o~o(~)) - 1 (Oav~(s) + O~va(s)).
Hence in order to establish the convergence announced in (d), it remains to establish that
{r~,(~)~(~) - r~,(o)~(~)} -~ 0 in L~(~) ~s ~ -~ 0, where the functions r ~ ( 0 ) := r ~af3 and r af~ 3 (0) := bar3 are independent of x3. But the behavior of the functions r : , ( ~ ) as : -, 0 (Thm. 3.3-1) and the inequality !:10,o~ < ~21vl0,n of part (i) together imply that
I r ~ ( ~ ) ~ ( ~ ) - r~(0)~(~)lo,~ -Ir~,(~)v~(~)- r~(0),~(:)10,~ f f . . _
1
(0)~(~)10
_
i~
and the conclusion follows from the assumed boundedness of the sequence (v(s))e>o in the space L2(f~). This proves (d). II Remark. It is easily verified that the result of part (d) still holds if O E C2(~; I~s) ( it was assumed that 0 E C3(~; I~3) in Thm. 3.3-1, but then sharper asymptotic behaviors of the scaled Christoffel symbols r : , (~) were sought). II
Sect. 4.3]
4.3.
A three-dimensional inequality of Korn's type
205
A THREE-DIMENSIONAL INEQUALITY OF KORN'S TYPE FOR A FAMILY OF LINEARLY ELASTIC ELLIPTIC MEMBRANE SHELLS
Let 70 be a subset of 7 with length 70 > 0. If no particular assumption is made on the geometry of the surface S, it will be shown in the next chapter (Thm. 5.3-1) that there exist constants el > 0 and C > 0 such that, for all 0 < e < ez and all fields v - (v~) E H1(12) vanishing on the set 3'0 • [-1, 1], 1/2
. Ilvili *
C
2
-<
i ltj( ; v)io,, *,3
This relation is a three-dimensional inequality of Korn's type for a family of "general " shells, the scaled strains eil]j(e; v) "in curvilinear coordinates" replacing the classical strains 89 + Oivj) "in Cartesian coordinates". It is remarkable that, for a family of elliptic membrane shells, the "constant" C/e may be replaced by a constant that is independent of e, at the expense, however, of replacing I]v311z,~ by ]v310,~ in the lefthand side. More specifically, another three-dimensional inequality of Korn's type holds for such a family, which plays a key rSle in the proof of the convergence theorem (Thm. 4.4-1): It is used there to establish the fundamental a priori bounds that the family (u(e))e>0 satisfies. The following result is due to Ciarlet & Lods [1996b, Thm. 4.1]. T h e o r e m 4.3-1. Assume that 0 E C3(~; I~3) and let eo > 0 be defined as in Thm. 3.1-1. Consider a family of linearly elastic elliptic membrane shells with thickness 2e with each having the same elliptic middle surface S = 0(-~). Define the space
V ( ~ ) " - {v - (vi) e H I ( ~ ) ; v - 0 on 7 • [-1, 1]}.
Then there exist a constant el satisfying 0 < ex <_ eo and a constant C such that, for all 0 < e <_ ez, the following t h r e e d i m e n s i o n a l i n e q u a l i t y of K o r n ' s t y p e for a f a m i l y of l i n e a r l y
206
[Ch. 4
Linearly elastic elliptic memb~'ane shells
elastic elliptic m e m b r a n e shells holds:
{
2) 1/2 _< c
~ $,3 o
r
2 I~iiJ(~; vllo, a
}1/2
,
where the scaled linearized strains eill./(e; v) are defined by
~o,,,(~; .) := ~1 (Of3va +
Oavf3) -
1(1 ) eall3(e; v) := ~ 03va + O,~v3
P r~,(~),~, -
r~3(c)v~,
1 esllz(e; v) :-- --03V3.
Proof. For the sake of brevity, we set the following rule governing the usage of constants in this proof; Whenever cz, c2, etc., appears in an inequality, it means that there ezists a constant, denoted by this symbol, that is > 0 and independent of all the "variables" that enter the inequality such that this inequality holds.
(i) We first establish that, for all 0 < e < so,
{
~ iiv,~ll21,n§
2 }
1,2{ ~__C1
2
i,j
O:
2 }1,
E ieillJ(g; VlIo, f ~ I E IvilO,.
i
for all v - ( v i ) E Hl(f~).
Given v = (vi) E Hl(f~), let eij(v) := ~1(Ojvi + Oivj) and v(e) := (vl, v2, eva) for e _< eo.
Then
,~(',,(,:)/-
eall~(':; v) + :L" (':/",',,
~33(~(e)) - e2~3113(e; ~),
A three-dimensional inequality of Korn's type
Sect. 4.3]
207
and consequently, the estimates on the functions FPai(e) established in Thm. 3.3-1 show that 1/2 z,3
z,3
i
since we may assume without loss of generality that ~0 < 1. By the "classical" three-dimensional Korn inequality in Cartesian coordinates (see, e.g., Vol. II, Thm. 1.1-2 or let 19 = ides in Thm. 1.7-4 in this volume), }z/2 z/2
o,a + Iv
_
10,a
$,3
and the assertion follows from the last two inequalities. (ii) We nezt establish that there ezists 0 < cz <_ ~o such that, for allO < ~ <<_~1,
{
}"' (
) .
~/2 for all v E V(ft).
~
z,3
z
The proof will then be complete, since this inequality and that of (i) together imply that
c~ 2 ~
II~ll 2, . + I~s[2O, f l
<- - (1 + 2c~) E leilIJ(e; v)12O , f ~
a
i,j
for all v E V(f~). If the announced inequality is false, there exist em > 0 and a sequence (Vrn)m~__1 of vector fields v rn = (v~) C V(f~) such that (the Latin letters m and n are used here for indexing sequences)" ~m --+ 0 a s m - - + oo~
eillj(Sm; v rn) --+ 0 in L2(12) as m -+ oo, 2 i
208
Linearly elastic elliptic membrane shells
[Ch. 4
Both sequences tt-vam~cr )m=o are then bounded in HI(f~) by (i). Hence there exist subsequences (V~)n~176and there exist functions va C H 1(f~) satisfying va - 0 on 7 • [-1, 1] and a function v3 E L2(f~) such that van ~ va in
H1
(f~) and v an --+ va in L 2 (f~),
v~ ---" v3 in L2(f~) as n --~ oo, where --~ and ---" denote strong and weak convergences, respectively. The remainder of the proof consists in showing that the sequence (v~)n~176 converges strongly in L2(f~) and that the three limit functions vi vanish, in contradiction with ~ i Ivm[2,n = 1 for all m. To these ends, we proceed in three stages: First, we show that (averages such as ~n are defined in Sect. 4.2) ~n ~ 0 in the space VM(Og) as n -+ cr
To see this, observe that, by Thm. 4.2-1 (a), eallf3(Sn; vn) -+ 0 in L2(~) ~
eallf3(en; vn) -+ 0 in L2(w),
and that, by Thin. 4.2-1 (d), eall~(Sn; v n) --+ 0 in L2(w) ~
7a~(~ n) --~ 0 in L2(w),
as n -+ cr Hence the convergence ~n _+ 0 in VM(W) follows from the inequality of Korn's type on an elliptic surface (Thm. 2.7-3). Second, we show that v,~ ~ 0 in L2(n).
The estimates on the functions r~3(s ) established in Thin. 3.3-1 OO OO and the assumed properties of the sequences (Sn)n=l and ( v )n n = l imply that 03v~ q- ~nOaV~ -- 2enea[]3(~n; V n) + 2~nr~3(6n)V n -+ 0 in L2(~). Let 7~ e V(f~); since the sequence (v~')n~176is bounded in L2(f~),
Sect. 4 . 4 ]
Convergence of the scaled displacements as e ~ 0
209
we have (recall that va := limn-~oo van i n L 2 (f~)) Oava r dx = - L vaOacp dx
= - lira { f n v~no3 ~ rt--+ o o
{f.
= n-+oolim
+ en fnv' O' ~
(o3v: + e.oov~)~odx} - O,
and thus 03ca - 0 in L2(f~). Hence va can be identified with va by Thm. 4.2-1 (a); but ~o- -~ ~o i=
L2
(n) ~
-~n
-~ _,~ i~
L2
(~)
on the one hand and va - 0 on the other. Hence van _+ 0 in L 2(f~). Finally, we show that
By the assumed convergence ezlla(~n; v") --~ 0 in L2(~2), 03vr~ -- ene3113(en; v n) --+ 0 in L2(~)
on the one hand and ~ --+ 0 in L2(w) on the other. The convergence v~ -+ 0 in L2(f~) then follows from Whm. 4.2-1 (c). We have therefore reached a contradiction, and the proof is complete. II 4.4.
CONVERGENCE DISPLACEMENTS
OF THE SCALED A S ~ --+ 0
We now establish the main results of this chapter: Consider a family of linearly elastic elliptic membrane shells with thickness 2e > 0 with each having the same middle surface S = 0(~). Then the solutions u(e) of the associated scaled three-dimensional problems 79(e; f~) (ram. 4.1-1) c o ~ ~ g ~ i~ H~(n) • H~(n) • L2(n) ~ ~ -+ 0 toward a limit u and this limit, which is independent of the transverse variable x3, can be identified with the solution ~ of a two-dimensional variational problem posed over the set w. This limit problem will be
Linearly elastic elliptic membrane shells
210
[Ch. 4
later identified (Thm. 4.5-1) as the scaled two-dimensional variational problem of an elliptic membrane shell. The functions 7aft(r/) and a af3~r used in the next theorem respectively represent the covariant components of the l i n e a r i z e d c h a n g e of m e t r i c t e n s o r associated with a displacement field 71iai of the middle surface S (Sect. 2.4) and the contravariant components of the sealed t w o - d i m e n s i o n a l elasticity t e n s o r of t h e shell; we recall that so > 0 is defined in Thm. 3.1-1. The following result is due to Ciarlet & Lods [1996b, Thm. 5.1]. T h e o r e m 4.4-1. Assume that 0 E C3(~; R3). Consider a family of linearly elastic elliptic membrane shells with thickness 2s approaching zero and with each having the same elliptic middle surface S = 8(-~), and let the assumptions on the data be as in Sect. 4.1. Let u(s) denote for 0 < s < so the solution of the associated scaled three-dimensional problems 79(s; f~) (Thin. 4.1-1). Then there exist functions ua E Hl(f~) satisfying ua = 0 on q' • [-1, 1] and a function us E L2(~) such that: ua(s) -+ ua in H I ( a ) and u3(s) -+ u3 in L2(~) as s -+ 0, u "- (ui) is independent of the transverse variable x3.
Furthermore, the average
-- "--
1;
udz3
satisfies the following scaled v a r i a t i o n a l t w o - d i m e n s i o n a l prob-
lem
~M(W)of -
a linearly elastic elliptic m e m b r a n e shell:
e
•
•
fw aa*So"r')'a'r(~)')/a'8 ( " ) %~ dY -- fw pi ~TiV/-~dy for aU ) / = (r/i) E VM(W),
Sect. 4.4]
Convergence of the scaled displacements as e ~ 0
where (the definitions of the functions I ~ called in Sect. 3.1)"
~(~)
1 := ~ ( 0 ~ ~ + 0 o ~ ) - r ~ , ~
211
bar3, a a~ and a are re-
- ba~rl3,
aa~ ar ._.-- 4)~tz aa~ a ar + 21z(a aa a ~r + aar a~a),
A+2/~ pi :=
l
f i dzs + h i+ + h i_ and h~: " "- h i (., +l). 1
Proof. For the sake of clarity, the proof is divided into ten parts, numbered (i) to (x). As it essentially involves technicalities, the consideration of surface forces is postponed until part (ix); in other words, we assume in parts (i) to (viii) that the scaled unknown u(e) satisfies the following variational problem T~(e; n):
u(~) e V ( ~ ) : = {~ - (~) e n~(~); ~ - 0 o~ v • [-1, 1]}, AiJk~(~)~kll~(~; ~(~))~,iij(~; ~) v/g(~) ~ - f , I% V/g(~ ) ~ for all v E V(~2). For notational brevity, we let
throughout the proof. (i) A priori bounds and extraction of weakly convergent sequences: The norms leillj(e)10,a, Ilua(e)lll, c~, and ius(e)10,c~ are bounded independently of 0 < e < el, where el > 0 is given by Thm. 4.3-1. Consequently, there exists a subsequence, still denoted (u(e))e>0 ]or convenience, and there exist functions eil]j E L2(~), ua E H I ( ~ ) sati s l i n g ua - 0 on 7 • [-1, 1], and us e L2(~) such that (we recall that ~ and ~ respectively denote strong and weak convergences)" eillJ(e) __.xeillj in L2(~), ua(e) ~ ua in H1(~2) and thus ua(e) ~ ua in t ~ ( ~ ) , us(e) ~ us in L2(~2).
From the variational equations satisfied by the scaled unknown u(s) (Thm. 4.1-1), the asymptotic behavior of the function g(e)
Linearly elastic elliptic membrane shells
212
[Ch. 4
(Thin. 3.3-1), the uniform positive definiteness of the scaled threedimensional elasticity tensor (Thin. 3.3-2), and the three-dimensional inequality of Korn's type for an elliptic membrane shell (Thm. 4.3-1), we infer that
Cff ~ [ui(6)10,n _
z,a + lu3(~)10,a
i
<- ~
a
leillJ(e)lo,2 n _< Cego x/2 f• AiJm(e)ekllt(e)eillJ(e)v/g(e)dz
i,j
= <- C~9o
fo fizti(g)4g(g ) dz 1/2 1//2
~x
~1
fi 2
1o,a
~lui(ell02,a
i
;
i
hence all the assertions of (i) follow. (ii) The nmits ui found in (i) are independent of x3. By (i),
03ua(e) + e0au3(e) = 2e {eall3(e ) + r~3(e)u~(e)} ~ 0 in L2(f~), since the functions r~3(e ) converge in C~ (Thin. 3.3-1). get there be given ~o E :D(f~); since ua(e) ~ ua in H i ( a ) and (u3(e))~>o is bounded in L2(f~) by (i),
03ua ~ dx - lim
03ua(e) qodx,
lim { faeO,,u3(e)~odx} = - lim { faeu3(e)O~odx} - 0 ;
e~O
e~O
hence
which means that 03ua - 0 in L2(f~). Part (i) likewise implies that -
3113( )
0 in/:2(a).
Let ~0 E T~(a); since u3(e) ~ u3 in L2(a) by (i),
f f ~z303~odo~ = e--+olimfau3(e)O3~~
- - nm
03u3(~) ~~
= ~
Sect. 4.4]
Convergence of the scaled displacements as e --+ 0
213
which means that 03u3 = 0 in the sense of distributions; it then suffices to apply Thin. 4.2-1 (a). (iii) The limits eillJ f o u n d in (i) are i n d e p e n d e n t of z3; they are m o r e o v e r related to the limit u := (ui) by X
tr
eall3 --- O, e3tl3 -- --~ +)~2/~ a ,,,~e o,iI~"
The convergences eall~(e ) ---" ec,]i/3 in L2(fl), ua(e) ~ ua in Ht(F~), u 3 ( s ) - u3 in L 2 ( a ) ( p a r t (i)) , r ~~(~) -~ r ~ ~ a r~(~) -~ b~ i~ C~
(Thm. 3.3-1) together imply that
1 (o~=z(~) + o ~ ( ~ ) ) - r~z(~)=~(~) -~ ~z(=) - ~ilz in L2(~). Besides, the functions eall~ are independent of x3 since the functions ui are independent of z3 by (ii). Let v = (vi) be an arbitrary function in the space V ( ~ ) = {v - (vi) C H t ( ~ ) ; v - 0 on 3' • [-1, 1]}. The following relations are immediate consequences of the definitions of the scaled strains eilij (e; v)"
~eali~(e; v) ~ 0 in L2(~), 1
i.,2
~e3113(~; "O) -- 03"03 for all ~ > 0.
Using the variational equations of the three-dimensional problem P(e; ~) (Thm. 4.1-1) and the relations A a ~ a ( e ) - Aaaaa(e) - 0 (Thm. 3.3-1), we obtain
/A'Jk'(~) {~kll~(~)~,llj(~; ~)} ~ ( ~ ) ~ = /~ {A'~(~)~ll~(~) + A~'~(~)~ll~(~) } {~oll,(~; ~)} Cg(~)d~
214
Linearly elastic elliptic membrane shells
[Ch. 4
+ fa {4Aa3~3(e)%l13(s)} {eealla(e; v)} ~ ) d z
+ f. {A~(~)~.II~(~) + A~(~)~II~(~) } {~11~(~;~)} v/g(~)d.
Keep v E V(f~) fized and let s --+ 0. The asymptotic behavior of the functions eeillJ(e; v), the asymptotic behavior of the functions Aifl't(s) and g(s) (Thins. 3.3-1 and 3.3-2), and the weak convergences eillJ(6) - - eillj in LZ(f~) (part (i)) together imply that:
~fn{ 2~aa're,rllzOzva
+
+ (a +
-
0.
Letting v vary in V(f~) then yields the relations satisfied by the limits eil13 (if w E L2(f~) satisfies fa wO3vdx = 0 for all v E HZ(f~) that vanish on 3' • [-1, 1], then w = 0; cf. Thm. 3.4-1). (iv) The function ~ := (ui) satisfies the two-dimensional variational problem 7:'M(w) described in the statement of the theorem, with pi := f_l z fi dxa (recall that h i = 0 for the time being). Since the solution to this problem is unique (Thms. 2.7-2 and 3.3-2), the convergences of part (i) hold in/act for the whole family (u(s))e>0 (if the functions ui are unique, so are the functions ui and eillJ by parts (ii) and (iii)). That ~ e VM(w) follows from Thm. 4.2-1 (parts (a) and (b)). Let v = (vi) E V(f~) be independent of the variable x3; then the asymptotic behavior of the functions r pal3(s) and ra3(s ) (Thm. 3.3-1) shows that
1 (0.~, + o~,.) - r2",. - b.,,3 i. L2(a). ~113(~; ~)-~ {1~0~3 + bT~ } i= L~(a). e3lj3(e; v) -- O,
as e --+ 0. Keep such a function v E V(f~) fixed in the variational equations of problem 79(s; f~) and let e ~ 0. The asymptotic behavior of the functions eillJ(s; v), the asymptotic behavior of the functions
Convergence of the scaled displacements as e --4 0
Sect. 4.4]
AiJkt(e)
215
and g(e) (Thms. 3.3-1 and 3.3-2), the weak convergences
e~llj(e ) ~ e~llj in L2(~) (part (i)), and the relations satisfied by the limits e, llj (part (iii)) together yield the equations f f t { X+2#2X/'taa#a,rr + p(aatraf3r + aara#t~) }e,rllrTaf3(v)~,-ddx -- ffl f i v i v Z a d z '
which we may also write as (both functions u and v are independent
of z3):
~ aa/3trr'ytrr(U)TalJ(v)V/'ddY -- f~ { f_l1 fi da~3}viv/-d dY where the functions "),a~(rl) and Given r / - (?'/i) e H~(w), let
aa~'rr are defined in the v -(vi) be defined by
theorem.
Then v E V(f~) and v is independent of z3; hence by Thm. 4.2-1 (b), the above variational equations are satisfied with ~ = r/. Since both sides of these equations are continuous linear forms with respect to ~3 r]3 E L2(w) for fixed va E H~(w), these equations are valid for all rl E VM(co) = H I ( c o ) X H01(co) x L2(co), since Hlo(co) is dense in (v) The part (i) ave
weak convevgences eillJ(e) ~ eillJ in in fact strong: eillJ(e) +
L2(f~)
established in
eillJ in LZ(ft).
Combining the uniform positive definiteness of the scaled elasticity tensor and the inequalities 0 < g0 ___g(e)(x), x E f~ (Thms. 3.3-1 and 3.3-2) with the variational equations of problem ?(e; fl) (again, recall that h i = 0 for the time being) where we let v = u(e), we first infer that
i/2
i,j
where -
.- fa
-
Linearly elastic elliptic membrane shells
216
[Ch. 4
-- fft fiui(E)V/g(~)dx - fft Aijkl(c)(2eklll(E)- eklll)eillJ~/g(~)dx, Using the weak convergences established in part (i) and the asymptotic behavior of the functions AiJkz(s) and g(e), we next have A "= e~01imA ( s ) =
fa fiuiC~ddz -- fft Aifld(O)ekllleillJV/-ddx"
Using the expressions of the functions eill3 (part (iii)), we finally obtain
A~Jkt(O) (Thin. 3.3-2) and
n AiJkt (O)ekllteillJ~/a dx - fn {)~aa#a'rr + #(aa~a[3~ + aa~a#~)}%il~e~ll/3C'd dx
+ ~ {)~aareall r + ()~ +
2plealla}eall3v/-adx
aa#aer+p(aaa#r+aara#)}%lleallv/-ddx.
-
Letting r / = ~ in the variational equations satisfied by ~ (part (iv)) and using the expression of the function eall# found in part (iii), we conclude that A=0. Consequently, the strong convergences established.
eillJ(S) --~ eillJ in L2(f~) are
VM(~), i.e.,
~(~)-~ ~ i~ H~(~)~na ~3(~)-~ ~3 i= t~(~). By virtue of the inequality of Korn's type on an elliptic surface (Thin. 2.7-3), proving these strong cortvergences is equivalent r proving that "ya~3(u(e)) --+ 3'a/3(u) - Call[3in L2(w), since 7a/3(u) -
Call# by (iii). But, since eall~3(e) --+ Call# in L2(f~) by
part (~), we i=fe~ from Thin. 4.2-1 (~) that eall~3(e) --+ eall~ in L2(w),
Sect. 4 . 4 ]
217
Convergence of the scaled displacements as e --~ 0
on the one hand, and we infer from Thm. 4.2-1 (d) that (eallf3(e) -,Taf3(u(e))) -+ o in L2(w), on the other (recall that eatlf3(e) : : ea[if3(e; u(e))). Hence the announced strong convergences hold. (vii) The weak convergence u3(~) --~ u3 in L2(~t) established in part (i) is in fact strong: U3(~ ) -+ U3 in L 2 ( ~ ) .
First, we have 0huh(e) = ~ealla(e) -+ 0 in L2(~2); second, we have already shown in (vi) that ~3(e) ~ u3 in L2(w). Hence the conclusion follows from Thin. 4.2-1 (c) and from the independence of the function u3 with respect to the transverse variable x3 (cf. (ii)). (viii) It remains to show that the weak convergences ua(~) ~ ua in H z ( ~ ) established in part (i) are strong, i.e., that ua(e) -+ u~ in Hi(fl). To this end, we observe that, by the classical three-dimensional Korn inequality in Cartesian coordinates, proving these strong convergences is equivalent to proving that eij(u'(e)) --+ eij(u') in L2(fl),
where eij(v) := 89
u '(~) Since eal[~(r) that
:=
+ Oivj) and
(,,~(~), ,,2(~), o),
,.,'
:=
(uz, u2,
o).
ea~(u'(e))-r~,(~)~(~),parts (iii)and
~oli,(~) -* {~o~("') - r ~ ~
- bo~}
(v)imply
- ~oli~ i. L~(~).
The asymptotic behavior of the functions rPa~(e) and the strong convergences ui(~) -+ ui in L2(~)established in parts (i) and (vii) together imply that
e.~(,~'(~)) ~ ~.~(,,')in L~(n).
218
Linearly elastic elliptic membrane shells
[Ch. 4
Notice in passing that we do not need here the strong convergences ~a(e) --+ ~a in Hi(w), but that we definitely need the strong convergence ~3(e) -+ us (all these convergences have been established in part (vi)), which in turn implied in (vii) the strong convergence Since e33(u'(e)) - e33(u') = O, it suffices to show that (Oaua - 0 by (ii)) 03ua(e) = 2ea3(ut(e)) --+ 0 -- 2ea3(u') in .L2(~/), or equivalently, that 03ua(e) ~ 0 in H-l(f~) and Oi03ua(e) -+ 0 in H - l ( f t ) .
The equivalence between the last two relations is yet another consequence of the lemma of J.L. Lions (Thin. 1.7-1), which, together With the closed graph theorem, implies that the mapping V E jTj2(~'~) --'> (V, 01 v, ~2 v, O3t;) E H - I ( G ) is an isomorphism.
Sin~ o3~(~) = 2~(~,13(~) + r~3(~)~.(~)) - ~o~3(~), w. n~.t have, for all 7~ E ~9(ft),
and consequently, there exists by (i) a constant cl independent of e such that
whe~ ll" I1-~,. denotes th~ norm in H-~(a). Hen~ 0s~o(~) -~ 0 in H-~(a). We next have the identity
- ~ o (~ts~(~) + ~r~(~)~.(~)) imply that
Sect. 4.4]
Convergence of the sealed displacements as e ~ 0
219
since OaOoua = 0 in ~'(12). Denoting by < . , 9> the duality between 7)'(12) and T)(12), we thus have, for all qo e T)(12),
< o, (~o,l~(~) + ~r~(~)~.(~)), ~ >
= -~ f. {~ol,~(~) + r~(~)~.(~)} 0,~ a~, and consequently, there exists c2 independent of e such that
II0, (~,i~(~) + ~r~(~)~(~))II-x,n ~ ~ , by (i). The last term in the expression of O~Osua(e) is treated analogously. Hence O~Osua(e) -+ 0 in H-l(12). Finally, we have, for all ~o E :D(12),
< 0303ua(e), ~o
>=- fn 03ua(e)Os~odz
_- _2e fm {e~llS(e) + r:s(e)u~(e)} Os~odm+ e2 fa e3113(e)Oa~dx,, and consequently, there exists by (i) a constant ca independent of e such that ll0303u~(e)ll-x,n
< cse,
Hence 0303ua (e) -+ 0 in H - 1 (f~), and the proof is complete when
only body forces are considered. (ix) Let X ( f ~ ) ' = {v e L2(f~); Oar e L2(f~)} (Osv is a derivative in the sense of distributions). Then the trace v(., s) of any function v e X(f~) is well defined as a function in L2(w) for all s e [-1, 1] and the trace operator defined in this fashion is continuous. In particular, there exists a constant c4 such that 2
2
1/2
for all v C X(f~); as a consequence, there exists a constant c5 such that
IlvsllL,(r+ur_)
< cs ~ leillj(e; V)lo, f z .
.
for all v E V(ft) (these inequalities will afford the consideration of
surface forces in part (x)):
220
[Ch. 4
Linearly elastic elliptic membrane shells
Let v e X(f~). For almost ally e co and alls C [-1, 1], we can write (Ex. 4.1)
v(y, - 1 ) - v(y, s ) -
F
Osv(y,z3) dz3.
1
Consequently, Iv(.,-1)12,
<2
~
<_ 2
Iv(y, s)[ 2 dy + 2
f lf
03v(y, z3) dzz dy
1
Iv(y,
s)l 2 dy + 2
(1 + s)
]Oar(y, za)l 2 dza dy 1
<_ 2 f~ Iv(y, s)l 2 dy
+ 4103vl 20 ,
f~'
Integrating the last inequality with respect to s E [-1, 1] yields
IlvliL,(r_) _< {ivlo2,a + 4[0avlo2,n} ~/2. Since IlVllz2(r+) is likewise bounded, the first inequality is proved. L~t ~e~t ~ = ( ~ ) e V ( ~ ) . T h e ~
IIv311L'(r+~r_) _< c4 Iv31o2. + 103v31~,.
}1/2
1~3120, f~ + 1e3113(c; V)Jo, 2 f~
< c4(6' + 1) ~ leill~(e; o
,,)lg,.
~
by the three-dimensional inequality of Korn's type for an elliptic membrane shell (Thm. 4.3-1), and the second inequality is proved. (x) Consideration of surface forces:
We henceforth assume that only surface forces act on the shells, i.e., the variational problem 79(e, f~) satisfied by the scaled unknown u(e) now takes the form (cf. Thin. 4.1-1 where fi _ 0; recall that
~kll,(~) := ~klt~(~; ~'(~)))" ~(~) e V(~)
-- {~ -- (Vi) e H I ( ~ ) ;
~ -" 0 on "), x [ - 1 , 1 ] } ,
f AiJkt(e)ekll,(e)eiliJ(e;v)v/g(e)dz=fr
+UP_
hiviv/g(e)dr
for all v E V(f~).
Sect. 4.4]
Convergence of the scaled displacements as e ~ 0
221
Carrying out an asymptotic analysis of u(e) as e -+ 0 involves the same eight steps as those corresponding to body forces (cf. parts (i) to (viii)). We only indicate the modifications needed for handling the new right-hand side in the variational equations of 7~(e; ~). The chain of inequalities that led in part (i) to the a priori bounds now reads, thanks to the inequality proved in (ix) and to the threedimensional inequality of Korn~s type for an elliptic membrane shell:
{
2}
c~ ~ ~ Ilu~(~)ll~,n + i~(~)lo,~
2
_< ~ l~,llj(~)lo,n
a
i,j
<- C~g~1/2( f~ +ur_ h ~ o ( ~ ) ~ ( ~ )
dr
<-C~go 1/2,.,1/2 ~1 ( II h a IIL~(r+ur_)llu~ (e)JlL~(r+ur)
+ IIh311~-cr +,r_ )11~3(~)il L~cr+~r_ ))
_
~1
~6 ~ II~(~)ll~,n +~llh~ll~"C~+,~_) ~ I~,llj(~)lo,~
< Cegol/2,,1/2
--
Yl
( C6CM -'1-C7)
{
Ii~llj(~lllo,2 a
E 9
.
'i
,
and thus the conclusions are the same. The constant c6 depends on the norms JJhaJJL2(r+ur_) and on the norm of the trace operator acting from HI(f~) into g2(r+ u r_); the constant c7 depends on jJh3JJL2(r+ur_) and on the constant c~ (cf. part (ix)). Part (ii) is the same. In (iii), the right-hand side of the relation
/ A'J~(~)~ll,(~){~,llj(~; ~)}~(~) d~ - ~f~+ur_ h',, Vg(~) dr again converges to 0 as e --~ 0 if v E V(f~) is fixed, since g(e) -~ a in C~ (Tam. 3.3-1). In part (iv), let again v = (vi) e V(12) be independent of xs; then, thanks again to the convergence g(e) -~ a
;Linearly elastic elliptic membrane shells
222
[Ch. 4
in g0(~) and to Lebesgue's dominated convergence theorem, limfr hi vi ~ e--+O +uP_
dr - fr +ur_ hi vi v/-ady =
+ h/)
ay.
The same denseness argument as in (iv) then shows that ~ satisfies
f aa~rT~T(~)Ta~(17)v~dy=f(hi++hi)~?i~/-ddy fo~ ~i: n = (~) e v ~ ( ~ ) .
In part (v), we now have
A(~)-- f r
+ur_
hiui(~)4ff(g)dr
-- ~ Aijkl(g)(2ekJJl(g ) -- ekl]l)eijlj4ff(g ) aT,. Since a linear mapping that is strongly continuous is also continuous with respect to the weak topologies (see, e.g., Brezis [1983, p. 39]), u~(e) ~ u~ in H:(f~) implies u~(s) ~ u~ in L2(r+ u r _ ) , and thus lim ~r haua(e)v/g(e) dr e--+O +ur_
= e-+O lim{fr+or_
haua(e)(~g(e)-v~)dF +UP_
+UP_
on the one hand. For the same reason, the weak convergences u3(e) --~ u3 in L2(12) and 03u3(s) - eeslls(e) ~ 0 = Osus in L2(12) (cf. (i) and (ii)) and the first inequality in (ix) together imply that u3(e)(., + l ) ~ u3(., 5=1) in L9 (w). Therefore, lime~o
+ur_
+or_
on the other.
v u3 (r v/g(e) dr
Sect. 4.5]
The two-dimensional equations
223
In this fashion, we again conclude as in (v) that A = 0. The remaining steps (vi), (vii), and (viii) are unaltered. The proof is thus complete, m Remark. The treatment of surface forces given here (cf. parts (ix) and (x)) is based on an idea of Lods [1995]. It affords a substantially simpler proof than the one originally given in Ciarlet & Lods [1996b, Sect. 6]. m
4.5.
THE TWO-DIMENSIONAL EQUATIONS OF A LINEARLY ELASTIC ELLIPTIC MEMBRANE SHELL; EXISTENCE~ UNIQUENESS~ AND REGULARITY OF SOLUTIONS; FORMULATION AS A B O U N D A R Y V A L U E P R O B L E M
The next theorem recapitulates the definition and assembles the main properties of the limit two-dimensional problem PM(W) found at the outcome of the asymptotic analysis carried out in Thm. 4.4-1. Note that 7~M(W) is an atypical variational problem in that one of the unknowns, viz., ~3 "only" lies in the space L2(w). The existence and uniqueness theory, which is quickly reviewed in this theorem, is expounded in detail in Sect. 2.7, where ad hoc references are also provided. T h e o r e m 4.5-1. Let w be a domain in I~2 and let 8 E C2' 1(~; i~3) be an injective mapping such that the two vectors a a = OaO are linearly independent at all points of-~ and such that the surface S = 8(~) is elliptic. (a) The associated scaled t w o - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 7)M(W) of a l i n e a r l y elastic elliptic m e m b r a n e shell: Given functions pi E L2(w), find ~ - (~i) that satisfies
r e
•
•
an
=
e
224
Linearly elastic elliptic membrane shells
[Ch. 4
where 1
:= ~ ( o ~
~(~)
aa~ r .
+ o~)
O"
- r o , ~ - b.,~s,
4A# -- A -4- 2# aa/saar -4- 2# ( aaa a/3r + aar a/3~) ,
has one and only one solution, which is also the unique solution of the minimization problem: Find ~ such that r E VM(W) and jM(r
1
jM(rl) := ~
inf jM(rl), where neVM(~)
7ar(rl)Ta~(rl)vrddy-
L"p'~?iv~dy,
where the functional jM " VM(0)) is called the s e a l e d t w o - d i m e n sional e n e r g y of a l i n e a r l y elastic elliptic m e m b r a n e shell. (b) If the solution ~ - (~i) of 7:'M(W) is smooth enough, it also satisfies the boundary value problem: -n~f~if 3 - p~ in w, _ba~naf3 _ p3 in w, i~
-
0
onT,
where the functions
are the c o n t r a v a r i a n t c o m p o n e n t s of the scaled s t r e s s r e s u l t a n t t e n s o r field, and the ]'unctions
nO'l~ . - 0~o~ + r~,~ ~ + r ~ n ~ are f i r s t - o r d e r c o v a r i a n t d e r i v a t i v e s of this tensor field. (c) Assume that there exist an integer m ~ 0 and a real number q > 1 such that 7 is of class Cm+3, 0 C cm+3(~; I~3),pa E w m ' q ( w ) , and p3 E Wm+l'q(w). Then
r .
(r ,
w~+"~(~) • w~+2'~(~) • w~+~'~(~).
Sect. 4.5]
The two-dimensional equations
225
Proof. The existence and uniqueness of a solution to the variational problem 7~M(W), or to its equivalent minimization problem~ is a consequence of the inequality of Korn's type on an elliptic surface (Thm. 2.7-3), of the uniform positive definiteness of the scaled two-dimensional elasticity tensor of the shell (Thm. 3.3-2 (a)), of the inequality a(y) > ao > 0, y ~ ~ (Thm. 3.3-1), and of the Lax-Milgram lemma.
In view of finding the associated boundary value problem announced in part (b), we first prove that
o~~-
v~rL.
To this end, we note that the function v/~ = det V O and the Christoffel symbols riP/associated with the canonical extension 0 of the mapping 8, defined for e0 > 0 small enough by (Thm. 3.1-1): O(y, X3) -- 0(y) -~ x3a3(y) for y e ~ and Ix31 ~ e0,
satisfy
(ojv~)(y, ~ ) -
v/g(y, ~ ) r ~ ( y , ~ )
for all y E ~ and Ix31 ~ eo (see part (i) of the proof of Thm. 1.6-1). Letting x3 = 0 and noting that
~/g(y, 0) - ~/a(y), r ~ ( y , 0) - r ~ ( y ) , r L ( y , 0) - 0 for all y E ~, we obtain
( o . ~ ) ( y ) - ~/a(y)r;o(y) fo~ all y e ~. Using the Green formula in Sobolev spaces (see, e.g., Vol. I, Thm. 6.1-9) and assuming that the functions n a/3 - n ~a are in Hi(w), we next obtain:
-
f V~n~(~(O~+O~)-r:~-b~3)dy
Linearly elastic elliptic membrane shells
226
[Ch. 4
= - ~ v/-a(O~naf3+ Fraf3nrf} + F~rnar)r/ady-~ v/a n a/3bar3713dy
for all v / = (~/i) E VM(w). Hence the variational equations imply that
f
{
+ p~
+ (bo, n~ +
} dy- o
for all (7/~) C VM(og), and thus n~f31~ - p a and barn a~ - p S in w. The regularity result of part (c), which is due to Genevey [1996], is left as a problem (Ex. 4.2). II
Remarks. (1) As shown by Ciarlet & Sanchez-Palencia [1996], the existence and uniqueness of a solution to the variational problem 7~M(W) (cf. part (a)) may be also obtained after an ad hoc elimination of the unknown ~s has been performed (this elimination is described in Ex. 4.2 (1)). (2) The proof of part (b) is highly reminiscent of, and should thus be profitably compared to, that of Thm. 1.6-1. (3) The boundary value problem found in (b) is another instance of a uniformly elliptic system in the sense of Agmon, Douglis & Nirenberg [1964]; See Ex. 4.3. (4) First-order covariant derivatives b~l~ of a tensor (the curvature tensor) defined by means of its mized components b~ have been defined in Thm. 2.5-1. II In order to get physically meaningful formulas, it remains to "descale" the unknowns ~i that satisfy the limit "scaled" problem 7)M(W) found in Thm. 4.4-1. In view of the scalings
ui(~)(x) - u~(x ~) for all x ~ - 7rex e ~ made on the covariant components of the displacement field (Sect. 4.1), we are led to defining for each e > 0 the c o v a r i a n t c o m p o n e n t s ~ 9~ -+ I~ of the l i m i t d i s p l a c e m e n t field ~e .-w__+ I~3 of the middle surface S of the shell by letting (the vectors a i form the contravariant basis at each point of S): :=
.=
Sect. 4.5]
The two-dimensional equations
227
A w o r d of c a u t i o n . Naturally, the field ~e := (~{) and the field ~e _ ~ a i must be carefiully distinguished! The former is essentially a convenient mathematical "intermediary", but only the latter has physical significance, m
Remark. Conceivably, the limit scaled displacement field across the thickness of the shell could be also de-scaled, resulting into the limit displacement field ~e(0) 9~e Ra inside the shell defined by (the vectors g2,e form the contravariant basis at each point of the reference configuration O(~e); cf. Sect. 3.1)" .
.
.
-
since the scaled limit u = (ui) is independent of the transverse variable x3 (Thm. 4.4-1). For the same reason, however, the de-scaled field does not inherit any remarkable structure as x~ varies across the thickness of the shell. By contrast, the limit displacement field across a plate is a Kirchhoff-Love one (Vol. II, Thm. 1.7-1): It does inherit this richer structure because different scalings can be made in this case on the "horizontal" and "vertical" components of the displacement field, m Recall that fi, e C L2(fl e) and h i,~ C L2(p~_ U pc_) represent the contravariant components of the applied body and surface forces actually acting on the shell and that M and #~ denote the actual Lam6 constants of its constituting material. We then have the following immediate corollary to Thms. 4.4-1 and 4.5-1; naturally, the existence, uniqueness and regularity results of Thm. 4.5-1 apply verbatim to the solution of the "de-scaled" problem 7 ~ ( w ) (for this reason, they are not reproduced here): T h e o r e m 4.5-2. Let the assumptions on the data be as in Thin.
4.4-1. The
eZd
:=
Io m d by the
omvo-
n e n t s of the limit displacement field ~ a i of the middle surface S sat-
isfies the following t w o - d i m e n s i o n a l variational p r o b l e m 7 ~ ( w ) of a l i n e a r l y elastic elliptic m e m b r a n e shell:
~.e E VM(W):-- H](w) X Hi(w) X ~2(W),
for all U = (T/i) E VM(W),
Linearly elastic elliptic membrane shells
228
[Ch. 4
where IT
aaf~ITr,~ ._.-- 4Ae# e aaf~aITr + 2t.te(aaITa~T + aara~r), Ae + 2p e p,,e := j _ ft, e dx~ + h ; e + h i'e and h~ e := h t'e(. +r 9
f~'
~
~
~
~
e
Equivalently, the field ~e satisfies the following minimization problem: ~e e VM(W) and jeM(r ) --
inf J~(~7), where n~VM(~)
eL e a a ~ r ' e T ~ , ( n ) T a ~ ( n ) C - a d y -
jeM(n) := -~
/
f,e~?ix~dy.
If the field ~ (i~) is smooth enough, it also satisfies the following boundary value problem: -
_na[3,el~ _ pa, e in w, _ba~na~, e _ p3, e in w iae = 0 o n e ,
where
~,,,
.= ~ o ~ , ~ w ~ ( r
na[3,,lz .- OiTna[J,e + r ~ ~ ,
' + r~~,
, II
Each one of the three formulations found in Thm. 4.5-2 constitutes one version of the t w o - d i m e n s i o n a l e q u a t i o n s o f a l i n e a r l y e l a s t i c e l l i p t i c m e m b r a n e shell. The functions 7a/3(Y) are the covariant components of the l i n e a r i z e d c h a n g e o f m e t r i c t e n s o r associated with a displacement field yia / of the middle surface S, the functions a a[3~*'e are the contravariant components of the t w o d i m e n s i o n a l e l a s t i c i t y t e n s o r o f t h e shell, and the functions
S e c t . 4.5]
The two-dimensional equations
229
n af3' ~ are the contravariant components of the s t r e s s r e s u l t a n t t e n s o r field ("resultant" means "after integration of the limit stresses across the thickness of the shell"; cf. Ex. 4.4). The functional j ~ 9 VM(03) -+ ] ~ is the t w o - d i m e n s i o n a l ene r g y , and the functional
r / e VM(W) --+ ~1 f~ eaaf~r, e'y~(r/)3~af~(r/)v/-ady
is the t w o - d i m e n s i o n a l s t r a i n e n e r g y , of a l i n e a r l y e l a s t i c ell i p t i c m e m b r a n e shell. Finally, the equations - n aft, elf3 - pa, e and - b a r n a~' e = p3, in w constitute the t w o - d i m e n s i o n a l e q u a t i o n s of e q u i l i b r i u m , and the relations n ac~,e = eaaf~ar'eT~r(~e ) constitute the t w o - d i m e n s i o n a l c o n s t i t u t i v e e q u a t i o n , of a l i n e a r l y e l a s t i c elliptic m e m b r a n e shell. Note that the same two-dimensional problem :P~(w) is evidently obtained if the scalings on the unknowns are the same as before, i.e., u~(x e) - ui(~)(x) for all x e - 7rex e ~ e but the following more general a s s u m p t i o n s on t h e d a t a are made: Ae - e ~A
I~,~(~ ~) = ~f~(~) h~.~(~ ~) - ~+~h~(~)
and
#e _ ~tp,
fo~ ~H ~ - . ~ ~o~ aU ~ - . ~
~ ~. C r ~ u r~_.
where the constants A > 0 and # > 0 and the functions f/ E L2(12) and h i C L 2 ( r + U r _ ) are independent of e and t is an arbitrary real number. Besides, the analysis of Sect. 3.4 shows that these assumptions on the data are the only ones possible for elliptic membrane shells.
Remark. A different kind of generalization is possible. For definiteness, assume that the Lam~ constants are independent of ~. Then it is easily verified that the same scaled limit problem 7~M(W) is obtained under the assumptions that fi'~(x~) - fi(e; x) for all x e - 7r~x e ~2~,
Linearly elastic elliptic membrane shells
230
[Ch. 4
hi'e(x e) = ehi(s; x) for all x e = 7rex e F~_ t2 F~, and fi(e; .) __4 fi in L2(~) as s -+ 0,
hi(e; .) -+ h i in
L~(r+ u r_) as e --4 0.
The functions p~ appearing in the right-hand sides of problem 7 ~ ( w ) are then defined by
p~ "- e
dx3 + h+ +
_ .
1
1 4.6.
JUSTIFICATION OF THE TWO-DIMENSIONAL EQUATIONS OF A LINEARLY ELASTIC E L L I P T I C M E M B R A N E SHELL~ C O M M E N T A R Y AND REFINEMENTS
It remains to convert in terms of de-scaled unknowns the fundamental convergence theorem established in Sect. 4.4. As the "originar' unknowns u ei are defined over a domain that varies with e (the set fte), their averages ~ f ~ u~ dx~ are more appropriate for this purpose, since they are defined over a fixed domain (the set w). The convergences ua(s) -4 ua in Ht(ft) and ua(s) -4 u3 in L2(ft) (Thin.
4.4-1),
the
-
an
-
e
(Sect. 4.1), the de-scalings ~ = ~i - 89f1_ I uidx3 (Sect. 4 . 5 ) a n d Thm. 4.2-1 together yield the following convergences for the averages ~ f~-e u~ dx~ of the covariant components of the original threedimensional displacement:
2e
e
Itae dx~ --4 ~ in H i ( w ) a n d
u~ dx~ --4 ~ in L2(w). 6
However, these eonvergences can be further improved and given a more intrinsic character by considering instead the averages of the tangential component ueaoha' ~ and of the normal component u ~ g 3' e of the three-dimensional displacement vector itself; note in this respect that, along a given normal direction to the surface S, the vectors ga, and gS, e remain respectively parallel to the tangent plane and normal j More to S since g~ - a a - x~baacr, g3e' - - a 3 , and g i e . g j , ~ - - (~i"
Sect. 4.6]
231
Justification of the two-dimensional equations
specifically, the above convergences combined with the behavior as --~ 0 of the vectors 9i, e lead to the following result: Theorem 4.6-1. Assume that O E C3(~; IR3). Consider a family of linearly elastic elliptic membrane shells with thickness approaching zero and with each having the same elliptic middle surface S - O(-~), and let the assumptions on the data be as in Sect. 4.1. Let (u~) e Ht(~2 ~) and ( ~ ) e Hi(w)• H i ( w ) • L2(w) respectively denote for each s > 0 the solutions to the three-dimensional and twodimensional problems p ( ~ e ) and ~ ( w ) (Sect. 3.1 and T h m . 4.5-2). Fi,~a@, t~t (~) c H~(~) • H~(~) • L2(~) d~note th~ ,otution to problem T'M(W) (Thin. 4.5-1), which is thus independent of ~. Then
e a = ~ a ~ in ~ -- ~a and thus Caa
H 1 (w)
for all s > 0,
u e . a , e dz~ -~ {aa a in H i ( w ) as s --+ 0, e a,-
_ _
2s and
s - ~3 and thus ~ a 3 -
~3a 3 in L2(w) for all e > O,
If_': -
28
e
u~9 ~'~ dx~ --+ ~3a 3 in L2(w) as s --+ O.
Proof. As is easily seen (a similar a r g u m e n t was used in the proof of T h m . 3.3-1), the a s s u m p t i o n 8 E C3(~; I~3) implies t h a t the vector fields g a ( s ) 9~ -+ I~3 defined by 9 a ( s ) ( x ) "- 9 a'e (x ~) for all are such t h a t (the vector fields a n 9-~ -+ I~3 are identified here with vector fields defined over the set fl)" go(~)
-~o
o ( ~ ) i~ e ~ ( ~ ) .
-
Since
i F uen~o a'~ dx~
--
2~
e
-
~e
a
_
i F ~(~lg~(~l
~
d~s -
~~
I
= -
2
1
u ~ ( ~ ) ( g ~ ( ~ ) - a ~ ) e~3 - ( u ~ ( e ) -
~)a~,
232
Linearly elastic elliptic membrane shells
[Ch. 4
the convergences ua(e) ~ ua in Hl(fl) and ga(e) --4 a a in C1(~) imply that ua(e) (ga(e) - a a) -+ 0 in Hl(ft); hence 89f l 1 u a ( s ) ( g a ( s ) - a a) dza -+ 0 in Hi(w) by Thm. 4.2-1 (b). The same theorem also shows that ( u a ( e ) - ~ a ) a a -~ 0 in Hi(w). The proof is even simpler for the normal components" Since ga,6 = aa,
1;
2e
u~g 3,e dz i - ~ a 3 - (ua(e) - ~3)a 3
e
and Thm. 4.2-1 (a) then shows that (ua(e) - ~3)a 3 --~ 0 in L2(w). m The fields ~;" ~ --+ I~3 and ~ " ~T
: = i a~a
~
~ --+ I~3 defined by
a n d ~~~N ' - - i ~ a3,
which appear in Thm. 4.6-1, are respectively called the limit t a n gential d i s p l a c e m e n t field, and the limit n o r m a l d i s p l a c e m e n t field, of the middle surface S of the shell. Naturally, they are related to the limit displacement field ~e _ ~ai of S (Sect. 4.5) by
=
Under the essential assumptions that 70 = 7 and that the surface S is elliptic, we have therefore justified by a convergence result (Thm. 4.6-1) two-dimensional equations that are called those of a linearly elastic "membrane" shell in the literature (which, however, usually ig-
nores the distinction between "elliptic" and "generalized" membrane shells); see, e.g., Koiter [1966, eqs. (9.14) and (9.15)], Green & Zerna [1968, Sect. 11.1], Dikmen [1982, eqs. (7.10)], or Niordson [1985, eq. (10.3)]. In so doing, we have also justified the formal asymptotic approach of Sanchez-Palencia [1990] (see also Miara & Sanchez-Palencia [1996] and Caillerie & Sanchez-Palencia [1995b]) when "bending is well-inhibited", according to the terminology of E. Sanchez-Palencia. A w o r d of caution. In an elliptic membrane shell, body forces of order O(1) with respect to e thus produce a limit displacement field that is also O(1). By contrast, body forces must be of order O(e 2) in
Sect. 4.6]
Justification o/ the two-dimensional equations
233
order to produce an O(1) limit displacement field in a flezural shell. See Chap. 6. II The first convergence results for linearly elastic membrane shells have been obtained by Destuynder [1980] in his Doctoral Dissertation. In particular, the convergences established there in Thm. 7.9 (p. 305) under the assumption that the surface S is elliptic are almost identical to those established in Thm. 4.4-1 for the components ua(e), but "weaker" for the component u3(e), since P. Destuynder only established that eus(e) --+ 0 in L2(f~). Besides, his justification of the membrane shell equations remained partially formal as it still relied on an assumed asymptotic expansion of us(e). Using D-convergence theory (see, e.g., Vol. II, Sect. 1.11, and the references therein), Acerbi, Buttazzo gz Percivale [1988] were able to obtain convergence theorems for linearly elastic shells viewed as "thin inclusions" in a larger, surrounding elastic body. As a consequence, the distinction between membrane shells and flexural shells (Chap. 6) is no longer related to the geometry of the middle surface and to the boundary conditions as here, but instead to the ratio (as a power of e) between the Lam~ constants of the two elastic materials in presence. This asymptotic analysis is thus more reminiscent of that of Ciarlet, Le Dret & Nzengwa [1989] (described at length in Vol. II, Chap. 2), who considered an elastic multi-structure consisting of a plate partly inserted in an elastic body; if the shell were a plate, the approach of Acerbi, Buttazzo & Percivale [1988] would only apply to the inserted portion, however. After the original work of Ciarlet & Lods [1996b] described in this chapter, the asymptotic analysis of linearly elastic membrane shells underwent several refinements and generalizations: First, Genevey [1999] has shown that the convergence result of Thm. 4.4-1 can be also obtained by resorting to r-convergence theory. Using the techniques of Lions [1973], Mardare [1998a] was then able to compute a corrector, so as to obtain in this fashion the following remarkable error estimate: In addition to the hypotheses made in Thm. 4.4-1, assume that the boundary o/ the domain w is o~ class d 2, that Oaf a E L2(f~) and h i --O, and that
,f,
u dz~ e I-I2(w) r] VM(W).
234
Linearly elastic elliptic membrane shells
[Ch. 4
Then there exists a constant C -- C(w, 0, fi, ~) independent of e such that ]lu(~) - uliH,(n)•215
<_ C~~/~,
and moreover, the exponent 1/6 is the best possible. Other useful extensions include the justification by an asymptotic analysis of linearly elastic membrane shells with variable thickness (Busse [1998]; see also Ex. 4.5) or made with a nonhomogeneous and anisotropic material (Giroud [1998]), the convergence o/the (scaled) stresses and the explicit forms of the limit stresses (Collard & Miara [1999]; see also Ex. 4.4), an asymptotic analysis of the associated time-dependent problem (Xiao Li-ming [1998]), and the extension of the present analysis to "membrane shells" whose middle surface S is elliptic but has "no boundary", such as an entire ellipsoid (Ramos
[1995] a.d Snca u [1997]). The surprising phenomena appearing when a linearly elastic elliptic membrane shell "becomes a plate" are investigated in Ciarlet [1992a, 1992b], Sanchez-Palencia [1994], and Lods [1995]. The variational formulation of the limit two-dimensional problem of a linearly elastic membrane shell (Thm. 4.5-2) possesses the unusual feature that its third unknown ~ "only" belongs to the space L2(w). This explains why the averaged three-dimensional boundary conditions
1;
~ "- ~
u~ dx~ - 0 on 7
e
are "lost" as 6 --+ 0, since ~ - 0 on V does not make sense. As expected, this loss is compensated by the appearance of a boundary layer in the unknown ~ . The remaining boundary conditions ~a~ - 0 on-), constitute t w o - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s of w e a k simple support. Again because the third unknown ~J is only in L2(w), the linear operator associated with the variational problem of a linearly elastic elliptic membrane shell is not compact (Ex. 4.6) and thus the analysis of the corresponding eigenvalue problem requires special care; see Sanchez-Hubert & Sanchez-Palencia [1997, Chap. 10] and, for the finite element approximations, Rappaz, Sanchez-Hubert, Sanchez-Palencia & Vassiliev [1997]. The controllability and homogenization of elliptic membrane shells likewise pose special difficulties; see Valente [1997] and Hamdache & Sanchez-Palencia [1997].
235
Ezercises
EXERCISES
4.1. The results of this exercise are needed in the proofs of Thms. 4.2-1 and 4.4-1. Let w be a domain in ~2, and let f~ - wx] - 1, 1[. (1) Let v E L 2 (f~) be such that 03v = 0 in f~ in the sense of distributions. Show that there exists r/E L2(w) such that v(y, x3) - r/(y) for almost all (y, x3) C f~. (2) Let v E L2(f~) be such that its derivative 03v in the sense of distributions belongs to L2(ft). Show that its trace v(., s) is well defined as a function in L2(w) for all s c [-1, 1] and that
v(y, s) - v(y, - 1 ) + / _ ~ Oav(y, x3) dx3 1
for almost all y E w and all s E [-1, 1]. Remarks. This exercise is solved in, e.g., Le Dret [1991, Lemmas 1.3 and 4.1]. The result of (1) was also needed in the asymptotic analysis of plates; cf. Vol. II, Ex. 1.1. 4.2. The following problem is based on Genevey [1996]; it shows how to establish the regularity (announced in Tam. 4.5-1 (c)) of the solution to the two-dimensional equations of a linearly elastic elliptic membrane shell. Unless otherwise specified, the notations and assumptions are those of Thm. 4.5-1. (1) Assume that 0 C C2(~; I~3). Show that ~ - (~/) satisfies the
two-dimensional problem 79M(W) of a linearly elastic membrane shell if and only if ~ " - (~a) satisfies the following variational problem
e
.-
•
A
for all ~ / - (r/a) e VM(W), and
~3 - daa~b~rZYa~(~) + dp 3, where (note that the assumption of ellipticity of S ensures that the function d is well defined),
236
Linearly elastic elliptic membrane shells ~,(~)
1
.'= ~ ( 0 ~
+ 0~)
[Ch. 4
- r~,~,
:-- aaf3~r _ cb.r6a~af3a~arb~, d :- (aa/3~rb~,baf3) -1 . Remark. As shown by Ciarlet & Sanchez-Palencia [1996], this elimination of the unknown ~a is the basis of another proof of the existence and uniqueness of a solution to the two-dimensional problem 7~M(W) of an elliptic membrane shell (Thm. 4.5-1). (2) Assuming that the data and the solution ~ to 75M(W) are smooth enough, write the boundary value problem that ~ - (~a) satisfies, as a system of two second-order partial differential equations in w, together with the two boundary conditions ca - 0 on 7. (3) Assume from now on that the boundary 7 is of class C3 and that 0 E C3(~; ]~3). Show that the second order system found in (2) is "uniformly elliptic" and satisfies the "supplementary condition on L" and the "complementing boundary condition", in the sense of Agmon, Douglis & Nirenberg [1964]. Remark. Similar verifications were made in the proof of Thm. 2.7-2 for the system ~af3(Y) - 0 in w. (4) Show that the second-order system found in (2) is "strongly elliptic" in the sense of Ne~as [1967, p. 185]. Using Ne~as [1967, Lemma 3.2, p. 260]), infer from this property that ~a e H2(w)NHlo(W) if pa E L2(w) and pa E Hi(w). (5) Using Geymonat [1965, Thm. 3.5], deduce from (4) that Ca e W2'q(w) M H~(w) if pa e Lq(w) and p3 e Wl'q(w) for some q>l. (6) Assume that there exist an integer m >_ 1 and a real number q > 1 such that 7 is of class Cre+a, 8 C Cm+a(~; I~a), pa C wm'q(w), and p3 E Wm+l'q(w). Using Agmon, Douglis & Nirenberg [1964, Thm. 10.5], show that Ca e Wm+2'q(w)M H~(w). Remark. An analogous proof of regularity holds for the equations of three-dimensional linearized elasticity; cf. Vol. I, Thm. 6.3-6 and Ex. 1.11. 4.B. Let the notations and assumptions be as in Thm. 4.5-1. (1) Show that the three partial differential equations _naf31f3 _ pa and - na~baf3 = pa in w
Ezercises
237
found in Thm. 4.5-1 (b) constitute a system of partial differential equations, of the second order with respect to the unknowns ~a and of the first order with respect to the unknowns ~z, that is "uniformly elliptic" in the sense of Agmon, Douglis & Nirenberg [1964]. (2) Show that, when the two boundary conditions ~a - 0 on 3' are appended as in Thm. 4.5-1 (b) to the three partial differential equations of (1), the "complementing boundary condition" of ibid. holds. Remarks. Another uniformly elliptic system (then of the first order with respect to the first and second unknowns and of order zero with respect to the third unknown) satisfying the complementing boundary condition has already been encountered in Thm. 2.7-2. Incidentally, checking whether the complementing boundary condition is satisfied may require substantial algebraic manipulations; algorithms of formal computation overcome this difficulty in some cases; see Conn~table [1995]. (3) Show that the boundary value problem in Thm. 4.5-1 (b) is an instance of the elliptic systems studied by Grubb & Geymonat [1977], whose approach thus provides another proof of the existence result of Thm. 4.5-1 (a). Hint: See Sanchez-Hubert & Sanchez-Palencia [1997, Chap. 7, Props. 3.8 and 3.9]. 4.4. We established in Thm. 4.4-1 that the scaled displacements converge and we also saw there how their limits can be identified. The present problem shows that, under the same assumptions, the convergence of the (scaled) stresses can be likewise established and that the corresponding limit stresses can be similarly explicitly computed. These results are due to Collard & Miara [1999, Thm. 7]. Given a family of linearly elastic membrane shells that satisfies the assumptions of Thm. 4.4-1, let criJ, ~ -- Aijkl, ~eklli(ue )
denote for each e > 0 the contravariant components of the linearized stress tensor field (Thin. 1.6-1) inside each shell and define the scaled stresses o'iJ(~) "-~ --~ ]~ by letting cr'3,~(x~) - . cr'3(e)(x) for all x ~ - 7r~x C ~2. Note that the scaled stresses then satisfy -
238
[Ch. 4
Linearly elastic elliptic membrane shells
(1) Show that
~ ( ~ ) -~ ~ ( o ) i~ L2(a), 1 i~(~) -~ ~ , ~ i~ H~(-1, 1; H-I(~)) as e --+ 0, where the limits are given by a ~ (0) - a ~ ora3,1 __ _ h a- _ fz__.~ f a d y 3
- ( 1 + xa)(0~crat3(0) + F~.aa#(0) +
as3, ~ = - h ~ - f'__[ f3 dys - (1 + xs)a~a
r~.~'*(o)),
(0)b~.
(2) Let the limit stresses aa#'e(0) be defined by the de-scalings a~t3"(0)(x ") := aa~(0)(x) for all x ~ - ~r~x e ~. Show that the functions n a#, e appearing in the boundary value problem found in Thra. 4.5-2 satisfy the relations
,,~,~(o) a ~ ,
n ~,~ e
thus justifying their definition as components of the stress resultant tensor field. 4.5. Consider a family of linearly elastic elliptic membrane shells (70 = 7 and S is elliptic) with variable thickness, i.e., whose reference configurations are the sets o e ( ~ e), where the mapping {9e 9~ ~ IR3 is defined for each e > 0 by
|
~i)"= o(y) + ~i~(y)~s(y)
, w • [-e, e] the given function e E W 2' (w) satisfying 0 < e0 < e(y) for all y E ~. For such a family of elliptic membrane shells, carry out an asymptotic analysis analogous to that of Thm. 4.4-1. Assuming for simplicity that there are no surface forces~ show in particular that the limit scaled two-dimensional problem takes the form
L
a
afiltrr e
e
7[rr(~)Th/3(rl)eVrddy =
L{f_l1 fi dx3 }~Tie~ dy
Ezercises
239
for a11 y E VM(W) -- Hi(w) • Hi(w) x L2(w), where the covariant components of the linearized change of metric tensor are now replaced by the more general expressions: I
~ ( ~ ) '- ~(0~~ + 0 ~ )
Remark.
~
-
1
r~.
-b~~.
-
e
This asymptotic analysis is carried out in Busse [1998,
Thm. 4.2]. 4.6. Let the notations and assumptions be as in Thm. 4.5-1. Show that the linear operator A : p e {VM(w)}' -+ A p e VM(W) defined by
f a~'~7a~(AP)'Yag(~l)v/ady- P(~I) for
all r / e VM(W)
is not compact. Hint (based on an idea ofD. Coutand): Assume that A is compact and consider a sequence ( P k) koo = l of elements of {VM(w)}' of the form pk _ (0, 0, pk), where (pk)~=l is a bounded sequence in L2(w) that admits no strongly convergent subsequence (such a sequence exists since dim L2(w) = +00). Then use the relation I3 - d ~ ' ~ b ~ . , ( ~ ) +
dp ~
established in Ex. 4.2 (1) to reach a contraction.
This Page Intentionally Left Blank
CHAPTER 5 LINEARLY SHELLS
ELASTIC
GENERALIZED
MEMBRANE
INTRODUCTION
Consider a linearly elastic shell with middle surface S = 8(~), subjected to a boundary condition of place along a portion of its lateral face with 8(3'0) as its middle curve, where 70 C 3'. Such a shell is a linearly elastic generalized membrane shell if it is not an elliptic membrane shell according to the definition given in Chap. 4, yet its associated space
v ~ ( ~ ) - {n -(n~) e H~(~) • H~(~) • H~(~); Ui-
OrU3 - 0 on 70,7a/3(~7)- 0 in w}.
still reduces to {0} (Sect. 5.1). As shown later in this chapter (Sect. 5.8), examples of linearly elastic generalized membrane shells abound. The elliptic membrane shells studied in Chap. 4 provide a first instance of linearly elastic membrane shells, i.e., those for which V f ( w ) = {0}. The generalized membrane shells studied in the present chapter thus exhaust all the remaining linearly elastic membrane shells. The purpose of this chapter is to identify and to mathematically justify the two-dimensional equations of a linearly elastic generalized membrane shell, by establishing the convergence in ad hoc functional
spaces of the three-dimensional displacements as the thickness of such a shell approaches zero. More specifically, consider a family of linearly elastic generalized membrane shells with thickness 2e approaching zero, with each having the same middle surface S - 0(~), and with each subjected to a boundary condition of place along a portion of its lateral face having the same set O(T0) as its middle curve. The associated threedimensional problems, posed in curvilinear coordinates over the sets 12~ = w • - e, e[, are first transformed for each s > 0 into equivalent problems, but now posed over the set f~ = w • 1, 1[, which
242
Linearly elastic generalized membrane shells
[Ch. 5
is independent of c . This transformation relies in a crucial way on appropriate scalings of the unknowns u~ (the covariant components of the displacement field) and assumptions on the Lamd constants
)~e and It ~ and on the contravariant components fz, e of the applied body forces (for simplicity, we assume in this introduction that there are no applied surface forces). More specifically, we define the scaled unknown u(e) = (ui(e)) by letting u~(x e) - ui(e)(x) for all x e : rex e ~e, where 7re(xl, z2, xa) = (Xl, Z2, eX3). Guided by the formal asymptotic analysis of Sect. 3.4, we then assume that there exist constants ,k > 0 and # > 0 and functions fi E L2(f~) independent of e such that
Ae =
A,
and #e = p ,
fi'e(x') - fi(x) for all x e = 7rex e f~e It is found in this fashion that the scaledunknown u(e) = (ui(e)) satisfies a variational problem of the form (Thm. 5.1-1): u(e) e V(f~) -- {v -- (vi) C H : ( f t ) ; v = 0 on 70 • [-1, 1]},
f f AiJkl(~)ek[.l(e; u(~))ei[[j(e; v ) ~ ) =
dx
f ~ i v / a ( ~ ) d~ fo~ all ~ e V ( a ) ,
where, for any vector fieldv = (vi) E H:(f~), the scaled linearized strains eillj(e; v) - ejlli(6; v) E L2(f~) are defined by:
e,~ll~ (~; v) - ~i (o,~o e~ll3(e;
v)
e3113(~; v)
-
+ ao~,)
-
+ Oar3) ~.I (103va e
r:,(~)~, F:3(s)vz
-- 1 03v3.
After establishing several analytical preliminaries (Sect. 5.2), we prove a crucial three-dimensional inequality of Korn's type (Thm. 5.3-1): Given a family of linearly elastic shells, w i t h each having the same middle surface S = 0(~) and with each subjected to a boundary condition of place along a portion of its lateral face having
243
Introduction
the same set 0(70) as its middle curve, there exists a constant C such that
leillJ(~; v)]0,a
~
2
Ilvll ,a _< c .
,
$,3
for e > 0 small enough and for all v E V(f~). Note that this inequality holds whether the space VF(w) reduces to {0} (as in this chapter or in Chap. 4) or not (as in the next chapter). Recall that another three-dimensional inequality of Korn's type holds, this time with a constant independent of e, but only for a family of elliptic membrane shells; cf. Sect. 4.3. A last preliminary is needed, in the form of the following definition (Sect. 5.5): The applied forces are said to be admissible if there exists a constant n0 such that
I{
,
,
z,3
for e > 0 small enough and for all v = (vi) E V(f~). Equipped with these preliminaries, we then establish the main result of this chapter: Consider a generalized membrane shell "of the first kind", i.e., one for which the space Vo(w) - { r / - (r/i) E Hi(w); rl - 0 on 70, 7,,~(r/) - 0 in w} "already" reduces to {0} (Sect. 5.4). [. [M defined by
Equivalently, the semi-norm
0,1
O,w a,/J
becomes a norm over the space V(od) -- { , :
(?]i) e Hi(w); r / = 0 on 70}.
This assumption is satisfied for instances if 70 ~ Ow and S is elliptic, or if S is a portion of a hyperboloid of revolution and 70 is "large enough" (these and other examples are reviewed in Sect. 5.8). We also assume that the applied forces are admissible. We then show (Thin. 5.6-1) that the averages 89f l 1 u(e) dx3 converge as e --+ 0 in the space V~M(f~) -- completion of V(w) with respect to I" IM,
244
[Ch. 5
Linearly elastic generalized membrane shells
and that the limit r C V~M(W) satisfies the following (scaled) twodimensional equations of a linearly elastic generalized membrane shell of the first kind: BL(r
r / ) - L~M(r/) for all r/C V~M(W),
where B L is the unique continuous extension to V~M(W) • V~M(W) of the bilinear form already encountered in the (scaled) two-dimensional variational problem of a membrane shell (Thm. 4.5-1), viz., (~, r/) e V(w) • V(w)-+ f~ aa~arT~r(~)Ta~(y)~dy , and the continuous linear form L~M 9V~M(W) ~ I~ reflects the limit behavior of the admissible applied forces as e ~ 0 (a convergence result also holds for the fields u(6) themselves; cf. again Thm. 5.6-1). We recall that ~(~)
~.~
1
-
~(0~o
_
4~,
+ 0o~) - ro~
~.,~
+ 2,(~.~,
- b~3,
~+
~.~~),
A+2/~ the functions 7a~(r/) and a a~ar being respectively the covariant components of the linearized change of metric tensor associated with a displacement field r/ia i of the middle surface S and the contravariant components of the (scaled) two-dimensional elasticity tensor of the shell. We also treat generalized membrane shells "of the second kind", i.e., tho~r fo~ winch L. lY i~ no longe~ a norm o~r
the space V ( ~ )
(this is simply another way of saying that V0(w) ~ {0}), but is a norm over the smaller space VK(~)-
{~ = (,~) e H~(~) • H~ ( ~ ) • H~(~);
r/i = 0~?s = 0 on 70} A convergence theorem can still be established in this case, but only in the completion of the quotient space V ( w ) / V o ( w ) with respect to I" IM (Thm. 5.6-2). We conclude this chapter by reviewing the existence and uniqueness properties of the solution to the above variational problem. We
Sect. 5.1]
245
Definition and assumptions on the data
also rewrite the fundamental convergence theorem in terms of descaled unknowns and data, thus providing a justification in terms of "physical" quantities of the two-dimensional equations of a linearly elastic generalized membrane shell (Thin. 5.8-1). These two-dimensional equations often provide intriguing examples of "sensitive" variational problems, according to the terminology recently introduced by J.L. Lions and E. Sanchez-Palencia, in the sense that they may possess two unusual features: They are posed in spaces that are not necessarily contained in spaces of distributions and their solutions may be "highly sensitive" to arbitrarily small smooth perturbations on the data (Sect. 5.8). In Sect. 5.8, we also review several examples of linearly elastic generalized membrane shells. 5.1.
LINEARLY ELASTIC GENERALIZED MEMBRANE SHELLS: DEFINITION AND ASSUMPTIONS ON THE DATA; THE THREE-DIMENSIONAL EQUATIONS OVER A DOMAIN INDEPENDENT OF g
Let w be a domain in It~2 with boundary 7 and let 8 E C2(~; R s) be an injective mapping such that the two vectors OaS(y) are linearly independent at all points y C ~. A linearly elastic shell with middle surface S = 8(~) is called a l i n e a r l y elastic g e n e r a l i z e d m e m b r a n e shell if the following three conditions are simultaneously satisfied (the definitions and notations are those of Sect. 3.1): (i) The shell is subjected to a boundary condition of place along a portion of its lateral face with 0(70) as its middle curve, where the subset 70 C 7 satisfies
length 70 > 0. (ii) Define the space (av denotes the outer normal derivative operator along 7):
vv(~) :--
{n-
(,7,) e H~(,,,) • n ~ ( , ~ ) • n ~ ( ~ ) ; r/i - 0,,~:3 - 0 on 70, 7a~(r/) - 0 in
w}.
Linearly elastic generalized membrane shells
246
[Ch. 5
Then vF(
) = {o}.
We recall that 1
tr
denote the covariant components of the linearized change of metric tensor associated with a displacement field yia i of the surface S. The subscript " F " announces the central rSle that the space VF(w) plays in the definition of flexural shells; cf. Sect. 6.1. (iii) The shell is not an elliptic membrane shell. We recall that a linearly elastic shell is an elliptic membrane shell if V0 - V and S is elliptic (Sect. 4.1) and that an elliptic membrane shell also provides an instance where the space VF(w), which in this case is the space H](w) • H~(w) • H2(w), reduces to {0} (if O e C2'1(w; Rz); cf. Thm. 2.7-2). Generalized membrane shells thus ezhaust all the remaining cases of l i n e a r l y elastic m e m b r a n e shells, i.e., those for which vF(
) = {o}.
Examples of linearly elastic generalized membrane shells are given in Sect. 5.8 A w o r d of c a u t i o n . Like the definition of a linearly elastic elliptic membrane shell, that of a linearly elastic generalized membrane shell depends only on the subset of the lateral face where the shell is subjected to a boundary condition of place (via the set V0) and on the "geometry" of the middle curve of the shell; cf. Ex. 5.1. i The formal analysis of Sect. 3.4 then naturally leads us to make the following scalings of t h e u n k n o w n s and a s s u m p t i o n s on t h e d a t a for a family of linearly elastic generalized membrane shells, with each having the same middle surface S = 0(~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(V0) as its middle curve, as their thickness 2e approaches zero. First, we define the scaled u n k n o w n u(e) = (ui(s)) by letting
=.
-
e
Sect. 5.1]
247
Definition and assumptions on the data
Next, we require that the Lamd constants and the applied body and surface force densities be such that
and
#~ - #,
for all
x e -7r~x C ~2e,
for all
xe-Tr~xEr~_Ur
e
where the constants A > 0 and # > 0 and the functions f i C L2(~) and h i E L2(r+ U r_) are independent of e (Fig. 3.2-1 recapitulates the definitions of the sets f~e, f~, F~, r + , r e_, and r _ ) . Remark. For notational brevity, the functions fi and h i stand (as in the preceding chapter) for the functions that were respectively denoted fi, o and h i' t in Sect. 3.4. m
A w o r d of c a u t i o n . In order to carry out our asymptotic analysis of generalized membrane shells, we will have to make in addition a rather stringent assumption on the applied forces, which supersedes in fact the present ones, in such a way that the linear form appearing in the variational problem 7~(e; ~) described in Thm. 5.1-1 becomes continuous with respect to an ad hoc norm, and uniformly so with respect to ~; cf. Sect. 5.5. m As an immediate corollary to Thm. 3.2-1 (simply corresponding to p - 0), we obtain the problems satisfied by the scaled unknown over the set ~, thus over a domain that is independent of ~: T h e o r e m 5.1-1. Let w be a domain in R 2, let 0 E C2(~; ~3) be an injective mapping such that the two vectors aa - OaO are linearly independent at all points of-~, and let eo > 0 be as in Thm. 3.1-1. Consider a family of linearly elastic generalized membrane shells with thickness 2e, with each having the same middle surface S - 8(~) and with each subjected to a boundary condition of place along a portion o] its lateral face having the same set 8('/0) as its middle curve. Let the assumptions on the data be as above. Then, for each 0 < e ~ ~o, the scaled unknown u(s) - (ui(s)) satisfies the following s c a l e d t h r e e - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m :P(s; ~) of a
248
Linearly elastic generalizedmembrane shells
[Ch. 5
linearly elastic generalized m e m b r a n e shell: ~(8) C V ( ~ ) : = {~ = (~)i) E Hl(f~); v - 0 on to},
fflAijkl(g)eklll(8; ~(s))eillj(s; ~)r -- ff~ fir i ~ )
dx -~-fr
+UP-
dx
hil)i C g ( g ) d r
for all~ C V ( ~ ) ,
where ro "- 70 • [-1, 1], the scaled linearized strains eitlj(e; v) are given by
.)
-
1
+
-
e,~ll3(e;v) = ~l ( l oe 3 v ~ + c9,~v3)
P
(e)
p,
r:3(e)v~
eslls(e; v) = _103133, g and the functions AiJkt(e),
g(e), ri~(e )
are defined as in Thm. 3.2-1.
II Our main objective in this chapter consists in analyzing the behavior of the solutions u(e) C H I ( o ) of problems 79(e; f~) as e ~ O. To this end, we begin by proving in Sects. 5.2 and 5.3 various "analytical" preliminaries, which complement the "geometrical" and "mechanical" preliminaries proved in Sect. 3.3. Note that these preliminaries are common to all types of linearly elastic shells. In particular, they likewise play an essential r61e in the asymptotic analysis of linearly elastic flexural shells carried out in the next chapter. 5.2.
ANALYTICAL PRELIMINARIES
To begin with, we analyze the asymptotic behavior as e ---> 0 of the scaled linearized strains
Sect. 5.2]
249
Analytical preliminaries
appearing in the definition of the scaled three-dimensional problem T~(s; ~) (Thm. 5.1-1). To this end, we are naturally led to introduce the "three-dimensional analogs" Vaf~(v) and pa~(v) of the covariant components ~'af~(~) and Paf~(Y) of the "two-dimensional" linearized change of metric and change of curvature tensors (Sects. 2.4 and 2.5). The following result is due to Ciarlet, Lods & Miara [1996, Lemma 3.2]. T h e o r e m 5.2-1. Let the ]unctions r~a~, ba# , ~ ~ go (-~) be identified with functions in C~ and let for any v = (vi) C tI~(~2) the functions 7af~(v) ~ L2(~) and PaO(v) e H - ~ ( ~ ) be defined by: 7 . , ( ~ ) := 1 ( o ~ o + o ~ )
- r 7 ~ , ~ - bo..~,
p . , ( ~ ) := 0 . , , ~ - r ~ , o ~ +b~(0,,~
- r~.~)
+ b$(0.,~ - r ~ , . ) + ~1~
- ~7~,~,
~ h ~ b~lo - 0ob~ + r ~ b $ - r af3b~ ~ ~ - baif~ ~ (Thm. 2.5-1) Then there exists a constant Ct such that,/or all 0 < e ~_ ~o and all v E I-It(~), the scaled linearized strains ea{{f3(e; v) satisfiy 1 I~-~oll,(~; - ) - ~ ~l l , (~ ; -){0, . -< c ~ ~
I~o{0,., O~
where
1
1
and (11" I1-1,~ denotes the norm in H - I ( ~ ) ) : 1 0 3 ~ l l , ( ~ ; ~) + p ~ , ( ~ ) l i - ~ , n
Proof. The constants cl, c2 and c3 appearing in this proof are meant to be independent of e and of v C I-I1(~). The first inequality is a consequence of the definitions of the scaled strains Calif,(e; v) and of the asymptotic behavior of the functions riaf3 (~) (Tam. 3.3-1). Next, let the functions e~ v) e L2(~) and
250
[Ch. 5
Linearly elastic generalized membrane shells
tSaf~(v) e H - 1(12) be defined by
e~
,(
,
)
v ) : - - ~ tOotv3 Jr--oq3vc~ Jr-b~vtr, c ~a~(v) := Oaf3v3 + O~(b~vr) + O~(b~v~) - r ~ (O~v3 + 2b~v~)
-~;1~ - ~g~~. Then a simple computation shows that
0 (e ; V) - ba~e 3113(~; V) -b eX3 b~b~r~e3113(e; V) -2F,~%113 v) - OcrV3 - 2b~vr).
q-~x,3b~la(2e~
Consequently,
llO~llz,(~; v)§
,~,:,~(~,)ll-~,n
on the one hand. On the other, the first inequality implies that
11~1a=~,~l,~(~; v) - o=~,l~(~; v)ll-~,~ _< ~ ~ Ivalo,~, the definitions of the scaled strains eall3(e; v) and of the functions e~ v) together with the asymptotic behavior of the functions r~3(e ) imply that
le,:,l13(~; v)
o - ~oll=(~; ~)1o,~ _< ~3~ ~ I~1o,~,
and, finally, an easy computation shows that, for v E Hl(f~),
p~,(,,)- ~,~(,,). Hence the second inequality follows from the above relations.
II
The next theorem, which is due to Ciarlet, Lods & Miara Lemma 3.3], is crucial, as it plays an essential r61e in the of a three-dimensional inequality of Korn's type for a family early elastic shells (Thm. 5.3-1) and of the convergence of the
[1996, proofs of linscaled
Sect. 5.2]
Analytical preliminaries
251
unknown as e --~ 0 (Thms. 5.6-1 and 5.6-2): Consider a sequence (u(e))e>0 of functions in the space V(f~) that converges weakly in Ht(f~), thus also strongly in L2(f~), and let u be its limit. We first show that, if it so happens that the corresponding sequences (eillJ(e; u(e)))e>0 weakly converge in L2(f~), considerable information can then be gathered about the limit u and the functions 7a~3(u) and pa#(u). We also show that, if in addition the corresponding sequences (pa#(u(e)))e>o strongly converge in H-X(f~), then in fact the sequence (u(e))e>0 strongly converges in Ht(f~). In the following statement, the symbols -+ and ~ respectively denote strong and weak convergences and 0v denotes the outer normal derivative operator along the boundary of w. Theorem
5.2-2. Let v(~)
. - {~ - ( , , ) e a ~ ( u ) ;
~ - o on r 0 } ,
and let eillj(Z; "V) E L2(~), "/'a/3('V) E L2(~), and pa~('V) E I-]'-l(~) be defined for any function v E V(f~) as in Thms. 5.1-1 and 5.2-1. Let (u(e))~>0 be a sequence of functions u(e) E V(f~) that satisfies u(e) ~ u in Ht(f~), 1 -eeillJ(e; u(e)) --' elillj in L 2(f~), as e ~ O. Then u-
(ui) is independent of the transverse variable x3,
1//
-~ - ( ~ ) := ~
~ ~ d~3 e H ~(~) • H ~(~) • H ~(~),
ui -- 0~u3 - - 0 on 70, -y~(,,) - o,
pa,(u) E L2(n) and pa,(u)- -03e~ll,. If in addition there exist ]unctions Xa# E H-t(f~) such that p ~ ( ~ ( ~ ) ) -+ x ~
i~ a - ~ ( n )
~s ~ -~ o,
then .(~) -+.
i . r I l ( a ) ~s ~ -+ o,
pa~(u)- xa~ and thus Xa~ E L2(fl).
Linearly elastic generalized membrane shells
252
[Ch. 5
Proof. For the sake of clarity, the proof is divided into six parts, numbered (i) to (vi). We first recall that (Thm. 4.2-1 (a)), given v E L2(f~), ~ ( y ) : = 89f~i v(y, z3)dz3 is finite for almost all y E w and the average
1/:
-
V :-- "~
1
v dx3,
defined in this fashion is in L2(w), or in HI(w) if v E Hi(f~); in particular, the functions ~/are in H i (w) and they vanish on 70 (Thm. 4.2-1 (b)). For notational brevity, we let ei[ij(e ) --- eill:/(e; u(e)) throughout the proof. Finally, observe that the assumption u(~) --" u in Hl(f~) implies that u(e) --+ u in L2(f~) as the imbedding from Hi(f~) into L2(f~)is compact. (i) We first show that u - (ui) is independent of x3 and that - 0 on 70- Since the sequence (u(e))e>o is bounded in Hi(f~) and eillj(e ) --+ 0 in L2(f~) (in a Hilbert space, a weakly convergent sequence is bounded), 03u
(e) -
O u3(e) +
-
2rX3(
03u3(e) Hence 03Ui
"- 0,
in L2(~2), ce3113(c) -+ 0 in L2(f~). )u
-
(6)}
0
and consequently, by Thin. 4.2-1 (a),
u(y, x 3 ) - - ~ ( y ) for almost all (y, x3) e ~2, where the function ~ - (~/) = 89f_i i u d z 3 is in the space Hi(w) and satisfies ~ - 0 on 70. Thus u - (ui) is independent of the transverse variable z3. (ii) We next show that ~3 E H2(w) and Ov~3 v E Ci (~), an integration by parts shows that
0 on 3'0. If
17
x3v(y, z3) d$3 = -~
i (1 - x2)O3v(y, x3)dx3 for all y E ~.
By Thm. 4.2-1, the mappings w E L2(f~) ~ ~ E L2(w) and w E Hi(f~) -+ ~ E Hi(w)
Sect. 5.2]
253
Analytical preliminaries
are both continuous, so that the above relation remains valid (for almost all y E w) if v E Hi(f~). Let
1/:
~(~) - (~(~)) := ~ Hence we may also write
1;
~(~)- ~
(1 - x~)03u(s)dm3.
a:3U(~) dx3,
and, by Thm. 4.2-1 (b), ~i(s) E Hi(w) and ~i(s) - 0 on %. By assumption, e3[[3(e ) -- 103u3(~ ) ~
0 in L2(f~), and thus, by
definition of ~3i (s), The assumptions made on the sequence (u(s))e>0 combined with the asymptotic behavior of the functions r~3(s ) (Thm. 3.3-1) imply that 103ua(s)
-
2eall3(S)
-
O~u3(s) + 2 r ~ 3 ( s ) u ~ ( s ) ~
{-Oau3 2b~u~} -
in L2(f~), and thus (a strongly continuous linear mapping is also continuous for the weak topologies; see, e.g., Brezis [1983, Thm. III.9], by definition of ~ (~),
~(~)
1;
~
Since the functions again by Thm. 4.2-1 that -1 ~(~) -
(x~ -- 1)(Oau3 + 2b~u~)dx3 in L2(w).
u/ are independent of z3, since Oau3 = Oau3, (b), and since _~__f-ii (x 2 - 1)dx3 - 2,_ it follows -1 ~ := - ~2 (0a~3 + 2 ~ ~ )
in L2(w)
Our next objective consists in showing that, in ]act, the sequences (~(~))~>0 weakly converge in Hi(w). To this end, it suffices to establish that the sequences ( i ( o a ~ ( s ) + 0~l~ia(s)))e>0 weakly converge in L2(w) (by the two-dimensional Korn inequality, the norm I1" Ili,~ is equivalent to the norm (~?a) --~ {~-~a,~1 89 q- 0~Ta)l~,~} i/2 over
Linearly elastic generalized membrane shells
254
[Ch. 5
thespace {(r/a) E Hi(w); r/a - 0 on V0}). A simple computation, based in particular on the equality ~i(e) =
-
i
zSu(e) dz3, shows
that
-~ (oo~(~1 + o ~ ( ~ 1 ) - 2~~,,~(~) 2
",,~"
-
-
2babo.B;r,3"tt3(~) "4-
where ~ (~; eall/3(e) := eall~
(~) +
r~z=~
,,(~))
and the functions eall~(e; i v) are defined as in Thm. 5.2-1. The first inequality established in this theorem together with the assumptions us(g) --+ u s in L2(~) then imply that
1 } -+ o ~ii~(~)- ~li~(~)
in
.
(a),
1 i and the assumptions -chilD(s) --~ ealll 3 in L 2 (f~) in turn imply that g
e~ Ii,(e) ~ call i # in L2 (f~) Consequently, x3el,~ll~,e, ('~ --~ z3 elall~ in L2(W) Since u(e) ~ u in L2(f~) as u(e) ~ u in H i ( n ) by assumption, 2 (e))~>o and (bab~zsU3(e))e>o ~ 2 strongly conboth sequences (b~31az3u~ verge in L2(w); since ~ ( e ) ~ 0 in L2(w) (part (ii)), the sequence (ba#g~(e))~>0 strongly converges (to 0) in L2(w) and since (gza(e))e>0 weakly converges in L 2(w) (part (ii)), the sequence (ra/3u~(e))~>0~ -i weakly converges in L2(w). Hence the sequence (~za(e))e>0 weakly converges in Hi(w); thus ~za(e ) ~ ~ in L2 (w) implies that ~za e Hi(w), and therefore that 0a~3 E Hi(w) since ~ ~ E Hi(w); we have shown in this fashion that u3 E H2(w). We know from (ii) that ~ ( e ) - - 0 on 3'0; therefore, ~za(e ) ~ -"/.Sot i in Hi(w) implies that -~tct i _ 0 on 70. Since ui - 0 on 70, the equality ~1~. _ - 23( ~0 ., .~~ + 2bP.~..~.. shows that 0a~s - 0 on 70; hence
Ovu3 = 0 on 3'0. We have thus established that (ui) E H i ( w ) x H i ( w ) x H2(w) and that ~i = 0vu3 = 0 on 3'0.
Analytical preliminaries
S e c t . 5.2]
255
(iii) We ne=t prove that 7af3(u(e)) -+ 0 in L2(~) as s --+ 0 and that Va~(u) - O. By definition, the functions ealj~(s z ) are given by (Thm. 5.2-1) 1 Hence 1
I'ya/3(~(~))- e~ll/3(~)lo, n _< ~1er1
~e~ll/3(~)lo,~
and thus, by the first inequality in Thm. 5.2-1 and the assumptions made on the sequence (u(s))e>0,
7~(u(~)) -~ 0 i= L2(~). By the same assumptions, 7a~(u(s)) ~ Va~(u) in L2(I2). Hence Va~(u) - 0, as was to be proved. (iv) We next prove that paf~(u) = -03elallf~ in L2(fl). From the same assumptions and from the second inequality in Thm. 5.2-1, we infer that
{lo,~ll,(~) C
+
p~,(,,(~))} --~ o i~ H-~(a).
The operator 0a " L2(f~) --+ H-Z(f~) being also continuous for the weak topologies, we deduce from the assumption u(s) --~ u in HZ(I2) that _
103easlf3(s) ~
1 03Call ~ in H - 1 (f~),
hence that eaill3
~ 0 in H - (f~).
Since, again by the assumptions on the sequence (u(e))e>o,
256
Linearly elastic generalized membrane shells
[Ch. 5
the desired equality pa~(u) - -03e alll3 i holds in H - i (ft); that it is in fact an equality in L2(ft) follows from the relation ~3 E H2(w) established in part (ii) which, together with the independence of u3 with respect of z3 established in part (i), implies that u3 E H2(f~). (v) We ne~t prove that, under the additional assumption that p~z(u(e)) ~ x ~ in H - ~ ( n ) as ~ ~ O, the sequence (u3(r strongly converges in H i (f~). We first note that 03~t3(E) -- ~e3113(~) -+ 0 -- 03~t3 in L2(~).
By the lemma of J.L. Lions (Thin. 1.7-1), the mapping v E L2(~) -+ (v, (Oiv)) E H - I ( ~ ) x H - I ( ~ ) is an isomorphism. In order to prove that Oau3(e) -+ Oau3 in L2(~),
it therefore suffices to prove that
0~3(~) ~ 0~3 i~ H-~(~), 0aiU3(~ ) ---} OaiU3ill H-I(~'~).
The first convergence is a consequence of the assumed convergence u3(e) --+ u3 in L2(ft). The relation 03u3(~) --+ 0 = 03u3 in L2(ft) likewise implies that 0a3U3(~) --+ 0 -- 0a3U3 in H-i(f~).
From the definition of the functions Pat3(v) (Thm. 5.2-1) and from the assumption u(6) ---" u in I-Ii (f~), we infer that
p~z(,,(~)) -~ p~z(=) i= H-~(n), and thus,
p~(=(~)) -~ x~z - p~z(~) i= H-~(n). This relation also shows that Xat3 E L2(f~) since pa[3(u) E L2(~t)
(p~t (iv)). Since ui(e) --+ ui in H - i ( f t ) and Oaui(e) -+ Oaui in g - i ( ~ t ) by assumption, and since O~u3(e)
-
p ~ ( u ( e ) ) + r~,t30~u3(e ) - b~(O~u~(e) - r L = ~ ( ~ ) ) -bX(Oz=~(~)
-
r~=~(~)) - bXb=~(~) + bXbzu~(~),
257
Analytical preliminaries
Sect. 5.2]
the convergellces Oau3(e) ~ Oau3 in H - I ( ~ ) imply that
o~3(~)
~ o~3
i~ H-~(~).
Hence Oaiu3(e) --+ Oaiu3 in H-l(f~) and consequently,
Oau3(~) -+ Oau3 C L 2 ( ~ ) . (vi) Finally, we prove that the sequences (ua(e))e>o strongly converge in HZ(f~). By virtue of Korll's inequality applied to functions in the space V(f~), this is equivalent to proving that
eij(ut(e)) ~ eij(ut) ill L 2 ( ~ ) , where 1
~'(~) := (~(~), ~(~), 0),
U I := (~tl, ~t2, 0).
In part (iii), we have shown that 7a13(u(~)) -+ 0 in L2(n). Since
~.~(u'(~)) - v.~(u(e)) + r 2 ~ ( e ) + ba~u3(c), we conclude that eal3(ut(e)) ---+ { r ~ u t r
+ ba~u3} = ect~(U t) in L 2 ( ~ ) ,
by assumption. Next, 1
~ ( ~ ' ( ~ ) ) - ~o3~(~1 1
- ~e {2e~ 113(e) - O~u3(e) + 2r~3(e)u~(e)} --+ 0 - -~03u~ - e~3(u'),
-
by part (i); finally, e33(~t(c)) -- 0 -- e33(t/,'),
and the proof is complete.
II
258 5.3.
Linearly elastic generalized membrane shells
[Ch. 5
A THREE-DIMENSIONAL INEQUALITY OF KORN~S TYPE FOR A FAMILY OF LINEARLY ELASTIC SHELLS
The key to the convergence theorems of Sect. 5.6 is a three-dimensional inequality of Korn's type (Thm. 5.3-1), which may be viewed as a "scaled" Korn's inequality in curvilinear coordinates (Thm. 1.7-4) ]or linearly elastic shells. The "constant" C/6 appearing in this inequality, together with ad hoc assumptions on the applied forces (Sect. 5.5), will yield the fundamental a priori bounds that the family (u(c))s>0 satisfies (cf. parts (i) and (ii) in the proofs of Thms. 5.6-1 and 5.6-2). We emphasize that this inequality holds for an arbitrary "geometry" of the sur]ace S -- 8(~) and for an arbitrary subset "yo of 3~ with length V0 > 0, whether the space VF(w) introduced in Sect. 5.1 reduces to {0}, as in this chapter or the previous one, or not. Indeed, this inequality is likewise put to intensive use in the convergence theorem of the next chapter (Thm. 6.2-1)~ where we consider flexural shells, i.e., those for which Vv(w) ~ {0}.
Remarks. (1) We have seen in Thm. 4.3-1 that, if "Y0 = ~' and the surface S is elliptic, the "constant" C / s can be replaced by a "genuine" constant, i.e., that is independent of s; however, the norm Ilvslll, a appearing in the left-hand side must then be replaced by the norm I 10, (2) Another Korn inequality with a "constant" also of the form C//s was established in Kohn & Vogelius [1985] (see also Acerbi, Buttazzo & Percivale [1988]). It holds, however, over the "variable" domain f~ and besides, it involves the "usual" functions 89 (O~v~ + O~v~). II The following inequality of Korn's type is due to Ciarlet, Lods & Miara [1996, Thm. 4.1]; we recall that e0 > 0 is defined in Thm. 3.1-1. T h e o r e m 5.3-1. Assume that 8 E C3(~; I~S). Consider a family of linearly elastic shells with thickness 2e, with each having the same middle surface S : 0(-~) and with each subjected to a boundary condition of place along a portion of its lateral ]ace having the same
Sect. 5.3]
A three-dimensional inequality of Korn's type
259
set 0(7o) as its middle curve, where the subset 7o C 7 satisfies length 70 > 0. Define the space v(n)
: = {~ - ( , , ) e a ~ ( n ) ;
~ - o o. r0},
where Fo - 70 x [ - 1, 1]. Then there exist a constant ~1 satisfying 0 < el < so and a constant C such that, ]or all 0 < ~ < ~1, the following t h r e e -
d i m e n s i o n a l i n e q u a l i t y of Korn~s t y p e for a f a m i l y o f l i n e a r l y elastic shells holds:
~/2 Ilvlll~ < , _ - c { ~ l e.i l. l J ( ~ ; ~ v)l 2'n}
for all v C V(f~),
z~J
where the scaled linearized strains eillJ(e; v) are defined by
eol.,(~;
v) :=
1
~(0,vo + O.v,)- r:,(~)~.
e~lls(e; v) "= ~l(103va+Oav3) e
- r:~(~)~.
1 e3ll3(e; v ) :-- - 0 3 v 3 .
II Proof. Assume that this inequality is false. Then there exist em > 0 a n d v m - (vr~) e V ( f ~ ) , m - O , 1 , . . . (the Latin l e t t e r s m and n are used here for indexing sequences), such that
em -+ 0 as m - + cr 1 ~m
eillJ(em; v m) -+ 0 in L2(Ft) as m --+ c~.
Linearly elaztic generalized membrane shells
260
[Ch. 5
Since the sequence (Vm)mc~=O is bounded in H i ( ~ ) , there exist a subsequence ( v " ) ,oo = o and a function v E V(f~) such that v " ~ v in H i ( f ~ ) a n d v" ~ v in L2(f~)as n ~ c~.
Since
1 eallfl.(em; vm ) ~ 0 in L2(f~), em
10~eatlfl(e,; v n) --+ 0 in H - l ( a ) , gn
and, by the second inequality in Thin. 5.2-1,
+
1
v"
~n
_
)11 z,a
Hence the assumptions on the sequences (em)m~-_o and (v m)m~=o imply that We may thus apply Thm. 5.2-2. This gives: v - (vi) is independent of za,
-v - ( v i ) : =
1/:
~
i v dza E H i (w) • H i (w) • H 2 (w),
vi - Ovva - 0 on 70,
7af3(v) - pail(v) - 0 in ft. Consequently, 7af3 (v) - Pail (v) = 0 in w. The i n f i n i t e s i m a l rigid displacement l e m m a on a general surface (Thm. 2.6-3 (b)) then implies that ~ = 0 in w; hence v - 0 in fl since v is independent of xa. Thm. 5.2-2 also shows that v" -+ v in Hi(f~) as n --+ c~, so that v" --+ 0 in I-Ii(f~). But this contradicts [[v"l]i,a - 1 for all n, and the proof is complete, m If the mapping O is of the form 8(yl, y2) - (yi, y2, 0) for all (yi, Y2) E ~, the above inequality of Korn's type reduces for e - 1 to the t h r e e - d i m e n s i o n a l K o r n inequality in C a r t e s i a n coordinates over the set ~ = w • 1, 1[. m Remark.
Sect. 5.4] 5.4.
Generalized membrane shells of the first and second kinds
GENERALIZED MEMBRANE FIRST AND SECOND KINDS
261
SHELLS OF THE
According to the definition given in Sect. 5.1, if a linearly elastic shell is a generalized membrane shell, the space
VF(W) -- {W --(~7i) e HI(w) • HI(w) • H2(w); 7]i -- Or,~3 - - 0
on 7o, 7a~(17) - 0
in w)
reduces to {0). This condition is clearly equivalent to stating that
2
1/2
a,f~ for all ~1 - (Yi) e H i ( w ) • H i ( w ) • L2(w), becomes a n o r m over the space
v~(~)
:= {~ = (~i) e H~(~) • H~(~) • H~(~); r / / - O,,r/3 = 0 on 70 }.
The subscript " K " reminds that this space is central to the study of the two-dimensional Koiter equations for a linearly elastic shell; cf. Sect. 2.6 and Chap. 7. As already noted in Sect. 5.1, a linearly elastic elliptic membrane shell also provides an instance of such an occurrence since, by the inequality of Korn's type on an elliptic surface (Thm. 2.7-3), there exists a constant c such that 1/2
I1,o 1112,,,,+ J,~ 10,0,
< clv/IM for all v/E VM(w), --
O)
O~
where
v ~ ( . ~ ) . - Ho~(.~) • H~(~) • L~(~).
262
Linearly elastic generalized membrane shells
[Ch. 5
Hence if the shell is an elliptic membrane shell, the semi-norm I 9IM is already a norm over the space VM(W), thus a [ortiori over the space V g ( w ) , which in this case is equal to H~(w) x Hto(w) x H2(w). But this inequality of Korn's type shows much more, namely that the space VM(W) is complete when it is endowed with the norm I" !M (the "other" inequality clearly holds), a fact that sets elliptic membrane shells apart from generalized membrane shells. More specifically, the functional spaces in which the limit two-dimensional equations of a generalized membrane shell are well posed will turn out to be abstract completions, which are no longer immediately apprehensible! It seems in this respect that Destuynder [1985, p. 37] was the first to realize the need of considering such abstract completions in linearized membrane shell theory. As shown in the proof of the convergence theorem of Sect. 5.5, an additional precaution must be exercized, as generalized membrane shells need themselves to be subdivided into two different categories. To describe these, we introduce the two spaces v ( ~ ) .= {n = (v~) e E~(~); n = o o~ 70), v 0 ( ~ ) := {n e v ( ~ ) ; 7 ~ ( n ) = 0 i~ ~).
Remark. The space V0(w) thus plays the same r61e with respect to the space V(w) as does VF(w) with respect to r E ( w ) . In both cases, they constitute instances of spaces of "linearized inextensional displacements"; cf. Sect. 6.1. II Then a linearly elastic generalized membrane shell is of t h e first k i n d if
Vo(~) = {o), or equivalently, if the semi-norm 1" [M is "already" a norm over the space V(w) (hence a fortiori over the space VK(W) C V(w)). Otherwise, i.e., if
VF(w) = {0} but Vo(w)# {0}, or equivalently, if t 9IM is a norm over Vg(w) but not over V(w), the shell is a generalized membrane shell of t h e s e c o n d kind.
Sect. 5.4]
Generalized membrane shells of the first and second kinds
263
E z a m p l e s of linearly elastic generalized membrane shells of the
first kind abound; cf. Sect. 5.8. A w o r d of c a u t i o n . Whether a generalized membrane shell is of the first or second kind again depends only on the set V0 and on the "geometry" of the middle surface S; cf. Ex. 5.1. II An appraisal of the difficulties inherent to the analysis of generalized membrane shells lies in the definition of the ad hoc completions alluded to above: For a generalized membrane shell of the first kind, the limit two-dimensional equations are well posed in the space (Thm. 5.6-1) V~M(W) "-- completion of V(w) with respect to I" M, while for a generalized membrane shell of the second kind, these equations are well posed in the space (Thm. 5.6-2) V~M(W) := completion of v ( ~ ) / v 0 ( ~ ) with respect to I" MIdentifying such completions requires in each instance a careful analysis, which leads at times to surprising conclusions; the examples described in Sect. 5.8 are eloquent in this respect. Observe in this context how easier was the asymptotic analysis of a family of elliptic membrane shells, since V(w) - H01(w) in this case, so that the corresponding completion V~M(W) is simply the space -
•
•
To carry out our asymptotic analysis of a family of generalized membrane shells, we shall also need to consider the space (as usual, denotes the average of v with respect to the transverse variable) V0(12) : - {v E Hi(12); v - O on r0, Osv - O in 12, 7a~(v) - 0 in w},
which is the "three-dimensional analog" of the space V0(w). We shall likewise need the semi-norm I" IM defined by
iv12 .'-
2 + IO vlo,.
2}
for all v C V(12),
Linearly elastic generalized membrane shells
264
[Ch. 5
which is the "three-dimensional analog" of the semi-norm I" IM Observe in passing that 0g
"
Vo(.o) = {o) ~ Vo(n) = {o), or equivalently~ that l" IM i~ ~ ~o,m o~ the ~ p ~ v(~) if and only i f [ . IM is a norm over the space V ( ~ ) . In other words, a linearly elastic generalized membrane shell is of the first kind if and only if
Vo(a) = {o). 5.5.
ADMISSIBLE
APPLIED
FORCES
In order to derive the fundamental a priori estimates that the family (u(s))e>0 of scaled unknowns satisfies, we need to assume that the applied forces contribute in a special way to the variational problem T'(s; ~2)~ according to the following considerations. Consider a family of linearly elastic generalized membrane shells of the first or second kind, with thickness 2s, with each having the same middle surface S - 0(~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(70) as its middle curve, and let the assumptions on the data be as in Sect. 5.1. For each s > 0, let the linear form L(s) : V ( ~ ) --+ R be defined by
L(~)(,,) .= fn
+uP_
for aU v E V(~2).
In other words, L(s)(v) is the right-hand side in problem :P(s; ~) (Thin. 5.1-1)~ which takes into account the applied forces through the functions ]i E L2(~) and h i C L2(r+ u r_). Then each linear form L(s) 9V(~2) -+ I~ is clearly continuous with respect to the norm I[" I!1,~ and uniformly so with respect to 0 < s ~ ~o (recall that g(s)(x) <_ gl for all x E ~ and an 0 < 6 _< 60; of. Thm. 3.3-1). By the three-dimensional inequality of Korn's type for a family of linearly elastic shells (Thin. 5.3-1), there thus exists a constant a-1 such that
L(~)(v)l _< '~-_A~ ~ g
i,j
le~jjj(~; vlto,~
)1/2for ~n v c v ( a ) .
Admissible applied forces
Sect. 5.5]
265
This inequality shows that the linear forms L(s) 9V(f~) --~ I~ are also continuous with respect to the norm (itself dependent on s)
i,j but no longer uniformly so with respect to s if no specific assumption is made on the applied forces. As we shall see, carrying out the asymptotic analysis of a family of generalized membrane shells of either type crucially hinges on the uniform continuity with respect to s of the linear forms with respect to this norm (cf. part (ii) in the proofs of both Thins. 5.6-1 and 5.6-2). In view of fulfilling this requirement in a concrete manner, we observe that, by the Hahn-Banach extension theorem (see, e.g., Yosida [1966, Chap. 4]), if a linear form L ( s ) : V(f~) ~ R is continuous with respect to this norm, there exist functions F i J ( s ) -- FJi(s) E L 2 ( ~ ) (which are not necessarily uniquely determined) such that
: fn FiJ(~)eillj(s;
v)~(~)dx
for all v e V(f~).
The required uniform continuity is thus guaranteed if we impose that the norms [FiJ(s)[0,a be uniformly bounded with respect to s. But we shall also see that the identification of the limit twodimensional problems found at the outcome of the asymptotic analysis further requires that the functions FiJ(s) have limits in L2(~) as -~ 0 (cf. part (v) in the proofs of both Thms. 5.6-1 and 5.6-2). Combining these two requirements, we are therefore naturally led to the following definition: Applied forces acting on a family of linearly elastic generalized membrane shells are said to be a d m i s s i b l e if there exist for each s > 0 functions FiJ(s) - FJi(s) E L2(f~) and there exist functions F ij = F ji E L2(f~) such that
- fn for aU 0 < s _< so and for all v E V(f~),
FiJ(e) --~ F ij in L2(~) as s ~ 0.
266
Linearly elastic generalized membrane shells
[Ch. 5
If the applied forces are admissible, there thus exists a constant n0 such that (recall that g(e)(x) < gl, x E ~; cf. Thm. 3.3-1)"
i,j
for all 0 < e _< e0 and for aU v C V(~2). As announced supra, this inequality will be put to an essential use in part (ii) of both Thms. 5.6-1 and 5.6-2.
Remark. The more convenient definition of admissible forces given here, which is due to Mardare [1998c], slightly differs from that originally given by Ciarlet & Lods [1996d]; cf. Ex. 5.2. 1 This definition is in effect an assumption about the contravariant components f~'e E L2(~ e) and h ~,e E of the applied body and surface forces actually acting on the shells of the family under consideration: It means that the right-hand side in the variational equations of problem ~(~e) (Sect. 3.1) can also be written ]or each
~>Oas
f~e
f$1 ev ie ~ / ~
d x e -4-
-
+ur 9 ~_
where vi(x) "- v~(x e) for all x e - 1rex e ~6 and L(e)(v) is of the above form. As such, it may be again understood as an assumption on the orders of the applied forces as 6 -+ 0. Naturally, admissible forces have to be identified for each type of generalized membrane shells; see in this respect the examples described in Sect. 5.8. 5.6.
C O N V E R G E N C E OF T H E S C A L E D D I S P L A C E M E N T S AS g --+ 0
We now establish the main results of this chapter: To begin with, we consider a family of linearly elastic generalized membrane shells o] the first kind, with thickness 2e > 0, with each having the same middle surface S = 8(W) and with each subjected to a boundary condition of place along a portion of its lateral face having the same
Sect. 5 . 6 ]
Convergence of the scaled displacements as e --4 0
267
set 0(V0) as its middle curve. Then the solutions u(e) of the associated scaled three-dimensional problems T'(e; [2) (Thm. 5.1-1) converge in an "abstract" completion V~M(~2) as e -+ 0. In addition, the averages
~(~)
1/
2
u(e)dxs
1
of the scaled unknowns likewise converge in an "abstract" completion V~M(W) as e -+ 0 and their limit satisfies an "abstract" variational problem posed over the same space VUM(W). The functions 7af3(v/) and a a ~ r used in the next theorem respectively represent the covariant components of the change of metric tensor associated with a displacement field yia i of the surface S and the contravariant components of the (scaled) two-dimensional elasticity tensor of the "limit" shell. Hence the bilinear form BM defined in/ra is precisely that found in the scaled variational problem of a linearly elastic elliptic membrane shell (Thm. 4.5-1). We recall that e0 > 0 is defined in Thm. 3.1-1. For convenience, we recapitulate the definitions of some spaces and semi-norms appearing in Thm. 5.6-1 and its proof:
v(a)
- {~ - (,~) e I ~ ( a ) ;
103rIo,2 • "J- ([~lwM)2 1/2
IVI~
v(~)
~ - o o . r0 - -~o • [-1, 1]}, where ~ -
1
- {~ - (~i) e H~(~); ~ = 0 o~ ~o},
Io1~ -
{
~
17,~,~(~)1~,,,,
}~/~
9
Notice that the assumption that is central to this chapter, namely that VF(w) -- {0} (Sect. 5.1), is not needed until part (vi) of the proof. The following result is due to Ciarlet & Lods [1996d, Thm. 5.1]; only the proof had to be slightly modified to take into account the new definition of admissible applied forces given in Sect. 5.5. T h e o r e m 5.6-1. Assume that 0 C C3(~; I~3). Consider a family of linearly elastic generalized membrane shells of the first kind (Sect. 5.4), with thickness 2s approaching zero, with each having the same middle surface S = 0(-~), with each subjected to a boundary condition
268
Linearly elastic generalized membrane shells
[Ch. 5
of place along a portion of its lateral face having the same set 0(')'0) as its middle curve, and subjected to applied forces that are admissible (Sect. 5.5). Let u(e) denote for 0 < e < eo the solution of the associated scaled three-dimensional problems 79(e; f~) (Thin. 5.1-1). Define the spaces
V~M(fl) "= completion of V(~t) with respect to l" IM, V~M(w) "-- completion of V(w) with respect to l" IM.
:=
1;
u(e) -+ u in V~M(f~) as e --+ 0, u(e) dx3 -+ r in V~M(w) as e -+ 0.
Let 4,~1~ aaf3a~ r , + 2p(aa~af3 r + a ~ a ~ ) ,
a a ~ ~ .__
)~+2p
1
7 ~ ( ~ ) "- i(0Z~. + 0o~Z) - r,zn~ - b,z~3, BM(r n) "- f~ a " Z ~ ~ ( r
fo~ r n e V(~), for n e V(w),
LM(rl) "= L ~~
1
A+2#
where the functions F ij C L2(f~) are those used in the definition of admissible forces (Sect. 5.5), and let B~M and L~M denote the unique continuous extensions from V(w) to V~M(W) of the bilinear form BM and linear form LM. Then the limit ~ satisfies the following sealed
two-dimensional variational p r o b l e m 7~M(W) of a linearly elastic generalized m e m b r a n e shell of the first kind: r E V~M(w) and B~M((, 1/) - L~(r/) for aU 7/E V~M(w).
Sect. 5.6]
Convergence of the scaled displacements as s ---> 0
269
Proof. The proof is divided into eleven parts. Throughout the proof, we let
~llj(~) := e~ll~(~; u(~)), and we let -~ and ~ respectively denote strong and weak convergences. Whenever Cl, c2, etc., appears in an inequality, it means that there exists a constant, denoted by this symbol, that is > 0 and independent of all the "variables" entering the inequality such that this inequality holds. (i) There exist 0 < el <_ s0 and co such that, for all 0 < ~ < el, Ivl M ~ ~o
{
,
Let ~(v)
'}
~
i/2
I~,llj(~; v)lo,~
for all v 6 V(f~).
o
1 "- ~(O~v~ + 0~,~) - r ~ , v ~ - b ~ v s
for all v - (vi) E V(f~). The relations
~ . , ( . ) - ~.ll~(~; .) + (r~,(~) - r~,)v~ - ~ b ~ b ~ , ~ , Oava -- 2seall3(s; v ) -
sOav3 +
2sr23(s)v~,
Oava - sealla(s; v),
IIr~(~) - rE~llo,~,~ = o(~),
Ilr~(~)llo,~,~ = 0(1),
the last ones being consequences of Thm. 3.3-1, imply that, for all 0 < s < s0 (without loss of generality, we may assume that so _ 1) and for all v C V(f~),
a,~
i
i,j
Hence the desired inequality is obtained by combining the inequalities 1
i'Ya~(~)lo,~ _< ~l'~(,,)lo, a,
which fonow from T h m . 4.2-1 (~) ~nd the ~el~tion~ ~,,~(~) - ~,~(,,), with the three-dimensional inequality of Korn's type for a family of linearly elastic shells ( T h m . 5.3-1).
270
Linearly elastic generalizedmembrane shells
[Ch. 5
(ii) A priori bounds and extraction of weakly convergent sequences: The semi-norms lu(e)l~ and lu(~)l~ and the norms Iteu(~)il~,, and leillJ(~)10,n are bounded independently of 0 < ~ <_ e~. Define the spaces V~M(f~) "= completion of V(f~)with respect to t" IM V~M(W) "= completion of V(w) with respect to I" IMThen there is a subsequence, still denoted (u(e))e>0 for convenience, ~nd t h ~ e~i,t,, e V ~ ( ~ ) , u -~ = (~i x) e V(U), ~llJ e L~(~),
~(~) -~ ~ ~n V ~ ( . ) , ~ ( ~ ) -~ ~-~ in H~(~) a~d t a ~ ~ ( ~ ) -~ ~-x in L~(~),
eill/(e) ~ eillj in L2(f~),
0~3(~) - ~e3113(~) -~ 0 in L2(~), u(e) ~ r in V~M(W) as e --+ O.
Since the applied forces are admissible, there exists a constant n0 such that (Sect. 5.5) 2 IL(~)(v)l < ~0{ ~ leillJ(~;v)i0,.} 1/2
i,j for all v E V(f~) and all 0 < e _< e0. Combining the variational equations of problem 79(e; f~) (Thin. 5.1-1) with the asymptotic behavior of the function g(e) (Thin. 3.3-1), the uniform positive definiteness of the scaled three-dimensional elasticity tensor (Thin. 3.3-2), the assumption that the applied forces are admissible, and the inequality obtained in part (i), we thus obtain
%2 (lu(~)lM)2 < ~ leill~(~)12 --
O,n
i,j
Cego 1/2 f~ Aijkl(g)eklll(e)eillj(g)v/g(g) dx = Cegol/2L(e)(u(e))
2 i,j
Sect. 5 . 6 ]
Convergence o1: the scaled displacements as e --+ 0
for all 0 < e < el.
271
The boundedness of ]eillJ(e)[o,~, hence that
of i~(6)[~ and of iu(~)lY < lu(e)l~, fonow from this inequality; the boundedness of II~u(~)lll, a fonows from the three-dimensional inequality of Korn's type for a family of linearly elastic shells (Thm. 5.3-1). Note that the assumption that the applied forces are "admissible" is crucially needed here. (iii) The limits eillJ found in part (ii) satisfy 1 /7/33 Call3- 2--~aa~ , F33
e3113 - - A +A2p aaf3eal I~ + A + 2 p the functions F ij E L2(f~) being those used in the definition of admissible forces (Sect. 5.5).
Let v = (vi) be an arbitrary function in the space V(f~). Then ee~ll~(s; v) -~ 0 in LZ(f~), 1 eeallz(e; v) -+ ~03va in L2 (f~), eealla(e; v) - 03v3 for all e > 0, by definition of the functions eillJ(e; v). The variational equations of problem :P(e; 12) (Thm. 5.1-1) may also be written as (note that Aaf3~z(e)- AaZZz(e)= 0; cf. Thin. 3.3-2 (b)):
+ fa 2F~
~)}v/g(~)d~
Linearly elastic generalized membrane shells
272
[Ch. 5
Keep v E V(f~) fized and let e --> 0. The asymptotic behavior of the functions eeillj(e; v), the asymptotic behavior of the functions Aqkl(e) and g(e) (Thms. 3.3-1 and 3.3-2), and the weak convergences eillJ(z) ~ eillj in L2(fl) (part (ii)) together imply that
•
{2/IaaCre~r[]303Va +
()kaCrre~l[r + ()k +
2/t)e3113)O3v3} ~fadx
-"/a { fa3 Oava "t- F a a 0 3 v 3 }
V/-a d;c.
Letting v vary in V(f~) then yields the relations satisfied by the limits eill3 (if w E L2(f~) satisfies fa wO3vdx = 0 for all v C H1(~2) that vanish on r0, then w = 0; cf. Thm. 3.4-1). (iv) The whole family (u(e))e>0 satisfies: {eall/J(e) - 7 a ~ ( u ( e ) ) } ~ 0 in L2(fl) as e --+ 0; consequently, the subsequence considered in part (ii) satisfies: "~(~(~))
~
~-II~ ~ L 2 ( ~ ) 9
The definitions of the functions eall/3(e) and 3'a/3(u(e)) and the asymptotic behavior of the functions rPa~(e) (Thm. 3.3-1) together imply that __ p ~ , 2
where the functions ro.~ 2
a.
o.
satisfy ~~ 2 (E) [[0 '
for all 0 < e < e0. Since [v[0,~ <
< C2E2
~21~10,~(Thm.
4.2-1 (a)) and
x3v - 8 9 x2)O3v (this relation, which is easily seen to hold if v e CZ(~), is then extended to v e Hl(~) by Tam. 4.2-1 (b)),
1
Ix--~[o,o., - ~[(1- x2)O3vlo,,,, ~ c3]03vlo, n,
Sect. 5.6]
Convergence o1:the scaled displacements as e --->0
273
and thus
a
i
The announced convergence for the whole family thus follows from the boundedness of the norms Ileu(e)lil, n and semi-norms In(e)! M f~ established in part (ii). (v) The limits eall~ found in part (ii) satisfy
fw a a~Crre~ll;Ta~(17)v/-ddy - fw qoa#7~(17)~dy for all r/E V(w), where ~aa~F
ss } dx3 E L2(w),
A+2#
the functions F ij E L2(fl) being those used in the definition of admissible forces (Sect. 5.5). Let v = (vi) E V(f~) be independent of the transverse variable x3. For such a function v, eall/3(e; v) + %/3(v)in L=(f~),
,,)
+
e3113(e; v) = O. Keep such a function v E V(f~) fixed in the variational equations of problem ?(e; f~) and let e + 0. The asymptotic behavior of the functions AiJkt(e) and g(e) (Whms. 3.3-1 and 3.3-2), the above asymptotic behavior of the functions eillJ(e; v), the weak convergences eillJ(e) ~ eitlj in L2(f~) (part (ii)), and the relations satisfied by the limits e~llj (part (iii)) together imply that
~--~0
A
2Fa3
1
Linearly elastic generalized membrane shells
274
[Ch. 5
The assumption that the applied forces are admissible implies that, for such a function v,
lim L(e)(v)
e:--+O
: e~olimfa (Fa#(e)eall#(e; v)+ 2FaS(e)eails(e; v))v/g(e)dz
/.o
1
The conjunction of the above relations then implies that jf
2A/~ aaflaar
{ A + 2it
+ #(aa~ai3r +
aar
a'~) }%1t,7a/3 ( v ) x ~ dz
[,/w aa/3~r %lirTa~ (W)x/~ dy
~Aa+a2/.t ~FSS}dzs)
7a~(~)V~dy'
where ~ "= 89f_i 1 v dx3. Given any function ~/ E V(w), there exists a function v E V(f~) independent of x s such that ~ = r/. Hence the announced relations hold for all r/E V(w). (vi) The subsequence (u(e))e>0 found in part (ii) is such that ~,,(~) -~ 0 in H I ( ~ ) , 03uct(g) .._x 0 in L2(~),
as e --+ O. Furthermore, eall~ is independent of the transverse variable z s. The functions
~-~(~)-= ~u(~) e v(~) associated with the subsequence found in part (ii) satisfy u - i ( e ) ~ u -i in Hi(f/) and u - i ( e ) -+ u -1 in L2(f~), 1
Sect. 5.6]
275
Convergence of the scaled displacements as e ~ 0
Hence u -1 E VF(w) by Thm. 5.2-2 and consequently u -1 -- 0, since VF(w) -- {0} by assumption. By the same theorem, u -1 is independent of x a; hence U -1 -- 0~
and the convergence eu(e) ~ 0 in H i ( n ) , hence also the convergence su(s) --+ 0 in L2(~), are established. These, combined with the relations (which follow from the definition of the scaled strains
Osu~(e) -
2ee~lts(e)
-
eO~us(e) + 2erXs(e)u~(e)
and the boundedness of the sequence (r~3(e))~>0 in C0(~) (Thin. 3.3-1), in turn yield the weak convergences Oaua(e) ~ 0 in L2(f~). Again by Thm. 5.2-2,
o3~alw - -p,~z (u -~) - o, since u -1 - 0. The functions ealll3 are thus independent of x3. (vii) The following strong convergences hold as e -+ 0 (so far, they were only known to be weak; cf. parts (ii), (iv), and (vi)):
eillJ(e) ~ eillj in L2(~), ~'u(~) -+ 0 in H I ( n ) ,
-r,~,(u(~)) -~ ~otl, m z 2(,,,), u(e) -+ r in V~M(W). Letting v - u(e) in the variational equations of problem :P(e; f~) and using the asymptotic behavior of the function g(e) (Whm. 3.3-1) and the uniform positive definiteness of the scaled elasticity tensor (Thm. 3.3-2), we obtain
i,j
where
AiJm(e)(2eklll(6) -- eklll)eillj X f ~
dx.
276
[Ch. 5
Linearly elastic generalized membrane shells
The strong convergences FiJ(e) --+ F ij in L2(f~), the weak convergences eillj(s ) ---, eillJ in L2(f~) (part (ii)), and the relations satisfied by the limits eillJ (parts (iii) and (vi)) together imply that
limL(s)(u(e))
e--+O
= e~olim{faFa~(e)eall~(e)v/g(e)dz
:
~
~~162
+
fa
X { ;~~176
+ ~ +1 2----~F33F33}v~d x
as e --+ 0, on the one hand. The asymptotic behavior of the functions Aifl~z(e) and g(e) (Thms. 3.3-1 and 3.3-2), the weak convergences eill.i(e ) ---" eiltJ in L2(f~) (part (ii)), and the relations satisfied by the limits eillJ (parts (iii) and (vi)) together imply that lim ~ A/Jm(s)(2ekllt(s) -
e--+O
eklll)eiljjv/g(e)dx
= f~ AiJkZ(O)ekllteilljV/-ddz
~A +a2a#l 3 F33)Call/3~
dm
+/n { 4paaZ%lj3eal]3+ F33e3113} V~ dx =
~ aa~ar%11~ eall~v/-ady 1 Fa3F~3 + l + fn {;~o~ A+2p
F 33F 33 } v/-a dz,
on the other. Hence
lim A(e) - A "- { ~ ~oa/3ea,,/3v/~dy-- ~ aa[3~ e~llr eajl~v/-ddY } " e-+O Since u(e) E V(w) for each s > 0, we have in particular
f aa~Te,rllr3,a~(u(e))V/'ddy - f~ r
(u(e))x/~dy'
Sect. 5 . 6 ]
Convergence of the scaled displacements as e ~ 0
277
by part (v). Using the weak convergences "ya/3(u(e)) ~ eall/3 in L2(w) (part (iv)), we thus obtain A=0. In this fashion, the strong convergences eillJ(e) --y eillJ in L2(ft) are established. In Thm. 5.2-1, we showed that 1 03eall~(e; v) + paD(v)[l_l,f~
for all v E Hi(f~), where I1" ]l--l, fl denotes the norm in g - l ( ~ ) . Letting v - " u - l ( e ) "-- e ' u ( e ) in this inequality thus yields
by part (ii). Since eall/3(e ) ~ chill3 in L2(ft) as shown above and since the limits eall~ are independent of the transverse variable (part (vi)), 03eall~(e ) -+ 03eal]~ -- 0 in H - l ( f t ) ;
therefore,
p~(~-~(~)) -~ 0 i~ H-~(a). An application of Thm. 5.2-2 thus shows that u-~(~) - ~,~(~) -+ o i~ H~(a).
By Thm. 4.2-1 (a), the strong convergences eillj(e ) -~ eillJ in L2(f~) imply in particular the strong convergences
~ll,(~) -* e~ll, i~ Z2(~), which, combined with those of part (iv), yield the strong convergenccs 7a~(u(e)) ~ eaf[~ in L2(fl). Hence (Ta~(u(e)))~>o is a Cauchy sequence in L2 (w). Since
278
Linearly elastic generalized membrane shells
[Ch. 5
the strong convergence u(e) ---4(~ in V~M(W) is established. (viii) The limit ~ E V~M(W)found in part (vii) satisfies the equations B~M(r 17)- L~M(y) for a l l . E V~M(w), which have a unique solution. Consequently, the convergence r
established in part (vii) holds for the whole family (u(e))e>o.
Let v/E V(w). Since u(s) E V(w), limBM(u(e) r/) = lim f~ aa~Ver(u(e))Ta~(rl)v~dy
e-+O
~
~-40
-- ~ aaf3areali~'7,~f~(n)V~dy = L(n),
by part (v) and the convergences 7af3(u(e)) --+ ea[[~ in L2(w) established in part (vii). But we also have lim BM(U(S) ~?) = B~M(r ,7) since u(e) -+ ~" in V~M(w) (part (vii)). Hence B~M(r O)
-
L(y) for all rl E V(w),
and thus B~M(~, y ) - L~M(y) for all I7 E V~M(W), by definition of the continuous extensions B~M and L~M. These variational equations have one and only one solution, for the space V~M(W), the bilinear form B~M, and the linear form L~M satisfy aU the assumptions of the Lax-Milgram lemma (the ellipticity of the bilinear form relies in particular on the uniform positive definiteness of the scaled two-dimensional elasticity tensor of the shell; cf. Thm. 3.3-2). We next prove a simple corollary of the lemma of J.L. Lions; this corollary will be put to use in part (x).
Sect. 5.6]
Convergence of the scaled displacements as e -.-+ 0
279
(ix) Let 12 be a domain in ~3. Given v - (vi) E L2(~), define the distributions
~ij(.) "= ~l (o~,j + %~) e H-~(~) Let there be given a sequence of functions v h = (v~) e L2(12) such that
v k --+ 0 in H-l(12) and eij(v k) -+ 0 in H-l(12) as k -+ oo. Then
v k --+ 0
in L2(~).
As shown by Amrouche & Girault [1994], the implication {w E H-2(12) and Oiw E H-2(12), 1 ~ i < n} =~ w E H-l(12) holds if 12 is a domain in I~n, where H-2(12) := dual space of Ho2(f~). Let then v = (vi) be such that v E H-l(12) and % ( v ) C H-l(12). Since w e H-l(12) implies Ojw e H-2(f~), we have
O~v~ e H-2(e), o k ( o j ~ ) - { o k ~ j ( ~ ) + oj~k(~) - o ~ j k ( ~ ) } e H - ~ ( a ) ,
so that Ojvi e H-l(12). Using the lemma of J.L. Lions (Thm. 1.7-1), we therefore conclude that v E L2(f~). In other words, {v - ( v i ) E H-l(12); eu(v ) E H-l(12)} = L2(12).
Then the closed graph theorem shows that
[V[O,~l< c6 { [[V[[-1, f~ -+-~ [[eij(v)[[-1,fl) i,j
for all v E L2(f~) and the asserted property follows from this inequality.
280
Linearly elastic generalized membrane shells
[Ch. 5
(x) The following strong convergences hold (so far, they were only known to be weak; cf. parts (ii) and (vi)): u(s) --+ u in V~M(fl), 03ua(s) --+ 0 in L2(fl). To establish the convergence u(s) --+ u in V~M(~), it suffices to show that (u(s))e>0 is a Cauchy sequence with respect to the norm !" IM. By definition,
(1,,(~)- ~(~')1~) ~ = ~
I ~ ( ~ , ( ~ ) ) -"Yo~(,,(~'))lo~,~ + Z
!o~,(~) - o~,(~')1o~ ~.
Consequently, in view of the convergences 03u3(s) -+ 0 in L2(~) and 7a~(u(s)) --+ ealll3 m L2(w) (parts (ii) and (vii)), it remains in fact to establish the convergences 03ua(s) -+ 0 in L2(~). Proving these is equivalent to showing that 03u'(e) --+ 0 in L2(fl), where u ' ( 6 ) : = (Ul(e), u2(s), 0). By part (ix), this is in turn equivalent to proving that 03u' (s) --+ 0 in H -1 (~),
together with eij(O3u'(s)) -+ 0 in H-Z(~),
where
~j(.) := ~1 (oj,~ + o~,j) By definition of the functions eallz(s),
0~=~(~) = 2e~ll3(~) - ~o~=3(e) + 2erx~(~)=.(~). Hence the boundedness of the sequences (I'~3(e))e>0 in C~ 3.3-1) and the convergences (parts (vi) and (vii)) seall3(s ) --+ 0 in L2(~),
~u~(e) -~ o in L2(n), imply that
(Thm.
Sect. 5 . 6 ]
Convergence of the scaled displacements as e --+ 0
281
this shows that 0su' (e) --+ 0 in H - 1(f~). In view of establishing that eij(O3u'(e)) --+ 0 in H-l(f~), we first notice that ~s(os,,'(~))
= o3~ss(~,'(~))
= o.
The asymptotic behavior of the functions r~,s(e ) (Thin. 3.3-1) and the convergences eillJ(e) -4 eillJ in L2(f~) (part (vii)) next imply that {03Uct(g ) "4- e0c~t3(g) -}- 2~bXu~(e) } --4 0 in Z2(n); consequently O33ua(e) -1- eOa3u3(e)
-t-
2eb~,O3uo.(e) ~ 0 in H - l ( f l ) .
Since 03u3(e) --+ 0 in L2(f~) (part (ii)) and eu,,(e) ~ 0 in L2(f~) (part (vi)), we have
2eas(Osu'(e))
-
Ossua(e) --+ 0 in H-1(~2).
From the relations ealll3(e) --+ Call/3 in L2(~2) and 0seall/3 - 0 (part (vi)), we next infer that 03eall/3(e) --+ 0 in H - I ( ~ ) , hence that
{Osea~(u(e))-03( r pa~(e)up(e))} --~ 0 in H -1 (~), by definition of the functions eall/3(e). Because the functions rPa/3(e) are in cl(-~), the functions rPa~(e)Up(e) are in HI(~); besides,
o~(r~z(~)=~(~))
-
(o~r ~~z(~))=~(~)+ r ~~
(~)os~(e),
and c#3
since 03up(e) ~ 0 in H-~(f~) (we have established that 03U3(g) ~ 0 in L2(f~) in part (ii) and we just established that 03ua(e) ~ 0 in H-l(f~)). Finally, the estimates 11o3r~(~)llo,~,~ < c ~ (Thm. 3.3-1) and the convergences eu(e) --+ 0 in L2(ft) (part (vi)) imply that
(o~r~(~))~,(~) -~ 0 i~ L~Ca).
We thus infer from these relations that
and all the required convergences eij(O3u'(e)) -~ 0 in H-I(~) are thus established.
282
Linearly elastic generalized membrane shells
[Ch. 5
(xi) The whole family (u(e))e>0 converges strongly to u in the space V~M(~). We have shown that the whole family (u(s))e>0 converges in the space V~M(W) (part (viii)) and that Osu(6) --+ 0 in L2(~) for a subsequence (parts (ii) and (x)). Since the limit of this subsequence is unique, the whole family (Oau(s))e>o converges in L2(~). Hence the family (u(s))e>0 is a Cauchy sequence in the Hilbert space V~M(~), and the proof is complete. 1 Remark. Thm. 5.6-1 applies afortiori to a family of linearly elastic membrane shells, in which case the space V~M(W) is simply H~(w) • H~(w) • L2(w). The convergence obtained in this fashion "over the set w" is identical to, but the convergence "over the set fl" is weaker than, that obtained in Thm. 4.4-1, where we were able to establish the convergence of the family (u(e))e>0 in the space H I ( ~ ) • H I ( ~ ) x L2(~). In addition, we did not have to assume in Thm. 4.4-1 that the forces were "admissible". 1
We next consider a family of linearly elastic generalized membrane shells of the second kind, in which case similar convergences hold, again in "abstract" completions, but these are now completions of quotient spaces. We use the following notations: The equivalence class of v E V(~) in the quotient space V ( ~ ) / V 0 ( ~ ) is noted/~; likewise, the equivalence class of ~ e V(w) in the quotient space V ( w ) / V o ( w ) is noted/1. Similar notations designate equivalence classes in the completions The foUowing result is due to Ciarlet & Lods [1996d, Thm. 5.1]. T h e o r e m 5.6-2. Assume that 0 E C3(~; R3). Consider a family of linearly elastic generalized membrane shells of the second kind (Sect. 5.4), with thickness 2e approaching zero, with each having the same middle surface S - O(-~), and with each subjected to a boundary condition of place along a portion o] its lateral face having the same set 8('/0) as its middle curve, and subjected to applied forces that are admissible (Sect. 5.5). Let u(e) denote for 0 < ~ ~_ ~o the solution of the associated scaled three-dimensional problems :P(e; ~) (Thm. 5.1-1).
Sect. 5.6]
Convergence of the scaled displacements as e -~ 0
283
Define the spaces V~M(f]) "-- completion of V(f~)/V0(~t) with respect to [-[M, V~M(W) "-- completion of V(w)/Vo(w) with respect to l" IM
where Vo(~2) := {v E V(ft); 03v - 0 in f/, %t3(~)= 0 in w} # {0}, Vo(w) := {I/E V(w); %t3(~/)- 0 in w} # {0}.
Then there exist it E V~M(~) and ~ E ~r~M(W) such that /,(~) ~ / ,
in V~M(ft) as 6 -+ 0,
u(e) dz3 --+ ~ in V~M(w) as e --+ O.
Let the forms BM and LM be defined as in Thm. 5.6-1.
r
For
o E
/3M(~, //) := BM(~', rl') for any r E ~ and any r/' E//, LM(//) := LM(rI') for any r/' E//,
bilinear form BM and linear form s from V(w)/Vo(w) to ~flM(W). Then the limit ~ satisfies the following scaled t w o - d i m e n s i o n a l variational p r o b l e m 7~(w) of a generalized m e m b r a n e shell of the second kind-
284
[Ch. 5
Linearly elastic generalized membrane shells
Proof. The proof closely follows that of Thm. 5.6-1. We simply indicate which parts need to be modified.
Part (ii) becomes (otherwise its proof is identical)"
The semi-
.o~m~ I"(~)lg . . d I"(~)!~, h e . ~ th~ .o~m~ la(~)l.~ . . d la(~)l.~ and the norms ]leu(e)iil, a and ]eillJ(e)lo, n , are bounded independently
o/O<e<e~. Define the spaces
v ~ ( ~ ) := ~ompletion of V ( n ) / V 0 ( ~ ) with ~e~pe~t to I" I~, V~M(W) := completion of V(w)/Vo(w) with respect to I" IM. Then there is a subsequence, still denoted (u(s))~>0 for convenience, and there exist/~ C ~r~M(f~), u -1 = (u~-1) C V(f~), eillJ E L2(f~), and E ~r~M(W) such that
i,(~) ~ i, ~n ~r~(~), ~,(~) -
~,-~ in a x ( n ) ~nd thu~ ~,(~) + ~,-~ i~ r.~(~),
eillJ(~) ___x eill j in L2(~'~),
03u3(~) - ~3113(~) + 0 in L2(~), uie) ~ t in *~M(W) as e -+ O. Part (vii) becomes" ~ ---~0:
The following strong convergences hold as eillJ(e) ~ eiltJ in L2(~),
~(~) + o i~ H~(a). ~(~(~11 + ~lle in L2(~). uie ) -+ ~ in *~M(W),
The only difference in the proof of part (vii) consists in noting that
and thus that the strong convergence u i e ) - ~ ~ in "V~(w) holds.
Sect. 5 . 6 ]
Convergenceof the scaled displacements as e --+ 0
285
Part (viii) and its proof become: The limit ~ E ~r~M(W) found in part (vii) satisfies the equations /)~M(~, //) -- L~(//) for all/1 E *~(w), which have a unique solution. Consequently, the convergence
established in part (vii) holds for the whole family (uie))e>0.
Let n E V(w). Since u(e) E V(w), P
lim BM(U(e) n) -- lim ]~, aa~Vrvvr(U(e))Va#(~l)x/~dy e--40
~
e-40
: f aa#~r e a li#'Va#(17)x/~ dy J~ :
L(n),
by part (v) and by the convergences Va#(u(e)) --+ call# in L2(w) established in part (vii). But we also have BM(U(e), i7) BM(U(e), ~1) and L(//) - L(r/) by definition of the forms/3 and s therefore, -
lim BM(uis ) il) : L(il)
e--~O
~
-
~
on the one hand. On the other, lim .BM(Uie), il)=-B~M(~, il),
e---~0
since ui-s ) --+ ~ in ~r~M(W) (part (vii)). Hence /~M(~, r
i(r
for all ~ E V(w),
and thus by definition of the continuous extensions/3~M and L~M. These variational equations have one and only one solution, for the space V~M(W), the bilinear form /)~M, and the linear form /~M, satisfy all the assumptions of the Lax-Milgram lemma (the ellipticity
286
Linearly elastic generalized membrane shells
[Ch. 5
of the bilinear form relies in particular on the uniform positive definiteness of the scaled two-dimensional elasticity tensor of the shell; cf. Thm. 3.3-2). Part (x) becomes: The/ollowing strong convergences hold:
Osua(e) --+ 0 in L2(a). To establish the convergence/~(e) -+/~ in V~M(~), it suffices to show that (/~(s))e>0 is a Cauchy sequence with respect to the norm I" IM. By definition, -
= ~ a,~
=
-
I~(~(~11 -~.~(~(~'111~,~ + ~
10~,(~) - o~(~'11~,.. i
The remainder of the proof of part (x) is the same. Finally, part (xi) and its proof become: The whole/amily (u(e))e>0 converges strongly to it in the space ~r~M(f~). We have shown that the whole family (uie))e>o converges in the space ~r~M(W) (part (viii)) and that 03u(e) --+ 0 in L2(~) for a subsequence (parts (ii) and (x)). Since the limit of this subsequence is unique, the whole family (03u(e))e>o converges in L2(~2). Hence the family (u(~))e>o is a Cauchy sequence in the Hilbert space ~r~M(~2), and the proof is complete. II A scrutiny of the proofs of Thms. 5.6-1 and 5.6-2 shows that they provide worthy information about the asymptotic behavior of the scaled three-dimensional solutions u(e) and of their averages u(e), viz.
eillJ(e; u(e)) -+ eillj in L2(~),
os,,(~) ~ o in L(~), ~,(~) ~ 0 in tt~(~).
Sect. 5.7]
The two-dimensional equations
287
A noticeable feature of these convergences is that they occur in spaces that are more "decent" that the somewhat "exotic" spaces V~M(W) For a family of linearly elastic elliptic membrane, or flezural, shells, the three-dimensional limit u was, or will be, found to be independent of ~3 (Thms. 4.4-1 or 6.2-1). The convergence 03u(e) -~ 0 in L 2(f~) is an attenuated form of this property for a family of generalized membrane shells.
5.7.
T H E T W O - D I M E N S I O N A L E Q U A T I O N S OF A LINEARLY ELASTIC GENERALIZED M E M B R A N E SHELL; E X I S T E N C E A N D U N I Q U E N E S S OF S O L U T I O N S
For brevity, we only consider in this section generalized membrane shells of the first kind, as those of the second kind are treated in a similar fashion. The next theorem recapitulates the definition and assembles the main properties of the limit two-dimensional problem found at the outcome of the asymptotic analysis carried out in Thin. 5.6-1.
T h e o r e m 5.7-1. Let w be a domain in IR2, let 70 be a subset of the boundary of w with length 70 > 0, and let 8 e C3(~; IR3) be an injective mapping such that the two vectors aa -- Oa8 are linearly independent at all points of ~. Assume that Vo(~)
=
{o}, wh~
v 0 ( ~ ) := {~ = (~) e H~(~); ~ = o o~ ~0, ~ ( ~ ) 1 ~o~(~) .= ~ ( o ~ o + o o ~ ) - r ~ ~ - bo~3,
= 0 i~ ~},
and define the spaces
V(w) "= { ~ / - (Yi) E Hi(w); 17 = 0 on 70}, V~M(W) "-- completion of V(w) with respect to [. ]M, where
a, fl
Linearly elastic generalized membrane shells
288
[Ch. 5
Let aaf3ar ._ _
BM(r
U) ' =
4)~lZ aaf3aar + 2D(aaaaf3r + aaraf3~), A+2~
L ~"~~(r
~o~ r U e V(~).
L(U) "- f V ~ . ~ ( u ) ~ d y ~o~ U e V(~). J~
where the functions qoa# E L2(w) ave given, and let B~M and L~M denote the unique continuous extensions from V(w) to V~M(W) of the bilinear form BM and linear form L. Then the scaled two-dimensional variational problem 7)~M(W) of a linearly elastic generalized membrane shell of the first kind: Find that satisfies r C V~M(w) and B~M(r y) = L~M(y) for all y E V~M(w),
has one and only one solution. Proof. The assumption V0(w) = {0} means that the semi-norm [. ]M is a norm over the space V(w). The linear form L 9V(w) --+ R and the bilinear form BM : V(w) • V(w) --+ Ii~ are clearly continuous with respect to this norm. Besides, BM(~, ~)~_ cel v:~(l~l M ) 2 for all rl C V(w), since the scaled two-dimensional elasticity tensor of the shell is uniformly positive definite (Tam. 3.3-2 (a)) and a(y) > a0 > 0 for all y E ~ (Thm. 3.3-1). These properties remain valid on the space V~M(W) since V ( w ) i s by construction dense in V~M(W), again with respect to I" ]M~ . The conclusion thus follows from the La~-Milgram lemma. I Thm. 5.7-1 is established under the assumption that V0(w) = {0}. Therefore it applies not only to generalized membrane shells but also to elliptic membrane shells, since these satisfy the even stronger assumption (established in Thm. 2.7-2; recall that 70 = "Yin this case): {,I - (,7~)e H~(~,) • Ho~(~,) • L2(,,); "y~(,7) = o i~ ~,} - {o}.
Sect. 5.7]
289
The t w o - d i m e n s i o n a l equations
However, Thm. 5.7-1 alone does not say anything about the corresponding space V~M(W), which in this case turns out to be the space VM(W) -- H~(w) • H~(w) • L2(w). To prove this constitutes the "hard" part of the analysis, which rests on the inequality of Korn's type on an elliptic surface (Thm. 2.7-3). The same inequality is also crucially needed for establishing that, again in the case of an elliptic membrane shell, any continuous linear form L on the space VM(W) may be indeed expressed as in Thm. 5.7-1 by means of functions In order to get physically meaningful formulas, it remains to "descale" the unknown ~ that satisfies the limit "scaled" problem :P~M(W) found in Thm. 5.7-1. In view of the scaling -
-
-
e
made on the covariant components u~ of the displacement field (Sect. 5.1), we are naturally led to defining for each ~ > 0 the l i m i t v e c t o r field ~e by letting
A w o r d of c a u t i o n . For the elliptic membrane shells considered in the previous chapter, or for the flexural shells considered in the next chapter, each component of the field ~ = (~i) can be separately de-scaled, affording in turn the consideration of the limit tangential displacement field ~ a a and of the limit normal displacement field e 3 ~3a of the middle surface of the shell, where ~ := ~i (rE Sect. 4.6 or 6.4). By contrast, such a componentwise de-scaling is impossible for a generalized membrane shell, unless specific knowledge on the components of the "functions" in the completion V~M(W) allows to do so (the examples of Sect. 5.8 are particularly illuminating in this respect). In other words, the above de-scaling ~e = ~ is to be understood as an equality in the "abstract" completion V~M(W) and nothing else! 1 Recall that )e and #e denote for each e > 0 the actual Lam6 constants of the elastic material constituting the shell. We then have the following immediate corollary to Thm. 5.7-1; naturally, the existence and uniqueness results of Thm. 5.7-1 apply verbatim to the de-scaled problem T'~(w) (for this reason, they are not reproduced here)"
290
[Ch. 5
Linearly elastic generalized membrane shells
T h e o r e m 5.7-2. Let the assumptions and definitions not repeated here be as in Thm. 5.7-1, let aa/3'Tr, e ._.-- 4Ae# e aa/3a,Tr + 2l.te(aaaaf3r + aaraf3~r), A e + 2p e
Bi~(r V) := e f~ a " ~ " % ~ r ( r
for r
e V(~o).
L%(.7) := f~ ~o"~'~-y.~(.)~ey f o ~ . e v(~o). ~oar e := e~oa~, ~e I~e and let B M and L M denote the unique continuous extensions from V(w) to V~M(W) 4 the bilinear form B ~ and linear form L~M. Then the limit vector field ~e satisfies the following two-dimensional vari-
ational p r o b l e m 79~(w) of a linearly elastic generalized m e m b r a n e shell of the first kind: ~e ~, .) r e V~M(w) and BM(r
- L~
(.) for all
r/
e V~M(W).
Equivalently, the field ~e satisfies the following minimization problem:
~e E V~M(w) and j ~ ( ~ e ) _
inf.
Jg(,7) := II Each one of the two formulations found in Thm. 5.7-2 constitutes the two-dimensional equations of a linearly elastic generalized m e m b r a n e shell of the first kind. The functional j ~ . V~M(W) ~ R is the two-dimensional energy, and the functional 1 ~e-
is the two-dimensional strain energy, of a linearly elastic generalized m e m b r a n e shell of the first kind. The functions a a/3~T,e
Justification of the two-dimensional equations; ezamples
Sect. 5 . 8 ]
291
are the contravariant components of the t w o - d i m e n s i o n a l elasticity t e n s o r of t h e shell, already encountered in the two-dimensional equations of a linearly elastic elliptic membrane shell (Thm. 4.5-2). ~e Observe that the bilinear form B M found in the variational equations of a linearly elastic generalized membrane shell is an extension of the bilinear form B ~ already found in the variational equations of a linearly elastic elliptic membrane shell (Thm. 4.5-2). This is why both kinds constitute together the linearly elastic "membrane" shells. 5.8.
JUSTIFICATION OF THE TWO-DIMENSIONAL EQUATIONS OF A LINEARLY ELASTIC GENERALIZED MEMBRANE SHELL; EXAMPLES, COMMENTARY, AND REFINEMENTS
It remains to convert in terms of de-scaled unknowns the fundamental convergence result established in Sect. 5.6. As the "original" unknowns u ~ - (u~) are defined over a domain that varies with e (the set f~e), their averages 2~ fe-e u~ dz~ are more appropriate for this purpose, since they are defined over a fixed domain (the set w). For simplicity, we only consider generalized membrane shells of the first kind. T h e o r e m 5.8-1. Assume that 0 E C3(~; IR3). Consider a family of linearly elastic generalized membrane shells of the first kind (Sect. 5.4), with thickness 2~ approaching zero, with each having the same middle surface S = 0(-~), with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(7o) as its middle curve, and subjected to applied forces that are admissible (Sect. 5.5). For each e > O, let u e e ttl(f~ e) and Ce e V~M(W) respectively denote the solutions to the three-dimensional and two-dimensional
problems 79(f~e) and 79~(w) (Sect. 3.1 and
Thm. 5.7-2). Finally, let r e V~M(W) denote the solution to problem 7~M(W) (Thm. 5.7-1), which is thus independent of e. Then
~-~
1#
and ~
e
292
Linearly elastic generalized membrane shells
[Ch. 5
Proof. It suffices to combine the convergence
1; 2
1
established in Thm. 5.6-1, the relation
-
-
2s
e
u e dx~ - 1 -2
u(e) dx3, 1
which follows from the scaling u(s)(x) := u e (x e) for all x ~ - 7: x e Os (Sect. 5.1), and the de-scaling ~e := ~ (Sect. 5.7). II Remark. Without any specific information about the structure of the vector fields in the space V~M(W), these relations can no longer be refined (as in Sect. 4.6 for elliptic membrane shells) into ones involving separately the tangential and normal components of the three-dimensional displacement vector fields. II Under the essential assumptions that the space VF(W) reduces to {0) and that the forces are admissible, we have therefore justified by a convergence result (Thm. 5.8-1) the two-dimensional equations of a linearly elastic generalized membrane shell. In so doing, we have also justified the formal asymptotic approach of Caillerie & SanchezPalencia [1995b] when "bending is badly-inhibited", according to the terminology of E. Sanchez-Palencia. The asymptotic analysis of Ciarlet & Lods [1996d] described in this chapter has been extended by Slicaru [1998] to linearly elastic shells whose middle surface "has no boundary", such as a torus. Among linearly elastic shells, generalized membrane shells possess distinctive characteristics that set them apart: While forces applied to a family of elliptic membrane or flexural shells can be arbitrary (Sects. 4.1 and 6.1), body forces (for instance) applied to a family of generalized membrane shells can no longer be accounted for by an arbitrary linear form of the form v i v/fig dx ~, i.e., with arbitrary contravariant components fi,~ E L2(12~). They must be admissible "for the three-dimensional equations", in order that the associated scaled linear forms be continuous with respect to the norm v --+ { ~ i , j leillJ(6; v)l~,~} 1/2 and uniformly so with respect to s > 0 (Sect. 5.5).
Sect. 5.8]
Justificationof the two-dimensional equations; e~amples
293
The linear form found in the variational equations of the limit two-dimensional problem for such a shell is likewise subjected to a restriction: On the dense subspace V(w) of the space V~M(W), it must be of the form ~/--~ f~ 7~a~3'a~(r/)V~ dy (Thm. 5.6-1). In other words, the applied forces must be also admissible "for the two-dimensional equations "~ in such a way that the linear form appearing therein must be an element of the dual space of V~M(W). As this dual space may be quite "small'~ the limit variational problem, which otherwise satisfies all the assumptions of the LaxMilgram lemma (Thm. 5.7-1)~ possesses the unusual feature that its solution may no longer exist if the data undergo arbitrarily small, yet arbitrarily smooth, perturbations! Another unusual feature of this problem is that the space V~M(W) in which its solution is sought may not necessarily be a space of distributions! Such variational problems fall in the category of "sensitive problems" introduced by Lions & Sanchez-Palencia [1994]. Since then, such problems have been extensively studied~ either for their own sake or as regards their relations to the theory of generalized membrane shells, by Lions & Sanchez-Palencia [1996, 1997a, 1997b, 1998], Pitk/iranta & Sanchez-Palencia [1997], Sanchez-Palencia [1999], LeguiUon, Sanchez-Hubert & Sanchez-Palencia [1999]. Ezamples of linearly elastic generalized membrane shells are numerous. In each case~ however, the proof that the space VF(w) reduces to {0} and the identification of the corresponding space V~M(W) and of the admissible applied forces usually require delicate analyses. For this reason, these are not given here; only ad hoc references are provided. A measure of the difficulty inherent to showing that VF(w) -- {0} is given by the proof of the infinitesimal rigid displacement lemma when the surface S - 8(~) is elliptic and 70 = 7 (Whm. 2.7-2). All the following examples correspond to V0(w) - {0}, i.e., to generalized membrane shells of the first kind~ according to the definition of Sect. 5.4.
Remark. It seems that there are no known examples of generalized membrane shells of the second kind. According to C. Mardare, they should correspond to surfaces S "with little regularity", m A first ezample is provided by a shell whose middle surface is a portion of an elliptic surface S -- 8(-~), i.e., one whose Gaussian
294
Linearly elastic generalized membrane shells
[Ch. 5
Fig. 5.8-1: Two linearly elastic generalized membrane shells. A shell whose middle surface S = 0(~) is a portion of an ellipsoid and which is subjected to a b o u n d a r y condition of place (i.e., of vanishing displacement field) along a portion (darkened on the figure) of its lateral face whose middle curve 0(7o ) is such t h a t 0 < length 7o < length 7, provides an instance of a linearly elastic generalized m e m b r a n e shell, i.e., one which is not an elliptic membrane shell (Chap. 4), yet whose associated space V F ( w ) = {W = (~/,) e Hi(w) x HX(w) x H2(w); 17i = 0~7/s = 0 on 70} contains only W = 0 (like that associated with an elliptic m e m b r a n e shell). A comparison with Fig. 4.1-1 illustrates the crucial rSle played by the set 0(70) for determining the type of a shell! A shell whose middle surface S = 0(~) is a portion of a hyperboloid of revolution and which is subjected to a b o u n d a r y condition of place along its entire "lower" lateral face provides another instance of a linearly elastic generalized m e m b r a n e shell.
curvature is everywhere > 0 (Sect. 4.1), and which is subjected to a boundary condition of place along a portion of its lateral face whose middle curve 0(70) is such that 0 < length 7o < length 7 (Fig. 5.81). Using the unique continuation theorem of HSrmander [1983], Lods & Mardare [1998a] have shown that, under the assumption that 0 E C2' 1(~; ]R3), the space "- {,7 - (Vi) e
•
• L2(o );
- 0 o n 7o,
"already" reduces to {0}; hence afortiori Vo(w) = {0}!
Sect. 5 . 8 ]
Justificationof the two-dimensional equations; ezamples
295
This result extends an earlier result of Vekua [1962] who had shown that It(w) = {0} under the assumptions that O E W3'P(w; I~3), p > 2, and that 7 is of class C3. It also contains in the special case where 3'0 = 3' the infinitesimal rigid displacement lemma on an elliptic surface (Thin. 2.7-2) due to Ciarlet & Lods [1996a] and Ciarlet & Sanchez-Palencia [1996]. The two-dimensional equations for such a shell constitutes an example of a sensitive problem: As shown by Lions & Sanchez-Palencia [1996], the space V~M(W) is not a space of distributions and there exist functions pi E :D(w) such that applied forces corresponding to the linear form ~1 --+ f~ Pi~Ti~ dy are not admissible for the two-dimensional equations! Lods & Mardare [1998a] have also shown that the regularity of the mapping 0 and the assumption of ellipticity of the surface S can be both substantially weakened when S is a portion of a surface of
revolution. Remark. While the space V~M(W) corresponding to an elliptic membrane shell, i.e., one whose middle surface S is smooth and elliptic and which is subjected to a boundary condition of place along its entire lateral face (7o = 7), is a space of distributions (since V~M(W) -- Hio(w) x Hio(w) • L2(w) in this case; cf. Chap. 4), it ceases to be so when there is a fold, also called an edge, on S, in which case the two-dimensional problem becomes sensitive; see Lions & SanchezPalencia [1998], G~rard & Sanchez-Palencia [1999]. I
A second ezample is that of a shell whose middle surface is a portion of a hyperbolic surface S, i.e., one whose Gaussian curvature is everywhere < 0, provided the curve intersects all the asymptotic lines (Sect. 2.2) of S; for instance, a shell whose middle surface is a portion of a hyperboloid of revolution and which is subjected to a boundary condition of place along its entire "lower" lateral face (Fig. 5.8-1) fulfills these requirements. Under the assumption that O E c2'l(w; ]~3), Mardare [1998c] has shown that R(w) = {0} and that the space V~M(W) is a closed subspace of L2(w) x L2(w) x H - i ( w ) , extending in this fashion earlier results of Vekua [1962] and Sanchez-Palencia [1993]. Interesting complements about shells with hyperbolic middle surfaces are found in Sanchez-Hubert gz Sanchez-Palencia [1997, Chap. 7, Sect. 2], Karamian [199Sb].
296
Linearly elastic generalized membrane shells
[Ch. 5
i j
Fig. 5.8-2: Other ezample8 of linearly elastic generalized membrane shells. Consider a shell whose middle surface S = 0(~) is a portion of a cone or a cylinder and which is subjected to a b o u n d a r y condition of place (i.e., of vanishing displacement field) along a portion (darkened on the figure) of its lateral face with O(')'0) as its middle curve. If O(V0) intersects all the generatrices of S, such a shell is a linearly elastic generalized membrane shell.
Remark. If the hyperbolic surface S has a fold, the space V~M(W) may no longer be a space of distributions, thus yielding another example of sensitive problem; see Lions & Sanchez-Palencia [1997b]. II
A third ezample is that of a shell whose middle surface S is a portion of a parabolic surface, i.e., one consisting entirely of parabolic points (Sect. 2.2), again if the set 0(~'0) is "large enough". Such is the case when S is a portion of a cylinder or a portion of a cone and the curve O(~fo) intersects all the generatrices of S (which are also the asymptotic lines of S in this case; cf. Sect. 2.2); cf. Fig. 5.8-2. Under the assumption that 8 E W2'~176 I~a), Mardare [1998c] has shown that It(w) - {0} and that the space V~M(W)is a closed subspace of H - l ( w ) • H - l ( w ) • H-2(w); the regularity assumptions can again be
Ezercises
297
weakened when S is a portion of a surface of revolution. Interesting complements about shells with parabolic middle surfaces are found in Sanchez-Hubert & Sanchez-Palencia [1997, Chap. 7, Sect. 4]. A compact surface "without boundary" is infinitesimally rigid if any displacement field satisfying Va~(~/) = 0 everywhere on the surface (these relations must be first re-interpreted so as to make sense on such a surface) are infinitesimal rigid displacements in the sense of Thm. 2.6-5. The identification of such infinitesimally rigid surfaces is a classical problem in the differential geometry of surfaces; see, e.g., Nirenberg [1962]. A shell whose middle surface S has no boundary is a linearly elastic generalized membrane shell if S is infinitesimally rigid. Such shells have been extensively studied by Slicaru [1998], who notably extended the analysis of Metskhovrishvili [1957] when S is a torus. A related phenomenon is the infinitesimal rigidification induced by folds on a surface and its relations with linear shell theory. In particular, the asymptotic behavior as the thickness approaches zero of a shell with a smooth middle surface and that of one with an arbitrarily close middle surface, but with folds (such as a polyhedral surface, i.e., an assembly of planar facets), may be startlingly different! In this direction, see Akian & Sanchez-Palencia [1992], Choi [1993], Choi & Sanchez-Palencia [1993], Geymonat & SanchezPalencia [1995], Sanchez-Palencia [1995], G~rard & Sanchez-Palencia [1999].
EXERCISES
5.1. The notations used in (1) should be self-explanatory. (1) Let a surface S -- 0(~) -- 0 ( { ~ } - ) be equipped with two systems of curvilinear coordinates (ya) C ~ and (ya) C { ~ } - and assume that 0(V0) - 0(V0). Show that Vg(w) - {0} implies that V F ( ~ ) - {0}. This means that the definition of a linearly elastic generalized membrane shell (Sect. 5.1) is independent of the system of curvilinear coordinates employed for representing the surface S. (2) Show that whether such a shell is of the first or of the second kind (Sect. 5.4) is likewise independent of the particular system of curvilinear coordinates employed.
298
Linearly elastic generalized membrane shells
[Ch. 5
5.2. According to the definition originally given by Ciarlet & Lods [1996d], the applied forces are admissible if, for each e > 0, there exist functions ~oaf3(s) - q~oa(s) e L2(w) and Fi(s) e L2(~) such that L ( e ) ( v ) may be also expressed as
dy + fn for all v E V(~2), where ~ = ~1 f_l 1 v dxa, and if there exist functions ~oafJ C L2(w) and F i E L2(~) such that ~oa/3(e) -+ ~oa/3 in L2(w) and Fi(e) -+ F i in L2(f~) as e --> O. Using this definition (which differs from that of Sect. 5.5), carry out an asymptotic analysis analogous to that of Thm. 5.6-1. Hint: If the applied forces are admissible, there exists a constant C such that (recall that g(e) - a + O(e) in C~ cf. T a m . 3.3-1). IL(e)(v)[ < C3[vl M for all v e V(~2) and all 0 < e _< e0.
For further details, see Ciarlet & Lods [1996d, Thm. 5.1].
CHAPTER 6 LINEARLY
ELASTIC
FLEXURAL
SHELLS
INTRODUCTION
Consider a linearly elastic shell with middle surface S = 8(~), subjected to a b o u n d a r y condition of place along a portion of its lateral face with 8(70) as its middle curve, where 7o C 7. Such a shell is a linearly elastic flezural shell if its associated space
v~(~)
- {~ - (,~) e H ~ ( ~ ) • H ~ ( ~ ) • H ~ ( ~ ) ; yi - 0~ya - 0 o n 7 0 , 7 a ~ ( ' 1 ) - 0 in w }.
contains nonzero functions (Sect. 6.1). Examples, where S is either a portion of a cylinder (in which case 8(70) is included in one or two generatrices) or a portion of a cone (in which case ~(70) is included in one generatrix) are presented in Sect. 6.1. The purpose of this chapter is to identify and to mathematically justify the two-dimensional equations of a linearly elastic flexural shell, by establishing the convergence in ad hoc functional spaces of the three-dimensional displacements as the thickness of such a shell approaches zero. More specifically, consider a family of linearly elastic flexural shells with thickness 2c approaching zero, with each having the same middle surface S = 8(~), and with each subjected to a b o u n d a r y condition of place along a portion of its lateral face having the same set 8(70) as its middle curve. The associated three-dimensional problems, posed in curvilinear coordinates over the sets 12~ = w x ] - c, el, are first transformed for each ~ > 0 into equivalent problems, but now posed over the set 12 = w x] - 1,1[, which is independent of e. This transformation relies in a crucial way on appropriate scalings of the unknowns u i (the covariant components of the displacement field) and assumptions on the Lamd constants )~ and #e and on the contravariant components fi, e of the applied body forces (for simplicity, we assume in this
Linearly elastic flezural shells
300
[Ch. 6
introduction that there are no applied surface forces). More specifically, we define the scaled unknown u(s) = (ui(s)) by letting
Zt~(Xe) -- 'tti(g)(X ) for all z" = ~r'z E ~ ' , where 7r~(xl, x2, x3) = (Xl, x2, sx3). We then assume that there exist constants A > 0 and/z > 0 and f u n c t i o n s / i independent of e such that Ae
= A
and pe
=/z,
f i " ( x ' ) -- s2fi(x) for all z e - ,fez E f~e. It is found in this fashion that the scaled unknown u(e) satisfies a variational problem of the form (Thm. 6.1-1):
u(s) --(ui(s)) E V ( ~ ) -
{v - ( v i ) E H I ( ~ ) ; v-
f
0 on Vo • [-1, 1]},
AiJkZ(s)ekllt(s; u(s))eillJ(~; v ) ~ - ( s ) d x = ~2 f f i v i % ~ ) d x gft
for aU v E V ( ~ ) ,
where, for any vector field v = (vi) E Hl(f~), the scaled linearized strains e~llj(e; v) - ejll~(e; v) E L2(n) are defined by:
-
1
p
+
l(103Va+Oav3)
-
e3113(e; v) - 103v3. g
Using various analytical preliminaries established in the previous chapter, most notably the three-dimensional inequality o] Korn's type /or a family of linearly elastic shells, we then establish the main result of this chapter (Thm. 6.2-1), by showing that the family (u(e))e>o strongly converges in the space H I ( n ) as s -+ 0 and that u = lime--+ou(s) is obtained by solving a two-dimensional problem.
301
Introduction
More specifically, we show that the limit u is independent of the transverse variable and that ~ = 89f l 1 u dx3 satisfies the following (scaled) two dimensional variational problem of a linearly elastic flex-
ural shell: r E V F ( W ) - { r / - (r/i) e HI(w) x Hi(w) x H2(w); r / i - 0vy3 - 0 on 70, Vaf3(r/)- 0 in w},
1 1 -~ f aat3~rpar(')Pal3(rl)~dY: f ~ { f l ' i d x a } ~ i x ~ d Y for all v / = (7//) e Vv(w), where
=
p~(r/) -
aaf3~r _
+
-
-
ba
73,
0~73 - F~#0,W3 - b~b~/3~73+ b~(Oarl~ - r ~ , ~ , )
4Ap aaf3a,Tr+ 2#(aaCaf3r + aara~r )~+2#
the functions 7a/3(v/), pal3(v/), and a a/3~r being respectively the covariant components of the linearized change of metric and change of curvature tensors associated with a displacement field r//a / of the middle surface S (first encountered in Sects. 2.4 and 2.5) and the contravariant components of the (scaled) two-dimensional elasticity
tensor of the shell. We conclude this chapter by reviewing the existence and uniqueness properties of the solution to the above variational problem and to the associated minimization problem (Thm. 6.3-1). We also rewrite the fundamental convergence theorem in terms of de-scaled unknowns and data, thus providing a justification in terms of "physical" quantities of the two-dimensional equations of a linearly elastic flexural shell (Thm. 6.4-1). In particular, the limit displacement field ~ a z of the middle surface S is such that ~e _ ( ~ ) satisfies the following minimization
problem: ~e e VF(w) and j ~ ( ~ e ) _
inf
j~(r/),
Linearly elastic flezural shells
302
[Ch. 6
where the two-dimensional energy j~ 9Vy(w) --+ I~ is defined by
1f
e 3aa~r'ep~r(rl)pa~(rl)~/-a dy
J~,
and 4A~#~ Ae
6.1.
+ 2p ~
~~
+ 2,~(~~ ~+ ~~).
LINEARLY ELASTIC FLEXURAL SHELLS: DEFINITION~ EXAMPLES, AND ASSUMPTIONS ON THE DATA; THE THREE-DIMENSIONAL EQUATIONS OVER A DOMAIN INDEPENDENT OF
Let w be a domain in I~2 with boundary V and let 0 E C2(~; I~3) be an injective mapping such that the two vectors OaS(y) are linearly independent at all points y E ~. A linearly elastic shell with middle surface S = 0(~) is called a l i n e a r l y elastic f l e x u r a l shell if the following two conditions are simultaneously satisfied (the definitions and notations are those of Sect. 3.1): (i) The shell is subjected to a boundary condition of place along a portion of its lateral face with 0(7o) as its middle curve, where the subset V0 C V satisfies
length 7o > O. (ii) Define the space (0~ denotes the outer normal derivative operator along 7)" VF(W) := { ~ -
(~i) 6 H i ( w ) x / - / l ( w ) x H2(w); Yi - 0vy3 - 0 on 70, 7a#(Y) - 0 in w}.
Sect. 6 . 1 ]
Definition, ezamples, and assumption on the data
303
Then the space VF(w) contains nonzero functions; equivalently,
vF( ) # {o). We recall that the functions 1
O"
denote the covariant components of the linearized change of metric tensor associated with a displacement field yia i of the surface S. The subscript " F " announces that VF(w) is the functional space over which the limit two-dimensional equations of a linearly elastic flexural shell are posed (Thin. 6.2-1). In other words, there exist nonzero a d m i s s i b l e l i n e a r i z e d inext e n s i o n a l d i s p l a c e m e n t s y i a i of the middle surface S: "Admissible" means that they satisfy boundary conditions "of clamping" along the curve 8(70), expressed here by means of the boundary conditions yi - Ouy3 on 70 on the associated field r / - (yi) (these boundary conditions are interpreted later; cf. Sect. 6.4); "linearized inextensional" reflects that the functions 7a~(r/) are the linearizations with respect to W -- (rli) of the covariant components of the exact change of metric tensor associated with a displacement field y i a z of the surface S; cf. Sect. 2.4. Examples of linearly elastic flexural shells are given in Figs. 6.1-1 to 6.1-3; see also Exs. 6.1 to 6.5. A w o r d of c a u t i o n . Like those of linearly elastic elliptic membrane or generalized membrane shells, the definition of a linearly elastic flexural shell depends only on the subset of the lateral face where the shell is subjected to a boundary condition of place (via the set 70) and on the geometry of the middle surface of the shell; in this respect, see also Ex. 6.6. m The formal analysis of Sect. 3.4 then naturally leads us to make the following s c a l i n g s of t h e u n k n o w n s and a s s u m p t i o n s o n t h e d a t a for a family of linearly elastic flexural shells, with each having the same middle surface S - 0(~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(70) as its middle curve, as their thickness 2e approaches zero.
304
Linearly elastic flezural shells
[Ch. 6
/
Fig. 6.1-1: A linearly elastic flezural shell. A shell whose middle surface S = 8(~) is a portion of a cylinder and which is subjected to a boundary condition of place (i.e., of vanishing displacement field) along a portion (darkened on the figure) of its lateral face whose middle curve 8(70) is contained in one or two generatrices of S provides an instance of a linearly elastic flexural shell, i.e., one for which the space v.(~)
= ~ , ~ H~(~) • H~(~) • H~(~); ~, = o ~
= 0 o . ~o, ~ o ~ ( , ) = 0 i . ~}
contains nonzero functions ~/; cf. Ex. 6.1. The "two-dimensional boundary conditions of clamping" 7/i -- a~T/s - 0 on 70 inherited by the limit two-dimensional equations are so named because they mean that the points of, and the tangent spaces to, the deformed and undeformed middle surfaces coincide along the set 8(70) (as suggested in the "two-dimensional" figure); cf. Sect. 6.3. This example and that of the cylindrical shell from Fig. 5.8-2 illustrate the crucial rSle played by the set 8(7o) for determining the type of a shell.
Sect. 6 . 1 ]
Definition, ezamples, and assumption on the data
305
Fig. 6.1-2: Another ezample of a linearly elastic flezural shell. A shell whose middle surface S = 0(~) is a portion of a cone excluding its vertex and which is subjected to a boundary condition of place along a portion (darkened on the figure) of its lateral face whose middle curve 8(70) is contained in one generatrix of S provides another example where VF(w) r {0}; el. Ex. 6.2. The two-dimensional boundary conditions of clamping inherited by the limit two-dimensional equations are suggested on the "two-dimensional" figure. A comparison with Fig. 5.8-2 again illustrates the rSle played by the set 0(7o) for determining the type of shell.
u ( e ) = (ui($)) by l e t t i n g
F i r s t , we define t h e s e a l e d u n k n o w n
~Z~(Xe) --" Zti(8)(X ) for all x e - 7r~x C ~ . N e x t , we require t h a t the Lamd constants a n d t h e applied body and surface force densities be such t h a t
)~e = :k fi, e ( x c ) =
e2fi(x)
and
#e = / ~ ,
for all
x" - r e x
an
-
E f~e, c
u r'_,
where the constants ~ > 0 and # > 0 and the functions f i E L 2 ( ~ ) and h ~ E L 2 ( p + U r _ ) are independent of e (Fig. 3.2-1 r e c a p i t u l a t e s t h e definitions of t h e sets f~e f~, P~_, r
r e
and r_)
306
Linearly elastic flezural shells
[Ch. 6
t t x,~
..
Fig. 6.1-3: Another ezample of a linearly elastic flezural shell. A plate subjected to a boundary condition of place along any portion (darkened on the figure) of its lateral face whose middle line 70 satisfies length 70 > 0 provides an instance of a linearly elastic flexuxal shell since Vp(w) D ( r / = (0, 0, r/s); r/s E H~(w)) ~ ~0) in each case; cf. Ex. 6.3. The two-dimensional boundary conditions of clamping inherited by the limit two-dimensional equations are suggested on the "two-dimensional" figures (for visual clarity, only a portion of the plate is represented in the last case). Surprisingly, if 7o is "large enough", the same plate is no longer modeled as a flexuxal shell when it is viewed as a nonlinearly elastic body! See Fig. 9.1-2.
Sect. 6.1]
Definition, ezamples, and assumption on the data
307
Remark. For notational brevity, the functions f i and h i stand for the functions that were respectively denoted fi, 2 and h i' 3 in Sect. 3.4. II
As an immediate corollary to Thm. 3.2-1 (simply corresponding to p - 2), we obtain the problems satisfied by the scaled unknown over the set f~, thus over a domain that is independent o/e" T h e o r e m 6.1-1. Let w be a domain in IR2, let 0 E C2(~; IR3) be an injective mapping such that the two vectors aa = OaO are linearly independent at all points of-~, and let eo > 0 be as in Thm. 3.1-1. Consider a family of linearly elastic flexural shells with thickness 2e, with each having the same middle surface S = 0(-~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 8(')'0) as its middle curve. Let the assumptions on the data be as above. Then, for each 0 < e < co, the scaled unknown u(e) = (ui(e)) satisfies the following s c a l e d t h r e e - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 7~(e; f~) of a l i n e a r l y e l a s t i c f l e x u r a l shell: ~(~) e v ( ~ ) : =
{~ - (,~) e H ~ ( ~ ) ; ~ - o oil r0},
AiJkl(e)ekllZ(e; u(e))eillj(e; v ) X / ~
dx
for all v E V(~2),
where ro " - 7 0 • [-1, 1], the scaled strains eillj(e; v) are given by
e=ll~(e; v) = ~ (ae~= + oo,e) - r~~e (~),~, eall3(e;v)
-- ~
(10g o + 0o ,) -
ealla(e; V) -- e103v3,
and the functions AiJkz(e), g(e), and rPj(e) are defined as in Thin. 3.2-1. I
308
Linearly elastic flezural shells
[Ch. 6
Our main objective in this chapter consists in analyzing the behavior of the solutions u(e) E H I ( ~ ) of problems ~(e; ~) as r -+ 0. To this end, essential uses will be made of the analytical preliminaries and of the three-dimensional inequality of Korn's type for a family of linearly elastic shells (Sects. 5.2 and 5.3), already needed for the asymptotic analysis of linearly elastic generalized membrane shells carried out in the last chapter.
6.2.
CONVERGENCE DISPLACEMENTS
OF THE SCALED A S g --+ 0
We now establish the main results of this chapter: Consider a family of linearly elastic flezuval shells with thickness 2s > 0, with each having the same middle surface S = O(w) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(70) as its middle curve. Then the solutions u(s) of the associated scaled three-dimensional problems 79(s; ft) (Thin. 6.1-1) converge in Hl(f~) as s ~ 0 toward a limit u and this limit, which is independent of the transverse variable z3, can be identified with the solution ~ of a two-dimensional variational problem posed over the set w. This limit problem wiU be later identified (Thm. 6.3-1) as the scaled two-dimensional variational problem of a linearly elastic flezural shell. The functions 7a# (~/), pa#(~/), and a a~ar used in the next theorem respectively represent the covariant components of the l i n e a r i z e d c h a n g e of m e t r i c , and c h a n g e of c u r v a t u r e , t e n s o r s associated with a displacement field 71iai of the surface S and the contravariant components of the sealed t w o - d i m e n s i o n a l elasticity t e n s o r of t h e shell; we recall that e0 > 0 is defined in Thm. 3.1-1. The next result is due to Ciarlet, Lods & Miara [1996, Thm. 5.1]. T h e o r e m 6.2-1. Assume that 0 C C3(@ ~3). Consider a family of linearly elastic flezuval shells with thickness 2~ approaching zero, with each having the same middle surface S = 0(-~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(7o) as its middle curve, and let the assumptions on the data be as in Sect. 6.1.
Convergenceof the scaled displacements as e --+ 0
Sect. 6 . 2 ]
309
Let u ( e ) denote for 0 < e < so the solution of the associated scared three-dimensional problems 79(e; ~2) (Thin. 6.1-1). T h e n there exists u - (ui) E I-II(f~) satisfying u - 0 on ~o - 70 • [-1, 1] such that
U(Z) -+ u in HI(f~) as ~ -+ 0, u := (ui) is independent of the transverse variable x3. Furthermore,
-
-
~:=2
1F
i
udxa
satisfies the following s c a l e d t w o - d i m e n s i o n a l v a r i a t i o n a l p r o b -
l e m 7>F(w) of a l i n e a r l y elastic f l e x u r a l shell: -- (~i) e V F ( 0 g ) : - - {~--(~}i) e Hl(0g) x gl(0g) x H2(0g);
7/i - 0vT/s = 0 on 70, "Yaf3(r/) = 0 in w }, 1 for all y = (~i) e VF(w), where (the definitions of the functions r~f3, ball, ~ , a af3, and a are recalled in Sect. 3.1)
~.,(~)
1 .= ~ ( o ~ .
p.,(~)
:= o . , ~ 3 - r 2 , o ~ 3
+ o.,,)
(r - r.~
- b.,~3.
- b~b~,s + b~(O,,. - r~.,~)
~b~)r/~, aa/3~r ._.-
4)V z aaf3 a~r + 2#(aa~ a [3r + aar a ~ ) ,
A+2~ pi :=
f ? fi dx3 q-h+i + hi_ a n d h ~": = h i ( 1
9, •
Linearly elastic flezural shells
310
[Ch. 6
Proof. For the sake of clarity, the proof is divided into six parts,
numbered (i) to (vi). For notational brevity, we let eillj(8 ) "-- eillj(g; U(8)) throughout the proof. (i) A priori bounds and eztraction of weakly convergent sequences: The norms
1 I~e, llj(~)lo,,and II~(e)ll~,.a~e bounded independently
of
0 < e < el, where et > 0 is given by Thm. 5.3-1. Consequently, there exists a subsequence, still denoted (u(e))e>0 for convenience, and there exist functions e illJ 1 E L 2 (f~) and u E H 1(~) satisfying u - 0 on r0 - 70 x [-1, 1] such that (recall that --+ and ~ respectively denote strong and weak convergences)" u(e) ~ u in Hl(f~) and thus u(e) --+ u in L2(f~),
1
_
1 eill j (e) ~ eillj in
L2
(~).
From the variational equations satisfied by the scaled unknown ~,(e) (Thm. 6.1-1), the asymptotic behavior of the function g(e) (Thin. 3.3-1), the uniform positive definiteness of the scaled three-dimensional elasticity tensor (Thin. 3.3-2), and the three-dimensional inequality of Korn's type for a family of linearly elastic shells (Thm. 5.3-1), we infer that
~2c-211u(~)1121,,
_<~
2 leilaj(e)[o,,
i,j
< Ggo 1/2
~ AiJk~(~l~kll,(~l~,tlj(~lv/g(~)d~
Since there exists a constant el depending on gl, on the norms If/10,n and and on the norm of the trace operator from H I (~) into L z (r+ U r _ ) such that
lih'[[L,(r+ur_)
+ur_ for all 0 < e ___e0, the assertions follow.
Sect. 6 . 2 ]
Convergence of the scaled displacements as e -.-+0
311
(ii) The limit u and its average ~ -- 89fi_l u da~3 satisfy the following properties: u = (ui) is independent of the transverse variable x3,
-- (ui) E HZ(w) • HZ(w) • H2(w) and ui = OvU3 = 0 on 3'o, 7a~(U) = 0 in w. These properties are consequences of the convergences established in part (i) and of Thm. 5.2-2. z and u f o u n d in part (i) are related by: (iii) The limits eillj --03 el~ll~
- P~,(~)i~ L 2 (~),
e~l13 e3il3 --
O, ~ +
2#
1
The equality -03ela[if3 - pa~(u) in L2(f~) again follows from Thm. 5.2-2. Let v = (vi) be an arbitrary function in the space V ( f ~ ) -- { v E H I ( f ] ) ;
v - 0 on r0}.
Then seallf3(e; v) --+ 0 in L2(f~),
~tl3(~; ~) -~ ~1 o 3 ~ in L2 (~), ee31ls(S; v) = 03v3 for all ~ > 0. Using the variational equations of the scaled variational problem 7v(e; f~) (Thm. 6.1-1) and the relations Aaf3a3(e) - Aa333(s) - 0 (Thin. 3.3-2), we may write Aijkl(~)eklll(~)eitlj(~; V ) % / f ~
+c
dx
{4A~ (~)~ll~ (~)} {~oi1~(~; *)} v/g(~i ' ~,
312
Linearly elasticflezuralshells
+ -1 ~n {Aaa'r (e)e~llr(e) + A3333(e)ealla(e)}{eealla (e;
= e2{fNfiviv/g(g)dxff-fr+uP
[Ch. 6
v)} v/g(Z) dx
hiviv/g(E)dr }.
Keep v E V(f~) fixed and let ~ ~ 0. The asymptotic behavior of the functions eeillj(s; v), the asymptotic behavior of the functions AiJkt(e) and g(e) (Thms. 3.3-1 and 3.3-2), and the weak convergences 1 1 (part (i)) together imply that: eeill/(e) ~ eillJ
{21~aa~ le~llzO3va+ (Aa
e~llr + (A + 21z)e1,13)O3v3} v/adx -- O.
Letting v vary in V(f~) then yields the other relations satisfied by 1 (if w E L 2 (f~) satisfies fu wOav dx - 0 for all v E H l(f~) the limits cilia that vanish on 70 • [-1, 1], then w = 0; cf. Thin. 3.4-1). (iv) The function U := (~i) satisfies the two-dimensional variational problem Py(w) described in the statement of the theorem. Since this problem has a unique solution (Thins. 2.6-4 and 3.3-2), the convergences u(e) -~ u in H1(~2) and u(e) -+ u in L 2 ( ~ ) (part (i)) hold in fact for the whole family (u(s))e>0 (if the function U is unique, so is the function u as it is independent of x3; el. part (ii)). Note in pass[Ie ing that reaching the same conclusion for the families -~l~J illJ(e)]e>0_ has to be postponed until part (v). That U E VF(W) has been proved in part (ii). Given art arbitrary function r / = (r/i)in the space VF(W), let the function v(e) - (vi(e)) be defined almost everywhere in f~ for all e > 0 by (the idea of constructing such a function is borrowed from Miara & Sanchez-Palencia [1996]):
va(e) := Ya - ex30a with 0a := 0ay3 + 2b~y~, v3(6) :=
First, we clearly have v(r
E V(~) for a11 e > 0,
e3[[3(8; V(8)) -- 0 for a11 e > 0.
Sect. 6.2]
313
Convergence of the scaled displacements as e -.4 0
Next, we show that, for a fized element v/E VF(w), identified wherever needed with a function in the space H t (fl) • H 1(fl) • H 2(fl), v(e) -+ 17 in Ht(~2), 1 -eallfl(e; v(e)) ---->{-xsPat3(~/)}
(~ealls(e; v(e)))
in L2(f~) as e -+ O,
converges in L2(~). e>0
The first relation clearly holds. Using the relations 7af3(v/) - 0, we find after an easy computation that
1
~o,t,~(~; ~(~)) := -7o,(~(~))~ + x~t,Slav,,(~ ) + ~bob~, v s (e) = -~s
{1
~(o~o~ + o~o~)
= -~po,(,)-
-
r.~o~
-
b;l~,7~
-
b2b~ns
}
~b;31oo,~.
Then, by Thm. 5.2-1, 1
and thus
1 -eallfl(e;
v(e)) -+ {-xspaf3(v/)}
in L2(f~).
Another computation shows that
_1 eealls (e; v(e)) = --e1 (r23(~) + b2),~ + ~rx3(~)o~ This equality, combined with the asymptotic behavior of the functions F~3(~ ) (Thm. 3.3-1) , proves that (1_ sealls(e;
v(e)) ) e>0
converges in
L2(f~). Keep the function v/C VF(w) fixed, let v - v(e) in the variational equations of problem 7~(e; f~) (Thm. 6.1-1), where v(e) - (vi(e)) is defined as supra, and let e -+ 0. The asymptotic behavior of the v(e)), the asymptotic behavior of the functions v (e) and -ei[ij(e; 1
314
Linearly elastic flezural shells
[Ch. 6
functions AiJkt(e) and g(e) (Thms. 3.3-1 and 3.3-2), the weak conver1 gences -eillJ(e) ~ ei~lj (part (i)), and the relations satisfied by the c
t (part (iii))together yield limits eill3
lim 1 fn A~Jkl
{/o
= e-~olim Aa~r(s){~ +
/~A~
+ +
1~
---- --
1
1 eotJ,(~; ~)}V~(~)d~
4a~
/a
+ = -~
1 lit(S)} v/g(e) dx ,,)}4g(~)d~
1 A ~ ( ~ ) { ~11,(~)} 4g(~l d~ A~(~){~II~(~)}{~ -~11~(~; ~)} v / g ~ d~
~3
aa~O'r 1
~ll~,~(,7)v~d~
x3 ,~+ 2#
+ "(ha~ a'r +
) ecrllrpa/3(W)~dx
= e--+O lim {fa 'ivi(e)V/'g(e)d:r+fr + u r _ hivi(e)v/g(e)dF} : s =
+ ~ S~a~ + h~ + h' ~v~dy.
We have yet to take into account the relations pa/3(u) - -03e~ll ~.~ in L2(~) (part (iii)). Since the function u is independent of x3 (see i are of the form part (ii)), these relations show that the functions ea"~llp 1
%11~3- Ta~ - x3pa/3(U) with T ~ c L2(w).
Sect. 6.2]
Convergence of the scaled displacements as e --+ 0
315
Therefore
2
x3a
% l l r P a ~ ( r l ) ~ dx -- -~
"~3"*
= -~
t,~r(U)pa/3(rl)V/-d d=
a a~'rp~r(~)pa~(rl)V~ dy,
and thus we have established that the variational equations of problem 79F(w) are satisfied by ~. (v) The weak convergences -eillj(s 1 : ) ~ eillJ in L2 (f~) established
in part (i) are in fact strong: :
1
~eillJ(S ) -+ eillJ in
L2
(a).
Besides, the Kmits e~ltJ are unique; hence these convergences hold for / \ 1 the whole family , r /(-eilJ(e)}e> O. Combining the uniform positive definiteness of the scaled elasticity tensor and the asymptotic behavior of the function g(s) (Thms. 3.3-1 and 3.3-2) with the variational equations of problem 7~(e; [2) where we let v - u(e), we obtain
c : : go:/2 ~
i,j
1
I~,llj(~)
-
: 2 ~lljlo,~
1
1
< A(s) -
--,
where n
1
z
= f~/~,(~)v/g(~) ~+ f~+ur_ h~,(~)v/g(~)dr -
AiJkl(e){eekllZ(e )_ -
z ~klj~
1 v/g( } ~,llJ
~1
e~.
The asymptotic behavior of the functions A#m(e) and g(s) together with the convergences ui(s) --+ ui in L2(f~) and ui(e) ~ ui in Hl(f~) (the latter implying that tr ui(e) -~ tr ui in L2(r+ u r _ ) ) and
316
[Ch. 6
Linearly elastic flezural shells
1 1 the weak convergences ~eillJ(e ) --~ eillj
i n L 2 (if2)
A := l i m A ( e ) - f y ' u , ~ d ~ + f ~ e--+O
(part (i))imply that
+ur_
hiuivZddF
- f. Using the expressions of the functions A~Jk~(0) (Thm. 3.3-2) and eill s l (part (iii)) and recalling that u - (ui) is independent of xa (part (ii)), we next infer that A "-
fi
dxa + h i+ + h i-
1
}
Ui %/c~dy
- ~1
/o
dafter %llreallf3 1 1 ~
dx.
The relations pail(u) - -03elallfl in L2(~) (part (iii)) imply that eallf~l -- T a r 1 - X 3 P a f l ( ~ ) with Taf~ E L2(w); hence
1
a afl~re~llreallflV/-d dx - 2
f
+ -~
T ~ r T a f 3 ~ dy a afl~rpar(u)pafl(-u)V~ dy.
Since ~ satisfies the variational equations of problem 79F(w) (see part (iv)), we thus have A = - f~ aa/3~r T~r T aft V~ dy.
Noting that A > 0 (the numbers A(s) are >_ 0 by their definitions) and that the scaled two-dimensional elasticity tensor of the shell is uniformly positive definite in ~ (Thm. 3.3-2), we conclude that Taft - 0. These relations have two consequences" First,
1 1 hold in L 2( ~ ) . and thus the strong convergences ~eillj(e ) --+ eillj Second, the functions e ~11/3 1 are uniquely determined, since they are given by and the function ~ is unique (part (iv)). That the functions e/113 are uniquely determined then follows from the relations established in part (iii).
Sect. 6.3]
317
The two-dimensional equations
(vi) The weak convergence u(~) part (i) is in fact strong: u(e) ~ u in
u
in
H~(~)
established in
H~(n).
By Thm. 5.2-2, it remains to show that each family (paf3(u(e)))e>o strongly converges in H - i (~). Since -~eall~(e 1 i 3 in L 2(12) by ) -4 eallf part (v), we first have
103eallf3(g ) ~ 03elllfj in H-I(~'~). By Thm. 5.2-1, we next have 1
-~ 0 in H - i ( n ) .
Hence
--03eall~} in H and the proof is complete.
6.3.
m
THE TWO-DIMENSIONAL EQUATIONS OF A LINEARLY ELASTIC FLEXURAL SHELL; EXISTENCE AND UNIQUENESS OF SOLUTIONS
The next theorem recapitulates the definition and assembles the main features of the limit two-dimensional problem 7~F(w) found at the outcome of the asymptotic analysis carried out in Thm. 6.2-1. The existence and uniqueness theory, which is quickly reviewed in this theorem, is expounded in detail in Sect. 2.6, where ad hoc references are also provided. T h e o r e m 6.3-1. Let w be a domain in I~2, let 7o be a subset of the boundary of w with length Vo > 0, and let 0 E C3(~; R 3) be an injective mapping such that the two vectors aa - OaO are linearly
318
Linearly elastic flezural shells
[Ch. 6
independent at all points of-~ and such that V~(w) # {0}, where
VF(~)
(,i) e m(~) • HI(~) • H~(~);
'-- { ~ -
~i "-- 0t, r/3 -- 0 011 ")'0, "Ycq3(l?) -- 0 ill OJ},
1
The associated scaled two-dimensional variational p r o b l e m 79F(w) of a linearly elastic flexural shell: Given pi E L 2 ( w ) , find = (~i) that satisfies
r c vF(w), 1
for all r / = (r/i) 6 VF(w),
where p~,(,)
.= o~~
- r~,o~
+b~(O~,
aa/3~r .__
-
- b~b~,~3 + b ~ ( O , ~
r~,~) +
(Oob~ + r ~ b 3
-
r~,)
~b~).~, ~
- r~
4A# aa~a~r + 2# (a a~a[3r + aaragl~), A+2p
has one and only one solution, which is also the unique solution of the minimization problem: Find ~ such that e Vv(w) and j v ( ~ ) = iF(n) := g
inf
iF(Y), where
/.
p~,(n)pa/9(n)~dy- p'yix/-ady.
Proof. The existence and uniqueness of a solution to the variational problem :PF(w), or to its equivalent minimization problem, is a consequence of the inequality of Korn's type on a general surface (Thin. 2.6-4), of the uniform positive definiteness of the scaled
Sect. 6.3]
The two-dimensional equations
319
two-dimensional elasticity tensor of the shell (Thm. 3.3-2), of the inequalities a(y) > a0 > 0 for all y C ~ (Thin. 3.3-1), and of the Lax-Milgram lemma, m The minimization problem encountered in Thm. 6.3-1 (or that in Thm. 6.3-2 below in its "de-scaled" formulation) provides an interesting example of a minimization problem with "equality constraints", namely the relations "Vaf3(v/) = 0 in w" that the elements v/of the space over which the functional is to be minimized must satisfy. Normally, a Lagrange multiplier is then attached to such a problem (see, e.g., Ciarlet [1982, Thm. 7.2-2]), allowing in turn to write the boundary value problem that is (at least formally) equivalent to the minimization problem; this is the case for instance of the limit "multi-dimensional" equations found at the outcome of the asymptotic analysis of elastic multi-structures (see Vol. II, Chap. 2; especially Sect. 2.5). Finding the Lagrange multiplier in the present case seems more challenging. In this direction, see Sanchez-Hubert ~5 Sanchez-Palencia [1997, Chap. 8], Brezzi, G~rard & Sanchez-Palencia [1998], Sanchez-Valencia [1999]. In order to get physically meaningful formulas, it remains to "descale" the unknowns ~i that satisfy the limit scaled problem 7~F(w) found in Thm. 6.2-1. In view of the scalings
u~(e)(x) - u~(x ~) for all x ~ : 7rex e ~ made on the covariant components of the displacement field (Sect. 6.1), we are naturally led to defining for each e > 0 the eovaria n t c o m p o n e n t s ~ 9~ -+ I~ of the limit d i s p l a c e m e n t field 9 w --+ of the middle surface S of the shell by letting (the vectors a i form the contravariant basis at each point of S)
~'~ "- ~'i and ~e := ~.~ai" A w o r d of c a u t i o n . Like those found in the analysis of linearly elastic elliptic membrane shells (Sect. 4.5), the fields ~ := ( ~ ) and ~e = ~ a i must be carefully distinguished! The former is essentially a convenient mathematical "intermediary", but only the latter has physical significance. II
Linearly elastic flezural shells
320
[Ch. 6
Remark. Conceivably, the limit scaled displacement field across the thickness of the shell could also be de-scaled, resulting into a limit displacement field ire(O) 9-~e --+ ~3 inside the shell defined by (the vectors 9i, e form the contravariant basis at each point of the reference configuration O(~e); cf. Sect. 3.1)" .:
',*,
since the scaled limit u = (ui) is independent of the transverse variable x3 (Thin. 6.2-1). For the same reason, however, the de-scaled field does not inherit any remarkable structure as x~ varies across the thickness of the shell. By contrast, the limit displacement field across a plate, which is a linearly elastic flexural shell (Ex. 6.3), is a Kirchhoff-Love one (Vol. IX, Thm. 1.7-1). However, it does inherit this richer structure only because different scalings can be made in this case on the "horizontal" and "vertical" components of the displacement field. II Recall that fi, e C L2(n e) and h i'e C L2(r~_ kJ re_) represent the contravariant components of the applied body and surface forces actually acting on the shell and that Ae and #e denote the actual Lam6 constants of its constituting material. We then have the following immediate corollary to Thms. 6.2-1 and 6.3-1; naturally, the ezisfence and uniqueness results of Thm. 6.3-1 apply verbatim to problem 7~(w) (for this reason, they are not reproduced here)" T h e o r e m 6.3-2. Let the assumptions on the data and the definitions of the functions 7a~(r/) and Pa~(~7) be as in Tam. 6.2-1. Then the vector field ~e := (i~) formed by the covariant components of the limit displacement field ~ a i of the middle surface S satisfies the following t w o - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 7~(w) of a l i n e a r l y elastic f l e x u r a l shell:
~e E V F ( w ) : = {~/= (~i) E Hi(w) • Hi(w) • H2(w); ~i -- 0u~3 = 0 on 70, 3'ag~(r/) = 0 in w},
V
ePzr(~e)Pa;3(~l)v~dY =
P"e~lix~dY
=u n = (w) e
Sect. 6.3]
The two-dimensional equations
321
where a a ~ , ~ ._.-- 41he#e aa~a~r + 2#e(aa~a~r + aaraDCr), ,V + 2# e pi,~ : :
f,,E dx~ + h ; e + h ''e and h~ e "- h i'e(., :kr f~
E
"
~
~
.
Equivalently, the field ~ -- (i~) satisfies the following minimization problem: ~ e VF(w)
and j ~ ( ~ e ) _
inf j~(r/), where ncvF(~)
II Each one of the two formulations found in T h m . 6.3-2 constitutes the t w o - d i m e n s i o n a l e q u a t i o n s o f a l i n e a r l y e l a s t i c f l e x u r a l shell. We recall that the condition V e ( w ) # {0), which is the basis of the definition of a linearly elastic flexural shell, means that there exist nonzero a d m i s s i b l e l i n e a r i z e d i n e x t e n s i o n a l d i s p l a c e m e n t s of the middle surface (Sect. 6.1), since the functions "yaf3(r/) used in the definition of VF(w) are the covariant components of the l i n e a r i z e d c h a n g e o f m e t r i c t e n s o r associated with a displacement field yia i of the middle surface S. In order to interpret the boundary conditions ~ - Ov~ on ")'0 satisfied by the field ~e - ( ~ ) , let rlia i be a displacement field of the middle surface S - 8(~) with smooth enough, but otherwise arbitrary, covariant components r/i 9~ --+ R. The tangent plane at an a r b i t r a r y point O(y) + yi(y)ai(y), y E -~, of the deformed surface (O + rlia*)(-~) is thus spanned by the vectors o.(o +
- a.(y) +
+
if these are linearly independent. Since ~i - 0 ~ 3 - 0 on 70 ~
~i - O~r/3 - 0 on 70,
Linearly elastic flezural shells
322
[Ch. 6
it follows that
O(y) + 71i(y)ai(y) - O(y) for aU y e 70, Oa(O + z}iai)(y) - aa(y) + Oaz}/3(y)af3(y) for all y C 70. These relations thus show that the points of the deformed and unde[ormed middle surfaces, and their tangent spaces at those points where the vectors Oa(O+~Iia i) are linearly independent, coincide along the set 0(7o). Such t w o - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s of c l a m p i n g are suggested in Figs. 6.1-1 to 6.1-3. The functions Pa/3(W) are the covariant components of the line a r i z e d c h a n g e of c u r v a t u r e t e n s o r associated with a displacement field 7}iai of the middle surface S and the functions a afJav, e are the contravariant components of the t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of t h e sheU, already encountered in the two-dimensional equations of linearly elastic elliptic membrane and generalized membrane sheUs (Thins. 4.5-2 and 5.7-2). FinaUy, the functional j~ 9VF(w) -+ IR is the t w o - d i m e n s i o n a l e n e r g y , and the functional
is the two-dimensional strain energy, of a linearly elastic flexural shell. Note that the s a m e two-dimensional problem 7)~(w) is evidently obtained if the scalings on the unknowns are the same as before, i.e., ~ ( x ~) - u i ( ~ ) ( x ) for aU x e - ~ex E ~~,
but the following more general a s s u m p t i o n s on t h e d a t a are made:
Ae = etA _
and all
#e = et#, -
n
where the constants A > 0 and # > 0 and the functions fi C L2(~) and h i E L2(p+ U r _ ) are independent of e and t is an arbitrary real
Sect. 6.4]
Justification of the two-dimensional equations
323
number. Besides, the analysis of Sect. 3.4 shows that these assumptions on the data are the only ones possible for flexural shells. Remark. A different kind of generalization is possible. For definiteness, assume that the Lamd constants are independent of e. Then it is easily verified that the same scaled limit problem 7~F(w) is obtained under the assumptions that
fi, e(Xe ) _
e2fi(~; :g) for
all z e - 7r~z e f~e,
h~"(~ ~) = ?h~(~; ~) fo~ ~11 ~" - ~ ' ~ c r ~ u r'_, and
fi(e; .) ~ f i in L2(~2) as e --+ 0, hi(s; .) -+ h i in L2(r+ u r _ ) as s ~ 0. The functions p~ appearing in the right-hand sides of problem 7~(w) are then defined by
fi dx3 + h i+ + h i_~. 1
)
B
6.4.
JUSTIFICATION OF THE TWO-DIMENSIONAL EQUATIONS OF A LINEARLY ELASTIC FLEXURAL SHELL; COMMENTARY AND REFINEMENTS
It remains to convert in terms of de-scaled unknowns the fundamental convergence theorem established in Sect. 6.2. As the "originaP' unknowns u ie are defined over a domain that varies with e (the set ~t~) , their averages -~ f_~ u ie dx~ are more appropriate for this purpose, since they are defined over a fixed domain (the set w). The convergences ui(e) --+ ui in H I ( ~ ) (Thm. 6.2-1), the scalings ui(~)(x) - u~(x ~) for all x ~ - 7r~x e ~e (Sect. 6.1), the de-scalings
~ - ~i - 89fl_ 1 ui dx3 (Sect. 6.3) and Thm. 4.2-1 together yield the following convergences of the averages 2~ f-~ u~ dx~ of the covariant components of the "original" three-dimensional displacement:
1 F e~
2e
d:~ -~ r i~ H~(~).
Linearly elastic flezural shells
324
[Ch. 6
However, these convergences can be further improved and given a more intrinsic character by considering instead the averages of the tangential component Ueaona'e and of the normal component u~g 3,~ of the three-dimensional displacement vector itself; note in this respect that, along a given normal direction to the surface S, the vectors ga, e and g3, 9remain respectively parallel to the tangent plane and normal e ~ g3, e - a3, and gie " g~'" e : Ji" j More to S since gea - a a - x3baacr, specifically, the above convergences combined with the behavior as e --+ 0 of the vectors gi, e lead to the following result: 6.4-1. Assume that B E C3(~; I~3). Consider a family of linearly elastic flexural shells with thickness 2e approaching zero, with each having the same middle surface S - 8(~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(70) as its middle curve, and let the assumptions on the data be as in Sect. 6.1. Theorem
Let (u~) e H I ( ~ ~) and (i~) e H i ( w ) • H i ( w ) x H2(w) respectively denote for each e > 0 the solutions to the three-dimensional and twodimensional problems 7~(~ e) and 7~(w) (Sect. 3.1 and T a m . 4.6-1). Finally, let (ii) e H i ( w ) • H i ( w ) • denote the solution to problem T'e(w) (Thin. 6.3-1), which is thus independent of Th~
e a ~ae - Ca and thus Caa
~aa a in H 1(w) for all e > 0,
Hl( o)
--
2e
e a,,
o,
and
- ~3 and thus ~ a 3 -
~3a3 in H2(w) for all e > 0,
lf_ ~u~g 3'~ dx~
3 in H i ( w ) a s e ~ 0.
2e
~ r
e
Pro@ The proof is similar to that of T h m . 4.6-1. The assumption 8 E C3(~; I~3) implies that the vector fields ga(e) . ~ _+ IR3 defined by ga(e)(x):-" ga'e(xe) for all x e - r e x e ~e are such t h a t
Sect. 6.4]
Justification of the two-dimensional equations
325
g~(e) - a a - O(e) in CI(~). Since
2r
u~g a'~ dz~ - ~ a a = -
2
i
u ~ ( ~ ) ( g ~ ( ~ ) - a ~) a~3 - ( , , ~ ( ~ ) - r
the convergences ua (r imply that
~,
--+ ua in H I (12) and ga(e) ~ a a in Ci (~)
=~(~) ( a ~ ( ~ )
- a ~) -~ 0 in H ~ ( a ) ;
hence 1 fl_l u,~(e)(g,~(e)_ a a) dz3 --4 0 in HI(w) by Thin. 4.2-1. The same theorem also shows that (ua(e) - ( a ) a a --+ 0 in Hi(w). Since 2
2s
,,~g3,~ a ~
- C~,~ ~ - (,,3(~) - 6 ) a ~
e
the same theorem shows that ( u 3 ( e ) - ~z)a 3 --+ 0 in I-Ii(~).
II
-e _ --+ R 3 and ~N -~ _ --4 R 3 defined by The fields ~T "w "w
:= ~aa
and ~N := ~ a3'
which appear in Thm. 4.6-1, are respectively called the l i m i t t a n g e n t i a l d i s p l a c e m e n t field and the limit n o r m a l d i s p l a c e m e n t field of the middle surface S of the shell. Naturally, they are related to the limit displacement field ~e _ ~ a i of S (Sect. 6.3) by
Under the essential assumptions that the space VF(w) contains nonzero elements, we have therefore justified by a convergence result (Thm. 6.4-1) the two-dimensional equations of a linearly elastic flezural shell. In so doing, we have justified the formal asymptotic approach of Sanchez-Palencia [1990] (see also Miara & Sanchez-Palencia [1996] and Caillerie & Sanchez-Palencia [19955]) when "bending is not inhibited", according to the terminology of E. Sanchez-Palencia. Credit should be given in this respect to Sanchez-Palencia [1989a] for recognizing the central r61e played by the space VF(w) in the classification of linearly elastic shells.
326
Linearly elastic flezural shells
[Ch. 6
A w o r d of c a u t i o n . In a flexural shell, body forces of order O(e 2) thus produce an O(1) limit displacement field. By contrast, body forces of order O(1) are required to also produce O(1) limit displacement fields in an elliptic membrane shell. Both types of shells thus exhibit strikingly different limit behaviors! This conclusion of a mechanical nature simply reflects that strikingly different three-dimensional inequalities of Korn's type (compare Thm. 4.3-1 with Thm. 5.3-1) are needed for deriving in each case the fundamental a priori bounds in the convergence theorems (compare parts (i) of the proofs of Thms. 4.4-1 and 6.2-1). II After the original work of Ciarlet, Lods & Miara [1996] described in this chapter, the asymptotic analysis of linearly elastic flexural shells underwent several refinements and generalizations: First, Genevey [1999] has shown that the convergence result of Thm. 6.2-1 can be also obtained by resorting to P-convergence theory. Other useful extensions include the justification by an asymptotic analysis of linearly elastic flezural shells with variable thickness (Busse [1998]; see Ex. 6.7) or made with a nonhomogeneous and anisotropic material (Giroud [1998]), the convergence of the (scaled) stresses and the explicit forms of the limit stresses (Collard & Miara [1999]; see Ex. 6.8), an asymptotic analysis of the associated eigenvalue problem (Kesavan & Sabu [1999a]) and time-dependent problem (Xiao Li-ming [1999a]). The surprising phenomena appearing when a linearly elastic flexural shell "becomes a plate" are investigated in Ciarlet [1992a, 1992b], Sanchez-Palencia [1994], and Lods [1996]. An asymptotic analysis similar in its principles can be applied to linearly elastic curved rods, whose limit behavior is always "flexurar'; see Alvarez-Dios & Viafio [1997], Jamal [1998]. We conclude this commentary by examining linearly elastic plates, which constitute examples of linearly elastic flexural shells (as already observed; cf. Fig. 6.1-3), since
v F ( ~ ) - { ~ - (0, o, ~3); ~3 e H~(~); ~3 - 0~3 - 0 o . ~0} ~ {0} in this case; naturally, we assume that the mapping 0 is of the form O(y~, y2) = (y~, y~, o) for an (ys, y2) ~ ~. Assume that there exist constants A > 0 and # > 0 and functions f i e L2(~) and h i E L2(r+ u r_) independent of ~ such that, for
Justification of the two-dimensional equations
Sect. 6.4]
327
each e > 0, the Lam6 constants )~e a n d tte of the m a t e r i a l constit u t i n g the plates a n d the Cartesian components j::,e . ~t~ __+ ]R a n d h/,e 9F~_ U F ~_ -+ I~ of the applied b o d y a n d surface forces (the set ~e is now the reference configuration) satisfy o
)~e = )~
and
#e = #,
_ ~2/~(~)
fo~ an
~ - .~
z n~,
h~,~ ( ~ ) _ ~3 h~(~)
fo~ all
~ - .~
~ r ~ u r~_,
/~,~(~)
a n d let the scaled u n k n o w n u ( s ) - (ui(s)) e Hl(~t) be defined by
ui(s)(z) := u~(z e) for all z ~ - 7rex e ~e. where the functions u i -+ It~ are the Cartesian c o m p o n e n t s of the displacement field of the plate. T h e n T h m . 6.2-1 shows that, in this case, there exists a function U3 E H2(~t) vanishing on F0 - 7 0 • [ - 1 , 1] such t h a t
u(e) - (ui(e)) -+ (0, O, u3) in I-I~(~2) as ~ -+ 0, u3 is i n d e p e n d e n t of the transverse variable x3,
~3 "-- ~
I U3 dz3 ~ V3(w),
1- ~ ba~avO~T~3Oa3rl3dy= fwp3rl3 dy for all rl3 ~ V3(w)
3 where
b~
~. .-.- 4 ~ ~~. A+2/z
+ 2~(6~~.
+ 6~.6~),
p3 :._ l_ f dx 3 + (h3+ + h 3_) and h 3, :-- h 3(., --~-1). 1 However, this result is only a special case of the "classical" convergence t h e o r e m for a plate; see Vol. II, Thin. 1.4-1. For, w h e n the a s y m p t o t i c analysis is applied in ibid. to a linearly elastic plate,
more freedom is allowed, as regards both the choice of scalings of the displacement field and the choice of the asymptotic assumptions on the applied forces. More specifically, another scaled u n k n o w n ~(~) = (~2i(e)) is defined in this case by letting ~(~)(z)
- su~(z ~) a n d ~3(s)(z) - u~(z ~) for all z ~ e ~ ,
328
Linearly elastic flezural shells
[Ch. 6
and the applied forces are such that there exist functions ]i E L2(fl) and h / E L2(F+ U F_) independent of e such that fa'e(X~) -- e]~(X) and ]3'e(xe) ha'e(x e)
-
-
e2/3(X) for all m E ~,
g2ha(x) and h3,e(x ~) - e3h3(x) for all x e r + [J r _ .
The cruder scalings and assumptions made when the plate is considered as a special case of a shell have two consequences: First, the horizontal components fa, e and h a'e of the applied forces do not contribute to the definition of the limit two-dimensional problem satisfied by the unknown ~3. Second, the de-scaled components of the limit displacement field (0, 0, ~3) found in this fashion are all of order zero with respect to s. Therefore, the de-scaled horizontal components of the limit displacement field that are of order one with respect to s are necessarily "ignored" in this approach. These horizontal components, together with the transverse component of order zero, form a KirchhofJ-Love displacement field (Vol. II, Thm. 1.7-1) inside the plate. Another major difference is that the asymptotic analysis of a linearly elastic plate yields a limit problem that simultaneously includes flexural and membrane equations. The reason why limit membrane equations are in addition obtained is again that horizontal components of order one of the limit displacement field can be recovered, thanks again to the refined scalings and assumptions that are allowed for a plate. That cruder scalings and assumptions are unavoidable for "genuine" shells has been rigorously established in Sect. 3.4, after Micra & Sanchez-Palencia [1996].
EXERCISES
6.1. Let f = (fa) E C2([0, 1]; IR2) be an injective mapping such that f ' ( t ) # 0 for all t E [0, 1] and let S - 8(~), where w --]0, 1[• 1[ and 8(t, z) - fa(t)e a + ze 3 for (t, z) E ~. The surface S is thus a portion of a cylinder orthogonal to, and passing through, the planar curve f([0, 1]); cf. Fig. 6.1-1.
Ezercises
329
(1) Assume that
0'o C {(O,z) CI~2; O_
v~(~) - { . -
(~) e H ~ ( ~ ) • H ~ ( ~ ) • H ~ ( ~ ) ; r / / - 0v~73 - 0 on 70, ")'o/3(t7) -- 0 in w }
contains nonzero functions. (2) Is the space VF(w) infinite-dimensional? (3) Assume that the curve f([0, 1]) contains a straight segment and that "Yo = Ow. Show that dim VF(w) = +c~. (4) Assume that the curve f([0, 1]) does not contain any straight segment and that ~o = Ow. Does the space VF(W) still contain nonzero functions? 6.2. Let f - (fa) e C2([0, 1];]~ 2) be an injective mapping such that f ' ( t ) 7~ 0 for all t e [0, 1] and let S - 0(~), where w : ] 0 , l[• O(t, s) -
1[ for some 0 < so < 1,
s f a ( t ) e a + (1 - s ) e 3 for (t, s) E W.
The surface S is thus a portion of a cone with vertex e 3 and passing through the planar curve f([0, 1]); cf. Fig. 6.1-2. (1) Assume that 3'0 - {(0, s) E I~2; so _< s < 1}. Show that VF(~) : {0}. (2) Assume that
")'o - {(0, s) C IR2; so _< s <_ 1} U {(1, s) C IR2; so _< s _< 1}. Does the space VF(w) still contain nonzero functions? 6.3. Show that a plate subjected to a boundary condition of place along any portion of its lateral face whose middle line 70 satisfies length "Yo > 0 (Fig. 6.1-3) constitutes an instance of a linearly elastic flexural shell. Hint: Show that the space VF(w) consists in this case of vector fields of the form r / - (0, 0, ~73) with r/a E H2(w) and ya - 0v~73 - 0 on "Y0.
330
[Ch. 6
Linearly elastic flezural shells
6.4. In this exercise, it is assumed that the injective mappings 0 are in C2(~; I~3) and that 70 - 7 . (1) Show that, if the surface S = O(w) contains a planar portion, then VF(w) # {0}, so that the associated shell is a linearly elastic flexural shell. (2) Assume that the surface S does not contain any planar portion. Does the space VF(w) still contain nonzero functions? 6.5. Do there exist instances where the space VF(w) contains nonzero functions and is finite-dimensional? 6.6. The notations used below should be self-explanatory. (1) Let a surface S be equipped with two systems of curvilinear coordinates (ya) e ~ and (Ya) e {5)}-, i.e., S - O(~) - /~({5~}-), and assume that 0(70) = 0('~0). Show that VF(w) # {0} implies that VF(5~) r {0}. In other words, the definition of a linearly elastic flexural shell (Sect. 6.1) is independent of the system of curvilinear coordinates employed for representing the surface S. (2) Show that the dimensions (finite or infinite) of the spaces VF(w) and V F ( ~ ) are equal. 6.7. Consider a family of linearly elastic flezural shells (i.e., the space VF(w) contains nonzero functions) with variable thickness, i.e., whose reference configurations are the sets Oe(~e), where the mapping (9 e 9~e ~ IR3 is defined for each e > 0 by
Oe(y, x~) := O(y) + x~e(y)a3(y) for
(v,
E
- - w x [-e, e], the given function e e W 2'
(w)
satisfying 0 < e0 _< e(y) for all y E ~. For such a family of flexural shells, carry out an asymptotic analysis analogous to that of Thm. 6.2-1. Assuming for simplicity that there are no surface forces, show in particular that the limit scaled two-dimensional problem takes the form
if
g
p~r(~)peB(rl)eV~ dy --
L{fl
}
]i dz3 nieV/-ddy 1
for all r/ E VF(w), where the space VF(w) is the same as in Thm. 6.2-1, but the covariant components of the linearized change of metric
331
Ezercises
and change of curvature tensors are now replaced by the more general expressions: " - ~1 ( O ~ a + 0 ~ , ) - r ~~, ~
~(~)
- ~I ba~ r/3 , 1
pet~(~) :-- Oa~ T]3 -- feral30~73 - (b~b~# + -{Oa[3ee
+~{b~(o~
- r~~)
+ b~ ( o ~
- :Pat30~e})r/3
- r~~)
+~2 {OaeOt3e}~3 _ el {OaeOt3y3 + ~e~c~73}. Remark. This asymptotic analysis is treated in Busse [1998, Thm. 5.3]. 6.8. We established in Thm. 6.2-1 that the scaled displacements converge and that their limits can be fully identified. The present problem shows that, under the same assumptions, the convergence of the (scaled) stresses can be also established and that the corresponding limit stresses can be likewise explicitly computed. These results are due to Collard & Miara [1999, Thm. 8]. Given a family of linearly elastic flexural shells that satisfies the assumptions of Thm. 6.2-1, let
o-iJ, e-- AiJkt, ee~llt(Ue ) denote for each e > 0 the contravariant components of the linearized stress tensor field (Thm. 1.6-1) inside each shell and define the scaled stresses a '3 (e) 9f~ --+ R by letting . o
~iJ'~(x~) --: aiJ(~)(x) for all x ~ - 7r~x e ~. Note that the scaled stresses then satisfy
criJ(s) - AiJkl(s)eklll(S; u(s)). Show that l a a ~ (e) _+ aa~, 1 in L2(f/),
~
o'i3(e) --~ cri3'2 in H i ( - 1 , 1; H - l ( w ) )
332
Linearly elastic flezural shells
[Ch. 6
as ~ --+ 0, where the limits are given by (covariant derivatives na#[~ are defined in Thm. 4.5-1)
aa~, 1 _ _x:3aaf3ar par(u), aaa, 2 = 1 ( x 2 _ 1)(aaf3~rO~r(g))l~ , 2 aaa, 2 _ l(x~ - 1)a a ~ r
Remark. These results should be profitably compared with those established for linearly elastic membrane shells (Ex. 4.4) and plates (Vol. II, Sect. 1.6).
CHAPTER 7 KOITER'S
EQUATIONS
TWO-DIMENSIONAL
AND
LINEAR
OTHER
SHELL THEORIES
INTRODUCTION
Consider as in the previous chapters a linearly elastic shell with middle surface S - 8(~) and thickness 2e, subjected to a boundary condition of place along a portion of its lateral face with 8(70) as its middle curve, and subjected to applied forces. This problem can thus be modeled either by the equations of threedimensional linearized elasticity (Chap. 1) or by two-dimensional equations obtained by an asymptotic analysis of the three-dimensional equations. We showed that the form of these two-dimensional equations depends on, and only on, the geometry of the surface S and on the set ~'0 (Chaps. 4 to 6). Founding his approach on a priori assumptions of a geometrical and mechanical nature about the solution of the three-dimensional equations (these assumptions are described in Sect. 7.1), W.T. Koiter has devised in the sixties yet another means of modeling the same problem by two-dimensional equations. The resulting two-dimensional Koiter equations for a linearly elastic shell take the following form, when they are expressed as a vari~ho~ r a~e ational p~obl~m: The unknown r - (r the covariant components ~:,i " ~ --+ I~ of the displacement field ~ : , i a ~ of the points of the middle surface S, satisfies: o
(~: - ( ( g , i )
E Vg(w)-
{n-
(Yi) E Hi(w) • Hi(w) • H2(w); ~i = 0~3 = 0 on 70},
g3
/~p~'~n~vqdy
for ~H n - (n~) E V ~ ( ~ ) ,
where
a a13~r'e a r e
the components of the two-dimensional elasticity
334
Koiter's equations and other linear shell theories
[Ch. 7
tensor of the shell, 7 ~ ( ~ ) and Pal3(r/) are the components of the linearized change of metric and change of curvature tensors associated with a displacement field ~Tiai of S, and the given functions pi, e E L2(w) account for the applied forces. Equivalently, the unknown ~ satisfies the following minimization problem: ~
E VK(W) and j ~ ( ~ )
=
inf j~(v/), uev~(~)
where the two-dimensional Koiter energy j ~ 9VK(W) --+ R is defined by j EK( n ) = -~
{ s a a~ar' eVar (17) Vaf3( n ) S3 aa[3trr, +-~-
~
~
"
As already shown in Sect. 2.6, the existence and uniqueness of a solution to these equations essentially rely on an inequality of Korn's type on a general surface, itself a consequence of a crucial lemma of J.L. Lions and of an infinitesimal rigid displacement lemma on a general surface; this existence and uniqueness result is recalled in Sect. 7.1, together with its extension to shells whose middle surface has "little regularity" (Thms. 7.1-1 and 7.1-2). In addition, we identify the associated boundary value problem and we give sufficient conditions for the regularity of its solution (Thm. 7.1-3). It is remarkable that Koiter's equations can be fully justified for all types of shells, even though it is clear that these equations cannot be recovered as the outcome of an asymptotic analysis of the threedimensional equations, since Chaps. 4 to 6 exhaust all such possible outcomes! More specifically, we show that, for each category of linearly elastic shells (elliptic membrane, generalized membrane, or flexural), the 1 fields ~eg and 2ss
f2
e u e dx~, where u e denotes the three-dimensional
solution, have the same asymptotic behavior in ad hoc functional spaces as ~ --+ 0 (Thms. 7.2-1 to 7.2-3). So, even though Koiter's linear model is not a limit model, it is in a sense the "best" two-dimensional one for linearly elastic shells ! One can thus only marvel at the insight that led W.T. Koiter to conceive the "right" equations, whose versatility is indeed remark-
Sect. 7.1]
The two-dimensional Koiter equations
335
able (see the commentary in Sect. 7.3), out of purely mechanical and geometrical intuitions! While Koiter's equations belong to the family of Kirchhoff-Love theories (Sect. 7.1), two-dimensional shell equations that rely on the notion of one-director Cosserat surfaces have also been proposed by P.M. Naghdi, again in the sixties. After describing the associated twodimensional Naghdi equations for a linearly elastic shell (Sect. 7.4), we review the existence and uniqueness theory for these equations, which is akin to that for Koiter's equations (Thm. 7.4-1). Finally, we briefly review in Sects. 7.5 and 7.6 other two-dimensional equations that have also been proposed for modeling linearly elastic shells, notably "shallow" shells.
7.1.
THE TWO-DIMENSIONAL KOITER EQUATIONS F O R A L I N E A R L Y E L A S T I C SHELL: EXISTENCE~ UNIQUENESS, AND REGULARITY OF S O L U T I O N S ; F O R M U L A T I O N AS A BOUNDARY VALUE PROBLEM
Let w be a domain in I~2 with boundary "y, let O C Cs(~; I~s) be an injective mapping such that the two vectors aa = OaO are linearly independent at all points of ~, and let ~0 be a portion of "y that satisfies length ~/o > O. Consider as in the previous chapters a linearly elastic shell with middle surface S = O(~) and thickness 2s > 0, i.e., a linearly elastic body whose reference configuration is the set O(~e), where : =
o(y,
-
0(y) +
(y,
e
The material constituting the shell is homogeneous and isotropic and the reference configuration is a natural state, so that the material is characterized by its two Lamd constants )~e > 0 and #e > 0. The shell is subjected to a boundary condition of place along the portion o ( r ~ ) of its lateral face, where r~ . - -y0 • [-e, e], i.e., the three-dimensional displacement vanishes on | Finally, the shell is subjected to applied body forces in its interior | and to applied surface forces
Koiter's equations and other linear shell theories
336
[Ch. 7
on its "upper" and "lower" faces and | ~_), their densities being given by their contravariant components ]i,e C L2(f~ e) and h
e
u
where
-
•
In a seminal work on shells, John [1965, 1971] has shown that, if the thickness is small enough, the state of stress is "approximately" planar and the stresses parallel to the middle surface vary "approximately linearly" across the thickness, at least "away from the lateral face". In Koiter's approach (Koiter [1960, 1966, 1970]), these approximations are taken as an a priori assumption of a mechanical nature (a precise statement of this assumption is found in Sect. 11.1) and combined with another a priori assumption of a geometrical nature, called the Kirchhoff-Love assumption: Any point on a normal to the middle surface remains on the normal to the deformed middle surface after the deformation has taken place and the distance between such a point and the middle surface remains constant (Ex. 7.1(1)). In fact, this assumption is required to hold only "to within the first order" in the linearized theory considered in this chapter, in which case it is called the linearized Kirchhoff-Love assumption (Ex. 7.1(2)).
Remark. In a pioneering contribution, Oden, Wempner &: Kross [1968] have suggested to incorporate some discretized version of the linearized Kirchhoff-Love assumption into finite element codes, typically by requiring that it be only satisfied on a finite set of points. In myriad numerical simulations, this idea has since then proved to be highly judicious, m Taking these two a priori assumptions into account, W.T. Koiter then shows that the displacement field across the thickness of the shell can be completely determined from the sole knowledge of the displacement field of the points of the middle surface S, and he identifies the two-dimensional problem, i.e., posed over the two-dimensional set ~, that this displacement field should satisfy. As in the two-dimensional theories encountered so far, the unknown is a vector field, now denoted ~ : -- (~ie,g ) 9 ~ ~ ]~3, whose components iie, g " "~ --~ I~ are the covariant components of the displacement field of the middle surface S. This means that ii~,g(y)ai(y) is the displacement of the point 0(y); cf. Fig. 7.1-1. In their linearized version (the nonlinear one is given in Sect. 11.1), the equations found by W.T. Koiter consist in solving the following
The two-dimensional Koiter equations
Sect. 7.1]
337
F
Fig. 7.1-1: A linearly elastic shell modeled by Koiter's two-dimensional equations. T h e t h r e e u n k n o w n s are the covariant c o m p o n e n t s (i~,/c : w --+ IR of the displacem e n t field of t h e m i d d l e surface S; this m e a n s t h a t (ie,lc(y)ai(y) is t h e displacem e n t of t h e p o i n t O(y) of the middle surface S. T h e " two-dimensional b o u n d a r y c conditions of clamping" (i,9~c = 0 ~(s,/r = 0 on 70 m e a n t h a t the points of, a n d the t a n g e n t spaces to, the deformed and u n d e f o r m e d middle surfaces coincide along the set 0('1,0).
variational problem 7>~c(W)(the associated boundary value problem is given in Thm. 7.1-3)" Find ~ c - ((~c,i) such that
77i - 0~77~ -
0 on
70
},
= f pi'enix~dy for all r l - (rli) E V g ( w ) , d~
338
[Ch. 7
Koiter's equations and other linear shell theories
where (the functions a a~3, ba~3, b~, see, e.g., Sect. 3.1): a ~ , ~ ._ 9-
4Aepe A~ + 2# e
r:,,
and a are defined as usual;
a ~ a ~ + 2#~(a~a~ + a ~ a ~ )
1
po,(v) :=
oo, w -
- b~b~73
a~
+b~(O~w - r ; , w )
+(oob; + rLb pi , . _
2
+ b;(O~w - rLw
)
-
]i "da:~ + .0+ + h"
and h'_t'e "- h"e( ., +s).
e
The functions 7a# (r/) and pa#(r/) are the customary covariant components of the linearized c h a n g e of m e t r i c and c h a n g e of c u r v a t u r e t e n s o r s associated with a displacement field y i a i of the middle surface S and the functions a a/3~r,e are the customary contravariant components of the t w o - d i m e n s i o n a l elasticity t e n s o r of t h e shell. Note that Destuynder [1985] has found an illuminating way of deriving the same linear Koiter equations from three-dimensional elasticity, which uses a priori assumptions only of a geometrical nature. 1
The ezistence and uniqueness of a solution to problem T'~:(w), which essentially follow from the Vg(w)-ellipticity of the bilinear form, was first established by Bernadou & Ciarlet [1976]; a more natural proof was subsequently proposed by Ciarlet & Miara [1992b], then combined with the first one in Bernadou, Ciarlet & Miara [1994]. This proof, which has been already given and discussed at length in Sect. 2.6, is simply outlined in the next theorem. Observe that 0 C Ca (~; IR3) is precisely the assumption that will afford the justification of Koiter's equations in Sect. 7.2. T h e o r e m 7.1-1. Let w be a d o m a i n in I~2, let 70 be a subset of 7 - Ow with length 70 > 0, and let 0 E C3(~; ~3) be an injective m a p p i n g such that the two vectors an = OaO are linearly i n d e p e n d e n t at all points of -~.
Sect. 7.1]
339
The two-dimensional Koiter equations
Then the variational problem 79~(w) has one and only one solution, which is also the unique solution to the minimization problem: Find ~K -- ( ~ , i ) such that r
C Vg(w) and J~c(r
-
inf j~(rl), where ,ev~(~)
1 e 3 aa/3ar'
pi' e~Ti~ dy.
Proof. The assumptions fi, e C L2(~2~) and h i'~ C L2(r u r t ) imply that pi,~ C L2(w). The existence and uniqueness of a solution to the variational problem :P~:(w), or to its equivalent minimization problem, are consequences of the inequality of Korn's type on a general surface (Thm. 2.6-4), of the uniform positive definiteness of the two-dimensional elastic tensor of the shell (Thm. 3.3-2), of the existence of a0 such that a(y) >_ a0 > 0 for all y e ~ (Thm. 3.3-1), and of the Lax-Milgram lemma, m In Sect. 2.6, we also described how Blouza & L e Dret [1994a, 1994b, 1999] showed that the introduction of new expressions, denoted "~af3(O) and fbaf~(O) (see in]ra), for the functions 7af3(rl) and Pa~(~) affords to consider more general situations, where the mapping 8 need only be in the space W 2, ~176 1~3). For a linearly elastic shell, simply supported along its entire boundary (boundary conditions of clamping along a portion of its boundary can be handled as well, provided they are first re-interpreted in an ad hoc manner), the associated Koiter~s e q u a t i o n s for shells whose m i d d l e surface has little r e g u l a r i t y accordingly take the following form: The unknown ~g, which is now the displacement field of the middle surface, satisfies the variational problem 75~7(w) 9
= L l~e" ~ ~ d y
for all ~ E ~r~c(W),
340
Koiter's equations and other linear shell theories
[Ch. 7
where 1 :=
. a,, + 0 , , 0 . a , ) ,
the given function/5 e E L2(w) account for the applied forces, and a a ~ r , e a r e the usual contravariant components of the two-dimensional elasticity tensor of the shell. Recall that "~a#(0)- 7a#(r/) and tba#(0 ) : pa#(r/) if 0 - ~7iai is such that r / = (~/i) e H i ( w ) • Hi(w) • H2(w). A proof similar to that of Thm. 7.1-1, now based on the inequality of Korn's type on a general sur]ace with little regularity (Thm. 2.6-6), then produces the following existence and uniqueness result: T h e o r e m 7.1-2. Let there be given a domain w in I~2 and an injective mapping 8 E W2'~176 I~a) such that the two vectors aa - OaO are linearly independent at all points of-~. Then the variational problem 79~c(w) has one and only one solution, which is also the unique solution to the minimization problem: ~~ Find ~g such that
~K E ~rK(w ) and jeK( ~ K ) -
inf
~c(~/), where
~e . ~Tv ~ dy.
II
We emphasize that, in this approach, the unknown ~K and the fields ~ are displacement fields of the middle surface, no longer recovered as ~K = ~c,i a~ or ~ -- ~ia z by means of their covariant components ~f,i or r/i. We next derive the boundary value problem that is (at least formally) equivalent to Koiter's variational problem 7~c(w). We only consider here the case where 70 = 7; see Ex. 7.2 for the case where length 3'0 < length 7. ~,~
o
,
Sect. 7.1]
341
The two-dimensional Koiter equations
We also state a regularity result that provides instances where the weak solution (the solution of the variational problem) becomes a classical solution (a solution of the boundary value problem). Theorem
7.1-3. A s s u m e that 7 0 - 7, so that •
•
(a) If the solution ~ g o] the corresponding variational problem 7~c(w) (Thm. 7.1-1) is smooth enough, it also satisfies the boundary value problem: mafl, ela fl _ b~ba~ma~,e _ ba~nafl, e = pa, e in w, _(na/3, e + b~mr
_ ba(mCf3, elf3) _ pa, e in w,
r
~ - o o~ ~ - O~G,~
~,
where na/3, e := eaaf3~r'eTar(~eK) ,
~3 a aflcrr' epar(r real3, e := --if-
and, for an arbitrary tensor with twice differentiable covariant components n af3, n~l~
:= O ~ n ~ + r ~ n ~
+
na[31a[3 .-- Oa(na[3[[3) + r~(nafll/3 ). (b) A s s u m e that, for some integer m > 0 and some real number q > 1, 7 is of class Cm+4, 0 C cm+4(~; ~3), pa, e E w m + l ' q ( w ) , and pa,~ E w m ' q ( w ) . Then
Ce_K - - ( ~ )
e wm+3'q(~g) x w m + 3 ' q ( ~ ) X wm+4'q(&l).
Proof. For notational convenience, we omit the exponents "e" and the indices " K " throughout the proof, i.e., we let
(: "-- ~k, naft := aaflcrr')'ar(r
maf~ "-- laaflCrrpcrr(~), pi := pi, e 3
342
Koiter's equations and other linear shell theories
[Ch. 7
Assume that the solution ~ is "smooth enough" in the sense that n a/~ e Hi(w) and m a/~ e H2(w). We already saw in the proof of Thm. 4.5-1 that
f ~'='~=.(r
~ - - f~ 4~{(='1,)~. + b.,=',3} d~
for aU ~7 -- (r/i) e H](w) • H](w) • L2(w), hence a ~ortiori for all , E H~(w) • H~o(W)• H2(w). It thus remains to transform the other integral appearing in the left-hand side of the variational equations, viz.,
= ~ ~ real30a/3~3 dy + ~ v/-ama/3(2b~O/3y~ - r~,0~3)dy
+ f~ v~m~(-2b;r~,~ + b~l~ - b ~ ~ ) d ~ , where 17 -- (Yi) e H~(w) x Hlo(w) x H2o(W). Using the symmetry the relation (already established in the proof of Thm. 4.5-1): m a~ = m t3a,
0,v~- ~ r ~ , and the Green formula in Sobolev spaces (see, e.g., Vol. I, Thm. 6.1-9), we obtain
~ ma~pa/3(n)V~ dy - - ~ ~/a(O/3ma~ + r$~mo' + r~,m~')0~ dy
+ ~ v/ama~(-2b~F~,y~ + b~]ay~ - b~bz/3y3)dy.
Sect. 7.1]
343
The two-dimensional Koiter equations
The same Green formula shows that
- f~ ~ ( o ~ , ~ '
+ r~,~ " + r~,,~)o~
dy
= ~ V~( maf~ laf~)Ya dy,
~
~ ~ ~ o ~ , ~'~e ~
-
r'r ~"~'"a~}y'~ dy" -~ ~ ~ { o ~ ( ~ ~) +~,~,,o
Consequently~
Using in this relation the easily verified formula
and the symmetry b~]~ - b~al~ (Thm. 2.5-1), we finally obtain
- ~ ~ { b~b~ m~ - m~ Io~},~ ~y. Hence the variational equations
{~o~~(r
+ ~a ~ p~,(r
p'~i}v~dy - 0
for all y = (vii) E Vg(w) imply that
f
~{(~,
+ b~,~')l, + b~(,~'l,) +p~
+ f~ ~ { b o , ~ o , + b~b~,,~~ - m~
dy
+ p~},~ ~y = 0
for all (y~) C VK(w). The announced partial differential equations are thus satisfied in w.
Koiter's equations and other linear shell theories
344
[Ch. 7
The regularity result of part (b), which is due to Alexandrescu [1994], is left as a problem (Ex. 7.3). m Each one of the three formulations found in Thms. 7.1-1 and 7.13 constitutes the t w o - d i m e n s i o n a l K o i t e r e q u a t i o n s for a line a r l y elastic shell. We recall that the functions ~/a~(~?) and paf3(vl) are the covariant components of the l i n e a r i z e d c h a n g e of m e t r i c and c h a n g e of c u r v a t u r e t e n s o r s associated with a displacement field ~?iai of the middle surface S, the functions a a/3ar,~ are the contravariant components of the t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of t h e shell. The functions n af~,e and m af3,e are the contravariant components of the s t r e s s r e s u l t a n t and s t r e s s couple, or b e n d i n g m o m e n t , t e n s o r fields. As shown in Sect. 6.3, the t w o - d i m e n s i o n a l b o u n d a r y cond i t i o n s of c l a m p i n g ~:,i - 0 ~ : , 3 - 0 on 70 express that the points of, and the tangent spaces to, the deformed and undeformed middle surfaces coincide along the set 8(')'0); cf. Fig. 7.1-1. The functional j~: 9 Vg(w) -+ IR in Thm. 7.1-1 is the twod i m e n s i o n a l K o i t e r e n e r g y of a l i n e a r l y elastic shell. The associated K o i t e r s t r a i n e n e r g y :
n e
-+
~3 aaf3ar' + -~ eO,Tr( ~l) Oa, ( rl ) } v/-a dy
is thus the sum of the strain energy of a linearly elastic el@tic membrane shell (Sect. 4.5) and of the strain energy of a linearly elastic flezural shell (Sect. 6.3)l Showing that this uncanny and eerie "addition" is in fact of the utmost pertinence is the object of the next section. Remarks. (1) In the original derivation of Koiter's equations, the functions n af3'~ and m af3,e found in Thm. 7.1-3 are first defined by means of an ad hoc "two-dimensional variational principle" (Koiter [1970, Sect. 4]); they are then shown to be related to the components craf~'e of the linearized stress tensor field (Thm. 1.6-1) inside the shell by the equations (Koiter [1970, eqs. (4.4) and (4.5)]): n"~, ~ =
F
{a"~, ~ E
Justification of Koitev's equations
Sect. 7.2] m "z''=
S
+
1
345 a i.
e
These formulas justify the terminology "stress resultants" and "stress couples" attached to the functions n af3, ~ and m all' e. (2) Analogous relations can be rigorously derived from the twodimensional equations of a linearly elastic elliptic membrane shell, the functions n aft' e := eaafl~r,e.y~r (~e) being then related to the limit stresses aafl'~(O) by the formulas (Ex. 4.4)
naf3, E _
craf3, e (0) dz~"
(3) In the two-dimensional linear Kirchhoff-Love theory of plates, the "stress couples" are usually defined by the formulas (Vol. I, Thm. 1.7-3) m af3'e -
F z~a af3'e (0) dx~.
Note the change of sign! (4) The functions m aft, e laf3 are instances of second-order covariant
derivatives of a second-order tensor field. (5) Koiter's equations can also accommodate loads and couples applied along the remaining portion 0(7 - 7 0 ) of the boundary of S; see Koiter [1970, eqs. (3.8), (3.9), and (3.13) to (3.15)]. II
7.2.
JUSTIFICATION OF KOITER~S EQUATIONS FOR ALL TYPES OF LINEARLY ELASTIC SHELLS
Consider the same linearly elastic shell problem as in Sect. 7.1. Then the corresponding three-dimensional variational problem 79(f~e) consists in finding u e - (u~) such that (all notations used infra are
Koiter's equations and other linear shell theories
346
[Ch. 7
defined as in, e.g., Sect. 3.1): u e e V(f~ e) -- {v e -- (v~) e Hl(f~e); v e - 0 on r~),
f
AiJkl, eekllt(ue)eill e e j (v e )~/~ dx e e
----fn fi'ev~xfg-gdxe+fr 9
sur ~_
hie' viedre
for all v ~ e V(O~),
where
AiJht, e := )~egij, egkt, e + tze(gih, egfl, e + git, egjk, e), 1
e
,
e
e~llJ(V" ) " = -~(O~v i -]- 0 i v j ) - ri~
e(ve
).
The unknown functions u~ in problem 7~(f~e) represent the covarig g ant components of the displacement field uigZ' of the points of the reference configuration | the functions A ijm,~ denote the contravariant components of the three-dimensional elasticity tensor, the e ( r e ) denote the covariant components of the linearized functions ei]lj strain tensor, and d r ~ denote the area element along the boundary of the set f~e. Consider now a family of such shells, with each having the same middle surface S - 8(~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 8(V0) as its middle curve. All the linearly elastic shells in such a family are thus either elliptic membrane, or generalized membrane, or flexural, according to the definitions given in Chaps. 4, 5, and 6. We now show that, in each case, the asymptotic behaviors as ~ --+ 0 of the
average ~ f~-e ue dxl of the solution to the three-dimensional variational problem P(f~') and of the solution r to the two-dimensional Koiter equations formulated as the variational problem 79~c(w) (Thin. 7.1-1) are identical For elliptic membrane and flexural shells, we show in addition that the same property holds, separately for the tangential components ~ f~ u'ao-a'" dx~ and r a and for the normal come aa 3 of the associated displacement ponents 1 fe__eulg3, e dx i and CK, fields.
Justification of Koiter's equations
Sect. 7.2]
347
To this end, we use in an essential way the convergence theorems established in Chaps. 4, 5, and 6, together with slight improvements over former results of Destuynder [1985], Sanchez-Palencia [1989a, 19895, 1992], and Caillerie & Sanchez-Palencia [1995a] (see also Caillerie [1996]) about the asymptotic behavior of the solution of Koiter's equation as e approaches zero. The forthcoming analyses have been recently extended by Xiao Liming [1998, 1999a, 1999b], who likewise justified the time-dependent Koiter equations for elliptic membrane and flezural shells. To begin with, we consider elliptic membrane shells. We follow here Ciarlet & Lods [1996c, Thm. 2.1]. T h e o r e m 7.2-1. Assume that B C g3(~; I~3). Consider a family of linearly elastic elliptic membrane shells, according to the definition given in Sect. 4.1, with thickness 2~ approaching zero and with each having the same elliptic middle surface S = B(-~), and let the assumptions on the data be as in Sect. 4.1; in particular then, 3'0 = 3'. For each ~ > 0 let (u~) e H I ( ~ e) and ~
--(~i~,K) e H~(w) • H~(w) • H2(w)
respectively denote the solutions to the three-dimensional and twodimensional variational problems 7~(~ e) and 79~(w). Let also
denote the solution to the two-dimensional scaled variational problem T~M(W) (Thm. 4.5-1), solution which is thus independent of e. Then
Ii
u ae~ n a'e dz~3 ----+ ~aa a in Hi(w) as e --+ 0, ~ , a aa ~
Caaa in Hi(w) as e --+ 0,
and
1F ~ , g a 3 --+ ffza 3 in L2(w) as e ~ O.
348
[Ch. 7
Koiter's equations and other linear shell theories
Proof. Under the assumptions that there exist constants A > 0 and p > 0 and functions fi E L2(f~) and h i E L2(r+ u r_) independent of e such that Ae = A
and
p~ = p,
f i , ~ ( x ~ ) - fi(x)
for all
x ~ -7r~x E W,
h~,~(~ ~) - ~hi(~)
fo, an
~ - ~
e r ~ u r ~_
(these are the assumptions on the data for a family of linearly elastic elliptic membrane shells; cf. Sect. 4.1) and that 0 E C3(~; I~3), the convergences
if
2-~
ueaga'~ dxea ~
~aaa in Hi(w)
u~g 3 dx~ ~
~3a 3 in L2(w)
and
1 2e
e
as e --+ 0 have been established in Thm. 4.6-1, as corollaries to the fundamental convergence result of Thm. 4.4-1. Th. r r ~ r in H~(~) • H~(~) • L2(~) wa~ ~ t established by Destuynder [1985, Thm. 7.1]; it was also noted by Sanchez-Palencia [1989a, Thm. 4.1] (see also Caillerie & SanchezPalencia [1995a]), who observed that it is a consequence of general results in perturbation theory, as found for instance in Sanchez-Palencia [1980]). We give here a simple and "self-contained" proof. Let a a ~ r .--'-- 2Ap aa~aCrr + 2#(aa,ra~r + aara~Cr) '
~+2#
BF(r
1~ a a ~ r P,rr(r
n):= ~
L(rl) :=
L
fi dx3 + h+ + h i_,
P*~Tiv/~dy, where pi ._ 9
v ~ ( ~ ) := H~(~) • H~(~) • L~(~),
ll,oll 1,w -~" 1,7312,,,,} ~/2
:: Cr
i
1
Sect. 7.2]
Justification of Koiter's equations
349
Recall that there exists a constant ce > 0 such that (Thm. 3.3-2)
It~] ~ < ~ ~ ( y ) ~ ~ a,f3
for all y E ~ and all symmetric matrices (tar3) and that, according to the inequality of Korn's type on an elliptic surface (Thin. 2.7-3), there exists a constant CM such that
i1.11v=. .
a,fl
for all v / = (r/i) E VM(W). Finally, note that the space Vg(w), which here is simply H~(w) x H~(w) x H2(w), is contained in the space
v~(~). By virtue of the assumptions on the applied forces, the solution ~ : of the two-dimensional Koiter equations also satisfies the scaled Koiter equations for an elliptic membrane shell, viz.,
BM(~K, ~) + s 2 B F ( ~ : , v/) = L(~7) for all ~/e Vg(w). Hence letting v / - ~ : in these scaled equations yields the inequality 1
2
1
v~(~)
~
2 0,~ -
a,fl
where
ll il - { 9
Ip'lo, o}'" 2
i
9
This inequality shows that the family ( ~ ) e > 0 is bounded in VM(W) and the families (sPaf3(~:))e>o are bounded in L2(w). Consequently, there exists a subsequence, still denoted (~:)e>o for convenience, and there exist a field ~* G VM(W) and functions p J E L2(w) such that (as usual --+ and ~ respectively denote strong and weak convergences): ~ : --~ ~* in VM(w) and spaf~(~:) --~ p ~ in L2(w). Fix ~/ E Vg(w) in the scaled Koiter equations and let s ~ 0; then the above weak convergences yield BM(~*, ~l) = L(~l). Since the space Vg(w) is dense in VM(w), we conclude that BM(~*, ~l) -- L(~l) for all ~7 E VM(oJ). Hence
r162
Koiter's equations and other linear shell theories
350
[Ch. 7
where r E VM(W)is the unique solution to the scaled problem (Thm. 4.5-1), and the weak convergence
~)M(W)
then holds for the whole family (~r By the inequality of Korn's type on an elliptic surface and by the positive definiteness of the scaled two-dimensional elasticity tensor of the shell, establishing the strong convergence ~ : -+ ~ in VM(W) is equivalent to establishing the convergence
BM(r
- r r
- r -~ O.
Letting r / - ~ : in the scaled Koiter equations shows that
BM(r
r
_< L(r
Hence
o _< B~(r = B~(r
_< L(r
r ~:Ic r
-
-
r -
-
2B~(r
2B~(r
r + B~(r
r + L(r
r
since BM(~, r - - L(~). Noting that the weak convergence ~ in VM(W) implies that
BM(r
-~
r -+ BM((~, ~) and L(r162 -+ L(r
we have thus shown that BM(~eK- ~, ~eg - ~ ) ~ O, hence that ~
-~ ~ in
VM(W).
This convergence clearly implies that ~ , ~a ~ -~ ~aa ~ in H l(w) and ~ , 3a 3 -~ C3a 3 in L 2 (w). 1
The convergence results of Thm. 7.2-1 have been improved by Mardare [1998b, Thm. 5.1], who showed that I!r
-
r
=
0(81/5),
Justification of Koiter's equations
Sect. 7.2]
351
and by Lods & Mardare [1998b, 1999c], who showed that
1F
u e dx~
-
r215215
-
O(g:l/5)
9
Remark. Under the assumptions of Thm. 7.2-1, the function ~'~,K thus "looses its boundary condition" as ~ approaches zero. This phenomenon, together with its accompanying boundary layer, can be already observed on a simple one-dimensional model of Koiter's equations (Ex. 7.4). As already noted (Sect. 4.6), a similar "loss 1 of boundary condition" is shared by the average 2ee
e u~ dx~ as s
approaches zero.
I
We next turn our attention to generalized membrane shells of the first kind. We follow here Ciarlet & Lods [1996d, Thins. 6.1 and 6.2]. In the same way that we required the applied forces to be "admissible" (Sect. 5.5) in order to carry out our asymptotic analysis of the solutions of the corresponding three-dimensional equations (Sect. 5.6), we need to assume that the applied forces enter Koiter's equations in such a way that the corresponding scaled linear forms are continuous with respect to the norm l" IM of the "limit" space V~M(W) and uniformly so with respect to ~. More specifically, we set the following definition (notice the analogy with that given in Sect. 5.5): Applied forces are a d m i s s i b l e for t h e t w o - d i m e n s i o n a l K o i t e r e q u a t i o n s if there exist functions r _ ~ a E L2(w) such that, for each s > 0, the right-hand side in Koiter's equations cart also be written as
p i' e~Ti~ d dy - 6 f~
n-
e
Remark. For simplicity, we assume here that the scaled linear forms (denoted LM in the next proof) are independent of s. But the conclusions of the next theorem would be unaltered under the more general assumption that, for each s > 0,
pi'e~Ti%~dy - e fw qoa~(s)Va~(17)v~dy for all r / = (r//) e VK(w),
352
Koiter's equations and other linear shell theories
with functions ~ ( e ) ~ L2(w) converging strongly to ~ as e approaches zero.
[Ch. 7
in L2(w) 1
T h e o r e m '/.2-2. Assume that 0 E C3(~; ]R3). Consider a family o/linearly elastic generalized membrane shells of the first kind, according to the definitions given in Sect. 5.1 and 5.4, with thickness 2e approaching zero, with each having the same middle surface S = 0(-~), with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(7o) as its middle curve, and subjected to applied forces that are admissible for both the three-dimensional equations (Sect. 5.5) and the two-dimensional Koiter equations, the/'unctions ~oa# C L2(w) coinciding in addition with those found in Thm. 5.6-1. For each e > O, let
u e e Hl(f~ 6) and r
e Hi(w) x Hi(w) • H2(w)
respectively denote the solutions to the three-dimensional and twodimensional variational problems 7~(f~~) and 7~c(w). Let also
r ~ V~M(W) "-- completion of V(w) with respect to [. IM, where v(~)
:= { . = (w) e a l ( ~ ) ; .
= o on ~0},
a,/3
denote the solution to the two-dimensional scaled variational problem 7:'~M(W) (Thin. 5.7-1), solution which is thus independent of e. Then
u ' dx~ ~
r in VUM(W) as e ~ O,
r
r in V~M(a,) as e --+ O.
~
Proof. Under the assumptions that 0 E C3(~; I~3) and that the applied forces are admissible in the sense of Sect. 5.5, the convergence
1S e u ~ dx~ ~
2~
r in V~M(W)
Justification of Koiter's equations
Sect. 7.2]
353
as e -+ 0 has been established in Thm. 5.8-1, as a corollary to the fundamental convergence result of Thm. 5.6-1. The rest of the proof is an elaboration over Caillerie & SanchezPalencia [1995a, Thm. 4.5], who established the weak convergence ~ : --~ ~ in V~M(W) as e --+ 0. In particular, we establish here that this convergence is strong. Since the space Vg(w) is dense in the space V(w) with respect to the norm t1" l]z,~ and there exists c such that IrllM~ _< cllrlliz,~ for all rl e V(w), the space VK(W) is dense in V(w) with respect to I.I M and thus the space V~M(W) is also the completion of Vg(w) with respect
to I Let B~M and L~M denote the unique continuous extensions from V(w) to V~M(W) of the bilinear and linear forms BM and LM defined by
LM(17) "= L ~~ The applied forces being admissible for the two-dimensional Koiter equations, their solution ~ : satisfies the scaled Koiter equations for a generalized membrane shell, viz., BM(r
~ ) + s2Bv(r
Y) - LM(y) for all y E VK(w),
where
1 L aa~r Hence letting rI - ~ : in these scaled equations yields the inequality (the constant ce stems from the positive definiteness of the scaled two-dimensional elasticity tensor of the shell; cf. Thm. 3.3-2):
+ where
1 a,/3
2 }z/2 9 fILM11 :-- {Ea,~IWa~Io,~
ilLMIIICki M, This shows that the family
(~:)~>0 is bounded in the space V~M(W) and the families (spaf3(~eg))e>O are bounded in L2(w).
354
[Ch. 7
Koiter's equations and other linear shell theories
Consequently, there exists a subsequence, still denoted (r for convenience, and there exist r e V~M(W) and p ~ e L2(w) such that (as usual ~ denotes weak convergence): r
---' r in V~M(W) and ep,~(r
~ p-~ in L2(w).
Fix y E VK(W) in the scaled Koiter equations and let e --+ 0; then the above weak convergences yield B~M(r *, ~) -- LM(y). Since VK(W) is dense in V~M(W), we conclude that B~M(r *, y) -- L~M(y) for all y E V~M(W). Hence
r162
where ~ e V~M(W)is the unique solution to the scaled problem 7~M(W) (Thin. 5.7-1), and the weak convergence r
=
r in V~M(W)
then holds for the whole family (r162 By the positive definiteness of the scaled two-dimensional elasticity tensor of the shell and by the definitions of the norm I" ]M and of the bilinear form BM and of its extension B~M, establishing the strong convergence ~ --~ ~ in V~M(W) is equivalent to establishing the convergence
B~(r Letting rI
=
- r r
- r ~ o.
in the scaled Koiter equations shows that
~c
Bu(r
r
_ Lu(r
Hence
o <_ B~(r = Bu(r
- r r r
_< L ~ ( r
- r 2B~(r
2B~(r
r
B~(r
r
r + L~(r
since B~M(r r -- L~M(r Noting that the weak convergence r162--~ r in V~M(w) implies that B~(r
r
B~(r
(:)and L~(r
L~M(r
we have thus shown that r and the proof is complete,
~ ~ in V~M(W), m
Justification of Koiter's equations
Sect. 7.2]
355
Remark. The kind of erratic behavior that may be expected when the applied forces cease to be admissible is displayed by an ingenious one-dimensional example, due to Sanchez-Hubert & SanchezPalencia [1993, p. 55] and Leguillon, Sanchez-Hubert & SanchezPalencia [1999]; cf. Ex. 7.4. II
Finally, we consider flezural shells. We follow here Ciarlet & Lods [1996c, Thm. 2.2]. T h e o r e m 7.2-3. Assume that 0 E C3(~; R3). Consider a ]amily of linearly elastic flexural shells, according to the definition given in Sect. 6.1, with thickness 2e approaching zero, with each having the same middle surface S = 0(-~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(7o ) as its middle curve, and let the assumptions on the data be as in Sect. 6.1. For each e > O, let (u~) e Hi(12 ~) and r
- (~i~,K) e Hi(co)
X
Hi(co)
X
H2(co)
respectively denote the solutions to the three-dimensional and twodimensional variational problems ~'(f~) and ~'~c(w). Let also
denote the solution to the two-dimensional scaled variational problem 7~F(w) (Tam. 6.3-1), solution which is thus independent of e. Then
1F
Ueaona'~ dx~ ~
~aa a in Hi(w) as s --4 0,
~ae, K aa ----4 i a a a in HI(w ) as e --~ 0, and
1F
in
0,
~ , K a3 --"+ ~3a3 in I'I2(w) as e --~ 0.
356
Koiter's equations and other linear shell theories
[Ch. 7
Proof. Under the assumptions that there exist constants A > 0 and # > 0 and functions fi E L2(f~) and h i e L2(r+ u r_) independent of e such that )~e=A
f~,~(~)_
and
#e=p,
~2f~(~) fo~ an ~ - . ~
h"~(~ ~) - ~h'(~) fo~ ~n ~ - ~
~ ~, e r~+ u r ~_
(these are the assumptions on the data for a family of linearly elastic flexural shells; cf. Sect. 6.1) and that e E C3(~; ~3), the convergences
1 F e u~g a'e dx i ~
2e
Caa a in Hi(w)
and u~g 3,e dx~ ~
2-~
~3 a3 ill H i ( w )
e
as e ~ 0 have been established in Thin. 6.4-1, as coroUaries to the fundamental convergence result of Thin. 6.2-1. The weak convergence ~ : ~ ~ in H 1(w) • H 1(w) • H 2 (w) was first established by Sanchez-Palencia [1989a, Thin. 2.1], as a consequence of general results in perturbation theory. We show that, in fact, the strong convergence ~ -+ ~ in VK(w) holds. In addition to the notation introduced in the proof of Thm. 7.2-1, let
Vv('.,') := {U e VK('.,.'); ")"~.~(V) = 0 i:r,.,.,,} C:: V/C(,,,),
:: { Z: ll,,otL, + I1,:,112, w } " Recall that, according to the inequality of Korn's type on a general surface (Thin. 2.6-4), there exists a constant c such that
ll,,iiv/,(..,) _< ~{ ~ t~o~(.,)i ~0,~ + ~ t.oo,(,,)lo..., }1/~ a,f3
a, f3
for all ~7 E V g ( w ) (by assumption, length ~/o > 0). By virtue of the assumptions on the applied forces, the solution ~ : also satisfies the scaled Koiter equations for a flexural shell (not to be confused with those for a membrane shell introduced in the proof of Thm. 7.2-1!), viz., 1
e-~BM(~eK, ~) + BF(~e_K, Vl) -- L(~) for an '7 e VK(,.,,).
Sect. 7.2]
Justification of Koiter's equations
357
Hence letting vI - ~ : in these scaled equations shows that (without loss of generality, we may assume e < 1)" 1
1
2
1
a,/3
a,/3
llLllllr Consequently, there exists a subsequence, still denoted (r for convenience, and there exists a function ~* E VK(w) such that ~ : ~ ~* in Vg(w) and ")'af3(~:) --+ 0 in L2(w). The weak convergence ~ : ~ ~* in Vg(w) implies the weak convergences ~,~f3(~e) ~ ~af3(~*) in L2(w); hence 7af3(~*) - 0 and thus ~* E VF(w). Fix ~ E VF(W) in the scaled Koiter equations and let e -+ 0; then the weak convergence ~ --~ ~* in Vg(w) yields BF(~*, r l ) = L(y). Hence
r162
where ~ E VF(w) is the unique solution to the scaled problem PF(W) (Thm. 6.3-1) and the weak convergence then holds for the whole family (~e)e >0. By the inequality of Korn's type on a general surface combined with the strong convergence ~af3(~:) ~ 0 in L2(w) and the relations ~af3(~) = 0, and by the positive definiteness of the scaled two-dimensional elasticity tensor of the shell, establishing the strong convergence ~ : --+ ~ in Vg(w) is equivalent to establishing the convergence BF(r
-
r
-
r
0.
Letting rl = ~ : in the scaled Koiter equations shows that
BF(r
r
< L((,~).
Hence
0 <_B F ( ~ -
~, ~ k - r = Bv(~k, ~)2BF(r ~)+ BF(~, r <_L(~K) -- 2BF(~K, ~) + L(~),
since BF(~, ~) -- L(~). Noting that the weak convergence ~ : --~ in VK(W) implies that
BF((~eK, ~) -+ BF(~, ~) and L ( ~ : ) -+ L(~),
358
Koiter's equations and other linear shell theories
we have thus shown that B F ( ~ r
- ~, ~
~ r in
[Ch. 7
- ~) --+ 0, hence that
VK(w).
This convergence clearly implies that ~c, aa a ~ ~aa a in Hi(w) and ~,3 a3 --~ ~3a3 in H2(og).
II Remarks. (1) Lods [1997] has identified a specific class of applied forces for which the error estimate [ [ ~ r - Clive(z) = o(~) upgrades the convergence ~ --+ ~ in VK(w) established in Thm. 7.2-3. (2) For shells subjected to a boundary condition of place along their entire lateral face, Lods & Mardare [1998b, 1999c] have obtained error estimates between the scaled three-dimensional solution and the three-dimensional extension across the thickness~ by means of the linearized Kirchhoff-Love assumption (Sect. 7.1), of the solution of Koiter's equations, m A major conclusion emerging from Thms. 7.2-1 to 7.2-3 is that we have justified the two-dimensional linear Koiter equations when they are applied either to an elliptic membrane shell, or to a generalized membrane shell of the first kind subjected to admissible applied forces, or to a flezural shell For Thms. 7.2-1 to 7.2-3 show that the averages across the thickness of the three-dimensional solution and the solution of Koiter's equations have the same principal part, viz., in each case the solution ~ to the corresponding two-dimensional scaled problem, as the thickness approaches zero. Remarks. (1) Generalized membrane sheUs of the second kind are not covered by the present analysis. This restriction is compensated by the apparent lack of examples of such shells. (2) Koiter's equations retain another property of the three-dimensional equations: Their solution converges to the solution of the two-dimensional equations of a linearly elastic plate when "the shell becomes a plate" (Ex. 3.5). (3) The asymptotic analysis of the eigenvalue problem for Koiter's equations as ~ approaches zero is extensively studied in SanchezHubert & Sanchez-Palencia [1997, Chap. 10]. See also SanchezPalencia [1992], Sanchez-Palencia & Vassiliev [1992]. Special men-
Sect. 7.2]
Justification of Koiter's equations
359
tion must be also made in this direction of the pioneering works of Goldenveizer, Lidski & Wovstik [1979] and Vassiliev [1987, 1990]. m By virtue of the de-scalings~ which are in each case of the form ~e = ~ (see Sects. 4.5, 5.7, and 6.3), our asymptotic analyses also show that the solution ~ of Koiter's equations is asymptotically as good as the solution ~e obtained by solving either the two-dimensional problem 7 ~ ( w ) , or the two-dimensional problem 7)~(w), or the twodimensional problem 7~(w) (see Thms. 4.5-2, 5.7-2, and 6.3-2), according to which category the shell falls into. Compared to these limit two-dimensional equations~ Koiter~s equations thus possess two outstanding advantages: Not only does using Koiter's equations avoid a "preliminary" knowledge of the category in which a given shell problem falls into, but it also avoids the mathematical or numerical difficulties inherent to each such category, briefly summarized below (see Chaps. 4 to 6 for more details). If the shell is an elliptic membrane one, no boundary condition can be imposed on the normal component ~ of the displacement field since ~ is "only" in L2(w)! If the shell is a generalized membrane one, the solution ~6 belongs to an "abstract" completion V~M(W); the boundary conditions on ~e may thus be quite "exotic"! If the shell is a flexural one, the unknown ~e is subjected to the constraints Va~(~ ~) = 0 in w, which certainly hinder its numerical approximation! A w o r d of c a u t i o n . These conclusions could not be reached by an asymptotic analysis of Koiter's equations alone, for they definitely require an asymptotic analysis of the three-dimensional equations, which was the object of Chaps. 4 to 6. Cela va sans dire certes, mais cela va encore mieux en le disant ! m These truly remarkable virtues of Koiter~s equations certainly explain their "improbable success" in the manifold numerical simulations where they are so often blithely used.
360 7.3.
Koiter's equations and other linear shell theories
[Ch. 7
KOITER'S EQUATIONS: ADDITIONAL COMMENTARY AND BIBLIOGRAPHICAL NOTES
As indicated in Sect. 7.1, W.T. Koiter based his derivation on two a priori assumptions, one of a mechanical nature stemming from delicate estimates of John [1965, 1971], and one of a geometrical nature, the Kirchhoff-Love assumption. The linearized version of this geometrical assumption (its explicit form is given in Ex. 7.1(2)) has been a posteriori justified for linearly elastic elliptic membrane shells by Lods & Mardare [1998b, 1999c]. Their justification relies on earlier estimates of Mardare [1998a, 1998b] and on an error estimate between the three-dimensional solution and the three-dimensional extension across the thickness, by means of the linearized KirchhoffLove assumption, of the displacement field obtained by solving Koiter's equations; this error estimate is valid for any shell subjected to a boundary condition of place along its entire lateral face. As shown in Sect. 7.2, the solution of Koiter's equations possesses the virtue of mimicking the behavior of the three-dimensional solution as the thickness approaches zero. This virtue certainly explains why they are so often deemed adequate for modeling and approximating a variety of situations where three-dimensional equations would be otherwise untractable. For instance, the modeling of linearly elastic shells with periodically varying thickness can be accomplished by an homogenization of Koiter's equations (cf. Telega & Lewi/~ski [1998a, 199Sb]; for illuminating introductions to homogenization theory in general, see the books by Sanchez-Hubert & Sanchez-Palencia [1992] and by Cioranescu & Donato [1999]); shells made of anisotropic and nonhomogeneous linearly elastic materials can be modeled by ad hoc extensions of Koiter's equations, where additional terms couple the linearized change of metric and change of curvature tensors (Caillerie & Sanchez-Palencia [1995a], Figueiredo & Leal [1998]); Koiter's equations are likewise well-suited for the mathematical modeling and numerical simulation of junctions between linearly elastic shells (see Bernadou & Cubier [1998a, 1998b]; an asymptotic analysis of the three-dimensional equations in the spirit of Ciarlet, Le Dret & Nzengwa [1989] and Le Dret [1989, 1990], or of Kozlov, Maz'ya & Movchan [1999], is yet to be carried out, however).
Sect. 7.3]
Koiter's equations: Commentary and bibliographical notes
361
The controllability of a shell modeled by the time-dependent Koiter equations, in the form justified by Xiao Li-ming [1999b], has been established by Miara & Valente [1999], who showed that the Hilbert uniqueness method ("HUM") of Lions [1988a, 1988b, 1988c] could be applied if the middle surface meets ad hoc geometrical conditions (satisfied in particular if the shell is shallow; cf. Sect. 7.6). In this direction, see also Geymonat, Loreti & Valente [1993], Lasiecka, Triggiani & Valente [1996]. Koiter's equations are also well adapted for shape optimization problems. Typical problems consist in determining the "best" geometry of the shell (i.e., the surface S) or the "best" variation of its thickness, so as to minimize the total weight under ad hoc constraints (e.g., unilateral boundary conditions of place, or bounds on the thickness). In this direction, see Chenais [1987, 1994] (who considered the closely related Budiansky-Sanders equations; cf. Sect. 7.5), Bernadou, Palma & aousselet [1991], Rao & Sokolowski [1996], Khludnev & Sokolowski [1997], Sokolowski [1997], Lewifiski & Sokolowski [1998]. Engineers and experts in computational mechanics often base their classification of shells on the relative orders of magnitudes of the "membrane" and "flezural" strain energies, viz.,
2 and
f~ aat3~r' e
~3 aa~r'eP~r( Ck )Pa~ (r k )V/-ddy,
found in Koiter's energy jeK (Sect. 7.1) evaluated at a given solution ~ c (see, e.g., Leino & Pitk~iranta [1994], Chapene & Bathe [1998a]), rather than on an asymptotic analysis of the three-dimensional solution as here. This approach, in which the applied forces may thus also dictate either a "membrane-dominated", or a "flezural-dominated", behavior, has been recently given a mathematical basis by Blouza, Brezzi & Lovadina [1999]. In the same spirit, Koiter's equations have been often applied to benchmark problems for testing the influence of the geometry of the middle surface and of the boundary conditions for determining the behavior of shells; see Kirmse [1993], Piila & Pitkiiranta [1993a, 1993b, 1995], Piila [1994, 1996], Pitkiiranta, Leino, Ovaskainen & Piila [1995]. Such benchmark problems are also of interest for iden-
362
Koiter's equations and other linear shell theories
[Ch. 7
tifying and approximating boundary layers in shells; see Hakula & Pitk~iranta [1995], Hakula [1997], Gerdes, Matache & Schwab [1998]. By contrast with "boundary" layers, "interior" layers, i.e., "away from the lateral face", may appear inside shells with a hyperbolic middle surface. This challenging phenomenon seems to be again well modeled by Koiter's equations, as suggested by Sanchez-Palencia & Sanchez-Hubert [1998]. See also Karamian [1998a], Leguillon, Sanchez-Hubert & Sanchez-Palencia [1999], Pitk~iranta, Matache & Schwab [1999]. Koiter's equations are often favored for the numerical simulation of shells with "small" thickness, Naghdi's equations (Sect. 7.4) being preferred for that of shells with "moderate" thickness. Accordingly, there exist myriad references on the numerical analysis of shell problems modeled by either kind of equations, the list given below constituting only a brief sample. In particular, the book by Bernadou [1994] is highly recommended, as its treatment follows essentially the same lines as this chapter. For the same reason, the "numerical" chapters of the books by Destuynder [1990] and Sanchez-Hubert & Sanchez-Palencia [1997] and the articles by Bernadou, Mato-Eiroa & Trouv4 [1994], Sanchez-Palencia [1995] and Chapelle & Bathe [1998a, 199Sb, 1999] are likewise highly recommended. Of particular interest is also the approach of Le Tallec & Mani [1998] (see also Carrive [1995], Carrive, Le Tallec & Mouro [1995], Kerdid & Mato-Eiroa [1998]), where the Cartesian components of the displacement, rather than its covariant components, are approximated; in this case, the solution to be approximated is instead that of Koiter's equations ]or shells whose middle surface has little regularity, due to Slouza & Le Dret [1999] (Thm. 7.1-2). Unless some care is exercised, the numerical approximation of shell problems is bound to be blurred by the locking phenomenon, first identified by Stolarski & Belytschko [1982], then defined by Babu~ka & Suri [1992] as the lack of uniformity of convergence with respect to e as the discretization parameter approaches zero. While this phenomenon is thus inherent to the approximation of any physical problem with a "small" parameter, an even more serious phenomenon, called membrane locking, is specific to flexural shells: It reflects the inability of most finite element spaces to approximate "sufficiently well" the space VF(w) of linearized inextensional displacements where the solution of the limit problem ultimately lies in this case (Thms. 6.4-1 and 7.2-3).
Sect. 7.4]
The two-dimensional Naghdi equations
363
In view of their acute importance, these phenomena have generated a substantial literature; see in particular Kamoukalos [1991], Akian & Sanchez-Palencia [1992], Habbal & Chenais [1992], Pitk~iranta [1992], Chenais & Zerner [1993], Leino & Pitk~iranta [1994], Zerner [1994], Bathe [1996], Chapelle [1996], Hakula, Leino & Pitk~iranta [1996], Ovaskainen & Pitk~iranta [1996], Chapelle [1997], Sanchez-Hubert & Sanchez-Palencia [1997, Chap. 11], Chapelle & Bathe [1998a], Choi, Palma, Sanchez-Palencia & Vilarifio [1998], Chapelle & Stenberg [1998]. Further references are provided at the end of the next section. 7.4.
THE TWO-DIMENSIONAL NAGHDI E Q U A T I O N S F O R A L I N E A R L Y E L A S T I C SHELL~ E X I S T E N C E A N D U N I Q U E N E S S OF S O L U T I O N S
Another two-dimensional linear shell theory, which appeals as much to the engineering community as Koiter's, is due to P.M. Naghdi. In this approach, which seems to be quite effective in numerical simulations of shells with "moderate" thicknesses, the shell is identified with a one-director Cosserat Bur]ace, i.e., a surface endowed with a director field (the field denoted raaSa infra). In this sense, it is the analog for shells of the Reissner-Mindlin theory for plates (Vol. II, Sect. 1.9). Conclusive attempts to establish the superiority of such theories over Koiter's theory for shells and Kirchhoff-Love's theory for plates are scarce. See, however, Lods & Mardare [1999a] for flexural shells and specific applied forces, Lods & Mardare [1999b] for shells subjected to a boundary condition of place along their entire lateral face, and Rhssle [1999a] for plates. Consider as in Sect. 7.1 a shell with middle surface S = O(~) and thickness 2s > 0, constituted by a homogeneous and isotropic linear elastic material with Lam6 constants )~e > 0 and #e > 0~ and subjected to applied body and surface forces with contravariant components fi, e c L 2 ( ~ 6) a n d h i,e E L 2( + te2 r r e_). In Naghdi's approach (Naghdi [1963, 1972]), the a priori assumption of a mechanical nature about the stresses inside the shell is the same as in Koiter's approach (Sect. 7.1), but the a priori assumption of a geometrical nature is different: The points situated on a line normal to S remain on a line and the lengths are unmodified along this line after the deformation has taken place, as in Koiter's
364
Koiter's equations and other linear shell theories
[Ch. 7
S
Fig. 7.4-1: A linearly elastic shell modeled by Naghdi's two-dimensional equations. The five unknowns are the three covariant components ~ : ~ --~ I~ of the displacement field of the middle surface S and the two covariant components r~ : ~ --+ R of the linearized rotation field of the unit normal vector along S; this means that ~(y)ai(y) + z~r~(y)aa(y) is the displacement of the point (O(y) + z~aS(y)).
approach; however, this line need no longer remain normal to the
deformed middle surface. In the linearized version of this approach, there are five u n k n o w n s , the three covariant c o m p o n e n t s ~ 9 ~ --~ I~ of the displacement field ~ a i of the middle surface S a n d the two covariant c o m p o n e n t s r a 9w --~ I~ of the linearized rotation field r ae a a of the unit normal vector along S (see Ex. 7'.6 for the reason this field should be tangential in a linearized theory). This means t h a t the displacement of the point (O(y) + x~a3(y)) is the vector ( ~ ( y ) a i ( y ) + x~rea(y)aa(y)); cf. Fig. 7.4-1. In their weak formulation, N a g h d i ' s
equations
for a l i n e a r l y
Sect. 7.4]
365
The two-dimensional Naghdi equations
e l a s t i c shell consist in solving the following variational problem Find (~e, v~) - ((i~), (r~,)) such that (the notation I-II(F~) stands for the space (Hi(w)) 5 in the definition of the space VN(w))"
(r ~') e v~(~):= {(n, ,)= ((~), (~)) e I~(~); ~7i = sa --- 0 on 7o},
~~ {a-z~,~ 7~r(~'~)Taf~(v/) +
c/~eaaf~Taa(~e , re )Tf~a(v/, s ) } v / - a d y
/ for all (17, s) ~ VN(w), where (the functions a af3, ba/3, b~, Fa~ , and a are defined as usual; see, e.g., Sect. 3.1): a af~aT' ~ .--'-- 4Ae#e a a~a ar + 21~ ( aa~a ~ + a araf3cr), X e + 2~ e 1
7 ~ ( n ) :: ~ ( 0 ~
+ 0~)
1 7~3(n, 8) := ~ ( o ~ 3 + b ~ N(n,S)._
- r~~ +
- b~3,
~),
1 1 ,,
pi, e ._
1
fi,6 dx i + h;e + h,,e and h~ e " - h ''e(., +e), E
and c is a strictly positive constant (c - 8 is commonly assumed; however, the "best" choice for this constant seems to be an unresolved issue). The functions a a[3~r'e a r e the contravariant components of the t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of t h e shell and the functions ")'a/3(r/) are the covariant components of the lineariT.ed c h a n g e of m e t r i c t e n s o r associated with a displacement field 7/ia' of the middle .
366
Koiter's equations and other linear shell theories
[Ch. 7
surface S, as before. The new functions 7a3(r/, s) and N are the covariant components of the linearized t r a n s v e r s e s h e a r s t r a i n t e n s o r and N a g h d i ' s linearized c h a n g e of c u r v a t u r e t e n s o r associated with displacement and linearized rotation fields yia i and 8 a a a of S; for 3. justification of these definitions, see, e.g., Bernadou [1994, Part I, Chap. 3]. The next e~istence and uniqueness result for the solution to the variational problem ~ v (w) is due to Bernadou, Ciarlet & Miara [1994, Thm. 3.1]. As its proof is akin to that of Thm. 7.1-1, it is left as a problem (Ex. 7.7). T h e o r e m 7.4-1. Let w be a domain in I~2, let 70 be a subset of Ow with length 70 > 0, and let 0 E e3(~; R 3) be an injective mapping such that the two vectors aa = OaO are linearly independent at all points of -~.
Then problem 79~r
has one and only one solution.
II
The variational problem 7~v(w) is, at least formally, equivalent to a boundary value problem, which is again a "uniformly" and "strongly elliptic" system in the sense of Agmon, Douglis & Nirenberg [1964]. Accordingly, the regularity of its solution can be established as the regularity of the solution of Koiter's equations (Ex. 7.3); see Iosifescu [1999]. In the same manner that Blouza & Le Dret [1994a, 1994b, 1999] have generalized Thm. 7.1-1 to "Koiter's equations for shells whose middle surface has little regularity" (Thm. 7.1-2), Slouza [1997] has extended Thm. 7.4-1 to Naghdi's equations ]or shells whose middle surface has little regularity (the mapping 0 need only be in the space
R3)).
Nonhomogeneous and anisotropic linearly elastic materials are likewise amenable to Naghdi's approach. The existence and uniqueness of the corresponding two-dimensional Naghdi's equations can be treated as in Thin. 7.3-1; see Figueiredo & Leal [1998]. The corresponding infinitesimal rigid displacement lemma is due to Coutris [1978].
Naghdi's equations are as much favored as Koiter's for analyzing the locking phenomenon and membrane locking, briefly described
Sect. 7.5]
Other linear shell theories
367
in Sect. 7.3; see Hughes & Franca [1988], Bathe, Brezzi & Fortin [1989], Pitk~iranta [1992], Lyly, Stenberg & Vihinen [1993], Arnold & Brezzi [1997a, 1997b], Bramble & Sun [1997], Suri [1997], Gerdes, Matache & Schwab [1998], Chapelle & Bathe [1998a], Chapelle & Stenberg [1998]. 7.5.
OTI-IER LINEAR
SHELL THEORIES
Linear "shallow" shell theories are treated separately (Sect. 7.6). B u d i a n s k y - S a n d e r s theory. Sanders [1959] and Koiter [1960] have proposed a linear shell theory akin to Koiter's, where the covariant components Pat3(r/) of the linearized change of curvature tensor BS (r/) of the B u d i a n s k y are replaced by the covariant components Pat3 S a n d e r s linearized change of c u r v a t u r e tensor, defined by BS
1
The remaining terms in the equations are otherwise identical to those in Koiter's equations (Sect. 7.1). BS(r/) , rather than The interest of using the modified functions Pa[3 the "genuine" functions pail(r/), has been discussed at length in Budiansky & Sanders [1967] and, for this reason, the resulting theory has become known as the Budiansky-Sanders theory (see also the commentary in Koiter [1970, Sect. 7.5]). More recently, Destuynder [1985] has clearly shown how this theory can be derived from three-dimensional linearized elasticity, again on the basis of two a priori assumptions, both of a geometrical nature (one of them being the linearized Kirchoff-Love assumption; cf. Sect. 7.1), thus providing a mathematical justification of this theory. Remarks. (1) A nonlinear Budiansky-Sanders theory, proposed by Destuynder [1982, 1985], is briefly presented in Sect. 11.2. Bs (r/) and the equivalence (2) The definition of the functions Pail between the equations
(,1)
Bs
= Pa~ (r/) - 0 in
and "fail(r/) = Pat3(r/) = 0 in w"
together imply that the ezistence and uniqueness theory for Koiter's equations (Sects. 2.6 and 7.1) extends almost verbatim to the Budiansky-Sanders equations. II
368
Koiter's equations and other linear shell
theories
[Ch. 7
H i e r a r c h i c s h e l l t h e o r i e s . The underlying principle in a hierarchic shell theory consists in minimizing the three-dimensional energy of a linearly elastic shell, expressed in terms of its "natural" curvilinear coordinates (yl, y2, x~), over a subspace of displacement fields of a special form: Each one of their covariant components is a finite sum of products of functions of (Yl, y2) (to be determined through the minimization process) times linearly independent functions of x~ (typically, ad hoc orthogonal polynomials). Increasing the number of linearly independent functions of the transverse variable produces a "hierarchy" of presumably increasingly accurate approximations. Due credit should be given in this respect to Vekua [1986, Chap. 1], for one of the earliest attempts to put such an approach for shells on a sound mathematical basis. Hierarchic theories are especially advocated as efficient numerical methods: In particular, they have proved to be quite successful for the numerical approximation of plate problems; see Szabo & Sahrmann [1988], Babu~ka & Li [1991], Babu~ka, Szabo & Actis [1992], Sabu~ka, d'Harcourt & Schwab [1993], Schwab [1994, 1996], Schwab & Wright [1995], Babu~ka & Schwab [1996]. They will certainly be increasingly used for the numerical approximation of shell problems~ notably for their abilities to handle multi-layered shells, also called composite or laminated shells, boundary layers, or the membrane locking phenomenon (Sect. 7.3); see notably Oden & Cho [1996], Actis, Szabo & Schwab [1999]. L i n e a r shell t h e o r i e s b a s e d o n t h e m e t h o d o f i n t e r n a l c o n s t r a i n t s . The linearized Kirchhoff-Love assumption (Sect. 7.1; see
also Ex. 7.1(2)) can be considered as an internal constraint imposed ab initio on the displacements that a shell can undergo. In this approach, first advocated by Podio-Guidugli [1989] for linearly elastic plates, then by Podio-Guidugli [1990] for linearly elastic shells~ special care must be exercised for the determination of the resulting two-dimensional constitutive equation, through a careful distinction between "reactive" and "active" stresses. " I n t r i n s i c ~ l i n e a r s h e l l t h e o r y . An "intrinsic" approach to linear shell theory has been proposed by Delfour & Zoldsio [19951 for modeling shells whose middle surface is the entire boundary r of a domain ~ C I~3. This approach makes an essential use of the "tangential differential calculus" developed in the book of Sokolowski
Linear shallow shell theories
Sect. 7.6]
369
& Zol6sio [1992] and of the "oriented distance function" A" ~3 -4 I~, defined by A ( x ) - d(x, r ) i f x e {I~a - f~} and A(x) = - d ( x , F) if x E 12, also used in Delfour & Zol6sio [1994, 1998] in a different context. After integrating the three-dimensional energy across the thickness and making ad hoc simplifying assumptions, M.C. Delfour and J.P. Zol6sio derive linear shell theories that bear strong resemblance with those of W.T. Koiter and P.M. Naghdi. An interesting feature of this approach is that local bases are not used. This approach has since then undergone manifold extensions, which often parallel the asymptotic analyses of the previous chapters. See notably Delfour & Zol6sio [1997a, 1997b], Delfour [1998,
1999]. 7.6.
LINEAR
SHALLOW
SHELL THEORIES
Linear shallow shell theories in Cartesian coordinates are treated at length in Vol. II, Chap. 3. Accordingly~ the additional commentary and bibliographical notes found in this section apply mostly to linear shallow shell theories expressed in curvilinear coordinates. According to the definition proposed by Ciarlet & Paumier [1986] in the nonlinear case, then justified by a convergence theorem as the thickness approaches zero by Ciarlet & Miara [1992a] in the linear case, a shell is s h a l l o w if the deviation of its middle surface S ~ from a plane is of the order of the thickness, i.e., if
S ~ -- Os (~), where
and 0 : ~ -4 R is a smooth enough function that is independent of e. A w o r d of c a u t i o n . This specific "variation of the middle surface with e" thus constitutes an additional assumption on the data, special to linear and nonlinear shallow shell theory, m
Koiter's equations and other linear shell theories
370
[Ch. 7
As shown by Busse, Ciarlet & Miara [1997], who use the same definition of "shallowness", the two-dimensional equations of a linearly elastic clamped shallow shell "in curvilinear coordinates" can be given a rigorous justification by means of a convergence theorem as the thickness goes to zero. As the proof essentially resembles that given in Cartesian coordinates by Ciarlet & Miara [1992a] (see also Vol. II, Sect. 3.5), it is not reproduced in this volume. We simply list the limit equations that are found in this fashion, when they are expressed as a minimization problem: Let baf3~r,e .-'- 4~e# e ~ a ~ r $e + 2#e p,,e :=
f
:=
sh,
+ 2tze(~a~Or + ~ar~f3~),
fi, e dx~ + h~_~ + h i, ~
"
e
4I
+
--
-
1
and let a i' e designate the vectors of the contravariant bases along the middle surface S e. Then the unknown is the vector field ~e = ( ~ ) , where the functions ~ : ~ --+ R are the covariant components of the displacement field ~ a i'e of S e, and (~e minimizes the energy jsh, e defined by
jsh ' e(rl) . - 2
/ eba~er, e_.h,e ecrr
. .h,e ( y ) - ] - -~- ba/3~r,e 0qry30afj~73 (y)ea/51
dy
-{fwPi'e?~idy-fwqa'e~adYl, over the space (the same as for Koiter's equations; cf. Sect. 7.1): V K ( 0 g ) :-- {O -- (~7i) 6 H l ( w ) x H l ( w ) x H 2 ( w ) ;
~7i -- 0v~3 -- 0 Oil 70}"
An inspection of this minimization problem reveals that, even though it is expressed in curvilinear coordinates, it "resembles more that of plate (Vol. II, Chap. 1) than that of shell"! For the contravariailt components of the metric tensor usually found in the twodimensional elasticity tensor are now replaced by Kronecker deltas, the area element along the middle surface is replaced by dy, and finally, the components of the linearized change of metric and change
Sect. 7.6]
371
Linear shallow shell theories $h~ s
of curvature tensors are replaced by the functions ear 3 (v/) and 0afar/3 where neither the Christoffel symbols nor any components of the curvature tensor of S e are to be found. The equations found in this fashion constitute Novozhilov's model of a shallow shell, so named after Novozhilov [1959]. These equations, which were first analyzed by Shoiket [1974], were given a first justification by Destuynder [1980] for special geometries. As shown by Andreoiu [1999a], it is a reassuring circumstance that the limit displacement fields found in either Cartesian or curvilinear coordinates, though not identical vector fields, are nevertheless "essentially the same", i.e., their components agree "to within their first orders", once they are expressed in a same basis. The asymptotic analysis of the corresponding eigenvalue problem has been carried out in Cartesian coordinates by Kesavan & Sabu [1999b]; there is no doubt that it could be similarly carried out in curvilinear coordinates. The exponential nature of the boundary layers that arise in linearly elastic shallow shells is analyzed in Pitk/iranta, Matache & Schwab [1999]. Models of multi-layered, or composite~ linearly elastic shallow shells, found in particular in hulls of sailboats, have been obtained by Kail [1994] by means of the method of formal asymptotic expansions. The control of vibrations in linearly elastic shallow shells by means of piezo-ceramic actuators is studied in detail in Banks, Smith & Wang [1995, 1996]. Other definitions of "shallowness" have been proposed, which often make explicit reference to the curvature of the middle surface. For instance, Destuynder [1985, Sect. 1] considers that a shell is "shallow" if 7/= ep for some p >_ 2, where the other "small" parameter is the ratio of the thickness 2e to the smallest absolute value of the radii of curvature along the middle surface, p = 2 corresponding to Novozhilov's model. In this direction, see also Vekua [1965], Green & Zerna [1968, p. 400], Gordeziani [1974], Dikmen [1982, p. 158].
372
[Ch. 7
Koiter's equations and other linear shell theories
EXERCISES
7.1. All the notations used in this problem should be self-explanatory. The Kirchhoff-Love assumption for a shell states that all points lying initially on a given normal to the middle surface remain on a single normal to the deformed middle surface and that the distance from any point of the shell to the middle surface remains unchanged. Consider a shell with a reference configuration O ( ~ e) equipped with its "natural" cuxvilinear coordinates (yl, y2, x~) C ~e and let g~,e denote the vectors of the corresponding contravariant bases in O(~e), where fF = w• - e, e[. Thus a displacement field vige i,~ of the set O ( ~ e) satisfies the Kirchhoff-Love assumption if and only if
where ~Tia~9 denotes the restriction to x~ = 0 of the field Ve-.i i Y' e and a3(r/) is the "usual" vector normal to the deformed middle surface
(0 + (1) Show that a displacement field v~g i,e of (9(fi e) satisfies the Kirehhoff-Love assumption if and only if the covariant components gi~(v e) of the metric tensor of the associated deformed configuration, assumed to be equipped with the same eurvilinear coordinates as those of | are of the form
g~/3(v e) -- aa/3(rl) - 2xlba/3(rl) + (xl)2b~(rl)bf3o.(17) and giea(ve)
--
~i3,
where aa~(r/), bad(r/), and b~,(r/) denote the covariant and mixed components of the metric and curvature tensors of the deformed middle surface (0 + ~Tiai)(~). (2) k displacement field ui "e..~,, of @( ~ ) satisfies the linearized Y Kirchhoff-Love assumption if only the linear terms with respect to r/ are retained in the difference { a z ( r / ) - a3}. Show that such a displacement is of the form (see, e.g., Bernadou & Boisserie [1982, eq. (1.3.19)]): veni, e --
a i -
i(O
v3 + b13a 71 )a a
7.2. Assume that the boundary 7 of w and the functions pi, e are smooth enough and that length 7o < length 7. Show that, if the solution ~ c - ( ~ , i ) to the variational problem 79~(w) (Thin. 7.1-1)
373
Ezercises
is smooth enough, it is also a solution of the following boundary value problem:
mafi}, e laj3 +
_
_
b~bat3ma[3,e _ ba~na[3, e = pa, e in co, -
iie, K
-- Ou~3,e K
-
i.
-- 0 on
70,
ma~'~uau~ --0 on 71, (m~Z'~l~)~,z + O ~ ( m ~ Z ' ~ r Z ) -- 0 on 7~,
(n az'e + 2ba~m~Z'6)vZ - 0 on ")'1, where 71 := 7 - 3'0, (ua) is the unit outer normal vector along 7, T1 := --u2, T2 := Ul, and 0~0 := ra0a0 denotes the tangential derivative of 0 in the direction of the vector (Ta). Remark. It is instructive to compare these equations with those of a linearly elastic plate (Vol. II, Thms. 1.5-1, 1.5-2, and 1.7-2). 7.3. The following problem shows, after Alexandrescu [1994], how to establish the regularity (announced in Thm. 7.1-3 (b)) of the solution of the two-dimensional Koiter equations for a linearly elastic shell. It is assumed throughout this problem that 70 = 3', and in (1), (2), and (3) that the boundary 7 is of class C4 and the mapping 0 is in the space C4(~; R 3). (1) Show that the linear system of partial differential equations found in Thm. 7.1-3, which is of the second order with respect to the unknowns ~ : , a and of the fourth order with respect to the unknown ~c, 3, is "uniformly elliptic" and satisfies the "supplementing condition on L " and the "complementing boundary condition" in the sense of Agmon, Douglis & Nirenberg [1964]. Remark. These properties are shared by the systems of partial differential equations found in Thm. 2.7-2 and in Ex. 4.2. (2) Show that the same system is "strongly elliptic" in the sense of Ne~as [1967, p. 185]. Using Ne~as [1967, Lemma 3.2, p. 260], infer from this property that the solution r - (i~,i) to the variational problem T'~c(w ) (Tam. 7.1-1), which belongs to the space Hlo(W) • H~(w) • H2o(W) since ")'0 = ~' by assumption, satisfies
374
Koiter's equations and other linear shell theories
[Ch. 7
if pa, e E H i ( w ) and pa, e C L2(w). (3) Using Geymonat [1965, Thm. 3.5], shows that
~K
--(~K,i)
e w3'q(o3) X w3'q(og) x w4'q(o3)
ifpa'e E w l ' q ( w ) and p3,e C Lq(w) for some q > 1. (4) Assume that, for some integer m >__ 1 and some real number q > 1, 7 is of class Cm+4, 8 E Cm+4(~; I~3), pa, e E Wm+l'q(w), and pa, e E wm'q(w). Show that
~eK "-(~K,i) e Wm+3'q(~g) x Wm+3'q(og) x wmT4'q(og). Hint: For (4), use Agmon, Douglis & Nirenberg [1964].
7.4. As shown by Sanchez-Hubert & Sanchez-Palencia [1993, p. 55] (see also Leguillon, Sanchez-Hubert & Sanchez-Palencia [1999]), the following simple, yet illuminating, two-point boundary value problem can be viewed as a convenient one-dimensional model of Koiter's equations, as regards in particular the behavior of their solution as approaches zero in the situations covered by Thms. 7.2-1 and 7.2-2. The space H - m ( 0 , 1) denotes the dual space of H~n(0, 1). Given f E H-2(0, 1), let u e E/-/2(0, 1) denote for each e > 0 the solution to the boundary value problem: ~2
d4u ~
d2u e
dx 4
dx 2
= f in ]0, 1[,
ue(0) - (ue)'(0) - ue(1) - (ue)'(1) - 0. (1) Assume that )e E L2(0, 1). Show that u ~ -+ u ~ in H~(0, 1) as e -+ 0, where u ~ E H~(0, 1) is the solution to the boundary value problem d2u o
dx2 = I in ]0, 1[, -0.
(2) Compute explicitly and plot the solutions u ~ corresponding to f = 1. Show that boundary layers appear in the derivative (ue) ~ as approaches zero. Remark. Such boundary layers reflect that the boundary conditions (ue)~(O)= (ue)~(1)= 0 are "lost" at the limit.
Ezercises
3'[5
(3) Let f r where r denotes the derivative of the Dirac distribution at x - ~. 1 Verify first that it e H -2(0 , 1) , but that f ~ H - 1 (0, 1); then show that -
-
u~(x)-+x
for each
0_<x<~
ue(x)-+l-x
for each
1 ~ <x
1
asr ase--+0,
so that an "interior" layer around x -- 89appears in the functions u e as e approaches v.ero. Remark. The choice it _ ~/2 provides an example of a right-hand side that is not "admissible", in the sense that it is in the dual of the space H~(w) (to which u e belongs for each e > 0), but not in the dual of the "limit" space H~(w) (to which the limit u ~ found in (1) belongs). This one-dimensional model of Koiter's equations thus suggests why a boundary layer can even appear "away from the boundary of the middle surface" in a generalized membrane shell subjected to seemingly innocuous, but not admissible, applied forces. 7.5. Let O - (~7/) C C3(~; R a) be an injective mapping such that the two vectors 0aB are linearly independent at all points of ~. Following Ciarlet [1992b], consider a one-parameter family of mappings O(t) -- (Oi(t)) 9 -+ I~~ defined for 0 < t <_ 1 and (yl, y2) E ~ by
03(t)(yl, y 2 ) - tOa(yl, y2). Assuming that the functions pi, e E L2(w) (which take into account the applied forces) are independent of t, let r e VK(W) denote for 0 < t _ 1 the solutions of the corresponding two-dimensional Koiter equations (Sect. 7.1), now parametrized by t. Show that the mapping t e [0, 1] ~ r e VK(W)is continuous. In particular, r162 -4 r in Hi(w) • Hi(w) • H2(w) when t ~ 0 +, i.e., whe
"tn
hal
pl t ", where
-
e Vzc(
)
satisfies the two-dimensional equations of a linearly elastic clamped plate (Vol. II, Sect. 1.7)"
r
O,
Oa rla } dy
"
dy
376
Koiter's equations and other linear shell theories
[Ch. 7
for all vl = (r/i) e V g ( w ) , where baf~#r, e =
4Ae/ze (Wf3~#r + 2#e((W~ f3r + ~ar(ff3~). Ae + 2/ze
Remark. This property of the solution of Koiter's equations is akin to a property of the three-dimensional solution (Ex. 3.5). 7.6. Show that the rotation vector of the unit normal vector associated with a displacement field of the middle surface is a tangential one "to within the first order". This explains why the "linearized" rotation vector found in Naghdi's equations (Sect. 7.4) has a zero component over the vector a s . Remark. Part of the problem consists in specifying what "first order" precisely means. 7.7. This problem establishes the existence and uniqueness of a solution to Naghdi's equations for a linearly elastic shell, after Bernadou, Ciarlet & Miara [1994, Thin. 3.1]; the notations are those of Sect. 7.4 and the assumptions are those of T h m . 7.4-1. Let 2
2
a~f~
a
a,/3
II(n..)11 :-
{~
, + ~ I.~1o2.~+ I(n. s)l 2 } 1/2 I~[o.~ i
I1(., ,)11
a
:: {I).11 o + I1,11
where v / = (Yi) and s = (sa). (1) Show that there exists a constant co such that
11(,7, ~)ll~,~ _< ~o11(,1, ~)11 fo~ aU ('7, ~) ~ ~(~)-
(H~(~)) ~
Hint" Use the Zemma of J.L. Lions as in Thin. 2.6-1.
(2) T,~t (,7, ~) e ~(~)
be ~uch that
= pa/3(~l, s) -- 0 in w.
= ~
Show t h a t the vector field ~?ia' is an infinitesimal rigid displacement
Ezercises
377
of the surface S - 0(~), in the sense that there exist two vectors ~, d E IR3 such that
N ~)Hint: Show that TI3 E H2(w), then that pa~(~l,
PaB ( . ) fo~
such fields (~/, s). The conclusion then follows from Thm. 2.6-3. (3) Let (~/, s) e I-II(w) be such that --
,
pa~(17,
s)
~?i - - s a
-- 0 in w, -- 0 on 70,
where 70 C ?, satisfies length 3'0 > 0. Show that (~/, s) - (0, 0). Hint: Use Thm. 2.6-3 again. (4) Show that there exists a constant c such that the following inequality of Korn's type on a general surface holds (compare with the inequality of Thm. 2.6-4):
I1(., 8)11.,,,,., ~ ~11(~, 8)11 for all (r/, s) e VN(w), where VN(~)
"-- { ( . ,
~) -- ( ( ~ ) ( ~ ) )
e ~I~(~); ~ -- ~
-- 0 0 ~ ~0}.
Hint: Argue by contradiction as in Thm. 2.6-4. (5) Let eBN 9VN(w) • VN(w) --~ I~ denote the bilinear form defined by the left-hand side of the variational equations in problem T'~v(w) (Sect. 7.4). Show that there exists a constant CN such that 11(17, s)[[ 2 _< cNBN((W, s), (17, s)) for all (r/, s) E V/v(w) and conclude that 7~V(W) has one and only one solution. Remark. Together, the results of (2) and (3) constitute an infinitesimal rigid displacement lemma on a general surface (compare with that of Tam. 2.6-3), which is due to Coutris [1978] (see also Coutris [1973, 1976]).
This Page Intentionally Left Blank
PART B
NONLINEAR
SHELL
THEORY
This Page Intentionally Left Blank
CHAPTER 8 ASYMPTOTIC ANALYSIS OF NONLINEARLY ELASTIC SHELLS: PRELIMINARIES
INTRODUCTION
In the second part of this volume 1, we give a detailed account of recent justifications of nonlinear shell theories that are also based on an asymptotic analysis of the three-dimensional solution with the thickness as the "small" parameter. A remarkable progress in the asymptotic analysis of nonlinearly elastic shells is due to B. Miara, then to B. Miara and V. Lods, who justified the two-dimensional equations of a nonlinearly elastic "membrane" shell and those of a nonlinearly elastic "flexural " shell, by means of the method of formal asymptotic expansions applied to the three-dimensional equations of a nonlinearly elastic shell modeled by a St Venant-Kirchhoff material. Another remarkable progress is due to H. Le Dret and A. Raoult, who gave the first proof of convergence of the three-dimensional solutions to a "two-dimensional" one as the thickness approaches zero. Their approach, which also leads to a nonlinear "membrane" shell theory, is described in the next chapter (Sect. 9.5). The purpose of this chapter is to lay the preliminary grounds for the formal approach. As a preparation, we examine how the equations of three-dimensional nonlinear elasticity expressed in terms of Cartesian coordinates in a reference configuration { ~ } - (these equations are reviewed in Sect. 8.1) are transformed when they are expressed in terms of curvilinear coordinates, i.e., in terms of coordinates in a set of the form _ | ( { ~ } - ) , where ~2 is a three-dimensional domain in IR~ and O is an injective mapping with other ad hoc properties. The vectors of the covariant and contravariant bases and the covariant and 1To a large extent, this second part can be read independently of the first one.
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
382
[Ch. 8
contravariant components of the metric tensor are then defined by the relations gi--OiE},
gi.gj_~,
gij--gi'gj,
gij_gi.gj.
More specifically, we show, by means of a direct computation (Thm. 8.2-3), that the variational, or weak, formulation of these equations takes the form: -
{.
(=~) e w ( n ) : =
-
(~,) e w ~ , ' ( ~ ) ;
. - o o= r0),
fa AiJkIEkllt(u)Fillj(u, v)v/gdz - fn fiviv/gdx -4- fr hiviv/g dF 1
for all v = (v~) E W ( f l ) , where uig i is the unknown displacement vector field inside the set {(~}-, r0 is a subset of r = 0 ~ with
area ro > O, AiJkl= )tgqgkl + #(gik gjt + git gik), and # are the Lamfi constants of the constituting St Venant-Kirchhoff material, r l - r - r0, the functions fi and h i are the contravariant components of the applied body and surface force densities, g - det(gij), the strains in curvilinear coordinates E/lij(v ) E L2(~) are defined for each v - (vi) e W1'4(12) by 1
E~llr ) - ~(v~llj + vjll~ + where Villi -
P p and r iPj Ojvi - rijv 1
g'~"v,,,il~v,,llj),
: g P " O i g j , and, finally, gmn
= Elllj(~)~.
This calculation introduces various fundamental notions, some familiar, such as the covariant derivatives villi of a vector field rig / (Chap. 1), and some new, such as the change of metric tensor associated with this vector field, found here by means of its covariant components Eilij ( v ). We then specialize the reference configuration in the usual way: Given a domain w in ~2 with boundary 7 and given a smooth enough injective mapping 8 9~ -+ R 3 such that the two vectors a a -" O a O
Introduction
383
are linearly independent at all points in ~, we consider a shell with middle surface S = 0(~) and thickness 2s > 0, i.e., a b o d y whose reference configuration is the set O ( ~ ) , where f~e _ w • s, e[ and
|
- a(y) + ~ i ~ 3 ( y ) fo~ an ~ - (y, ~ ) e
~.
The shell is subjected to applied body forces with contravariant components fz, e . f2e _+ I~ and is subjected to a boundary condition of place along a portion O(V0 • I - s , s]) of its lateral face O(V • I - s , s]), where V0 C V and length "Yo > O. Applied surface forces acting on the upper and lower faces O(w • ( s ) ) and O(w • { - s ) ) may be also considered; for simplicity, we assume in this introduction t h a t they vanish. Let )~e and p~ denote the Lam~ constants of the St VenantKirchhoff material constituting the shell. It thus follows t h a t the unknown u ~ = (u~), where u i -~ I~ are the covariant components of the displacement field of the points of the shell, satisfies the following three-dimensional equations in terms of the curvilinear coordinates x~ of the reference configuration O ( ~ e) (Sect. 8.3): ~t e e W ( ~
e) -- {V e -- (V~) e w l ' 4 ( ~ e ) ;
~)e __ 0 O12
r~),
A'J~' ~ill~(~)F,~lj (~ ~, .~)~/~ d~ ~
/o
v i V/~ dx ~ for all v e w ( ~ ) ,
e
where Aijkl, e _ )~egij, egkl, e +/~e (gik, egjl, e + gil, e gjk, e) ,
IJ
) -
~(~llJ + ~Jll, + 1
~
~lj( ~ , ~ ) - i(',~l~ + ~Jll, +
~ll,~-IIJ) , gmn, e
e
e
{~ll, V.llJ +
~ e UnlliVmlli )) 9
In Sect. 8.4, we transform this problem into art equivalent problem, but now posed over the set f~ = w • 1, 1[, which is i n d e p e n d e n t of ~. Mo~e ~pecific~ny, we d e , h e the ~ l ~ d ~ k ~ o ~ ~ ( ~ ) = (~(~)) by letting u~(x e) - u i ( s ) ( x ) for all x ~ - 7r~x E ~e, where 7re(x1, x2, x3) - (xl, x2, ex3), we assume t h a t there exist constants A > 0 a n d / z > 0 such that )~e = )~ and #e = #,
384
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
and finally, we define functions fi(e) 912 -+ I~ by letting
fi'e(xe)-- fi(~)(x) for
aU x e
- 7 r e x e I2e.
Note that, by contrast with the linear case, it can no longer be a priori assumed that the scaled displacement is "of order 0 with respect to ~". The main objective of this preliminary chapter precisely consists in proving that this is indeed the case! It is found in this fashion that the scaled unknown u(e) satisfies a variational problem of the form (Thm. 8.4-1):
~(~)
e w(~)
- { . - (~,) e w ~ , 4 ( ~ ) ;
. - o on r 0 } ,
faAiJkt(6)Ekll,(s; u(e))FillJ(S; u(e), v)V~(e ) dx = fa fi(s)viv/g(e)dx for all v e W(f~), where the scaled strains EillJ(e; u(e)) e L2(f~) in curvilinear coordinates are defined by EilIj(~; u(~))(x) - Ei~llj(u~)(x~ ) for all x ~ - 7r~x C ~ a~d, fo~ any . e W ~ ' 4 ( ~ ) , the functions F~llj(~; defined by F~llj(~;
~(~), .)(~)
- F~lj(~,
~(~), -)
,~)(~) fo~ an ~
e i2(~)
- ~r~ e
a~e
~.
Once this variational problem is written in extenso (Thin. 8.4-1), its specific form suggests that we use the method of formal asymptotic expansions: First, we replace the scaled unknown u(e) by a formal expansion:
~(~)-
1
~~-
N
1
+ ~N-~ ~
-N+I
+""
in the variational equations; then we equate to zero the factors of the successive powers of ~ found in the resulting equations until the order - N of the leading term is determined and (once this order is known) the leading term can be fully identified as the solution of an ad hoc variational problem, the "right" orders (in terms of e) to be assumed
on the applied forces being simultaneously determined. The "unreasonable efficiency" of this method (described in detail in Sect. 8.6) rests in particular on two essential guiding principles: No
385
Introduction
restriction should be put on the applied forces and the linearization of any nonlinear equation found in this process should produce an equation of the linear theory described in the first part of this volume. After recapitulating in Sect. 8.5 various "geometrical" and "mechanical" preliminaries (in fact, the same as in the linear case), we show that these two guiding principles already yield three preliminary
conclusions of paramount importance: First, N = 0, i.e., the formal expansion of the scaled unknown is of the form (Thm. 8.7-1): u(e)
-
u ~ + eu 1 +....
Second, the applied body forces are "at least of order 0"; this means that there exist functions fi, o E L2(f~) independent of ~ such that (Thm. 8.8-1): fi'e(x~) - fi'~
for all x e - ~'~m e ~e.
Third, the leading term u ~ 9~ ~ 1~3 is independent of xa and ~o _ (~o) _ 89 f ~l u~ dx3 satisfies the following two-dimensional variational problem (Thm. 8.8-1) ~0 E W ( w ) " - - {r/E Wl'4(w); r/-- 0 on 70},
f..a~crr ~llrFall~(n)v/-ddy ~.0 0 9 - f~ p',~ for all r / = (r/i) E W(w), where Eo
1
o
_mn
+ a
1
-- ~(?'/all/3 -q- ?]fllla -qT]all/~
,,O
amn
o
4As
+
~ a m . a n
o
{~mllannl113
- - ~1~7a -- r~ral~r]a" -- ba1~73, _
a mn
,,.0
~mll~%ll~)' + ffnll~nmlla}),
n311/51 :--
oq/51r]3-~- b;~cr,
+
A+2# pZ, 0 _
fi, 0 dxs.
1 As we shall see, there are then two essentially distinct possibilities: Either this problem constitutes the final step of the application of
386
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
the method of formal asymptotic expansions; this case corresponds to the nonlinearly elastic "membrane" shells studied in Chap. 9. Or the induction must be continued (in which case the functions fi,0 vanish); this case corresponds to the nonlinearly elastic "flezural" shells studied in Chap. 10. THREE-DIMENSIONAL NONLINEAR ELASTICITY IN CARTESIAN COORDINATES
8.1.
We briefly review in this section the equations of nonlinear elasticity, the constitutive equation being that of the "simplest" nonlinearly elastic material (more general elastic materials are considered in Sect. 9.5). Ample details about the various notions introduced may be found in Vol. 11, Chaps. 1 to 3. Let the "physical space" be endowed with a Cartesian frame and, for this reason, let it be henceforth identified with I~3. We let ~ = ~/ denote the basis vectors of the Cartesian frame (these notions are discussed in more detail at the beginning of Sect. 1.1). Let ~ be a domain in R 3 with boundary r , let d~ denote the volume element in ~, let dP denote the surface element along P, and let/t ni ~i denote the unit (lit[ - 1) outer normal vector along r . Finally, let r - P0 u P1 be a dP-measurable partition (P0 N P1 - r of the boundary that satisfies -
-
area Po > O.
The set { ~ } - is the r e f e r e n c e c o n f i g u r a t i o n occupied by an elastic b o d y in the absence of applied forces. The body is subjected to a p p l i e d b o d y forces in its interior, of density (]i) " ~ -+ R 3 per unit volume, and to a p p l i e d surface forces on the portion P1 of its boundary, of density (~ti) 9 P1 --+ ~3 per unit area. Since these densities do not depend on the unknown, the applied forces considered here are dead loads (Vol. I, Sect. 2.7). For definiteness, we assume that ]i e L2(~) and ~ti e L2(p1). 1We recall that "Vol. r' stands for "Ciarlet, P.G. [1988: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, North Holland, Amsterdam".
Sect. 8.1]
Three-dimensional nonlinear elasticity in Cartesian coordinates
387
The unknown is the vector field ,2 - (ui)" { f i ) - -+ I~3, where the three functions ui " {1~}- --+ ]~ are the C a r t e s i a n c o m p o n e n t s of the d i s p l a c e m e n t field fii&i" { ~ ) - -+ I~s that the body undergoes when it is subjected to the applied forces. This means that ~i(#.)& i is the displacement of the point ~ E { ~ ) - (Fig. 1.1-1). Notice that we identify fi~(~)&~ E I~3 with the vector/,(~) E I~3 and write accordingly the displacement field as
It is assumed that the displacement vanishes on the set r0, i.e., that it satisfies the b o u n d a r y c o n d i t i o n of place /~-- 0 on F0. Recall that a "boundary condition of place" is in this volume always meant to be homogeneous. The set ~({ ~ ) - ), where ~o " - ida3 + i, " ~ E { ~ ) - -+ ($ + / , ( ~ ) ) E I~3,
is called a d e f o r m e d c o n f i g u r a t i o n and the mapping ~ is called a d e f o r m a t i o n (of the reference configuration { ~ ) - ) . Since the approach in this section is essentially formal, we assume that the requirements that the deformation ~ should satisfy in order to be physically admissible (orientation-preserving character and injectivity; cf. Vol. I, Sect. 1.4) are satisfied. We let ~ - (xi) denote a generic point in the set ~ and we let The following e q u a t i o n s of e q u i l i b r i u m in C a r t e s i a n coord i n a t e s (Vol. I, Sect. 2.6) are then satisfied in the reference configuration {~}-" - 3 j ( & i j + ~kj&r +
- fi in ~, -
on
where ~ - (&ij) ' { ~ ) - -~ S 3 denotes the s e c o n d P i o l a - K i r c h h o f f s t r e s s t e n s o r field and its components &ij - &ji are called the
388
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
(second Piola-Kirchhoff) stresses (cf. Vol. I, Sect. 2.5; S 3 denotes the space of all symmetric matrices of order three). The functions are the components of the first P i o l a - K i r c h h o f f stress t e n s o r field ~ 7 - (~ij)" { ~ ) - --+ M[3 (cf. again Vol. I, Sect. 2.5; 1VII3 denotes the space of all matrices of order three). The boundary conditions on the set r l constitute a b o u n d a r y c o n d i t i o n of traction. Let W(I~) denote a space of sufficiently smooth vector fields - (vi)" { ~ ) - -+ R 3 that vanish on ~0. Then, if the unknown displacement/L belongs to the space W ( ~ ) , it can be easily established (by integration by parts; cf. Vol. I, Thm. 2.6-2) that the equations of equilibrium are formally equivalent to the p r i n c i p l e of v i r t u a l w o r k in C a r t e s i a n c o o r d i n a t e s , which states that:
1
for all
W(fi).
Note that the principle of virtual work is nothing but the weak, or variational, form of the equations of equilibrium and that the functions fi E W ( ~ ) that enter it are "variations" around the actual deformation 9 ~ - ides + i~ (Vol. I, Sect. 2.6). We finally assume that the constituting material is elastic, hom o g e n e o u s , and isotropic and that the reference configuration { ~ ) - is "stress-free", i.e., is a natural state (these notions are defined in Vol. I, Chap. 3). We consider here the simplest n o n l i n e a r l y elastic m a t e r i a l that satisfies these assumptions. More specifically, let the s t r a i n s 1
denote the C a r t e s i a n c o m p o n e n t s of the c h a n g e of m e t r i c or G r e e n - S t Venant strain tensor (Vol. I, Sect. 1.8):
1
}T
Sect. 8.1]
Three.dimensional nonlinear elasticity in Cartesian coordinates
389
associated with an arbitrary displacement field ~ - (fii) 9{ ~ ) - -+ I~3 of the reference configuration { ~ } - , where the matrix
01V2 02 V2 03 V2 31V3 02 ~3 03 V3 denotes the corresponding d i s p l a c e m e n t g r a d i e n t . Note in passing that the linearized strain tensor ~ ( i ~ ) - (~ij(~))introduced in Sect. 1.1 is indeed the linear part with respect to ~ in the strain tensor We assume that the material constituting the elastic body is a St V e n a n t - K i r c h h o f f m a t e r i a l . This means (Vol. I, Sect. 3.9) that there exist two constants A and # such that the second PiolaKirchhoff stress tensor is expressed in terms of the Green-St Venant strain tensor through a c o n s t i t u t i v e e q u a t i o n of the form: -- )~(tr 1~(~))I + 2#1~(/~), or, componentwise,
The two constants )~ and # are the L a m 6 c o n s t a n t s of the material; in accordance to experimental evidence, we assume that they satisfy A>Oand#>O.
Remarks. (1) The constitutive equation may be also expressed in terms of the Poisson ratio and the Young modulus of the same material (Vol. I, Sect. 3.8). (2) The same nonlinearly elastic material was used in Vol. II t, Chaps. 4 and 5, for justifying, again by the method of formal asymptotic expansions, the two-dimensional equations of nonlinearly elastic plates. II We thus realize, simply by inspecting the equations, that for a St Venant-Kirchhoff material the minimum regularity needed on the
1We recall that "Vol. II" stands for "Ciarlet, P.G. [1997]: Mathematical ElasNorth Holland, Amsterdam".
ticity, Volume II: Theory of Plates,
390
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
components ~i of the fields ~ = (vi) e W ( ~ ) in order that all integrals appearing in the left-hand sides of the principle of virtual work make sense is that they belong to the Sobolev space
.=
e L4(fi);
e L4(fi)).
Hence the space W ( ~ ) may be defined in the present case as: W ( ~ ) := {@ = (vi) e w l ' 4 ( ~ ) ; / ' = 0 on to}. If we assume t h a t / t
e
W l ' 4 ( ~ ) , we also have
A(tr l~(/t))I + 2#]~(/t) where
- ~
e L2(~),
L2(~) := {(#ij) e L2(~); l"ij --~'ji}.
To sum up, the displacement field it = (4ti) satisfies the following nonlinear d i s p l a c e m e n t - t r a c t i o n p r o b l e m ("displacementtraction" refers to the kind of boundary conditions considered here; el. Vol. I, Sect. 5.1): -0i(&ij + &kj0kui) -- ]i in ~, ui -- 0 o n r 0 ,
(~j + ~kj 0 k ~ ) ~ - h~ on ~ , which is in turn equivalent to the variational problem: /~ E W ( ~ ) : - - (6 -- (vi) C W1'4(~); 0 -- 0 on r o),
1
where
1
Sect. 8.1]
Three-dimensionalnonlinear elasticity in Cartesian coordinates 391
Either formulation constitutes the e q u a t i o n s of t h r e e - d i m e n sional n o n l i n e a r e l a s t i c i t y in C a r t e s i a n c o o r d i n a t e s for a displacement-traction problem associated with a St Venant-Kirchhoff material. Note in passing that the linearization of these equations yields precisely the linear equations considered in Sect. 1.1. Because a St Venant-Kirchhoff material is h y p e r e l a s t i c (Vol. I, Thm. 4.4-3), solving the above variational problem is equivalent to finding the stationary points of an associated functional J (defined below), i.e., those points where the derivative of J vanishes (this equivalence is proved in the next section in the more general setting of nonlinear elasticity in curvilinear coordinates, which include the Cartesian ones as a special case; cf. Thm. 8.2-3). Particular stationary points are thus obtained by solving a minimization problem, viz., find ~ such that /~ e W(fi) and 3(/,) --
inf 3(~),
~ew(h)
where the t h r e e - d i m e n s i o n a l e n e r g y J " W ( ~ ) -+ IR in C a r t e sian c o o r d i n a t e s associated with a St Venant-Kirchhoff material is defined for all ~ - (fii) 6 W ( ~ ) by
Let S 3 denote the space of all symmetric matrices of order three. The mapping
EES~-+ {~A (tr E) 2 + #tr E 2 }ER that appears in a self-explanatory way in the first integrand of the energy 3 is called the s t o r e d e n e r g y f u n c t i o n of a St VenantKirchhoff material. A w o r d of c a u t i o n . There is no available result guaranteeing the existence of a solution to this minimization problem (all that can
392
Asympto$ic analysis of nonlinearly elastic shelb: Preliminaries
[Ch. 8
be proved is that 3 is coercive on W ( ~ ) ; cf. Ex. 8.1). The only available existence result valid for a St Venant-Kirchhoff material is based on the implicit function theorem and is for this reason restricted to smooth boundaries, to "small enough" forces, and to special classes of boundary conditions, which do not include those considered here (see the discussion given in Vol. I, Sect. 6.7). In a landmark paper, Ball [1977] has developed a powerful existence theory for hyperelastic materials whose stored energy is polyconvex and satisfies ad hoc growth conditions (a detailed account of this theory is given in Vol. I, Chap. 7). This theory accommodates non-smooth boundaries and boundary conditions of the type considered here and is not restricted to "small enough" forces. However, even within the class of elastic materials to which it applies, which does not include St Venant-Kirchhoff materials (their stored energy functions are not polyconvex, cf. Raoult [1986]), it neither provides the existence of a solution to the corresponding variational problem, because the energy is not differentiable in the spaces where the minimizers are found (Vol. I, Sect. 7.10). m Detailed expositions of the modeling of three-dimensional nonlinear elasticity are found in TmesdeU & Noll [1965], Germain [1972], Wang & Truesdell [1973], Gurtin [1981a], Marsden & Hughes [1983, Chaps. 1-5], and Vol. I, Chaps. 1-5. Its mathematical theory is exposed in Ball [1977], Marsden & Hughes [1983], Valent [1988], and Vol. I, Chaps. 6 and 7.
8.2.
THREE-DIMENSIONAL NONLINEAR ELASTICITY IN CURVILINEAR COORDINATES
To begin with, we slightly modify some of the notations of Sect. 8.1~ so that the definitions and equations in Cartesian coordinates recalled there now obey the proper rules regarding covariant indices and contravariant exponents. Once so rewritten, they will more evidently appear as special cases of the more general ones in curvilinear coordinates found in this section. We now denote by ~ij the Kronecker symbol and b y / i . _ / i and hi := hi the components of the applied forces. We also define the components ~ijkl of the t h r e e - d i m e n s i o n a l elasticity t e n s o r in
Sect. 8.2]
Three-dimensional nonlinear elasticity in curvilinear coordinates 393
Cartesian
c o o r d i n a t e s by
Remark. The same changes of notations were performed on the equations of three-dimensional linearized elasticity in Cartesian coordinates (Sect. 1.1). II W i t h these notations, the minimization problem described at the end of Sect. 8.1 equivalently consists in finding f i - (~2i) such t h a t a e W(h)"--
J(/t) -
Eij(~)
{'V -- (Vi) C w l ' 4 ( h ) ;
~ -- 0 on F0} and
inf j ( ~ ) , where ocw(fi)
"-- ~
This minimization problem is formulated in terms of the Cartesian coordinates ~i of the points ~ - ($i) E { ~ } - and of the Cartesian components vi, ]2, h z of the displacement and force densities. Assume t h a t we are also given a domain f~ in I~~ and a smooth enough injective m a p p i n g 19 9 ~ --+ II~3 such that @(~) - { ~ } and the three vectors g/(x) - 0i| are linearly independent at all points x E f~. Our objective consists in reformulating this problem in terms of the curvilinear coordinates xi of the points ~ - @(x) e { ~ } - , where x - (x/) E f~. In other words, we wish to carry out a change of variables, from the "old" variables $/ to the "new" variables x/, in each one of the integrals appearing in 3(~), integrals which we thus wish to write as ^ .
^
~
where r l C r and @ ( r l ) - r l . Thanks to the various formulas t h a t were established in Chap. 1 for performing an analogous transformation on the equations of linearized elasticity, this change of variables becomes a simple task, which leads to the following result:
394
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
T h e o r e m 8.2-1. Let fi be a domain in IRa, let ]i E L2(~) and ]Zi E L2(F1) be given functions, and let it - (r E W(fi) denote a
minimizer of the energy ,] over the space W(fi). Let f~ be a domain in IR3 and let O be a C2-diffeomorphism of -0 onto {~}- - | so that the three vectors g i ( x ) -- Oi{~(X) are linearly independent at all points x C -~. Let the vectors gi(x) be defined by the relations o i ( w , ) . g j ( x ) -- ~}, let g(z) -- det(gi(x).gi(x)), and let gi~(~e) - g i ( x ) . gJ(x), x ~ -~. Then the vector field u - (Ui) "-~ ~ ]1~3 defined by ~i(~,)e, i : : ~i(~,)gi(x) for all ~ - O(x), x C ~,
satis[~es the following minimization problem:
u C W ( f ~ ) : = {v - (vi) C wl'4(f~); v - 0 on ro}, J(u)J ( ~ ) :=
inf
lfn
J(v), where AiJk*Ekll,(~)F~llj(~)~d~
- { fafiviv~dx where ro "- |
+ f r hiviv/gcIF} ,
r l : : {~-l(rl), the functions fi C L2(f~) and
h i E L2(rl) are defined by: ]i(~,)e i d~, - " v / g ( x ) f i ( x ) g i ( x )
hi(~,)e i d~(&) : :
v/g(x)hi(x)gi(x)
dx, ~, - O ( x ) , x E ~ ,
dr(x),
~ -
|
x ~ r~,
the functions A ijkt - A jikt - A ktij E C1(-~) are defined by: Aiikt ._ )tgij gkt + p(gik gjt + git gjk), and finally, the ]'unctions Eilli(v) - Ej[Ii(v ) C L 2 ( ~ ) are defined for all v - (vi) C wX'4(f~) by: 1
Ei[Ij(v ) := ~(vil]j + villi +
gmn
P p and Villi := Ojvi - riiv
vm[livnl[j), where
rijP
. _ gP . O~g~ - r jPi ~ cO (-0).
Sect. 8.2] Three-dimensional nonlinear elasticity in curvilinear coordinates 395 Proof. (i) Let [gj(x)] i "- g j ( x ) . ~i and [gJ(x)]i " - 9J(x) 9ei denote for each x E f~ the i-th component of the vectors gj(x) and 9J(z) over the basis {~1, ~2, ~3} = {~1, e2, es}. With any field ~ = (~3i) E W((~), we associate as in Sect. 1.3 a field v - (vi) 9 --+ I~3 by letting (see in particular Pigs. 1.1-1 and 1.3-1, which illustrate the special case where ~3i~* is the displacement field ~i ~i) 9 e ft.
~ ( ~ ) ~ i =: ~(~)g~(~) fo~ aU ~ = |
Since both (9 and its inverse mapping are Lipschitz-continuous by assumption, the functions ~3j o O are in the space Wl'4(f~) (see, e.g., Ne~as [1967, Chap. 2, Sect. 3] or Adams [1975, Chap. 3]); consequently, the functions vi = (~3j o | are also in Wl'4(f~) since [gi]j E C1(~) by assumption. Hence 6 6 W ( f i ) implies that ~ w(~). (ii) Again as in Sect. 1.3, we infer from the definitions of the functions fi E L2(f~) and h i E L 2 ( r l ) that
ffi/ividx+f~thivic~-fnfivi~/gdx.+frlhivi~/g
dF
for all ~ - (~3i) E W((~) and v - (vi) E W(f~) in the correspondence set in (i). ostabl
h.d
oe
t at
= (vkllt[gk]i[gt]j) (x) for aU & = 19(x), x for all ~ - (~3i) ~ W((~) and v = (vi) ~ W(f~) in the correspondence set in (i), where V~llt(x ) "- Otv~(x) - rP~t(x)vp(x). These relations in turn imply that
Eij(~))(~.)-- ~l (oj'Vi + Oi~)j "at- Oi~)mOj~)m)(X,) 1 : ~ ((vkllI + Vlllk)[gk]i[gl]j + VrllkVslll[gk]i[gr]m[gl]j[gS]m) (X)
1( {v~llg+ vgll~ + gr'Vrll~V"ilg}[g~]i[gl]J ) (~)
=-2
396
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
since [gr]m[gs]m = g r . g ' = g r'. Hence Eij(O)(~) - (Eklll(v)[gk]i[gt]j) (m) for all ~ -- O(x), x E ~, where the functions Ekllt(v ) E L2(f~) are those defined in the statement of the theorem. (iv) Since the relations between the functions /~ij(v) e L2(~) and Eillj(v) E L2(f~) are similar to those between the functions ~ij(6) E L2(~) and eillj(v ) C L2(f~)established in part ( i v ) o f the proof of Thm. 1.3-1, it likewise follows that
- (AiJktEkllt(v)Eillj(v))(x) for all ~ - @(x), x E f~, where the functions A ijkl E Ct(~) are those defined in the statement of the theorem. (v) Conclusions: Since d ~ -
v/g(x)dx, part (iv)shows that
f.
AijklEkllt(v)Eillj(v)V~ dx.
Together with that of part (ii), this relation shows that
J(e) - j(. ) for all 6 e W ( ~ ) and v e W ( ~ ) i n the correspondence set in (i), and thus the proof is complete, m Naturally, if @ = idR3, each vector gi(x) is equal to &i and thus gi(x ) _ ~.i, g(x) -- 1, gij(x) -- ~ij, and Villi - Ojvi; in addition, the fields (vi) and (vi), (fi) and (]i), (h i) and (hi), and (A ijkl) and (.~ijkl) coincide. The functional J : W(~2) -+ R found in Thin. 8.2-1 is caUed the t h r e e - d i m e n s i o n a l e n e r g y in c u r v i l i n e a r c o o r d i n a t e s associated with a St Venant-Kirchhoff material. The functions A ijkt 9 -~ --+ R are the c o n t r a v a r i a n t c o m p o n e n t s of the t h r e e - d i m e n s i o n a l e l a s t i c i t y t e n s o r in c u r v i l i n e a r c o o r d i n a t e s (the same tensor appeared in linearized elasticity; cf. Sect. 1.3).
Sect. 8.2] Three.dimensional nonlinear elasticity in curviIinear coordinates 397 Let there be given a vector field r i g z o n ~ interpreted as an arbitrary d i s p l a c e m e n t field of e(f~) given by means of its e o v a r i a n t c o m p o n e n t s vi 9~ -+ I~; this means that vi(x)gi(x) is the displacement of the point ~ - e ( x ) . We now show that the matrix 1
(EillJ(V)) -- (~(VillJ + Villi +
gmn
)) VmlliVnllJ
found in Thm. 8.2-1 generalizes to arbitrary curvilinear coordinates the Green-St Venant strain tensor
in Cartesian coordinates (Sect. 8.1). More specifically, we show that the matri$ (Eillj(v)) likewise measures (half of) the difference between the "new" metric in the de/o med configuration (| + the "otd" metric in the reference configuration e(f~), but now expressed by means of their covariant components in terms of the curvilinear coordinates xi. We also record the relation between the functions/~ij (~3) and Eillj(V) that was established in part (iii) of the proof of Thin. 8.2-1. T h e o r e m 8.2-2 Let f~ be a domain in R 3 let (9"-~--+ R 3 be a smooth enough injective mapping such that the three vectors gi - o i e are linearly independent at all points of ~ and let the vectors gi be defined by g i . g j _ ~ . Given a vector field vig i on-~ with smooth enough covariant components vi 9f~ -+ I~, let the c o v a r i a n t c o m p o n e n t s of the G r e e n - S t V e n a n t s t r a i n t e n s o r in c u r v i l i n e a r c o o r d i n a t e s associated with this vector field be defined by 1
EillJ(V) "- -2(Villi + Vfflli + gmnvmlliVnllJ),
where Villi
-
-
OjVi - - r i Pj V p. Then they also satisfy
E tlj(
) -
1
- g j),
where gij and gij(v) denote the covariant components of the metric tensors respectively attached to the sets e(-~) and ((9 + vigi)(-~).
398
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
For this reason, the functions EillJ(V) are also called the covaria n t components of the c h a n g e of m e t r i c t e n s o r in c u r v i l i n e a r coordinates, or simply the s t r a i n s in c u r v i l i n e a r coordinates, associated with this vector field. The strains in curvilinear coordinates EillJ(v) are related to the strains in Cartesian coordinates Eij(i~) by:
~,j(~)(~)
= (E~ll,(~)[g~],[g']j)
(~), ~ =
o(~).
Proof. The covariant components gij(v) are defined in f~ by g i j ( ~ ) -- Oi(l~) Jr_ Vm g m )" Oj(l~) q- v n g n ) .
Using the relations Oi (1~) -t- v m g m) = Oi 0 -q- OiVmg m -t- Vm Oig m = gi + (Oi~3m -- r ~ m V q ) g m = gi "~- Vmllig m
(recall that Oig q = - rqimP1 _m ; cf. Thin. 1.4-1), we obtain
g~j(v) - (g~ + v~ll~g~) . (gj + v.lljg") :
gi " gJ -t- Vnlljg i " g n + Vmllig
-" gij q- viii j + Villi q- g
mn
m
V m gn " g j + Vmlli nlljg "
VmlliVnllJ,
II
as was to be proved.
W e next show that the minimization problem of Thin. 8.2-1 also admits a variational formulation, like its Cartesian counterpart (Sect. 8.1). W e refer to, e.g., Vol. I, Sect. 1.2, for the definition and main properties of the Frdchet and G~teaux derivatives. T h e o r e m 8.2-3. As in Thin. 8.2-1, let the space W(f~) and the energy J : W(f~) ~ I~ be defined as
W(f~) - {v--(Vd) E w l ' 4 ( 1 2 ) ; v - 0 o n r 0 ) ,
1/o
-{f fivi~/'gdx+frh'V~dr}
Sect. 8.2]
Three-dimensional nonlinear elasticity in curvilinear coordinates 399
where the mappings Eillj " W1'4(f~) --~ L2(f~) are defined for each v e Wl'4(f~) by (Thm. 8.2-2):
1
~illj(V) :~ ~ ( g i j ( V ) -- gij)
1 gmn -- ~(VillJ + Villi + vmlliVnllJ)" Then the energy J is differentiable over the space W 1' 4(f~), hence over W(fl), and its Gdteaux derivatives J'(u)v are given by
J'(u)v -- fa AiJktEkllt(u)Fillj(U' - (~
v)v/g dx
f i v i v ~ d ~ h- frx h i v i ~ d r } ,
for all u, v C W 1' 4(f~), where llj ( , v ) : =
E illJ ' (,,),, 1
and E~llj(U) E s 14' (fl); L2(f~)) denotes the Frdchet derivative at any u E Wl'4(fl) of each mapping EillJ " W1'4(f~) ~ L2(f~) 9 Consequently, u E W(ft) is a stationary point of the energy J, i.e., u satisfies J~(u) = O, if and only if it satisfies the variational problem: u e W(fl) and, for all v e W(f~), /fl AiJktEkllt(u)Fitlj(u, v)v/g dx = ~ fiviv/g dx + f r hivi~/g dr. 1
This is in particular the case if u is a solution to the minimization problem of Thin. 8.2-1.
400
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
Proof. The linear form L defined by L(v) - ~ f i v i v ~ dx + fr hiviv/-g dr is continuous and differentiable over W l , a(f~), with
L'(u)v = f fiviv~dx + fr hiviv/gdr 1
for all u, v C w t ' a ( f ~ ) . For any v E Wl'4(f~), let j.(,,)
-
ira
The mapping J0 " W 1' 4(~) _.+ ]1~ defined in this fashion can be written as Jo - B o E, where the mappings E " W 1' 4(f~) _~ L2(f~; $3) and B : L2(f~; ~3) ~ I~ are defined by (~3 denotes the space of symmetric matrices of order three): ]~(V) "-- (.Ei[lj(V)) e
B ( E ) " - ~1
f.
L2(~; ~3) for all v
-(vi)
e
wl'4(~),
Aijk tEklliEillJv/-gdx for all E -- (EillJ) C L2(f~; $3).
As a sum of continuous linear and bilinear mappings, each mapping EillJ is differerttiable and its G~teaux derivatives are given by (see, e.g., Vol. I, Sect. 1.2): i
1
Eillj(u) v - -~(VillJ + Villi + gmn{UmlliVnllff + UnlljVmlli))" The mapping B is likewise differentiable and its G~teaux derivatives are given for all E - (EilIJ) and F = (Fill/) by B'(E)F :
fo
AiJktEklltFillJ V/-gdx.
By the chain rule (see, e.g., Vol. I, Thin. 1.2-1), we thus have -
-
A'Jk'EklI,(
)E
IIj(
).V
for all u, v E Wl'4(f~). Hence the proof is complete (the other statements in the theorem are immediate consequences of the expression
of J'(u)v).
II
Remarks. (1) As a sum of continuous k-linear forms, k = 1, 2, 3, 4, the energy J is in fact infinitely differentiable over the space W 1' 4(f~).
Sect. 8.2] Three-dimensional nonlinear elasticity in curvilineav coordinates 401 (2) Once it has been noticed that J is differentiable, a more direct way of computing J~ (u)v simply consists in identifying the linear part with respect to v in the difference J ( u + v) - J ( u ) . However, this procedure is less illuminating as it hinders the appearance of the derivatives E ~llJ ~ (u) " (3) Naturally, the variational equations in curvilinear coordinates found in Thm. 8.2-3 by differentiating the energy J can be also directly recovered from the variational equations in Cartesian coordinates described in Sect. 8.1 (Ex. 8.2); however, the method followed in Thm. 8.2-3 is simpler. II To conclude this section, we derive the boundary value problem that is (at least formally) equivalent to the variational problem found in Thm. 8.2-3. We recall that n i ( x ) e i denotes the unit outer normal vector at x E r . T h e o r e m 8.2-4. Let the assumptions and notations be as in Thms. 8.2-1 and 8.2-3. If a solution u = (ui) to the variational problem
e w(~)
-
{.
-
(.~) e w ~ , 4 ( ~ ) ;
. - 0 o~ r0},
f AiJklEkll,(u)F~ll~(u,v)~dz-f/iv~v~dZ+fr hivi~/~dr 1
for all v C W(~2),
is smooth enough, it also satisfies the following b o u n d a r y v a l u e p r o b l e m of t h r e e - d i m e n s i o n a l n o n l i n e a r e l a s t i c i t y in c u r v i linear coordinates:
-(a~J + akjgi~u~llk)l]j - A in f~, ui -- 0 on to, (a ij § akJgitutllk)nj -- hi on r l ,
where the functions ~ij . _ AijklEkllt(u)
402
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
are the contravariant components of the s e c o n d P i o l a - K i r e h h o t f and
s t r e s s t e n s o r field
~'Jllk := 0k~ ~j + r~k~ pj +
r~qa iq
denote the f i r s t - o r d e r e o v a r i a n t d e r i v a t i v e s of any tensor field with differentiable contravariant components Orij "-~ ---+I~. Proof. From the proof of Thm. 1.6-1, we infer that ff aiJeillj(V)~/g dx - - fn ~/'g(criJl'j)vi dx + fr v~riJnjvi clF, l f n a*j""gmn{umlliVnllJ + UnlljVmlli}~/g dx -~ -- fn aij gmnumllivnllj V/~ dx :--fflV~{(~rkJgilulllk)l,,}~3idx-~-fr~(orkJgil~lllk)njvid~ for all v - (vi) C W l ' 4 ( n ) . that
: [
Hence the variational equations imply
~ { ( ~ ' J + ~hJg"~,ll~)~ j - h'}~, ~r
01" 1
for all (vi) e W ( ~ ) . Letting (vi) vary first in (T~(~)) z, yields the announced boundary value problem,
then in W ( ~ ) , m
Naturally, the same boundary value problem in curvilinear coordinates can be derived from its Cartesian version (Sect. 8.1); cf. Ex. 8.3. It can also be directly derived in curvilinear coordinates from the axioms of continuum mechanics, as in Oden [1972, Chap. 11 or Washizu [1975, Chap. 4]. Either one of the formulations described in this section, i.e., as a minimization, or variational, or boundary value, problem constitutes the e q u a t i o n s of t h r e e - d i m e n s i o n a l n o n l i n e a r e l a s t i c i t y in curvilinear c o o r d i n a t e s for a displacement-traction problem associated with a St Venant-Kirchhoff material. Naturally, it contains as a special case the equations in Cartesian coordinates considered in Sect. 8.1.
The three-dimensional equations
Sect. 8.3]
403
The variational equations
/.
a'3 Fillj(u, v)x/~ d~, =
fivi ~/~ dz + fr hi vi v/g dr 1
for all v e W(f~) constitute the p r i n c i p l e of v i r t u a l w o r k in c u r v i l i n e a r coordin a t e s . By Thm. 8.2-4, this principle is (at least formally) equivalent to the e q u a t i o n s of e q u i l i b r i u m in c u r v i l i n e a r c o o r d i n a t e s :
_(orij _~_akjgitutllk)l[ j = fi in f~, (~ij + ~kjgilugll k)nj
=
hi
o n F1.
Finally, the functions
tij :__ criJ + akj gitutllk are the c o n t r a v a r i a n t c o m p o n e n t s of the first P i o l a - K i r c h h o f f stress tensor.
Remark. As their Cartesian special cases, the principle of virtual work and the equations of equilibrium are valid for any continuum (Vol. I, Chap. 2). I 8.3.
THE THREE-DIMENSIONAL EQUATIONS NONLINEARLY ELASTIC SHELL
OF A
To begin with, we reproduce those preliminaries from Sect. 3.1 that apply verbatim to nonlinearly elastic shells. Let w be a domain in ~2 with boundary 7. Let y - (ya) denote a generic point in the set ~ and let 0a := O/Oya. Let 8 E C2(~; ~3) be an injective mapping such that the two vectors := a, e(y)
are linearly independent at all points y E -~. They form the covariant basis of the tangent plane to the surface s
:=
404
Asymptoticanalysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
at the point O(y) and the two vectors aa(y) of the tangent plane at O(y) defined by the relations
form its contravariant basis. Let also
a~(y) A a2(y) a3(y) = a 3 ( y ) : = la,(y) A a~(y)l" Then la~(y)! = 1, the vector a3(y) is normal to S at the point O(y), and the three vectors ai(y) form the contravariant basis at 8(y); cs Fig. 2.3-1. Note that (Yt, Y2) constitutes a system of curvilinear coordinates (Sect. 2.1) for describing the surface S. Let 70 denote a dT-measurable subset of the boundary 3' of w satisfying
length 7o > O. For each e > 0, we define the sets ~ r ~ := ~ • {~},
rt
:= ~ , •
~, ~[,
:= ~ x { - ~ } ,
r~ := ~o • [-~, ~].
Let x e - (x~) denote a generic point in the set ~ and let 0~ : - O/Ox~; hence x ae - Y a and 0~ - 0a. Consider an e l a s t i c shell with m i d d l e s u r f a c e S - 8(~) and t h i c k n e s s 2e > 0, i.e., an elastic body whose reference configuration consists of all points within a distance < e from S (Fig. 1.2-1). In other words, the reference configuration of the shell is the image O ( ~ e) C IR3 of the s e t - ~ C R 3 through a mapping 0 9 ~ R3 given by
o ( ~ ~) .= 0(y) + ~la~(y) fo~ an ~ = (y, 4 ) - (y~, y~, ~g) e ~ . Thm. 3.1-1 is essential for describing the "geometry" of such a shell: It shows that, if the mapping 8 9W --+ I~s is injective and smooth enough, the mapping 0 9 -+ IRa is also injective for e > 0 small enough; consequently, (yl, Y2, x~) constitutes in this case a bona
Sect. 8.3]
The
three-dimensional
405
equations
fide system of curvilinear coordinates for describing the reference configuration O ( ~ e) of the shell, called the "natural" curvilinear coordinates of the shell (Sect. 1.2), and the physical problem is meaningful as the set 0(-~ e) does not "interpenetrate itself". The curvilinear coordinate x~ E I-e, e] is called the t r a n s v e r s e variable. The same theorem also shows that, again for e > 0 small enough, the three vectors
g~ (x~) :: 0~ O(x ~) form the covariant basis (of the tangent space, here IRa, to the manifold | at the point | and the three vectors gi,~(x~) defined by form the contravariant basis at O(x ~) (Sect. 1.2). The covariant and contravariant components gij and gij, e of the metric tensor (Sect. 1.2) and the Christoffel symbols riPj e (Sect. 1.4) of the manifold O ( ~ ~) are then defined by letting (the explicit dependence on x e is omitted): .__
gij
e
e
r 'j
.-
gi " gj ,
gij,
e .__
.o
"
g*'
9gi, e
gj.
Note the symmetries: gij
--
gji,
gii,
"
__ g 3 Z ,
,
ri I
__
Fj i
.
The volume element in the set O ( ~ e) is Vr~ dx e, where ge := det(g~j). We assume that the material constituting the shell is a St VenantKirchhoff material, with Lamd constants )~e > 0 and #~ > 0. The unknown of the problem is the vector field u ~ - (u~) 9~e R3 where the three functions ui -+ R are the covariant components of the displacement field u~g/,e 9~ ~ that the shell undergoes under the influence of applied forces; this means that u[ (x~)g/,e (x e) is the displacement of the point O(x6); see Fig. 3.1-1. Finally, we assume that the shell is subjected to a boundary condition of place along the portion e(r ) of its lateral face 0(3' x I-e, el), in the sense that the displacement vanishes there.
406
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
It then follows from Thm. 8.2-3 that the unknown u e -- (u~) satisfies the following t h r e e - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 79(fP) of a n o n l i n e a r l y e l a s t i c shell in c u r v i l i n e a r c o o r d i n a t e s , i.e., expressed in terms of the "natural" curvilinear coordinates x ie of the shell: u e e W(f~ e) "= {v e = (v~) e wl'4(f~e); v e -- 0 on r~},
e
-
f.
vi ~ - g dxe + 9
~ur ~_
h,,e v ie V c ~ d r e
for all v e E W(f~e), where AiJkt, e := )~egij, egkl, e -4-pe(gik, egfl, e + gil, egjh, e)
designate the contravariant components of the three-dimensional elasticity tensor, 1 s 9 E illj(v e e) := ~(villJ -4-villi + g
gmn, e e
VmlliVnllj), where
e
designate the strains in curvilinear coordinates associated with art arbitrary displacement field-e..~,e of the manifold O ( ~ e) (Thin. 8.2-2) "vi Y ::
1 = _ 2( ' lJ +
'
9
gmn, e
+
e
~ e v.lli + U.lljVmlli ) ) ,
and, finally, fi, e E L 2 ( ~ e) and h i,e E L2(r~_ U r ~ ) denote the contravariant components of the applied body and surface force densities, respectively applied to the interior O(f~ ~) of the shell and to its "upp e r " and "lower" faces | and O ( r e_), and dF e designates the area element along 0f~ e. We thus assume that there are no surface
Sect. 8.4] The three-dimensional equations over a domain independent of e 407 forces applied to the portion O ( ( - y - 70) • [ - e , e]) of the lateral face of the shell. We record in passing the symmetries Aijkl, e _ AJikl, e _ Aklij,
and the relations (satisfied because the mapping | form given supra) as r3,,
is of the special
= r ~ e = 0 and A a[3~3' 9 = Aa333, e ._ 0 in ~
Our final objective consists in showing by means of the method of formal asymptotic ezpansions that, if the d a t a are of an appropriate order with respect to e as e --+ 0, the above three-dimensional problems are "asymptotically equivalent" to a "two-dimensional problem posed over the middle surface of the shell". This means t h a t the new unknown should be ~e __ ( ~ ) , where ~ are the covariant components o] the displacement ~ a i 9-~ --+ I~s of the points of the middle surface S - 0(~). In other words, ~ (y)ai(y) is the displacement of the point O(y) E S; see Fig. 3.1-2.
8.4.
THE THREE-DIMENSIONAL A DOMAIN INDEPENDENT
EQUATIONS OF g
OVER
We describe in this section the basic preliminaries of the asymptotic analysis of a nonlinearly elastic shell. Since these are essentially the same as in the linear case, we freely borrow those parts of Sect. 3.2 t h a t apply verbatim to the nonlinear case. "Asymptotic analysis" means t h a t our objective is to study the behavior of the displacement field -~o u~.o i, e 9-~e ~ i~s as g --~ O~ all endeavor t h a t will be achieved by studying the behavior as e ~ 0 of the covariant components u i -+ R of the displacement field, i.e, the behavior of the unknown u e - (u~)" ~ ~ R s of the threedimensional shell problem p ( ~ e ) described in Sect. 8.3. Since these fields are defined on sets ~e that themselves vary with e, our first task naturally consists in transforming the threedimensional variational problems T~(~ e) into problems posed over a set that does not depend on e.
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
408
[Ch. 8
Let
f~ := w x ] F0 "-V0
x
[-1, 1],
1, l[,
F+ := w x {1},
r_
.-
• {-1}.
Let x - (xl, x2, x3) denote a generic point in the set ~ and let 0i "= 0 / 0 x i ; hence xa - ya (a generic point in the set ~ is denoted y - (yl, y2); cf. Sect. 8.3). The coordinate x3 E [ - 1 , 1] is called the t r a n s v e r s e v a r i a b l e , like x~ E [ - s , s] in Sect. 8.3 ("scaled" transverse variable could thus be preferred; however, no confusion should arise). W i t h each point x E ~, we associate the point x e E ~e t h r o u g h the bijection (Fig. 3.2-1) 71"e "X -- (Xl, X2, X3) E ~ ~
Xe -- (X~) -- (Xl, X2, ~X3) E ~e.
Consequently,
0~ - 0a and 0~ = _103.
W i t h the unknown u e - ( u [ ) " ~e --+ i~ 3 and the vector fields --+ appearing in the three-dimensional problem ~(fle), we associate the s c a l e d u n k n o w n u(e) - (ui(e)) "-~ -+ I~3 and the s c a l e d v e c t o r fields v - (vi) 9~ --+ R 3 defined by the scalings:
u~(x e) =: uiCs)(x) and v~(x e) - " vi(x) for all x e - r~x e ~ , and we call s c a l e d d i s p l a c e m e n t s the three components ui(s) of the scaled unknown u(s). We assume at the outset that the Lamd constants )~ and #e are independent of s, i.e., that there exist constants A > 0 and # > 0 such that )~e _ )~ and #e - # for all s > 0.
Sect. 8.4]
The t h r e e - d i m e n s i o n a l equations over a d o m a i n i n d e p e n d e n t of e 409
Remark. A multiplication of both Lam~ constants by a factor et, t E I~, is otherwise always possible, as we shall see in the next chapters (Sects. 9.1 and 10.2). The choice t - 0 is made here for definiteness. [] Then functions f i ( s ) . ft -+ IR and h i(e) defined by letting
9 F+ t2 F_ --+ I~ are
]i'e(xe) - " f i ( e ) ( x ) for all x e - 7rex E gte, hi'~(x e) - " hi(e)(x) for all x e - 1 r e x e F~_ U
A w o r d of c a u t i o n . At this stage, it cannot be a priori assumed as in Sect. 3.2 for the linear theory that the scaled displacements ui(e) are "of order 0 with respect to e". It so happens that this will be the case, in fact the only possible case, for the nonlinear theory, but proving this requires considerable efforts (Thm. 8.7-1)! Likewise, the "right" orders on the contravariant components of the applied forces are yet to be found (as in the linear theory). Finding these (which depend on the kinds of shells that are considered) again requires considerable efforts (Thms. 8.8-1 and 10.1-2)! m A simple computation then yields the variational problem that the scaled unknown u(e) satisfies over the set ft, thus over a domain that is independent of ~ (the Christoffel symbols ra'e ~'a3 and F ~ e vanish in ft e for the special class of mappings O considered here; consequently, the functions F~a(e ) and F~a(e) defined below likewise vanish in ft)" T h e o r e m 8.4-1. Let w be a domain in I~2, let 0 E C2(~; I~a) be an injective mapping such that the two vectors aa - OaO are linearly independent at all points of-~, and let eo > 0 be chosen as in Thm. 3.1-1. With the functions gij, e, rijP,e, ge, Aijkt, e . - ~ --+ R appearing in problem 7~(ft e) (Sect. 8.3), we associate for each 0 < e ~ ~o the AiJk ( ) 9
R
:=
.-
for all x e - lr~x E ~ ' .
by
410
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
Then for each 0 < e < co, the scaled unknown u(e) satisfies the following variational problem 79(e; f~), called the s e a l e d t h r e e - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m of a n o n l i n e a r l y e l a s t i c shell: ~(~) - (~i(~)) e w ( a ) . -
{~ - ( ~ ) e w l , 4 ( a ) ;
~ - o o~ r o } ,
ffl AiJU(e)Eklll(e; u(g))Fillj (g; u(e), v) v/g(e) dx g
+uP_ for all v C W ( ~ ) ,
where 1
lgmn
1
and ~aw~(e) . - o ~ ( ~ ) u311~(e) "- O~us(e) ~lls(~)
- r~(e)~(e), -
F~s(e)u~(e ),
"- - 1 0 ~ o ( ~1 - r ~ ( ~ ) ~ ( ~ ) ,
U3jl3(g ) "-- 1031~3(g), g
~,l~(~) . - o ~ Vstl~(e) "- 0~v3 ~oll3(~) "
10~
- r~(~)~p, F~3(e)v~,
-
- r~(~)~
V3JJ3(g) "-- _1C93V3. g gl
The functions AiJU(e) are called the c o n t r a v a r i a n t c o m p o n e n t s of the s c a l e d t h r e e - d i m e n s i o n a l e l a s t i c i t y t e n s o r o f t h e shell. The functions EillJ(e; u(e)) are called the s c a l e d s t r a i n s in c u r v i l i n e a r c o o r d i n a t e s because they satisfy EillJ(~; u(~))(x) - Ei~lj(U~)(xe ) for 511 x ~ - 7r~x E ~ .
Geometrical and mechanical preliminaries
S e c t . 8.5]
411
Note that the above definitions likewise imply that Fillj(r u(e), v)(x)
-
F/~lj(u~ , v~)(x ~) for a11 x ~
illj(x e) and VillJ(e)(x)
-
-
~ : x
e ~,
ve for all xe = ~r~x ~ ~e.
8.5.
GEOMETRICAL AND MECHANICAL PRELIMINARIES
Let w be a domain in ~2 and let 0 E C3(~; IR3) be an injective mapping such that the two vectors aa = 0a0 are linearly independent at all points of ~. Let ai and a i denote the vectors of the covariant and contravariant bases along the surface S = 0(~). The covariant and contravariant components as~ and a s~ of the metric tensor of the surface S (Sect. 2.1), the covariant and mixed components bs/3 and bfla of the curvature tensor of S (Sect. 2.2), and the Christoffel symbols r ~ of S (Sect. 2.3) are then defined by letting (whenever no confusion should arise, we henceforth drop the explicit dependence on the variable y E ~): as~ ::
as
bsf3
a 3 " Of~aa,
::
9a ~ ,
:=
a sf3 : = a s 9af3~
bfla "- afkrb~a,
a'.
Note the symmetries" -
a , . ,
-
-
The area element along S is v/a dy, where a : : det(as~). The covariant derivatives b~[s of the curvature tensor, defined in this case by means of its mixed components, are defined by (Thm. 2.5-1)
All the functions aaf3, a a~, ba~, ~ , continuous over the set ~.
r:,,.,
and b~la are thus at least
412
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
Our first result, which is essentially Thm. 3.3-1 rewritten in a slightly different form, gathers all the "geometrical" preliminaries needed in the sequel for the asymptotic analysis of nonlinearly elastic shells, regarding the behavior as e --+ 0 of the functions gij (e), r,~ (e), and g(e) (defined in Tam. 8.4-1). Note that, while these are functions of x - (y, x3) e ~2 - ~ • [-1, 1], their l i m i t s / o r e - 0 are functions of y E -~ only, i.e., the limits are independent of the transverse variable x3. T h e o r e m 8.5-1. Let w be a domain in R 2, let 0 E C3(~; iR3) be an injective mapping such that the two vectors aa - OaO are linearly independent at all points of-~ and let eo > 0 be chosen as in Thm. 3.1-1. Then
gij (e)
-
-
a ij + e x 3 g ij' 1 + O(e2),
aiJ .= a i . a j
r,5(~)
-
r ~,o
-o~
I
:__ 2aaCrb~ ~
gi3 , 1 = 0 ,
r,~ 0 + ~,~r,jp, 1 + o(~), where .
.__
gaf3,
,
where
~.o
~..o T~cr, 1
- b ; l o , - o ~ :=
-o~
1" cr
-=-b~b,,
3,0
up,0
r
a3 = ' 3 3
::
0~
r
3,1 rp, 1 a3 = ' 3 3 ::
0,
and finally, g(e) = a + O(e), for all 0 < e _< co, where the functions a #, ba~, b~, r ~af3, b~]a, and a are identified with functions in C~ and the order symbols O(e) and O(e 2) are meant with respect to the norm i[ " [[0,c~,~ defined by i1~110 oo ~
:=
sup{lw(m)l; m e n )
Proof. The asymptotic behavior of each function gZ3(e) immediately follows from the relations ~ a ( g ) _ act ~ gx3b~atr + O ( g 2 ) ,
established in the course of the proof of Thm. 3.3-1. All the other n relations were already proved in the same theorem.
Sect. 8.6]
The method of formal asymptotic ezpansions
413
Observe that the notation a i1 is consistent with that used for the contravariant components a af3 -- a a . a f3 of the metric tensor of the surface S; note also that a i3 = ~i3. Our second result, which is a restatement of Thm. 3.3-2 (b), gathers all the "mechanical" preliminaries needed in the sequel, in that it describes the behavior as ~ --+ 0 of the scaled contravariant components AiJkt(e) of the three-dimensional elasticity tensor (defined in Thm. 8.4-1). Note that their limits for e = 0 are again functions of y E ~ only. T h e o r e m 8.5-2. Let w be a domain in I~2, let 0 C g3(~; i~3) be an injective mapping such that the two vectors aa are linearly independent at all points of-~, and let ~o > 0 be chosen as in Thm. 3.1-1. Then the contravariant components AiJkt(e) - AJikt(e) -- AktiJ(e) of the scaled three-dimensional elasticity tensor satisfy
AiJkt(e)v/g(e ) = AiJkt(0)v/-d + 6B ijkl,1 + eZBiJkt,2 + o(62), where
Aa~'rr(0 ) : - Aaa~a ,rr + #(aa'ra ~r + aara~'r), Aaf333(0) := Aa aft, AaZ'r3(0) "- pa a'r, A3333(0) := )~ + 2/z, A
(0)
-
A
333 (0) " - 0,
for all 0 < ~ <_ co, where the functions a E C2(~) and a af3 E C 2 (-~) are identified with functions in C 2 (-0), the functions B ijkt' 1 and B ijkt' 2 are continuous over-~, and the order symbol o(e 2) is meant with respect to the norm II 9 II0,o , (defined in Thm. 8.5-1). II
8.6.
THE METHOD EXPANSIONS
OF FORMAL
ASYMPTOTIC
The specific dependence on s of the equations of problem 7~(s; f]) observed in Thm. 8.4-1 and the idea that s is a "small" parameter naturally lead to apply the m e t h o d of f o r m a l a s y m p t o t i c e x p a n s i o n s , briefly described infra. For general presentations, see Lions [1973] and Eckhaus [1979]. This method consists in using the following basic A n s a t z :
414
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
(i) Write a priori u(e) as a formal asymptotic expansion
u(s) -
1
-N
1 + s N ' l u --hr+l q - ' ' " , N E Z,
where the successive terms u -N, u -N+I, etc., in this expansion are independent of e. The "first" nonzero term u -N (nonzero if the applied forces are nonzero!) is caUed the l e a d i n g t e r m , and more generaUy u q, q ~_ - N , is called the t e r m of o r d e r q, of the formal expansion; the three dots " . . . " account for the fact that the number of successive terms that wiU be eventually needed is left unspecified at this stage; the expansion is formal in that it is not required to prove that the terms u q, q _~ - N + 1, do exist in the space W(fl), let alone that the above "series" converges! Note that, while in the linear case there was no loss of generality in assuming ab initio that the leading term is of order 0 (Sect. 3.4), this freedom is lost in the nonlinear case (see Thm. 8.7-1)! (ii) Equate to zero the factors of the successive powers e q, arranged by increasing values of q, found in problem 7~(6; f~) when u(6) is replaced by its formal ezpansion, solve the resulting variational equations and, assuming ad hoe properties on whichever successive terms are needed, pursue this procedure until the order - N of the leading term u - N is determined. In the present case, the "ad hoc properties" consist in assuming that each term needed in this process is in the space Wl'4(f~) and that it is required to satisfy the boundary condition on P0 only if this is necessary for continuing the recursion. (iii) Pursue likewise this procedure until the leading term can be fully identified, presumably as the solution of an ad hoc variational problem. (iv) The following two requirements constantly guide the procedures described in (ii) and (iii). The first requirement was systematized by Miara [1994a] for linearly elastic plates (see also Vol. I, Sect. 1.10) and Miara & Sanchez-Palertcia [1996] for linearly elastic shells (see also Sect. 3.4); the second was systematized by Miara [1994b, 1998] for nonlinearly elastic plates (see also Vol. I, Sect. 4.11) and shells (see Thins. 8.8-1 and 10.1-2). The first requirement, valid in both the linear and nonlinear cases, asserts that no restriction should be imposed on the applied forces
Sect. 8.7]
The leading term is of order zero
415
entering the right-hand side of the equations used for determining the leading term, since no such restriction is imposed on the "original" problem 7~(e; f/). The second requirement, valid in the nonlinear case only, asserts that, by retaining only the linear terms in any relation satisfied by terms of arbitrary order in the formal asymptotic expansion of the scaled unknown u(e), a relation of the linear theory should be recovered. Loosely speaking, we thus wish that the operations of taking "formal limits" as ~ goes to 0 and of linearizing should commute. For brevity, we shall call " l i n e a r i z a t l o n t r i c k " this second requirement. Applying the above Ansatz, we will first establish that N = 0, i.e., that the leading term is of order 0 (Thin. 8.7-1). We will then identify (first in Thin. 8.8-1, then in Thins. 9.2-1 and 10.3-4) the two classes of variational problems that the leading term can satisfy, according to two classes of specific assumptions bearing on the geometry of the middle surface of the shell and on the boundary conditions, together with the corresponding assumptions (in terms of powers of e) on the components of the applied forces. The proofs of these results, which are due to Miara [1998] and Lods & Miara [1998], are at times exceedingly delicate (they were already somewhat intricate in the linear case; cf. Sect. 3.4). They show the highly improbable efficiency of the method of formal asymptotic expansions: Seemingly endless and inextricable computations eventually lead to two-dimensional problems whose formulations exhibit a striking simplicity!
8.7.
THE LEADING
T E R M IS OF O R D E R Z E R O
As shown by Miara [1998, Thin. 1], whom we follow in this section, a first virtue of the Ansatz described in Sect. 8.6 is to afford the specification of the order of the leading term in the expansion of the scaled unknown: T h e o r e m 8.7-1. Assume that the scaled unknown satisfying problem 7'(~; f/) (Thin. 8.4-1) admits for each 0 < 6 < eo a formal
416
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
asymptotic expansion of the form 1
u(e) -
~eN u
-N +
1
eN_l u
-N+I + . .. , with
U - N , U - N + I E W(~'~) -- {V E wl'4(~'~), V - - 0 on F0}, u - N r 0,
for some integer N E Z. Then N -
O.
Proof. The proof is broken into seven parts. Before beginning the induction proper in (iv), we record several useful preliminaries: (i) Let the functions AiJm(O) be defined as in Thin. 8.5-2. Then, for any symmetric matrices (sin) and (tu) ,
AiJm(O)smtij -
(%aaf3a~r + lt{aaaa f~r + aaraf3~})s~rta~ +41za a~ sa3tz3 + )~aa/3s33ta13 -4-Aatrrs~rrt33 -4- (A -4- 2p)saata3.
This formula, which immediately follows from the definitions, will be constantly put to use in the ensuing arguments. (ii) Let a ij := a i . a j. Then, for any y E ~ and any matrix (tij),
a '~ (y)amn(y)timtjn >_ 0 and a'~(y)amn(y)timtjn = 0 r tij - O. Given any y E ~ and any matrix (tii), let ti(y) : - timam(y) and let [ti(y)] q denote the q-th Cartesian component of the vector ti(y). We thus have
aij(y)amn(y)tirntjn - aU(y) {(timam(y)) . (tjnan(y))} = =
tj(u)) ([tj(yll,
J(y))
-
E
p
Hence aU(y)am"(y)timtjn > 0 and, since the three vectors ai(y) are linearly independent,
a i j ( y ) a m " ( y ) t i m t j . - 0 =~ [ti(y)]Pai(y) - O, 1 <_ p <_ 3, ::~ ti(y) = timam(y) -- O, 1 _< i _< 3, =~ t i m = O , l <_i, m <_3.
Sect. 8.7]
417
The leading term is of order zero
(iii) Assume that the formal asymptotic expansion of the scaled unknown is of the form 1
~(~) = ~ w '~-
N
1 + ~N-~
U -N+I
for some integer N > 0,
+'''
with u - N E W(f~) and u -N+I E W(~2) (under these assumptions, it will be eventually shown that N = 0; thus the assumption N < 0 will be de facto ruled out). Together with the asymptotic behavior of the functions gZ~(e) a=d r,~ (~) as ~ -~ 0 ( T h e . S.5-1), such ~ e~pa=sio= i=duces specific formal asymptotic expansions of the various functions appearing in the formulation of problem :P(s; f~), viz., .
1
.
-N
umll.(~) - ~--~u~ll,~ + ' " , 1
1
-N-~
-N
-~- ~--NUmll3+ " "
Umi]3(~) -- ~N+l'/tmii3
'
1 E_2N E.II~(~; u ( ~ ) ) -
e2 N
-li~
+'"'
1
+
..,
Ectll3(~; u(~)) - ~2N+l 1 ~,-2N-2 E3il3(~; U ( ~ ) ) -- ~2N+2.~30i3 -+-''" ,
1
F.ii,(~; ~(~), ~) - ~ W F ~ g ( ~ ) + . . . , ~ +1~ F~I~-~ (~) + . . . ,
F.ll3(~; ~,(~) , ~) _
F3113(~; u(s), v) - sN+2F31131 _N_2(V)-~- "'" where, by definition, uq111'Eq/llj' and F/~lj(v ) designate for each q E Z the coefficient of s q in the induced expansions ofu/llj(s), EillJ(s; u(e)), ~ . d F~ltj(~; u ( ~ ) ,
~).
Note in passing that, while the functions factorizing the powers of s are by definition independent of e, they are dependent on one or several terms u q, q > - N . In this respect, particular caution should be exercised as regards this dependence. For instance, u -N
relic,
- 0~
N -r',~
-N a~d
-a,n-p
- N _ 03~&N+I
umlI3
..,,o
-N
-- l.'mSUp ,
418
A s y m p t o t i c analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
1 i.e., the factor of ~ in Umlla(e) depends only on u -N but the one in
Umll3(e) depends also on u -N+z (a special notation will be introduced whenever such different dependences could lead to confusion; see, e.g., part (ii) of the proof of Thin. 8.8-1). Likewise, it should be remembered that the expression of some factor may differ according to which value of N is considered; for instance, 2N 1 mn -N -N E~I ~ - ~a UmllaUnllO i f N >__1,
=
F ojj -N(v)
-
-
1
0
("~
+",llo +
mn
"
0
0
lio"-II,) if N = 0,
lamn { -N N + u -nllf~Vmlla } if N >- 1, UrnllaVn[l~ 1
- ~ (vall~ + V~lla + am" {U~
0
+ Unll~Vmlla}) if N - O,
where
Vmlla :-- OaVm -- rP'0" ~" otm '~P" We are now in a position to start the cancellation of the factors of the successive powers of e found in the variational equations of problem 7~(e; f~) when u(e) is replaced by its formal expansion. In what follows, L r designates for any integer r > - 3 N - 4 the linear form defined by
Lr(v) := fn f i ' r v i ~ d x
+ fr+ur_
hi, r + l v i v f a c ~ ,
and it is always understood that the functions/,,r and h i'r+l belong to the spaces L2(12) and L2(r+ U r _ ) and are independent of e. (iv) Assume that N >_ O. Since the lowest power of s in the left-hand side is e -3N-4, we are naturally led to first try
1
f i ( e ) __ e 3N+4
1h fi,-3N-4 and h~( e ) - ~e3N+ 3 i, -3N-3
Then the cancellation of the coefficient of e -3N-4 leads to the equation (the functions ~..,3113 and F3113 defined in (iii)):
f n ( )~ + 2p ) E 3-1123N- 2F3 t--1~- 2(v ) v/'d d x - L-3N-4(v )
Sect. 8.7]
The leading term is of order zero
419
for all v E W(fl). Since E -2N-2 3113
_
-
lamnOau~nN O3u-~N and F31-~3N-2(v)- amnOaUmN O3vn, 2
we must have
L - 3 N - 4 ( v ) -- / n f i ' - 3 N - 4 v i ~ d x
-t- ~ + u r - h i, -3N-3 Vi V~ dr - 0
for all v E W(f~) that are independent of x3. Consequently, the first requirement (that there be no restriction on the applied forces) implies that we must let fi,-3N-4 = 0 and h i'-3N-3 = O. Letting v - u - N in the resulting variational equations then shows that
/12 E3''3-2N-2F3,,3-N-2 ('tt -N) ~ dx - -~l / (amnO3u~O3unN)2~/a dx - O. Since the symmetric matrix (a ij) is positive definite, we conclude that
03 u - N - (03urn N) - 0 in ~, i.e., that u - N is independent of x3. We thus infer that E3~I2N-2 -- 0 ~nd F ~ l ~ - ~ ( ~ ) - 0 fo~ ~11 ~ e W ( a ) . Cancelling the coefficient of e -3N-4 thus yields the following relations (as usual, any function defined on ~ that is independent of x3 is identified with a function defined on ~)"
'U-N E W ( w ) : = E ~llJ -~-2
{71 = (~i) E wl'4(og); ~ = 0 on "Y0},
: 0 in a ~ d ~ ~ - 2 ( ~ )
_ 0 fo~ ~11 ~ c W ( a )
since ~3113~-2N-2= 0 and __F3113-N-2(v)-- 0 and the leading terms in the highe~ th~n ( - 2 N 2) ~na ( - N 2), ~especti~ly. Since E~-I~;-1 : 0 (the leading term in the formal expansion of E~ll~(~; ~(~)) ~s of o~de~ - 2 N ) ~ d ~1~] ~ - ~ : 0 ( s ~ 0 ~ - ~ - 0, 1 each factor of e2N+ 1 in the expansion of Eall3(e; u(e)) vanishes because it contains some derivative 03u~ N and the leading term in the expansion of E3ll3(e; u(e)) is of order strictly higher than ( - 2 N - 1)), we also have E -2N-1 = 0 in f~.
420
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch.
8
Our next try is thus
f i ( s ) _ s 3N+3 1 f , -93 N - 3 and hi (e) - saN+ 1 2 h "9- 3 N - 2 . The cancellation of the coefficient of s -3N-3 then yields the equations (the functions AiJm(O) are defined in Thin. 8.5-2)"
fft AiJkl (O)F~h-[]21N-1Fi~lT-2(v)v/-a dx
-
-
L - 3 N - 3 (V )
for all v E W(f~) 9 But since D?-2N-1 -"~klll - - 0, we must let f i , - 3 N - 3 __ 0 and
h i'-3N-2 = 0 (first requirement) and accordingly try ]i(s)
_
1 ~3N+2 f~'9 - 3 N - 2
1 h i, - 3 N - I and hi(e) = e 3 N + 1
in which case the cancellation of the coefficient of s - 3 N - 2 yields the equations
f
-
for all v E W ( ~ ) . But s i n c e F / H T - 2 C v ) -- 0 for all v C W ( ~ ) , we must l e t f i , - 3 N - 2 = 0 and h i,-3N-1 = 0 (first requirement). (v) Assume that N > 1. Our next try being thus
1 ft,-3N-1 9 f i ( e ) __ e 3N+1 and hi (e) - - -1~ h ' '9- 3 N the cancellation of the coefficient of S - 3 N - 1 in the variational equations of problem 7~(e; ~) then yields the equations
fn AiJkl(O)E-2NF-N-l(v)x/~dz kill i115
-
L-3N-I(v )
-
for all v E W(f~), where
1 mn - N ~.2 N
"o11
-2N E3113
-N
F-N-I(v)
~llf3
-- O,
iamnu-N u-N =
.,llo
iamn -- 2
1
mn
-N
-
.11 ,
-N -N ~tmll3UnJl3'
=
-N O am . Umlt3
v.
,
Sect. 8.7]
421
The leading term is of order zero
N being those defined in (iii); recall that 03Un N -- 0 the functions u -mtli by (iv) and that we assume N > 1 (otherwise some functions, such as E~I~;, have different expressions when N = 0; cf. (iii)). Letting v E W(f~) be independent of x3 then shows that we must let f i , - 3 N - 1 = 0 and h i ' - 3 N = 0; hence / n A i j k t l a ~ E - 2 N F - N - l (v )v/-d dx - 0 x"J kllg illJ
Let the field w N _ (w N) be defined for all
for all v C W(f~). (y, x3) C f~ by
Wm N .___ U~nlV-k-1 _
O -N (1 + X 3 ~) lp ,m3 Up
T h e n w N E W(f~) because both u - N and u - N + I are assumed to be in the space W(f~) (this double assumption is thus crucial!). Furthermore, 03w N = Umll3 - N , so that
F-fIN3-1( w N) - Ea-l~ ~
-2N
and F 3 ~ -I (w N) - 2E3113 .
Letting v - w N in the last variational equations thus shows that
faAijkl(o~E
- 2 N F - N - l (wN)v/-d dx - 0, J kilt illJ
where 2)~amnE-2N
Aijkl(a~"-2NF-N-I(wN)=
~''J~kll/
illJ
mlin
- 2 N -b 4 a t o n e - 2 N rz'-2N E3tl3 P roll3 "e'nll3 "
Since amnE_2N
-
1 a ija mnu -Nu -N
rnlln -- 2
> 0 in f~
(by (ii))and
1 mn -N -N _ m n r r t - 2 N rrw-2N -2N E3113 -- -~a Umll3Unll3 >_ 0 and a "~'ml13"C'nll3 >_ 0 in
(the matrix (a ran) is positive definite), we conclude that a m
hence that
n
rry - 2 N
rrv - 2 N
"~ml13 aWnll3
E-2N
-- 0 in f~,
roll3 = 0 i n f l .
422
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
1 mn -N -N In particular then, E3~122v- ~a umll3Unll3 = 0 and thus
-N
Umll3 - -
O,
(vi) Assume that N > 2 (the case N - 1 is considered separately; cf. (vii)). Our next try being thus f i ( e ) .._
_ _1 ~ ] , ,. - 3 N and
hi
~s 3 N1h_
(r
1
i, -32v+t
the cancellation of the coefficient of s -aN in the variational equations of problem :P(e; ~) then yields the equations (note that two terms are needed here from the expansions of the functions AiJkt(e)v/g(e); cf. Thin. 8.5-2)
E_2N+IF_N_ 1
~kllZ "~llJ
+ f• BiJkI'IE-2NF-N-lhIIt il1r
kilt
i11i
(v)dx = L-aN(v)
for all v E W(fl), where
1amnu_
~-~
= ~
~,l-i~ ~ -
o by
N _ -N
~llo~.ll, '
(~), 1 mn -N VOSVn,
F~l-i~-~(~) - o by (~), F -N(v) "11~
F___N (.
)
lamn
--
-IV
- N
(U..llaVnll~ + unll~ Vmlla)
a m n u - N + I dO V
the last expression of F~N3 (v) being valid only if N _> 2 (the expressions of Fa]IN(v ) are not needed since Aaaar(0) - 0 and E~I~N -- 0). N 1 Noting that Fa~la(v) -- ~3~3N ( v ) -- 0 if C93V -- 0 we thus conclude that the above variational equations reduce to
fo A~176
r
- L-~(~)
Sect. 8.7]
The leading term is of order zero
423
f o r all v E W(f~) t h a t are i n d e p e n d e n t
of x3. Since each t e r m in is c u b i c with respect to the func-
the s u m A a # ~ r ( O ~ E - 2 N F - N ( v )
N tions u -mlla' the l i n e a r i z a t i o n t r i c k (second requirement) implies t h a t
-- 0 for all v E W(f~) t h a t are i n d e p e n d e n t o f z a . Hence we m u s t let f i , - Z N = 0 and h i ' - a N + l -- O. Since u - N is independent of x3 by (iv), we may let v - u - N in the last variational equations. This gives L-aN(v)
~I-adz = O,
since F - g ( u - N ) _ 2 E - 2 N
But
A a#'rr (0) - )~a af~ a 'rr + # ( a a'r a ~r + a ar a fi~
( T h m . 8.5-2) and thus 2N
1 mn
E~-llfl -- ~ a
-N
-N
UmllaUnllf3 - 0
in f~
(to reach this conclusion, observe that aatrafart~rta~ >_ 0 and t h a t - 0 only if taf~ - 0 by (ii)); these relations in t u r n imply t h a t aa~afart~ta/3
u -N
mll~
-0
By definition (cf. (iii) and T h m . 8.5-1), N - ~pp, a/30up- N -- Oa u/3- g -- r ~a~ U~ N -- ba/aU3
u -~lla N _
Oau~N
-N uall~
0 ~ u ~ N _ r~,p, ~ 3 Oup- N __ OaU3 N .jr b ~ u ; N
Let ~i " - u i N I z s = O 9 T h e n ~i C Wl'4(w) since u~-N E Wl'4(f~) and 0 3 u [ N - 0 in f~ and ~i - 0 on q'0 since u~-N - 0 on r0. The above relations combined with the Gaul] and Weingarten formulas (Thm. 2.3-1) t h e n imply t h a t O a ( ~ i a i) - 0 in w, hence t h a t ~i - 0. We have thus shown t h a t u - N - 0 for all N > 2. o
(vii) F i n a l l y , a s s u m e t h a t N t h a t now
1. The o n l y difference from (vi) is mn
0
03 Vn"
424
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
But since the arguments that led in (vi) to the conclusion that u - N - 0 for N ~_ 2 only required the consideration of functions e w ( ~ ) that are i~d~p~d~t of ~, in which ca~e F~I~(~) - 0, they can be reproduced v e r b a t i m for N -- 1, thus showing that U -I -- 0~
and the proof is complete.
8.8.
m
I D E N T I F I C A T I O N OF A T W O - D I M E N S I O N A L VARIATIONAL PROBLEM SATISFIED BY THE LEADING TERM
It has just been shown (Thin. 8.7-1) that the asymptotic expansion of the scaled unknown is of the form u(e) - u ~ + eu I § According to the Ansatz of the method of formal asymptotic expansions (Sect. 8.6), there thus remains to continue the cancellation of the factors of the successive powers of e in the variational equations of problem 7)(e; ft) until the leading term u ~ can be fully identified, as the solutions of an a d hoc variational problem. In this direction, the following result, due to Miara [1998, Thin. 2], constitutes the key step. A w o r d of caution. The next theorem only states (in the form of a variational problem) a necessary condition that the leading term should satisfy. While there are cases (studied in Chap. 9) where this condition is also sufficient, i.e.,where this theorem also constitutes the final stage of the application of the method of formal asymptotic expansion, there are cases (studied in Chap. 10) where additional stages (which require considerable efforts!) are needed to complete the induction, m T h e o r e m 8.8-t. Assume that the scaled unknown u(c) = (u,(~))
satis~ing problem P(s; ft) (Thin. 8.4-1) admits a formal asymptotic ezpansion of the form + e2u 2 +-.. , with ~0 e w ( ~ ) and ~ , u 2 e W 1'4 (~). u(e) - u ~ + e u I
Sect. 8.8]
A two-dimensional problem satisfied by the leading term
425
Then in order that no restriction be put on the applied forces and that the linearization trick be satisfied (Sect. 8.6), the components of the applied forces must be of the form
fi, e(Xe ) -- fi, O(x )
for all
hi, e(xe) - ehi, l(x)
for all
x e : 7rex E ~e
where the functions fi, O E L2(~) and h i'1 E pendent of e.
L2(r+ u r_) are inde-
This being the case, the leading term u ~ is independent of the transverse variable x3 and r := ( r 89f~t u~ dxs satisfies the following two-dimensional variational problem:
~0 e W(w)"-- {n e wl'4(w); n -- 0 on 7o},
f a ,~.~-E~ll~Fallf3 o o (r/) C'd dy - f . p"~ fo~ ~n n - ( ~ )
e W(~),
where (recall that a mn -- a m . an) 9
Eo Fo
1
0
1
_ m n ,,O
~.~{
0
o
o
r/all/3 : - c9/~77a- r"a~rkr - ba/~rl3, r/sll/3 "= O/3r/s + b~rl~, aaf ~~ : :
pi, O :=
4A# aa~aar + 2#(aaaaf~r + a a r a ~ ) , A+2#
f
~i, 1 + hi,_ 1 and h~ t "- h ~'"t (', + 1 ) . fi, Odxs +,o+ 1
9
426
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
Pro@ The proof comprises three parts. (i) To begin with, we remark that the conclusions of part (iv) of the proof of Thm. 8.7-1 are valid for any N >_ 0 and under the sole assumption that u(s) can be expanded as u(e) - u - N + . . . , with u - N E W(f~). Letting N - 0, we thus infer that 03u ~ - 0 in f~ and that
r176 2
1
u ~ dxa e W ( w ) : = { r / e Wl'4(w); rl - 0 on 70},
and also that (recall that the functions Eq,~ and ~q,~(v) are by definition the factors of e q in the formal exp~ansions"of the functions EillJ(e; u(s)) and FiltJ(e; u(e), v) defined in Tam. 8.4-1). E/~Ij = 0 for all integers q < - 1 , F/~Ij(v) - 0 for all integers q < - 2 and all v E W(f~), and, finally, that we must let
fi,-2
__
0 and h i'-1 - O .
(ii) Our next try is thus f i ( ~ ) __
l fi,-1 a n d h i ( e ) - h i'O
where it is understood as in the proof of Thm. 8.7-1 that each function ]i,r E L2(f~) and each function h i'r+l E L2(r+ u r_), r >_ -1, appearing here and subsequently is independent of ~; likewise, we again let
Lr(v) . - f f fi, rviv/-adx + f r
+ur_
h i,r+l v v ddr, r >_ - 1 .
The cancellation of the coefficient of e-1 in the variational equations of problem "P(e; f~) (Thm. 8.4-1) then yields the equations (the functions AiJkt(O) are defined in Thm. 8.5-2):
il]j ffz Aijkl(o)EO[ll F-l(v)~/-ddx - L-l(v)
Sect. 8.8]
A two-dimensional problem satisfied by the leading term
427
for all v E W ( ~ ) , where Eoa
1
o
=
Eoa
E~ F
~ilt3
113 --
+
1 , (0)
mn 0
0
2(.Ual13 + U31la -4,(o)
0
11o .11 ) ,
+
amnu 0
u(O)
~II~ nil3)'
l amnu(Oml u(O)
- "~3[13A-~
13 ,,113'
= 0,
1 (Oava + a mn o
03Vn),
and (the functions riP~0 are defined in Thm. 8.5-1) 0 o _ pp, 0~,o and u (~ t ~.p, 0 0 U~ll~ := OaUm - ~ m - p roll3 "- 0 3 u ~ - rm3U p, The special notation u~l13 (which thus replaces the notation uOII 3 used in the proof of Thin. 8.7-1) emphasizes that, by contrast with the functions u~ which only depend on u ~ -(u~ the functions u(~
also depend on u 1 - ( u 1) (recall that u(m~ is by definition the
coefficient of s o in the formal asymptotic expansion of Umll3(e)). For this reason, the occurrence of the functions U(m~ implies that the formal asymptotic expansion of u(e) be "at least" of the form u(~)
-
u ~ + eu
1 +....
The expressions of the functions F / ~ ) ( v ) i m p l y that L - Z ( v ) - 0 for all v = W(12) that are i n d e p e n d e n t of z3. Hence we must let / i , - 1 _ 0 and h i'O - 0 (first requirement), so that we are left with the equations v/adz = 0
for all v = (vi) e W ( f l ) . When the functions F / ~ (v) are replaced -iby their expressions given supra, the integrand in the above integral takes the form (wrO3vr + w3Osv3) for ad hoc functions w i E L4/3(~). Then Thm. 3.4-1 shows that the functions w r and w 3 vanish
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
428
[Ch. 8
in fl, i.e., that ()~aa~E~ ~ + ()~ + 2#)E~
a
+2pE~ 3 (a ar + aaaa/3ru~ (Aaaf3E~
-4-()~ + 2p)E~
-- 0 in ~,
(o)
+ ~3113'
+2paa~E~
-Oinfl.
One obvious solution to this system of three equations is E~ 3 - 0 and Aaa/3E~
+ (A + 2p)E~ 3 = 0 in f~,
but there might be other solutions to this nonlinear system. Denoting by [... ]tin the linear part with respect to (any component of) u ~ or u 1 in the expression [... 1 and using the notations of the linear case (Sect. 3.4), we have
]lin ~l13J - e~ 2lz~E 0 ]lin _ X,,a/3o0 E
[
+
+
J 3113J
o as the coefficients of e ~ in by definition of the functions E/~I3 and eill3
the formal expansions of the functions Eilla(e; u(e)) and ei[[3(e; U(e)), the latter being precisely the linear parts in the former! Since it was found in the linear case (see step (ii) in Sect. 3.4) that e~ - 0 and e3113 0 = - A +)~2 # aagJeall~ 0 in f~, the linearization trick (second requirement) suggests that we only retain the "obvious" solution found above (as shown by CoUard [1999], this plausible argument can be made entirely rigorous; cf. Ex. 8.4). (iii) Our next try is thus
fi(e ) _ fi, O and hi(e) - eh i'l. The cancellation of the coefficient of e ~ in the variational equations of problem 7~(e; ~) then leads to the equations (two terms are needed here from the expansions of the functions AiJkl(e)v/-g(e); cf. Thin. 8.5-2): L
1 -1 AiJkl(O){E~lliFi~lj(v ) + EklItF/II j (v)}v/-d dx
+ ffl Bijkl'l EklliF/llj 0 -1 (v)dx - LO (v)
A two-dimensional problem satisfied by the leading term
Sect. 8.8]
429
for a11 v C W(K~), where the functions Ei~]j and F/~lj(v ) are defined by means of the formal expansions
0 + ~E~ IJ + . - . ~ ~llj(~; u(e)) - E~ltj 1 F~llj(~; u(e), v) - ~ ~ ) ( ~ )
0 + F~jlj(v)+....
Note that, while the functions E ~ F'7'1"()2113 "v'' and F~ (v) depend only on u ~ and u 1, the functions E/~lj depend also on u 2 (but not on u3; each term involving u 3 vanishes because it contains some derivative Oau~ as a factor). For this reason the formal asymptotic expansion of u(e) must be "at least" of the form 2 +....
q--O
In particular then, we must have
ff Aijkt(O)E~
dz -- L~
for all v E W(f~) that are independent of x3 since F~(v) -- 0 for "1" such functions; equivalently, after performing the usual identification, we must have
for all r / e W(w) - { r / e Wl'4(w); r / - 0 on ")'0}. Using the expressions of the functions Aijkl(O) and the relations satisfied by the functions E/~I3 (see part (ii)), we are left with
:
f.
()~aaf3a~r +
p(aaO.a#r+ aara#O.)) EallrFallf~(~l)~ 0 0 dx
4- ff~ (4paaaE~176 +
/o
()~aarEOll r + (X +
12foaa~E~176
4-AaaBE~176
) x/adx
2#)E3113 o ) F~)lla(r/)~dx dx -
L~
430
[Ch. 8
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
for all rl E W(w), where aO~f~ o-.r .__
4A# aa~a,7r + 2p(aaCra~.r + aar af3O.). A+2p
Since u ~ E W ( f t ) is independent of x3, it may be identified with a function ~0 C W(w). Consequently, the functions EOll~ = ~(~~ 1 o ) e L2(~ ) , , + U~lla~+ ~mnum10laUnllf3
F211~('~)=
1 ~('7,,11~ + '7~11,~+
amn
0 0 {",,,11,~'7,,11~ + ",,I1~'7,,,11o})e
L2
(~),
which are thus also independent of x3, may be likewise identified with functions (denoted for convenience by the same symbols)" Eo II~ = ~(~oll~ 1 0 + ~ll~ + ~_mn.o ~0 ~ll~-It~)
FOal
1
amn
0
e L 2(~1, 0
L2
where r/ail/9 - O~rla -r~D~Ta --ba~9~73 and ~73[I/3-- 0~9z}3-{-b~r/~ for all ~ - (r/~) E W(w). The last variational problem is thus indeed two-dimensional, m
The functions a a~r found in Thm. 8.8-1 are the contravariant components of the scaled two-dimensional elasticity tensor of the shell, already encountered in the linear theory (see, e.g., Sect.
3.4).
EXERCISES
8.1. The three-dimensional energy J in Cartesian coordinates associated with a St Venant-Kirchhoff material is defined by (the notations are those of Sects. 8.1 and 8.2)" 1
Ezercises
431
Show that there exist constants a > 0 and/3 ~ I~ such that -
'Ca) +
for all ~ e W ( ~ ) - {~ = (~3i) e W~'4(h); ~ = 0 on r 0 ) . Remark. The energy .] is thus coercive on the space W ( ~ ) . It cart be shown, however, that it is not weakly lower semi-continuous on W ( ~ ) ; cf. Ball & Murat [1984], Raoult [1986], Dacorogna [1989], Le Dret & Raoult [1995a]. 8.2. (1) Using the notations introduced at the beginning of Sect. 8.2, show that the variational equations in Cartesian coordinates of Sect. 8.1 may be equivalently rewritten as
O) d~ - /f /i~)id~ + f hivi aT, 1
where
(2) Using the notations and definitions of Thms. 8.2-1 and 8.2-3, show directly, i.e., without resorting to the functionals .] and J found in these theorems, that
~) d~ - f a AiJkIEkllt(u)Fillj(u' v ) v / g dx.
8.3. Show that the boundary value problem of three-dimensional nonlinear elasticity in curvilinear coordinates (Thm. 8.2-4), viz.,
-(~/J + crkJg/~u~llk)llj -- A in KZ, ui = 0 on to, (aij + ahJgitulllk)nj -- hi on r l , where a ij - AiJklEkllt(u), can be directly derived from its Cartesian counterpart (Sect. 8.1), viz.,
~i -
0 on to,
432
Asymptotic analysis of nonlinearly elastic shells: Preliminaries
[Ch. 8
where a ~j - .4iJkZ/~kZ(/L) ("directly" means without recourse to the variational equations, as in Thm. 8.2-4). 8.4. It was observed in part (ii) of the proof of Thm. 8.8-1 that an obvious solution to the nonlinear system of equations: (Aaaf~E~
+ (A -4- 2p)E~
3
-4-2pEa~ 3 (a ar A-- a atra/3ruO/3tia) - 0 in [2, (o)
+2paa~EOll 3u311 ~0 -- 0 in [2 is
_a~O E~ s -- 0 and ~~,~, ~ail~ + ()~ + 2p)EOIi 3 - - 0 in ~.
The objective of this problem is to find all the other solutions. (1) Show that another solution is u(~l3 - 0 a n d
U~l~)3 - - 1 .
(2) Show that all the remaining solutions are of the form u(a~I3 -- (aa/3 + U~
(o)
o
~ and "3113 - u3iiaw
- 1,
where the functions w a depend only on the functions u~ Remark. These results are due to Collard [1999].
CHAPTER 9 NONLINEARLY
ELASTIC
MEMBRANE
SHELLS
INTRODUCTION
The purpose of this chapter is to identify and to mathematically justify the two-dimensional equations of nonlinearly elastic "membrane" shells. These equations are of two kinds, depending on whether they are obtained through the method of formal asymptotic expansions or by means of a convergence theorem. We begin with the formal approach, which is due to B. Miara. Given a surface S = O(w) and a displacement field yia i of S with smooth enough covariant components r/i : ~ --~ 1~, let aao(r/) denote the covariant components of the metric tensor of the associated deformed surface (0 + ~Tiai)(-~). A nonlinearly elastic shell with middle surface S subjected to a boundary condition of place along a portion of its lateral face with 0(3:0) as its middle curve, where 70 C 7, is a nonlinearly elastic "membrane" shell if the manifold ~ 0 ( w ) = { r / = (Yi) e w l ' 4 ( w ) ; r / = 0 on 70, aaB(rl)-aaB = 0 in w}
reduces to {0} (Sect. 9.1). This means that 0 is the only displacement field rlia i of the surface S with covariant components ~i in W1,4(w) that is admissible and ineztensional, i.e., that vanishes along the curve 0(70) and leaves invariant the metric of S. This definition is motivated by the crucial observation that the induction begun in the previous chapter ends when ~ 0 ( w ) = {0}. The conclusions reached in Thm. 8.8-1 can then be reformulated as follows in this case (Thm. 9.2-1): Assume that ~ 0 ( w ) -- {0}. Then the contravariant components of the applied body forces must be of the form
/i'e(Xe)
: fi'0(X) for all x' -- 7r'x E f~',
where the functions ]i,0 are independent of e (for simplicity, we assume in this introduction that there are no surface forces), the
Nonlinearly elastic membrane shells
434
[Ch. 9
leading term u ~ 9-~ --+ I~3 o/ the formal asymptotic expansion of the scaled unknown is independent of the transverse variable, and r ~ f~_~ u ~ dz3 satisfies the following (scaled) two-dimensional variational problem o/ a nonlinearly elastic "membrane" shell: r
e w~(~)
- { ~ e w ~ , 4 ( ~ ) ; ~ = o oil ~0},
aaf3a'rGa'r(~o)(Gla/3(~~ ~/~dy - ~ pi'O~Ti~rady for all r / - (r/i) C WM(w), where, for any ~, r/E w l ' 4 ( 0 g ) , 1
1
G o ~ ( o ) = ~ ( a . ~ ( O ) - a.~) = ~(U-il, + ~,li- + I
am"~mll"~nll')'
1
aO~f~ O'T _
f,
p,, o _
4)~# aa~ aO.T + 2#(aaCraf3r + aar a/3O.), )~+2#
1
/,, o dza.
The functions Gaf3(r/) and a a/3~r are respectively the covariant components of the change of metric tensor associated with a displacement field 77iai of the middle surface S and the contravariant components of the (scaled) two-dimensional elasticity tensor o/ the shell. These equations also express that ~0 is a stationary point o / a certain functional over the space WM(W), so that particular solutions may be obtained by solving a minimization problem (Thm. 9.3-1), written here directly in its "de-scaled" version (Thm. 9.4-1): The vector field ~e = ( ~ ) formed by the covariant components of the limit displacement field ~ a i of the middle surface S satisfies" ~e e WM(W) and j ~ ( ~ e ) =
inf j~f(r/), n~WM(~)
where the two-dimensional energy jeM of a nonlinearly elastic membrane shell is defined by a a~Crr'e(a,Tr (17) - aar)(aa~(~) - aa~)V~ dy f pi, e~Tiv/-ddy,
Introduction
435
where 4 A e p ~ aa~SaSt" + 2/.te (aaO-alSr + aar a/SO-),
As + 2p ~ p,,e =
f,,e dx~. e
The functions a a~vr are the contravariant components of the twodimensional elasticity tensor of the shell, Ae and #e are the Lamfi constants of the material constituting the shell, and the functions fi, are the contravariant components of the body force density applied to the shell. The stored energy function of a nonlinearly elastic membrane shell is thus remarkably simple: It is a quadratic and positive definite expression in terms of the ezact difference between the metric tensor of the deformed middle surface and that of the undeformed one. Incidentally, this shows that the equations of the linear theory of "membrane" shells are immediately recovered (at least formally) under linearization of the nonlinear equations. We also briefly discuss in Sect. 9.4 approaches that could lead to an existence result for the above minimization problem or for the associated boundary value problem (Thin. 9.4-2). We conclude this chapter by reviewing in Sect. 9.5 a justification of yet another two-dimensional "membrane" shell theory, this time by means of a convergence theorem, the first of its kind for shells. More specifically, another two-dimensional minimization problem for a nonlinearly elastic shell has been derived from nonlinear threedimensional elasticity by H. Le Dret and A. Raoult for a realistic class of hyperelastic materials. Using F-convergence theory, they have shown that the minimizing deformations of the three-dimensional energies, once appropriately scaled over a fixed domain f~ C I~3, weakly converge in the space Wl,V(f~) as the thickness of the shell approaches zero (cf. Thm. 9.5-1; the exponent p > 1 is "governed" by the hyperelastic material considered). The limit deformation found in this fashion is independent of the transverse variable and minimizes a limit energy obtained by computing the F-limit of the three-dimensional energies; hence the existence of a minimizer of this limit energy is de facto established.
Nonlinearly elastic membrane shells
436
[Ch. 9
The limit stored energy function is again that of a nonlinearly elastic "membrane" shell, in the sense that it contains only first derivatives of the unknown deformation. For a St Venant-Kirchhoff material, it does not coincide, however, with the energy found via the method of formal asymptotic expansions, save when the singular values of the limit deformation gradients belong to a specific subset of the plane. For this reason, it does not always reduce under linearization to the linear theory of "membrane" shells. The origin of such differences may lie in that the latter approach models shells made with "soft" elastic materials (like the sails of a sailboat or a balloon), while the former models shells made with "rigid" elastic materials (like the hull of a ship or the roof of the Hong Kong Convention and Exhibition Centre). But this affirmation is yet to be substantiated. 9.1.
N O N L I N E A R L Y E L A S T I C M E M B R A N E SHELLS: DEFINITION, EXAMPLES, AND ASSUMPTIONS ON THE DATA
The analysis of Chap. 8 culminated in Thm. 8.8-1, where it was shown that the leading term u ~ of the formal asymptotic expansion of the scaled unknown u(e) is independent of x a and that ~0 := 1 f_l 1 u 0 dxs satisfies:
r = (r
e
- o on 7o}
e
and f~
aaf~arEOl[ r alif3(n)v/-ddy pZ'~ F~ f~ "
for all r / = (r/i) E W(w), where
EOll~ "-
1(C~
+ C llo +
amn o
o C, ll Cnll )"
An inspection of the functions E~ ~ appearing in the left-hand side of the variational equations thus immediately reveals that two fundamentally distinct situations must be considered, depending on whether there are non-zero fields r/ = (r/i) in the space W(w) that satisfy r/all~ + ~lla + amnr/mllar/n[lD - 0 in w.
Sect. 9 . 1 ]
Definition, ezamples, and assumptions on the data
437
If there are no such fields, the induction is terminated (this situation is studied in this chapter), while the induction may be continued if there are such fields (see Sect. 10.1). In this direction, the first task consists in recognizing in the functions E all~ ~ (half of) the differences between the covariant components of the metric tensor of the "deformed" surface (0 + ~~ and those of the metric tensor of the "undeformed" surface S = 8(~) and then in showing that the functions F~ are simply G~teaux derivatives at ~0 of these differences. Together, these expressions will in turn provide an illuminating interpretation of the variational problem satisfied by ~0 (Thm. 9.2-1) (the definitions of the covariant and mixed components baf~ and b~ of the curvature tensor and the Christoffel symbols I~f~ of S appearing in the next theorem are recalled in Sect. 8.5). T h e o r e m 9.1-1. Let w be a domain in ~2 and let 8 E g2(~; I~3) be an injective mapping such that the two vectors aa = Oa8 are final A a 2 early independent at all points of-~, let a3 - [al A a2[' and let the vectors a i be defined by a i . a j - ~ . Given a displacement field ~Tiai of the surface S 8(-~) with smooth enough covariant components Yi " -w -+ R, let the covariant components of the c h a n g e of m e t r i c t e n s o r associated with this displacement field be defined by 1
a . ~ ( n ) := ~ (a.~(n) - aa~) , where aa~ and aaf3 ( y ) are the covariant components of the metric
of
(O + 1
-
i(n-lJ
+ n il- + m n ll-n it ),
r/allf3 := 0/3~7a - F ~af}Tla.- baDr]3 and ~7311f3:= Ofjr/3 + b~?a. The Gdteaux derivatives of each function Gaf3" W 1' 4(w) ~ L2(w) are given by 1
am n
Nonlinearly elastic membrane shells
438
for all ~, r/ C W~'4(w)
9
[Ch. 9
In particular then, the functions E ~
all~
and
F~ appearing in the variational problem found in Thin. 8.8-1 are also given by
E~
= a a # ( r ~ and F~
= a~af3(r176
Proof. Let 0 "--yia'. By definition,
a,e(n) - (a, + o~,)). (ae + oe#) = a,~+o~O.a,~ +O,~O'a~+O,,O'O~O. By the formulas of Gaufl and Weingarten (Thm. 2.3-1), OaO - ~ l l a a ~ + r/311~a3, Since a i . a a - ~ia and a m . a n - a ran, it thus follows that
aa~(rl) - aa# -- r/all# + ~Z311a+ amn~mlla~nll~" Hence E~ - Ga/s(r176 That F~ - G'a#(r176 results from the way derivatives of continuous linear and bilinear mappings are computed (see, e.g., Vol. I, Sect. 1.2). II
Remarks. (1) The functions ~?illft may be also expressed in terms of the "two-dimensional" covariant derivatives rlil~ of the vector field 0 = rlia i (Thin. 2.3-1) as ~llz3 - ~1~ - ba[3~73 and ~7311~- r/31~l - b~r/~. The double bars used in the functions r/i[]f3 remind that they are restrictions for x3 = 0 of "three-dimensional" covariant derivatives
(Sect. 1.4). (2) The components Gaf3(r/) of the change of metric tensor are the restrictions for x~ - 0 of the "three-dimensional" strains E s ~llj(v~) with r / : = velz3=0. This was to be expected in view of their respective significances (Thms. 8.2-2 and 9.1-1). i
Sect. 9 . 1 ]
Definition, ezamples, and assumptions on the data
439
In this chapter, we consider the case where the only field ~1 = (71i) in the space W ( w ) - { r / E W 1' 4(w); r / - 0 in w} that satisfies 7/all~ -f- ~?/31la-4- amn~lmlta~Tnll/3 -- aa#(~l) -- aa# = 0 in w
is r/ = 0. This assumption, combined with Thin. 8.8-1, then leads to the following fundamental definition: Let w be a domain in I~2 with b o u n d a r y 7 and let 0 E g2(~; i~3) be an injective mapping such that the two vectors 0a0 are linearly independent at all points of ~. A nonlinearly elastic shell with middle surface S = 0(~) is called a n o n l i n e a r l y e l a s t i c m e m b r a n e shell if the following two conditions are simultaneously satisfied: (i) The shell is subjected to a boundary condition of place along a portion of its lateral face with 0(70) as its middle curve, where the subset 70 C 7 satisfies length 70 > 0. (ii) Define the manifold: := {n
-
c
,1 a,
z(n) -
0 o= a, a -
o i=
Then = {o).
A displacement field ~Tia i of the middle surface S is an i n e x t e n d i s p l a c e m e n t of S if aa~(~/) - aa~ = 0 in w. These relations imply t h a t the surfaces S - O(~) and (O + ~iai)(-~) are i s o m e t r i c , i.e., t h a t their metrics are the same (in particular then, the lengths of curves are the same; cf. Sect. 2.1). A displacement field ~Tia i of the middle surface S is a d m i s s i b l e if it satisfies ad hoc b o u n d a r y conditions along the curve 8(70), in the present case "boundary conditions of simple support", expressed by means of the b o u n d a r y conditions ~ / - 0 on 7o on the associated field 17 = (~/i) (these b o u n d a r y conditions are interpreted later; cf. Sect. 9.4). The assumption ~ t 0 ( w ) -- {0} thus expresses that the only admissible (i.e., such t h a t r/i - 0 on 3'0) inextensional displacement yia i of the middle surface S with covariant components Yi in the space W l'4(w) is zero. sional
440
Nonlinearly elastic membrane shells
[Ch. 9
e(z0• I-a, a])
Fig. 9.1-1: A nonlinearly elastic membrane shell. A shell whose middle surface S = 0(~) is a portion of a cylinder and which is subjected to a boundary condition of place (i.e., of vanishing displacement field) along a portion (darkened on the figure) of its lateral face whose middle curve 0(70 ) contains the two "end-curves" of S provides an instance of a nonlinearly elastic membrane shell, i.e., one for which the manifold M O ( W ) = { ~ = (17i) E W l' 4(O./); 71 = 0 o n 7 o , a a o ( r / ) - a a o = 0 i n w }
reduces to {0}; cf. Ex. 9.1.
E z a m p l e s of nonlinearly elastic m e m b r a n e shells are shown on
Figs. 9.1-1 a n d 9.1-2 a n d analyzed in Exs. 9.1 a n d 9.2.
A w o r d o f c a u t i o n . T h e definition of a nordinearly elastic memb r a n e shell depends only on the subset of the lateral face where the sheU is subjected to a b o u n d a r y condition of place (via the set 70) a n d on the geometry of the middle surface of the sheU (via its two fundam e n t a l forms a n d Christoffel symbols, which a p p e a r in the functions ~i[l~); see Ex. 9.3. m
A n o t h e r w o r d o f c a u t i o n . A l t h o u g h the above definition is strongly suggested by the a s y m p t o t i c analysis, it is a d m i t t e d l y perfectible, as it leaves aside nonlinearly elastic shells t h a t should also be d e e m e d " m e m b r a n e " ones.
Sect. 9 . 1 ]
Definition, ezamples, and assumptions on the data
441
r
Fig. 9.1-2: Another ezample of a nonlinearly elastic membrane shell. Consider a plate subjected to a boundary condition of place on a portion of its lateral face whose middle line 70 has the following property: There exists (at least) one direction such that the intersection of ~ with any line parallel to this direction is a finite union of segments whose end-points belong to 70 (this is thus the case if 7o = 7)- Then such a plate is a nonlinearly elastic membrane shell; el. Ex. 9.2. By contrast, such a plate is always modeled as a flex~al shell (even if 70 = 7) when it is viewed as a linearly elastic body! See Fig. 6.1-3.
C o n s i d e r for i n s t a n c e a shell whose m i d d l e surface is a half-sphere, s u b j e c t e d to a b o u n d a r y c o n d i t i o n of place along its entire l a t e r a l face (i.e., 70 = 7). T h e n :~40(w) ~ {0}, since s y m m e t r i e s w i t h r e s p e c t to planes o r t h o g o n a l to the axis of r e v o l u t i o n p r o d u c e a family of continuously varying isometric surfaces (O + ~Tiai)(-w) w i t h rl - (yi) e W l ' ~ 1 7 6 a n d rl r 0. Hence such a n o n l i n e a r l y elastic shell is not a " m e m b r a n e " one a c c o r d i n g to the above definition. Yet it is n e i t h e r a "flexural" one according to the definition given in Sect. 10.2, as it seems plausible t h a t the m a n i f o l d ~'t~,(w) i n t r o d u c e d in ibid. contains only two elements (the covariant c o m p o n e n t s of t h e a d m i s s i b l e i n e x t e n s i o n a l d i s p l a c e m e n t s in ~4~,(w) are t h e n r e q u i r e d to be in W2'4(w); see a g a i n Sect. 10.2). As h i n t e d by this e x a m p l e , w h e t h e r or not a m a n i f o l d of i n e x t e n s i o n a l d i s p l a c e m e n t s reduces to a finite set or not critically d e p e n d s on the smoothness allowed on the admissible inextensional displacements.
442
Nonlinearly elastic membrane shells
[Ch. 9
This challenging question is related to that of the rigidity of surfaces, a classical problem in differential geometry: A compact surface "without boundary" is rigid if any other such surface in IR3 with the same metric differs from it by a rigid deformation, i.e., a mapping of the form x E I~3 --~ a + Q o z , where a E I~3 and Q is an orthogonal matrix of order three; again, the allowed smoothness on the surfaces play a critical r61e. See, e.g., Pogorelov [1956], Nirenberg [1962], Klingenberg [1973, Thin. 6.2.8], Spivak [1975, Vol. V, Chap. 12], Berger & Gostiaux [1992, Thin. 11.14.1]. The same question can also be viewed as one about the existence of everted states under vanishing applied forces. For details about such eversion problems for shells (examples are given in Vol. I, Figs. 5.8-1 and 5.8-2), see in particular Srubshchik [1968, 1972, 1980], Vorovich [1969], Mel'nik & Srubshchik [1973], Antman [1979], PodioGuidugli, Rosati, Schiaffino & Valente [1989], Szeri [1990], Geymonat & L~ger [1994], Antman [1995, pp. 501-503], Antman & Srubshchik
[1998].
m
Since the formal asymptotic analysis ends with Thm. 8.8-1 when ~vt0(w) = {0), we are naturally led to make the following a s s u m p t i o n s on t h e d a t a for a family of nonlinearly elastic membrane shells, with each having the same middle surface S = 0(~) and with each subjected to a boundary condition of place along a portion of its lateral face having the same set 0(70) as its middle curve, as their thickness 2e approaches zero: We require that the Lamd constants and the applied body and surface force densities be such that
)~e = A and #e = #, fi'e(xe) = fi'O(x) for all x e = 7rex E f~',
hi'e(z e) - ehi'l(x)
for all
x' - 7re~ C I'~_ U r ~ ,
where the constants )~ > 0 and # > 0 and the functions fi, o E L2(f~) and h i'1 C L2(r+ U r _ ) are independent of e (Fig. 3.2-1 recapitulates the definitions of the sets f~e, f~, r ~ , r + , r e_, and r _ ) .
Note that the same limit two-dimensional equations are evidently obtained if the following more general a s s u m p t i o n s on t h e d a t a
Sect. 9.2]
The two.dimensional equations as a variational problem
443
are made: ,ke = ~t)~ fi'~(x') - etfi'~ ") -
and for all for an
pe = 6tp, x ~ - 7r'x C f~', -
u r'_
where the constants .k > 0 and p > 0 and the functions fi,0 E L2(f~) and h i' 1 C L 2 (I~+ (2r_) are again independent of 6 and t is an arbitrary real number. Besides, the analysis of Chap. 8 (see in particular Thm. 8.8-1) shows that these assumptions on the data are the only ones possible for nonlinearly elastic membrane shells. 9.2.
THE TWO-DIMENSIONAL VARIATIONAL PROBLEM
E Q U A T I O N S AS A
We know from Thm. 8.8-1 that the leading term u ~ of the formal asymptotic expansion of the scaled unknown u(6) is independent of the transverse variable x3 and that, once identified with a function ~0 of two variables, it satisfies a two-dimensional variational problem. When the manifold 2rio(w) reduces to {0} (Sect. 9.1), no inconsistency arises in the formulation of this problem, and thus the induction stops here. We also saw in Thin. 9.1-1 that the functions E~ ~ and F~ occurring in the formulation of this problem have more illuminating expressions than those found in Thm. 8.8-1; as a result, these new expressions shed a new light on the understanding of this variational problem, henceforth denoted 79M(W). The equivalent boundary value problem is given later; cf. Thm. 9.4-2. The following result, which is an immediate corollary to Thms. 8.8-1 and 9.1-1, is due to Miara [1998]. Notice that a new notation, viz., WM(W), is used from now on in this chapter for the space heretofore denoted W(w). T h e o r e m 9.2-1. Consider a family of nonlinearly elastic membrane shells according to the definition of Sect. 9.1, with thickness 2~ > O, with each having the same middle surface S = O(-~) and with each satisfying a boundary condition of place along a portion of its lateral face having the same set 8(70) as its middle curve, and let the assumptions on the data be as in Sect. 9.1. Finally, assume that 0
C3( ; R3).
Nonlinearly elastic membrane shells
444
[Ch. 9
Then the leading term u ~ 9-~ -+ IR3 of the formal asymptotic expansion of the scaled unknown u(e) is independent of the transverse variable and ~o .-'- 1_2s u~ dx3 satisfies the following scaled t w o - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m ~M(W) of a n o n l i n e a r l y elastic m e m b r a n e shell:
~0 e WM(W)"-- {r/e Wt'4(w); r/-- 0 on 70},
~ aaf3arGar('~(G~;3(,~
- L pi'~
dy
fo~ ~ n . = (,i) e W ~ ( ~ ) .
where,/or any r ~/ e Wt'4(w) (the functions aa~3(r/) are defined in Thm. 9.1-1),
1 1
= ~(~il~ + ,7~lt,~ + a"%~ll,~'7-1it3), ,
1
aat3(r
-
~('7ol1~ + '7~li,~ + a'"{c:,~lio'7,~li~ + c:,.li~'7,.ilo}),
r/all~ "= O~r/a aaO~r~. .=
F~t3~/~-
ba/3'q3
and ~3il/3 : - - 0/3'/'/3 + b~r/o-,
4X# aa~ a,~r + 2#(aa,~aOr + aa ~.aoa) ' $+2#
f , .o ._.
.
. ~i, 1 /,,o dz3 + h~ z + h ''z and ,o+ := h ~'" 1 (., +1),
1
and Ga~(~ I ) E s 1 4(w); L2(w)) denotes the Fr~chet derivative at E Wl'4(w) of each mapping Ga[3" w l ' 4 ( w ) -+ L2(w). II
Notice the analogy between the two-dimensional variational problem PM(W) and the variational problem of three-dimensional nonlinear elasticity we started from (Thm. 8.2-3).
Sect. 9.3]
9.3.
The two-dimensional equations as a minimization problem
THE TWO-DIMENSIONAL EQUATIONS MINIMIZATION PROBLEM
445
AS A
As noted by Miara [1998], the aforementioned analogy can be pursued further: First, the variational equations found in the twodimensional and three-dimensional problems simply express in each case that the Ggteaux derivatives of an ad hoc functional vanish; second, the integrand in the functional is in each case (apart from its linear term accounting for the applied forces) a quadratic and positive definite expression (via an elasticity tensor) in terms of a change o]
metric tensor. These crucial observations in turn allow to recast the two-dimensional variational problem 79M(W) as a minimization problem, which now exhibits a remarkable simplicity. In this problem lies the apex of the application of the method of formal asymptotic expansions to nonlinearly elastic membrane shells. T h e o r e m 9.3-1. Given a nonlinearly elastic membrane shell according to the definition given in Sect. 9.1, let the space WM(W) and the functional jM " WM(W) -+ I~ be defined by WM(W)
"-- { ~ -- (~7i) e W I ' 4 ( W ) ; ~ -- 0 o n ' ~ 0 } ,
1~ a a ~ r (aa.v(n) -
j M ( n ) "-- ~
a~r)(aa~(l?) - a a ~ ) v ~ d y -- f~ Pi' ~
dY'
where the functions aa#(17) are defined in Thin. 9.1-1 and the functions a a#~r and pi, 0 are defined in Thin. 9.2-1. Then the functional jM is differentiable over the space Wl'4(w), hence over WM(W), and r C WM(~) is a solution to the variational problem ~DM(W) Of Thm. 9.2-1 i] and only if it is a stationary point of the functional jM over the space WM(w), i.e., it satisfies j~M(~~ -- O. Particular solutions to problem 79M(W) are thus obtained by solving the minimization problem: Find ~ such that r E WM(W) and j M ( r
inf
neWM(.~)
jM(rl).
Nonlinearly elastic membrane shells
446
[Ch. 9
Proof. Since
jM(O) - ~l ~ a a f 3 t r r a~(O)a,,~(O)4"ddy-
f pt'~ "
where 1 G af3 Cvl ) -- "~( aaf~ ( vl ) -- aaf3 )
1 = ~(rh~ll~ + rl~lla + a'~n'q~ll,~r/,~ll~), an argument similar to that used in the proof of Thm. 8.2-3 shows that j M is differentiable over Wl,4(w) and that its G~teaux derivatives j ~ ( ~ ) y are given by
jtM(~) . -- ~aaf~a'r(~o.r(~)((~tafl(~),q)vfady- fwpi,~ for all ~, y E Wl'4(w), where 1
a',(r
= ~(~11, + ~,ll~ + a~"{r
+ i-li,~ll~})"
The variational equations satisfied by ~0 E WM(w) (Thm. 9.2-1) thus coincide with the equations
j~(r
_ 0 fo~ ~n ~ e w ~ ( ~ ) ,
which are themselves equivalent to the equation j~(~0) _ 0. That it makes sense to consider the above minimization problem stems from the observation that inf JM(~) >--OO, n~WM(~) an inequality itself a consequence of the coerciveness of the functional over the space WM(w) (Ex. 9.4). The existence theory proper is otherwise fragmentary at the present time; see the discussion in the next section, i
jM
The functional j M " WM(w) --~ ]~ is called the scaled twod i m e n s i o n a l e n e r g y of a n o n l i n e a r l y elastic m e m b r a n e shell.
Sect. 9.4] 9.4.
The two-dimensional equations derived by a formal analysis
447
TWO-DIMENSIONAL EQUATIONS OF A NONLINEARLY ELASTIC MEMBRANE SHELL THE
DERIVED
BY MEANS
ASYMPTOTIC
OF A FORMAL
ANALYSIS; COMMENTARY
In order to get physically meaningful formulas, it remains to descale the components (o of the vector field ~0 that satisfies the scaled two-dimensional problems found in Thms. 9.2-1 and 9.3-1. In view of the scalings ui(~)(x) - u~(x e) for all x e - vrex e ~e made on the covariant components of the displacement (Sect. 8.4), we are naturally led to defining for each ~ > 0 the c o v a r i a n t c o m p o n e n t s ~ 9~ --4 I~ of the l i m i t d i s p l a c e m e n t field ~e ._w__4 I~3 of the middle surface S of the shell by letting (the vectors a i form the contravariant basis at each point of S): r := (o and ~ := r
i.
Remark. I realize that, ]or once, my notations ( ~ vs. ~o) are somewhat misleading! 1 A w o r d of c a u t i o n . Naturally, the fields ( ~ ) and ~e _ ~ a i must be carefully distinguished! The former is essentially a convenient mathematical "intermediary", but only the latter has physical significance. 1 Recall that fi,~ E L2(~ e) and h i'c E L2(r u r!) represent the contravariant components of the applied body and surface forces actually acting on the shell and that Ae and p~ denote the actual Lamd constants of its constituting material. We then have the following immediate corollary to Thms. 9.2-1 and 9.3-1. T h e o r e m 9.4-1. Let the assumptions be as in Thm. 9.2-1. Then the vector field ~e := ( ~ ) formed by the covariant components of the limit displacement field ~ a i of the middle surface S satisfies the following t w o - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m 7)~(w) of a
448
Nonlinear'ly elastic membrane shells
nonlinearly
elastic m e m b r a n e
r e w.(~).=
[Ch. 9
shell:
{n e w~.4(~); n = o on ~0},
fo~ ~ n , = (~) e W u ( ~ ) , where 1 a . , ( , 7 ) "= 5 ( a . , ( ' l ) - a . , ) 1
= ~(~11~ + ~il,~ + a'~"'~-,li,~,,ll~), a'~(r
1
= ~(~oll~ + ~ll~ +
am n
{i~ll~,ii~ + ~ - t l ~ i l ~ } ) ,
r/allfj "-- Of3rla -- F~flrla -- ba~vl3 and v/31lf~"- Off'r/3 -4- b~vkr, aa~ar, e 9= p',~ . -
4)~ei~e aa/3a ~r + 21~e(aaea[3r + aara~e), A e + 2# e f
r
. . . ]',~ e ~ + h ;, ~ + h ',~ a=d h~. ~ . - h"~(., •
6
The field r _ ( ~ ) is a solution to problem 7)~(w) if and only if it is a stationary point of the functional j ~ " W M ( w ) ~ ~ defined by
jeM(~}) "= ~r f~ aa/3crr,e (a~r(~}) -- a~r)(aa~(~l) -- aa~)~fa dy -- f~ Pi' e~Tiv ~ dy"
Particular solutions to problem 7)~(w) may thus be obtained by solving the following minimization problem: Find ~e such that
~e E WM(W) and j ~ c ( r
inf j~c(r/). n~WM(.,) m
Sect. 9.4]
The two-dimensional equations derived by a formal analysis
449
We next derive the boundary value problem t h a t is (at least formally) equivalent to the variational p r o b l e m 7 ~ ( w ) found in T h m . 9.4-1. In the next theorem, (t,a) denotes the unit outer n o r m a l vector along 3' a n d 3'z := 3 ' - 3'0Theorem 9.4-2. Let the assumptions and notations be as in Whm. 9.4-1. If a solution ~e _ (i~) to the variational problem
e f aa~3~r'eGvr(,e)(G~a/3(,e)rl)v/-ddy-f pi'e~liV/-ddy for all r / :
(~i)
e
WM(60)
is smooth enough, it also satisfies the following boundary value problem: $
- ( n al3,e + n aG "aar~rlla)l~ + b~na/3'er
_ba/3(na/3, e + n~,eaars -~tl~) - (na~,e r e
~ _.._pOtg' in w,
: r ,3'' in w, ff~ = 0 on')'o,
(rta/3, e 4- n ~13' e u_ a r , 'qrll~ e ~/) ) D -- 0on71, e
na#(311avr -- 0 on 71,
where
n~,, ._ e~~,,~a~(r the functions Ga~(~e), a aoar'e, and pi, e are defined as in T h m . 9.4-1 and, for any vector field with differentiable contravariant components ~a : -~ __+ R and any tensor field with differentiable contravariant components n a[3 9 --+ R,
~~
-
o,~ ~ + r]~
~ ~na n ~
"= O~no~ + r ~ n ~ + r ~ n ~
Nonlinearly elastic membraneshells
450
[Ch. 9
Proof. For notational brevity, the dependence on e is omitted. Using Green's formula as in the proof of Thm. 4.5-1, we obtain l ~ ~na~(~Tall/3 + Y/311a)dY- - ~ ~{(na/sl/3)~?a + ba/3na~y3) + f~ v/'ana/3v/3yadT, 1_2~ V/-dna/3amn {~mlla~Tnll# +" ~nlt/3~Tmlla}dy =
f~ V/-dna/3amn~mlla~Tnll/3dy
=- f~ ~{(n~a~l~ll.)la -b~n~i311~}U~dy -
f~ ~/a{ba/3n~[3aar~rll~ + (na~311a)]~}y3 dy
+
+
}
for all (rli) e w l ' 4 ( w ) . Letting (Yi) vary first in (D(w)) 3, then in WM(W), yields the announced boundary value problem, m Each one of the three formulations found in Thms. 9.4-1 and 9.4-2 constitutes one version of the t w o - d i m e n s i o n a l e q u a t i o n s of a n o n l i n e a r l y elastic m e m b r a n e shell (the specific meaning conveyed by "membrane" is given below). 1 The functions Ga~(~/)- ~(aa~(~/)aa~) are the covariant components of the c h a n g e of m e t r i c t e n s o r associated with a displacement field yia i of the middle surface S (Thm. 9.1-1) and the functions a a~crr'e a r e the contravariant components of the t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of t h e shell. The functional j ~ 9WM(W) -+ I~ is the t w o - d i m e n s i o n a l energy, and the functional
r ~ aa~ar'e (a~r(rl)-
9/ e WM(W)-+ g
a~r)(aa~(17)- aa[3)x/ady
is the t w o - d i m e n s i o n a l s t r a i n energy, of a n o n l i n e a r l y elastic m e m b r a n e shell. Finally, the functions n a~,e are the contravariant components of the s t r e s s r e s u l t a n t t e n s o r field. The functions yal~ are instances
Sect. 9.4]
The two.dimensional equations derived by a formal analysis
451
of first-order covariant derivatives of a vector field defined by means of its contravariant components ya and the functions n af3 I~ are firstorder covariant derivatives of a tensor field defined by means of its contravariant components n af3 (such covariant derivatives have already been encountered in Thm. 4.5-1). Considered as relations with respect to the ]unctions n af3,e, the partial differential relations in w and the boundary conditions on 71 found in Thm. 9.4-2 constitute the t w o - d i m e n s i o n a l e q u a t i o n s of e q u i l i b r i u m of a n o n l i n e a r l y elastic m e m b r a n e shell. The relations n a~'~ - eaa~ar'eGaf3(~ e) constitute the t w o - d i m e n s i o n a l c o n s t i t u t i v e e q u a t i o n of a n o n l i n e a r l y elastic m e m b r a n e shell. Finally, the boundary conditions Yi - 0 on 70 express that the points of the undeformed and deformed middle surfaces 0(~) and (0 § ~iai)(-~) coincide along the curve 0(70). For this reason, they are called t w o - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s of s i m p l e support. A major conclusion is thus that, without any recourse to any a priori assumption of a geometrical or of a mechanical nature, the
method of formal asymptotic expansions provides a justification of the two-dimensional equations o] a nonlinearly elastic membrane shell, in the forms found in Thin. 9.4-1. This justification by Miara [1998] constitutes a generalization to shells of the formal analysis of Fox, Raoult & Simo [1993], who likewise identified and justified the twodimensional equations of a "nonlinearly elastic planar membrane", i.e., one for which the mapping O is of the form O(yl, Y2) -- (Yl, Y2, 0) for all y -: (yl, y2) C ~ (see also Vol. II, Sect. 4.12). Further important conclusions and comments are in order about the present justification: First and foremost, the resulting shell theory is a n o n l i n e a r " m e m b r a n e ~' t h e o r y in the sense that the s t o r e d e n e r g y funct i o n of a n o n l i n e a r l y elastic m e m b r a n e shell, defined by r/-
(~7i) e Wl'4(w) -4 g aa/3~rr' e (acrr(r/)- aar)(aao(rl)- aao),
is a quadratic and positive definite expression (via the two-dimensional elasticity tensor of the shell) in terms of the change o/metric tensor, i.e., of the exact difference between the metric tensor of the deformed middle surface and that of the undeformed one.
452
Nonlinearly elastic membrane shells
[Ch. 9
Note in passing the truly remarkable simplicity of the above stored energy function! Second, the resulting theory is f r a m e - i n d i f f e r e n t in the sense that the value of the above stored energy function is unaltered if r := 0 + ~ia' is replaced by Q~,, where Q is any orthogonal matrix of order three (Ex. 9.5). Third, the resulting theory is a large d i s p l a c e m e n t , or equivalently a large d e f o r m a t i o n , t h e o r y in the sense that the de-scaling produces a displacement field that is O(1) with respect to e, at least in a formal sense. Finally, a noteworthy characteristic of the equations found in Thm. 9.4-1 is that their formal linearization produces the equations of the linear "membrane" theory (Chaps. 4 and 5). This is perhaps best seen from the expression of the energy, in which the covariant components Ga~(~/) - 89 aa~) of the change of metric tensor become by definition the covariant components 7a~3(~/) of the linearized change of metric tensor (Thm. 4.5-2). An inspection of the partial differential equations in the associated boundary value problem (Thm. 9.4-2) likewise shows that they reduce to those of the linear theory when 70 = "Yand the middle surface S is elliptic (Thm. 4.5-2).
Remarks. (1) Otherwise the function spaces and the boundary conditions have to be appropriately modified in the linearization process, so that the two-dimensional equations of the linear membrane theory become well-posed over the resulting spaces VM(W) or V~M(W) (Thms. 4.5-1 and 5.7-1). (2) As observed by Miara [1998], another unexpected link with the linearized equations occur if higher-order forces are considered; cf. Ex. 9.6. II Note that Collard & Miara [1999] have shown that the formal analysis of Miara [1998] described supra also leads to the explicit computation of the limit stresses in a nonlinearly elastic membrane shell. The existence theory for the minimization problem is challenging: The energy j ~ is coercive on the space WM(W) (Ex. 9.4), but, as shown by Genevey [1997], j ~ is not sequentially weakly lower semicontinuous on WM(W) (essential to her proof is the non-polyconvexity of the stored energy function of a St Venant-Kirchhoff material es-
Sect. 9.5b]
The two-dimensional equations derived by F-convergence theory 453
tablished by Raoult [1986]). Hence the fundamental theorem of the calculus of variations cannot be applied (by contrast, this theorem can be successfully applied to nonlinearly elastic "flexural" shells; cf. Sect. 10.6). Another tool that is sometimes successful for proving existence theorems in nonlinear elasticity is the implicit ]unction theorem applied to the associated system of partial differential equations~ which in the present case is quasilinear (Thm. 9.4-2). But, even though the existence and regularity results for the linearized equations of a linearly elastic elliptic membrane shell are firmly established (Thm. 4.5-1), the implicit function theorem cannot be applied to the nonlinear membrane shell equations, expressed as a set of three nonlinear partial differential equations together with the boundary conditions ~ - 0 on 7, if only because the boundary condition ~ - 0 on -), is "lost" in the linearization. As a result~ the derivative at ~e _ 0 of the associated nonlinear operator is not an isomorphism between the "right" spaces, i.e., those of the linearized boundary value problem; cf. Genevey [1997]. As shown by Coutand [1997b, 1999b, 1999c, 1999d, 1999e] in the planar case~ the implicit function theorem can nevertheless be successfully used in particular situations, e.g., if the forces are themselves planar, or if the boundary conditions ~i = 0 on 9' are replaced by boundary conditions of "planar tension". Again in the planar case when ~ is a disk and V0 = ")'~ Genevey [1998] has likewise shown that the implicit function theorem provides the existence of radial solutions around an explicitly computable nonzero radial solution. Nonlinearly elastic membrane shells may thus be likewise amenable to this approach if the applied forces, or the boundary conditions, or the middle surface, are of special kinds.
9.5 ~.
THE TWO-DIMENSIONAL EQUATIONS OF A NONLINEAttLY ELASTIC MEMBRANE SHELL DERIVED BY MEANS OF r-CONVERGENCE THEORY~ COMMENTARY
A remarkable progress in the asymptotic analysis of nonlinearly elastic shells is due to Le Dret & Raoult [1996], who gave the first proof of convergence as the thickness approaches zero. In so doing, they extended to shells the analysis that they had successfully ap-
454
Nonlinearly elastic membrane shells
[Ch. 9
plied to "nonlinearly elastic planar membranes" in Le Dret & Raoult [1995a] (see also Vol. II, Sect. 4.13), where they had likewise given the first convergence result in the planar case. Note that their analysis has since then been extended by Ben Belgacem [1997! and E1 Bachari [1997, 1998] to stored energy functions such that W ( F ) -+ +cr as det F --~ 0 + and to junctions between shells and multi-layered shells. More specifically, H. Le Dret and A. Raoult showed that a subsequence of deformations that minimize (or rather "almost minimize" in a sense explained below) the scaled three-dimensional energies weakly converges in W I , P ( ~ ) as e -+ 0 (the number p E]I, cr is governed by the growth properties of the stored energy function). They showed in addition that the weak limit minimizes a "membrane" energy that is the r-limit of the scaled energies. We now give an abridged account of their analysis. Let w be a domain in I~ 2 with boundary 3' and let 0 E C2(~; I~3) be an injective mapping such that the two vectors an(y) = OaS(y) are linearly independent at all points y = (Ya) C ~. For each ~ > 0, we define the set :=
,,,x] -
e[,
we let z e -- (m~) denote a generic point in the set ~e and we let O~ "= O/Oz~; hence z ae - y a and 0~ - On. Consider as in Sect. 8.3 a family of elastic shells with the same middle surface S = 0(~) and whose thickness 2e > 0 approaches zero. The reference configuration of each shell is thus the image O ( ~ ~) C R 3 of the set ~e C li~3 through a mapping O 9~e -+ IR3 defined by O(x e) : : 8(y) + x~a3(y) for all x e - (y, x~) - (yl, y2, x~) e ~e. By Thm. 3.1-1, if the injective mapping 8 9~ -+ ~3 is smooth enough~ the mapping 0 9 -+ ~3 is also injective ]or e > 0 small enough and yl, y2, m~ then constitute the "natural" curvilinear coordinates for describing each reference configuration O ( ~ e). Assume that the shells are made for all ~ > 0 of the same hyperelastic homogeneous material, satisfying the following properties (hyperelastic materials are studied in detail in Vol. I, Chap. 4): Let 1~ 3 denote the space of all real square matrices of order three and let J. I denote any norm on 1~3; it is then assumed that the stored energy ]unction l~ 91~ 3 --+ I~ of the hyperelastic material satisfies the
Sect. 9.5 b]
The two.dimensional equations derived by P-convergence theory 455
following assumptions: There exist constants C > 0, a > 0, ~ E I~, and 1 < p < cx~ such that [W(F)[ ~ C(1 + [F[p) W(F) ~ ~IF[ p + / 3
for all F E 1~3, for a l l F E l ~ 3,
l~r(F) - ~7(G)[ ~ C ( I -~-IF[ p-1 + [G]p-1)[F - G[
for all F, G E M 3. It can be verified that the stored energy function of a St VenantKirchhoff material, which is given by I~(F)-
A {tr (FTF - I) }2 ~p tr ( F T F - I) 2 -{--~
satisfies such inequalities with p - 4; cf. Ex. 9.7. By contrast, the stored energy function of a linearly elastic material, which is given by
W ( F ) - P IIF + FT - 2III 2 + gA {tr (F T + F -
211} 2
where ]IF]] "- {tr F T F } 1/2, satisfiesthe first inequality with p - 2, but not the second one. It is further assumed that, for each s > 0, the shells are subjected in their interior to applied body forces of density ~ : (/~)" ~ -+ R 3 per unit volume, where f~ E Lq(~ e) and ~1 + ~1 : 1. Since these densities do not depend on the unknown, the applied forces are dead loads (Vol. I, Sect. 2.7). Applied surface forces on the "upper" and "lower" faces of the shells could be likewise considered, but are omitted for simplicity; see in this respect Le Dret & Raoult [1996] who consider a pressure load (cf. Appendix in ibid.), an example of applied surface force that is a live load (Vol. I, Sect. 2.7). Finally, it is assumed that each shell is subjected to a boundary condition of place along its entire lateral face 0 ( 7 • [-~, s]), i.e., that the displacement vanishes there. For each ~ > 0, let :=
let ~ - (~) denote a generic point in the reference configuration { ~ } - , let 0~ "- 0/05~, and let the deformation gradient associated with any deformation ~b~ - ( r { ~ } - --+ ]~3 of the reference
456
[Ch. 9
Nonlinearly elastic membrane shells
configuration be the matrix field ~e~be" { ~ } - -+ 1VIIs defined by ^~ ^~ 01~1 ^,^, 01~2 ^ ^
"--
^~ ^e 02~1 ^,^, 02 ~2 ^e ^e
o2r
03~b "e "e1 0^'3 ~^ 2' ^ ^
9
The three-dimensional problem is then posed as a minimization problem in terms of the unknown deformation field
~" "= i d ~ + ~
of the reference configuration, where ~e . {~e }- __+ IR3 is its displacement field: It consists in finding ~be such that
~
E ~(~e) and .~e(~e) _
inf
~e(~e), where
id]Ra o n |
• [ s s])},
This minimization problem may have n o solution; however, this is not a shortcoming as only the existence of a "diagonal infimizing family", whose existence is always guaranteed, is required in the ensuing analysis (Thin. 9.5-1). For each s > 0, this problem is first transformed into a problem posed over the set ~ e i.e., expressed in terms of the "natural" curvilinear coordinates of the shell, the unknown V~~ - (to~) 9 --+ IRs of this new problem being defined by ~o~(xe) :-- ~ ( ~ )
for &11 ~e __ O(xe), xe E ~E.
If s > 0 is small enough, the mapping | is a Ct-diffeomorphism of ~e onto its image O ( ~ e) C I~s and det V O ~ > 0 in ~e (Thm. 3.1-1). The formula for changing variables in multiple integrals then shows
Sect. 9.5 ~]
The two-dimensional equations derived by P-convergence theory 457
that ~oe satisfies the following minimization problem: ~oe e ~I'(fl e) and Ie(~o e) =
, ~ ( ~ ) . - {#,~ e w ~ , p ( ~ ) ;
inf I~(r V.,~~ ( ~ 9) #,~ - |
(v~r
where
on ,y • [-~, ~]},
- ~ ) det V ~ O d~ ~
- fa~ ~e. ~e det V e O d x e,
where the matrix field V e r e 9~e --~ 1V~3 is defined by (the matrix field V ~| ~ -+ M 3 is analogously defined)" 9
0ir 6
"=
e
01 r e
Ol r
02r 03r e
e
e
e
02 ~)2 03 r ~
e
02 r
e
9
03 r
' e
and the vector field f~ 9~e ~ R 3 is defined by ~ (x ~) . - .f~ ( ~ ) for a11 ~ - O(x~), x ~ e ~ . A w o r d of c a u t i o n . The function det r e | is equal to the function V/~, where ge _ det(g~j); cf. Sect. 8.3. However, the resemblance with the change of coordinates, also from Cartesian to curvilinear ones, performed in Sect. 8.3 stops here! For in Sect. 8.3, the new unknowns are the covariant components ui --~ IR of the displacement field ueoi'ei~, while in the present case the new unknowns are still Cartesian components ~o~ 9 --~ I~ of the deformation field ~o~e/. 1 Like the variational problem in Sect. 8.4, this minimization problem is then transformed into an analogous one, but now posed over the fixed domain f~ := w • 1, 1[. As in Sect. 8.4, we denote by x - (xl, x2, x3) a generic point in f~, we let 0k "- 0/0~i, and with each point x E ~, we associate the point x e C ~e through the bijection
71"e "X -- (Xl, X2, X3) e ~ ~ Xe = (Xe) = (Xl, X2, gX3) E ~e. Then, the unknown is scaled, by letting
~,~(~) - . ~(~)(~) ~o~ ~n ~ - ~ - ~ , 9 e ~,
458
[Ch. 9
Nonlinearly elastic membrane shells
and it is assumed that there exists a vector field ~ E L2(f~) independent of s such that f* (me) - fCm) for all x ~ : ~r~m, m e ~.
Remarks. (1) Inasmuch as the outcome of the asymptotic analysis carried out in this section will also be a nonlinear theory of "membrane" shells, the above assumption that the applied body forces are of order O(1) with respect to s is consistent with the assumption made in Sect. 9.1 on the applied body forces. (2) Should applied surface forces act on the upper and lower faces of the shells, they should be of order O(s) with respect to s, again as in Sect. 9.1. II These scalings and assumptions then imply that the s e a l e d def o r m a t i o n ~o(s) satisfies the following minimization problem: ~o(s) E 'I'(s; f~)and I ( s ) ( ~ o ( s ) ) -
inf
I(s)(r
where
9 (~; n) -= {r e w~,~(n); r = ~o(~) oil 7 • [-1, 1]}, /(~)(r
"- fn
( G ( s ) ) - I ) det G ( s ) d x ((0~r o2r 103r g
-
/o-I ' r
dx,
where the vector field ~o0(s) 9~ ~ I~3 is defined for each s > 0 by ~o0(s)(m) "- |
e) for all me - 7r~x, x E f~,
the matrix field G(s) 9~ --+ IVI[3 is defined by G(s)(z) "- V ' |
e) for all x" -7r~z, x E f~,
and the notation (bl; b2; b3) stands for the matrix in 1VII3 whose three column vectors are bl, b2, b3 (in this order).
Remark. The function det G(s) is equal to the function ~ ) , where g(e) is defined in Thm. 8.4-1. It The scaled displacement
Sect. 9.5 b]
The two-dimensional equations derived by r-convergence theory 459
therefore satisfies the following minimization problem: inf J(6)(~), where ~wca) ~V(~'/) :-- {~ e W:'P(~t); ~ - 0 on 7 x [-1, 1]}
~(e) c W(gt) and J ( e ) ( ~ ( ~ ) ) -
J(e)(v) : - / n
1 -/a
}" (V~
+ ~) det G(e)dx.
A w o r d of c a u t i o n . The notations ~(e) and ~ remind that, although this minimization problem is still expressed in terms of (scaled) curvilinear coordinates (the coordinates xi of the points x C f/), the unknowns are no longer (scaled) covariant components of the displacement field. Instead, the unknowns are now (scaled) Cartesian components, i.e., over a fixed Cartesian frame, of the (scaled) displacement field ~(e) 9~ --+ It(3. m Central to the subsequent analysis is the notion of quasiconvezity, due to Morrey [1952, 1966] (an account of its importance in the calculus of variations is provided in Dacorogna [1989, Chap. 5]): Let ~[m• denote the space of all real matrices with m rows and n columns; a function l~ 9M m• -+ R is quasiconvex if, for all bounded open subsets V C It(n, all F 6 1VIIre• and all ~ - ((i)~n=l C W 1, 0 oo (U), I~(F) <- measl U / g l~r(F + Vt~(x))dx, where V~ denotes the matrix (Oi(i) E M m• Given any function l~ 9M[m• --+ R, its quasiconvex envelope QI~r 9M m• --+ R is the function defined by QI~ "- sup{.~" M m•
-+ R; .~ is quasi-convex and X ~ l~}.
An illuminating instance of actual computation of a quasiconvex envelope is found in Le Dret & Raoult [1995b], who explicitly determine the quasiconvex envelope of the stored energy function of a St Venant-Kirchhoff material, m Remark.
Also central to the ensuing analysis is the notion of r-convergence, a powerful theory initiatedby De Giorgi [1975, 1977] (see
Nonlinearly elastic membrane shells
460
[Ch. 9
also De Giorgi g~ Franzoni [1975]); an illuminating introduction is found in De Giorgi & Dal Maso [1983] and thorough treatments are given in the books of Attouch [1984] and Dal Maso [1993]. As shown in the pioneering works of Acerbi, Buttaz~.o & Percivale [1991] for nonlinearly elastic strings, then of Le Dret & Raoult [1995a, 1996] for nonlinearly elastic planar membranes and membrane shells (reviewed here), this approach has thus far provided the only known convergence theorems for justifying lower-dimensional nonlinear theories of elastic bodies. We then recall a fundamental definition (various complements on F-convergence may be found in Vol. II, Sect. 1.11, where a detailed application of this theory is treated): Let V be a metric space and let J(e) : V --+ R be functionals defined for all e > 0. The family (J(e))e>0 is said to r - c o n v e r g e as e --+ 0 if there exists a functional J : V --+ i~ U {+c~}, called the r - l i m i t of the functionals J(c), such that ,(e) -+ -+ 0 j(,) < ~--+0
on the one hand and, given any v E V, there exist v(e) E V, ~ > 0, such that vie/.. --+ v as e --+ 0 a n d Jlvl.. = lim J/e~/v/e//,...... s--+0
on the other. As a preparation to the application of r-convergence theory, the scaled energies J ( s ) " ~7V(12) --+ I~ are first extended to the larger space L p (~2) by letting J(e)(O)
-
{
J(e)(O) if ~ e W(~2), +c~ if ~ e IF(l]) but ~ ~ W(~2).
Such an extension, customary in r-convergence theory, has inter alia the advantage of "incorporating" the boundary condition into the extended functional. Le Dret & Raoult [1996] then establish that the family (J(e))e>0 of extended energies r-converges as ~ -~ 0 in LP ( ~ ) and that its r-limit can be computed by means of quasiconvex envelopes. More precisely, their analysis leads to the following remarkable convergence theorem, where the limit minimization problems are directly posed as two-dimensional problems (part c)); this is licit since
Sect. 9.5 b]
The two-dimensional equations derived by F-convergence theory 461
the solutions of these limit problems do not depend on the transverse variable (part (b)). Note that, while minimizers of J(e) over ~r(f~) need not exist, the existence of a "diagonal infimizing family" in the sense understood below is always guaranteed because infvc~r(n ) J ( e ) ( v ) > - o o . In what follows, the notation (bl; b2) stands for the matrix in 1V[[3• with bl, b2 (in this order) as its column vectors and v/-ddy denotes as usual the area element along the surface S. T h e o r e m 9.5-1. Assume that there exist C > 0, c~ > 0,/3 E ~, and 1 < p < c~ such that the stored energy function l~ " 1V~3 -+ I~ satisfies the following growth conditions:
IW(F)I ~
C(1 + IF]p)
12V(F) ~ alF[P +/3
IlrC(F)- ff(G)l _< C(1
+
for all F e M 3, for all F C M 3,
IFIp-1 + IGIP-X)IF - GI for all F, G E M 3.
Let the space W(f~) be defined by
W(f~) :-- {~ e Wl'P(f~); ~ -- 0 on -y • [-1, 1]}, and let (s be a "diagonal infimizing family" of the scaled energies, i.e., a family that satisfies
~(~) E ~7~r(f~) and J(6)(~(e)) <
inf
J ( e ) ( O ) + h(e) for all ~ > O,
where h is any positive function that satisfies h(e) --~ 0 as e --~ O. Then: (a) The family (~(e))e>0 lies in a weakly compact subset of the space wl'p(f~).
(b) limit c W(n) sequence of (u(e))e>0 satisfies 0 3 ~ of the transverse variable.
0
e kly o ,e ge t
0 in f~ and is thus independent
462
[Ch. 9
Nonlinearly elastic membrane shells
(c) The vector field ~ "- 89ft_l ~da~3 satisfies the ]ollowing minimization problem (QI~tr0(y, .) denotes for each y E ~ the quasiconvex envelope of 15ro(y,-))" E W~'P(w) and 3M(~)=
~ ( 0 ) - 2 f Qr
inf
0ew~"(~)
3M(rl), where
(a~ + 010; ~2 + 020))Cddy
r
I~0(y, (bl; b2)):= inf lPtr((bt; b2; b3)G-t(9)) bsE~ s
for all (y, (bl; b2)) C w
X
~/~[3•
G ( y ) = ( a l ( y ) , a2(y), a3(y)) ,
the vectors ai(y) forming for each y E -~ the covariant basis at the point O(y) E S. I
It remains to de-scale the vector field ~. In view of the scalings performed on the deformations, we are naturally led to defining for each ~ > 0 the limit displacement field ~e 9_w --+ i~3 of the middle surface S by
It is then immediately verified that ~ satisfies the minimization problem (the notations are those of Thm. 9.5-1): ~e E W ~ ' P ( w ) a n d ' j ~ ( ~ e ) _
inf
0ew]"(~)
3~(0), where
~ ( 0 ) - 2~ [ QWo(y, ( ~ + 0~; ~2 + 02O))~dv r
{
A word of caution. As their three-dimensional counterparts fi(e) and ~, the notations ~e and 0 emphasize that, even though the limit minimization problem is expressed in terms of curvilinear
Sect. 9.5 b]
The two-dimensional equations derived by F-convergence theory 463
coordinates (the coordinates Yo of the points y E ~), the unknowns are no longer the covariant components of the limit displacement field. Instead, the unknowns are now the Cartesian components, i.e., over a fixed Cartesian frame, of the displacement field ~ 9--->w R3 of the middle surface. Likewise, ~ 9f~ -+ I~3 denotes the vector field formed by the Cartesian components of the applied body force density, m A natural question immediately arises: How does this de-scaled minimization problem compare with the de-scaled minimization problem derived in Sect. 9.4 by means of the method of formal asymptotic expansions? As we now explain, it presents similarities with, as well as differences from, the nonlinear membrane theory previously found by formal means. First, it shares with the nonlinear membrane theory of Sect. 9.4 the property that the unknown F/appears only by means of its firstorder partial derivatives Oa~7 in the stored energy function
found in the integrand of the e n e r g y j ~ . It is likewise a large displacement, or equivalently a large deformation, theory in the sense that the de-scaling produces a displacement field that is
0 (1) with respect to ~. Second, assume that the original stored energy function is frameindifferent in the sense that
r162 (RF)
- r162
R e
F e M
where 0 3+ denotes the set of all real orthogonal matrices t t of order three satisfying det t t = 1. This relation is stronger than the usual one, which holds only for F E 1~3 with d e t F > 0 (Vol. I, Thm. 4.2-1); it is, however, verified by the kinds of stored energy functions to which the present analysis applies, e.g., that of a St VenantKirchhoff material. Under this stronger assumption, Le Dret & Raoult [1996, Thm. 10] establish the crucial properties that the stored
energy function found in j ~ , once ezpressed as a function of the points of S, is frame-indifferent and that it depends only on the metric of the deformed middle sur]ace. Hence this theory is also a f r a m e i n d i f f e r e n t , n o n l i n e a r ~ m e m b r a n e " shell t h e o r y , like the one derived in Sect. 9.4 by means of a formal method.
464
Nonlinearly elastic membrane shells
[Ch. 9
It is remarkable that the stored energy function found in j ~ can be explicitly computed when the original three-dimensional stored energy function is that of a St Venant-Kirchhoff material; see Le Dret ~z Raoult [1996, Sect. 6]. It is no less remarkable that Genevey [1997, Thm. 2.2.1] has been able to show that, for such a material, this stored energy ]unction coincides with that found by Miara [1998] (Tam. 9.4-1), provided that the singular values of the limit deformation gradients belong to an ad hoc compact subset of I~2, which can be explicitly identified (Le Dret & Raoult [1995a, Prop. 16] had already made a similar observation in the planar case). As a result, the linearization of the present nonlinear membrane theory does not always produce the equations of the linear "membrane" theory, by contrast with the nonlinear membrane theory of Sect. 9.4 (see the commentary at the end of ibid.). A w o r d of c a u t i o n . These two approaches thus provide an intriguing instance where the limit equations found by a formal asymptotic analysis do not always coincide with those found by a convergence theorem. This is all the more puzzling, since the original threedimensional equations, the scalings, and the assumptions on the data are the same! m Le Dret & Raoult [1996, Sect. 6] have further shown that, if the stored energy function is frame-indifferent and satisfies I~(F) ~ l~(I) for all F E 1~3 (as does the stored energy function of a St VenantKirchhoff material), then the corresponding shell energy is constant under compression. This result has the striking consequence that "nonlinear membrane shells offer no resistance to crumpling. This is an empirical fact, witnessed by anyone who ever played with a deflated balloon" (to quote H. Le Dret and A. Raoult). This is why the equations found by Le Dret & Raoult [1996] seem to be especially appropriate for membrane shells made with a "soft" elastic material, but this assertion is yet to be rigorously established. Indeed, the modeling and numerical simulation of shells made of "soft", or "rubberlike", elastic materials is of paramount importance as they are so often encountered in practice; think of, e.g., a trampoline, a sail, or a tire! In this direction, see Muffin [1991], Schieck, Pietraszkiewicz ~ Stumpf [1992], Basar & Itskov [1998], Antman & Schuricht [1999].
465
Ezercise8
Another challenging open problem consists in determining whether I~-convergence theory could produce a two-dimensional "flexural" theory of nonlinearly elastic shells~ analogous to the theory (described in the next chapter) obtained through the formal approach.
EXERCISES
9.1. Let J~ - ( f a ) e C2([0, 1]; R 2) be an injective mapping such that f ' ( t ) ~ 0 for all t e [0, 1] and let S - 0(~), where w =]0, 1[• 1[ and 8(t, z) - f a ( t ) e a -F z e 3 for (t, z) e ~. The surface S is thus a portion of a cylinder orthogonal to~ and passing through, the planar curve f([0, 1]). Finally, let 70 C 7 be such that {(~, 0) ~ R2; 0 _< t _< 1} u {(~, 1) ~ R~; 0 ___ ~ _< 1} c ~0. Show that the manifold { ~ - ( ~ ) e I ~ ( ~ ) ; ~ - o on ~0, ~ , ( ~ )
- ~,
- 0 i~ ~ ) ,
which contains the manifold ~40(w) (Sect. 9.1), reduces to {0}. This exercise (due to D. Coutand) thus provides an instance of a nonlinearly elastic m e m b r a n e shell (Fig. 9.1-1). 9.2. Let w be a domain in R 2, let V0 be a subset of ~, - Ow, and let 0(yl, y2) - (yl, y2, 0) for all (yl, Y2) E ~. Show that the manifold {W - (r/i) E Hi(w); ~ - 0 on ~0, aa[3(~) - ~a[3 - 0 in w} reduces to ~0} if V0 = ~ or if V0 C ~ has the following property: There exists (at least) one direction such that the intersection of with any line parallel to this direction is a (finite, since w is a domain) union of segments whose end-points belong to 70. This exercise (due to C. Mardare) shows that a plate subjected to a boundary condition of place along a "sufficiently large" (in the above sense) portion of its lateral face constitutes an instance of a nonlinearly elastic membrane shell (Fig. 9.1-2). 9.3. The notations used in this exercise should be self-explanatory. Let a surface S - 0(~) - 0({5~}-) be equipped with two systems of
Nonlinearly elastic membrane shells
466
[Ch. 9
curvilinear coordinates (ya) e ~ and (Ya) e {~}- and assume that
O(yo) 0(~0). -
Show that 0~0(~o) = {0} implies that .A40(~) = {0} (the manifold .h,'t0(w) is defined in Sect. 9.1). This means that the definition of a nonlinearly elastic membrane shell (Sect. 9.1) is independent of the system of eurvilinear coordinates employed for representing the surface S.
9.4. The scaled two-dimensional energy jM of a nonlinearly elastic membrane shell is defined by (the notations are those of Thm.
9.3-1): 1 ~ aa~ r jM(rl) -- g (aar(rl) -- aar)(aa~(rl) -- aa~)~rddy
- ~ Pi'~ Show that there exist constants a > 0 and fl E IR such that _
, ~(~) + 3
for aU ~ e WM(W) -- {~ -- (~i) e Wl'4(w); ~ -- 0 on 70}. Remark. The energy jM is thus coercive on the space WM(W). Genevey [1997, Thin. 1.4.3] has shown that it is not weakly lower semi-continuous on WM(W), however (analogous properties hold in the three-dimensional case; cf. Ex. 8.1). 9.5. Show that the stored energy function of a nonlinearly elastic membrane shell, viz.,
-(~i) e wl.4(~) -+ ~ ~ ' ~ . ~ ( ~ ( ~ ) - ~ . ~ ) ( ~ ( ~ ) - ~,). is ]tame-indifferent, in the following sense: Given any ~ E w t ' 4 ( w ) , its value is unaltered if r := O + yia s is replaced by Q r where Q is any orthogonal matrix of order three. 9.6. This exercise shows that, if higher order forces are considered, the leading term of the formal expansion of the scaled unknown (then also of a higher order) satisfies the equations of a linearly elastic "membrane" shell found in the linear theory (Thm. 3.4-2). This observation is due to Miara [1998, p. 352].
467
Ezercises
Let the assumptions be as in Thm. 9.2-1, save that the applied body and surface force densities are now such that 9
/"e(x~) -- s r f " r ( x ) for all x ~ - 7r~x E ~ e hi'~(x e) = g l + r h i ' l + r ( x ) for all x ~ - 7rex E F~_ U F e
where r is an arbitrary integer ~ 1 and the functions fi, r E L2(~) and h i'l+r E L2(F+ U r _ ) are independent of s. (1) Show that the leading term of the formal asymptotic expansion of the scaled unknown u(s) is u r (in particular then, u ~ vanishes). (2) Show that u r 9 ~ ~ I~3 is independent of the transverse variable and that, once identified with a function ~r . ~ _~ ]~3, it satisfies the variational problem:
(~r E W M ( W )
-- {~7 E WI'4(w); W - 0 on 70},
for all ~ - (7/i) E WM(W), where the functions ")'a~(W) are the covariant components of the linearized change of metric tensor associated with a displacement field 7/ia ~ of S and pi, r _
/ f,,r " d~3 + h~ 1+~ +
h i_ , l+r
and h~ 1+~ - h"" 1 + , (., •
1
9.7. Let 1V~3 denote the set of all matrices of order three, let I" [ denote any norm on 1~3, and let the mapping T~r 9M 3 -+ R defined by W ( F ) - ~ P tr ( F T F - I )
2 § ~ {tr (F T F - I ) }
2 fora11FE M3
denote the stored energy function of a St Venant-Kirchhoff material. Show that there exist constants C > O, a > O, and 13 E ]R such that
JIV(F)J _< C(I
+ ]FI4)
II/V(F)I >__ alFI 4 -l-/3
for all F E l~ 3,
for all F E l~ 3,
[I~(F) - lSr(G)l ~_ C(1 + [FI 3 -$-IGI3)IF - GI for all F, G E M 3. Remark. Such a stored energy function thus satisfies all the assumptions needed in Thin. 9.5-1, with p = 4.
This Page Intentionally Left Blank
C H A P T E R 10 NONLINEARLY
ELASTIC
FLEXURAL
SHELLS
INTRODUCTION
The purpose of this chapter is to identify and mathematically justify the two-dimensional equations of a nonlinearly elastic "flexural" shell. To this end, we follow the approach of V. Lods and B. Miara, who showed how these may be obtained through the method of formal asymptotic expansions. Given a surface S - 0(~) and a displacement field rlia i of S with smooth enough covariant components ??i : w ~ I~, let aa#(~?) denote the covariant components of the metric tensor of the associated deformed surface (0 + ~iai)(-~) and let 1
denote those of the associated change of metric tensor. Consider a nonlinearly elastic shell with middle surface S, subjected to a boundary condition of place along a portion of its lateral face with 0(70) as its middle curve, where 70 C 7. Such a shell is a nonlinearly elastic "flezural" shell if the manifold ~F(60)
-- {?7 -- (77i) E W 2 ' 4 ( ~ g ) ;
F/ "-- O v ~ -~ 0 O12 "~/'0,
aa~(r/) - a a ~ = 0 in w}
contains nonzero elements and possesses nonzero tangent vectors at each one of its points (Sect. 10.2). The condition 2~4F(w) # {0} means that there exist nonzero displacement fields rli ai of the middle surface S that are admissible and ineztensional~ i.e., that satisfy ad hoc boundary conditions along the curve 0(70 ) (these boundary conditions are interpreted in Sect. 10.5) and that leave invariant the metric of S. This definition is motivated by a careful analysis (Thins. 10.1-1 and 10.1-2) of the remaining steps needed to complete the induction begun in Chap. 8 when the shell is not a "membrane" one according
470
[Ch. 10
Nonlinearly elastic flezural shells
to the definition given in the previous chapter. More precisely, the formal asymptotic analysis begun in Chap. 8 is concluded as follows when the shell is a "flexurar' one (Thm. 10.3-4): Assume that the manifold .A4F(w) does not reduce to {0} and that it possesses nonzero tangent vectors at each one of its points. Then the contravariant components of the applied body forces must be of the form f i ' e ( X e ) : ~2fi'2(X) for all x e - 7rex C ~ ,
where the functions fi, 2 a r e independent of s (for simplicity, we assume in this introduction that there are no surface forces), the
leading term u ~ 9-~ --+ R 3 of the formal asymptotic expansion of the scaled unknown is independent of the transverse variable, and ~o _- !2 f1-1 u~ dxs satisfies the following (scaled) two-dimensional variational problem of a nonlinearly elastic "flexural" shell: r
C ~4~v(w) - {~ E W2'4(w); U - 0vVl - 0 on Vo, Gaf3(y) - 0 in w},
3
a
Rar((:o) ( ( R b a ~ ) ( r 1 7 6
for all y - (Yi) E Tr V<0~F(~)
/
p"2~Tiv~dy
where
-- {~ E W2'4(W); ~ -- Ov~ -- 0 o n ~/0,
G~af3(~~ aa~ cry =
-- 0 in w},
4)~ aaf3a,Tr + 2#(aa,T a/3r + aar a~r) ' A+2p
pi, 2 _
/.
I ''2 dxa.
1
The functions R~f3(y) , which are well-defined as functions in L2(w) for any field vl E W 2' 4(w), have the crucial property that they satisfy
(Thin. 10.3-1).
where baf3(y) denote the covariant components of the curvature tensor of the deformed middle surface associated with any displacement
Introduction
471
field Viiai of S with smooth enough covariant components r/i : w --+ (in addition, the vectors Oa(O + yia i) should be linearly independent a.e. in w, to ensure that the normal vectors to the associated deformed surface are a.e. well defined). The above variational equations also express that ~0 is a stationary point of a certain functional over the manifold .AdF(w) (Thin. 10.4-1). Particular solutions are thus obtained by solving a minimization problem, which, by virtue of the relations R~f3(W) - baf3(W)-bag3 for all ~ C A4F(w), takes the foUowing remarkably simple form (directly written here in its "de-scaled" version): The vector field ~ - ( ~ ) formed by the covariant components of the limit displacement field ~ a i of the middle surface S satisfies (Thin. 10.5-1)-
~e E .h,4F(w) and j ~ ( ~ ) --
inf
j}(W),
where the two-dimensional energy j~ 92vtF(w) --+ IR of a nonlinearly elastic flexural shell is defined by Jl, ( , 7 ) -
e3 ~ a a~ar'e (bar(W) - bcrr)(baf3(v/) - ba/3)v/a dy - ~ Pi'erli~dY'
where _
4A
,
Ae +
pi,~ _
f
+
+
2# e
fi, e dxe3. g
The functions a a ~ r ' e are the contravariant components of the
two-dimensional elasticity tensor of the shell, Ae and #e are the Lamd constants of the material constituting the shell, and the functions fi, e are the contravariant components of the body force density applied to the shell. The functions (baf~(rl)- bar3) are the covariant components of the change of curvature tensor associated with a displacement field yia / of the middle surface S. The stored energy function of a nonlinearly elastic flexural shell is thus remarkably simple: It is a quadratic and positive definite expression in terms of the exact difference between the curvature tensor of the deformed middle surface and that of the undeformed one, on
472
Nonlinearly elastic flezural shells
[Ch. 10
the one hand. Since, on the other hand, the elements of the manifold 2viE(w) over which the energy j~ is to be minimized correspond to displacements that leave invariant the metric of the surface S, these two observations combined show that the equations of the linear theory of "flexural" shells are immediately recovered (at least formally) under linearization of the above nonlinear equations. Even if the formal approach has its inherent drawback (precisely that of being only "formal"!), it seems to be the only way to justify a nonlinear "flexural" shell theory, up to now at least. If only for this reason, there should be at least one nonlinear theory that reduces to the linear flexural theory (justified in Chap. 6) under linearization! We conclude this chapter by establishing the existence of a solution to the above minimization problem. The proof, which is based on the fundamental theorem of the calculus of variations, essentially hinges on a careful analysis of the properties of admissible inextensional displacements in H2(w) (Tam. 10.6-1). 10.1.
I D E N T I F I C A T I O N OF A T W O - D I M E N S I O N A L VARIATIONAL PROBLEM SATISFIED BY THE LEADING TERM WHEN THERE ARE NONZERO ADMISSIBLE INEXTENSIONAL DISPLACEMENTS
We consider in this chapter the case where there exist nonzero admissible inextensional displacements, according to the definition given in Sect. 9.1. It turns out that a stronger assumption is needed in order to pursue the analysis of this case, according to which the manifold ~t0(w) introduced in Sect. 9.1 (its definition is recalled in the next theorem) has nonzero tangent vectors at each one of its points, i.e., at each ~ E .~to(w), the tangent space Y e S , to(w) to J~,4o(w) contains nonzero ]unctions. Note that, by introducing this additional assumption, I depart here from Lods & Miara [1998]; however, the conclusions eventually reached will be identical. We begin the analysis of this case by showing that the linearization trick (the second requirement set in part (iv) of the Ansatz of the method of formal asymptotic expansions; cf. Sect. 8.6) leads to two important preliminary conclusions: First, the induction is not corn-
Sect. 10.1]
A two-dimensional problem satisfied by the leading term
473
pIete (whereas it terminates with Thm. 8.8-1 for a membrane shell) and, second, the leading term is itself an inextensional displacement.
W o r d s of caution. (1) It must be emphasized that .h40(w) is a manifold, not a vector space! (2) The conclusions of Thm. 10.1-1 are perfectly compatible with those reached in Thm. 8.8-1. Simply, the variational equations found in this theorem reduce to "0 = 0 for all v/ C W(w)" in the present case! II Note that, when the shell is a nonlinearly elastic "membrane" shell (Chap. 9), and only in this case, the space W ( ~ ) of the next theorem is denoted WM(~).
T h e o r e m 10.1-1. Assume that the scaled unknown u(e) = (ui(e)) appearing in problem :P(e; n) (Thin. 8.4-1) admits a formal asymptotic expansion of the form U ( g ) -- U 0 -4- g U 1 Jr- g2U2 - 4 - ' ' '
u ~ e w(a).-
e
, with
-
o on r0},
u 1, u 2 ~ W l , 4 ( ~ ) .
Define the space W(w) and the manifold .A/to(w) C W(w) by:
W(w) "- {v/e WX'4(w); v / - 0 on 70), )vt0(w) :-- {v/--(v/i) e W(w); G a 3 ( r / ) - 0 in w), where (Thm. 9.1-1) 1
1
G.~(,7) "- ~('Tall~ + '~,11- + a"n~'~il"'Tnll')- ~ ( a ~ , ( ' 7 ) - a-~)" Assume that
474
[Ch. 10
Nonlinearly elastic flezural shells
where G ~ ( ~ ) r I designate the Gdteauz derivatives of each function
c,~z. wx,4(,,,) ~ L2(,,,). Then the ]unctions fi, o C LZ(f~) and h i'1 C L2(r+ U r _ ) found in Thin. 8.8-1 must vanish, i.e., f i , 0 _ 0 in f~ a n d h i, 1 _ 0 on F + U r _ ,
and the leading term u ~ which is independent of the transverse variable x3 (Thin. 8.8-1), is such that its average r 9_ 89f1_1 u ~ dxs satisfies Ga~(~ ~ - 0 in w. Consequently,
r c A,to(w). Proof. The assumption about the tangent spaces to the manifold ~'to(w) implies in particular that
Vo~o(~) # {o}. But an inspection of the G~teaux derivatives (Thm. 9.1-1) I
1
am n
reveals that 1
O"
o',(o)n - ](oo~, + o,~) - r~,~
- b~,~,
i.e., that
a ' ~ ( o ) . - ~.,(~), where the functions q'a~(r/) are the covariant components of the linearized change of metric tensor (Sect. 2.4). In other words,
V o ~ 0 ( ~ ) = {n e w ( n ) ; 7 ~ ( n ) = 0 in ~ ) # {o), and thus the space V0(w) - {rl e HxCw); r / = 0 on "Y0,7~(rl) - 0 in ~} of the linear theory (Sect. 3.4) does not reduce to {0}.
Sect. 10.1]
A two-dimensional problem satisfied by the leading term
475
According to the linearization trick, we are thus compelled to let 1 __ 0 a s in the linear case (see step (v) in Sect. 3.4). These relations in turn imply that the leading term u ~ = ~o satisfies (Thm. 9.2-1) f i , O __ 0 a n d h i,
= 0 for all ~/E W(w), or equivalently (Thm. 9.3-1), that ~0 is a stationary point over the space W(w) of the functional j : W(w) -+ R defined by 1f J(n) -
I
J,,,
If ~0 E W(w) is such that Ga~(~ ~ = 0 in w, then ~0 is a stationary point of j over the space W(w) since then _._.i(C ~ --
inf j ( r / ) - 0 n~w(~)
(recall that the scaled two-dimensional elasticity tensor of a shell is uniformly positive definite; cf. Thm. 3.3-2). Since "-
-
where ['"]tin denotes the linear part with respect to r in the expression [.-. ] and since 7a#(~ ~ = 0 in w in the linear case (step (v) in Sect. 3.4), the linearization trick leads us to only retain those ~0 E W(w) that satisfy Ga~(~ ~ - 0 in w. I Equipped with the information contained in Thm. 10.1-1, about the applied forces and the leading term ~0, we are now in a position to continue the induction until ~0 can be fully identified as the solution of a variational problem when A40(w) ~ {0} and Tr ~ {0} at each ~ E ~40(w). The next theorem, which is due to Lods & Miara [1998, Thm. 1], constitutes the final stage of the induction in this case (more illuminating, and more concise, versions of this final stage will be presented in Sects. 10.3 and 10.4). Not the least stunning consequence of the assumptions made on the manifold ~d0(w) is an "automatic refinement" of the regularity of the leading term ~0 and of the boundary conditions it should satisfy (a similar phenomenon was already encountered in the linear theory; cL Thm. 3.4-3).
Nonlinearly elastic flezural shells
476
[Ch. 10
T h e o r e m 10.1-2. Assume that (the notations are those of Thm. 10.1-1):
A~0(W) 9 := {~/ = ( ~ i ) e W(co); G a / 9 ( ~ ) = 0 in oJ) # {o),
and that, at each ~ C .Alto(w), the tangent space qrr to .Alto(w) at ~ contains nonzero functions. Finally, assume that the scaled unknozon u(8) = (ui(8)) appearing i?z problem '~(~:; ~ ) ( T h m . S.4-1) admits a ]ormal asymptotic expansion of the form u(~) = u 0 + eu 1 -{- e2u 2 + e3u 3 -~ ~:4U4 -[- "'" , with u ~ u 1 E W(f~), u ~ r 0, and u q E W 1 ' 4 ( ~ ) , 2 _< q < 4.
Then in order that no restriction be put on the applied forces and that the linearization trick be satisfied (Sect. 8.6), the components of the applied forces must be of the form
/,i,,(~,) _ ~/,/,2(~) h~,,(~,) - ~3h~,~(~)
foraU for all
x e -- 7 r ~ x E f ~ , xe = 7rexcr~_ur ~
where the functions fi,2 E L2(f~) and h i'3 C L2(r+ u r_) are independent of e. This being the case, the leading term u ~ is independent of the transverse variable x3 and ~o .-'- !2 f~l u~ dx3 satisfies the following two-dimensional variational problem (0~ denotes the outer normal derivative operator along the boundary 7): ~0 E , ~ F ( W ) " - - {~ -- (T]i) E W 2 ' 4 ( w ) ; ~ -- ~v~ - 0 on ")'0,
G ~ ( n ) : o i~ ~}, 1 L ~,,,a/3o'r-zll~Fai]~(17)v/-ddy ~0 "0 - J~wp*,"2~Tiv/-ddY
fo~ all n -
(n,) ~ Vr
"where
V(o~F(~)
:= {n C W2'4(~); n -- O~n - - " Oil 70, G ' ~ ( r 1 7 6 - 0 i~ ~ )
A two-dimensional problem satisfied by the leading term
S e c t . 10.1]
477
designates the tangent space to the manifold .~tF(w) at ~o, the ]unc^0 tions ~o~11/~ depend on ~o the functions Fall~(O ) depend on r and linearly on ~ (the complete, but lengthy, definitions of these functions are given respectively in parts (iv) and (vi) of the subsequent proof and also repeated in e~tenso in Thms. 10.3-1 and 10.3-3), and
finally, aaf3,7r .--'-- 4Art aa~a,7r + 2#(aa,raf3 r + aaraf~,,.), A+2# 9 := p~,2
" ~i, f~,2dxs+,o+
3
+ hi,_ 3 a n d , ,I •i ,
3
: = h ''" 3 (., +1).
1
Proof. Let us first specify some notations frequently used below. By Ei~lj , q _> 0, and Fi~[j(v), q > - 1 , we denote the coefficient of e q in the formal asymptotxc expansions
~,llJ(~; ~(~)) - ~lJ + ~ l J F~llj(e; ,,(~) , v) = ~F,~](~) 1
+ ~2 E,2IIJ +
g3
F'~IJ + ' " ,
+ F?Ij(~)+ ~F,IIj(~)+...,
induced by that of the scaled unknown u(e). We recall that the functions Eillj(e; u(e)) and Fillj(e; u(e), v) are those appearing in the formulation of problem ~(e; 12) (Thm. 8.4-1); hence E illj q and Fi~lj(v ) are themselves functions defined over the set 12. For any integer r :> 0, L r designates the linear form defined by
Lr(v) :- f
fi, rviv/-ddX + f r
+ur_
hi, r+lvi~rddr
where it is understood that the functions fi, r and h i'r+t respectively belong to the spaces L2(12) and L2(r+ u r_) and that they are independent of e. Finally, recall that ~?alt/3 - 0/3~?a - F ~ / 3 y ~ - ba/3y3 and Y311/3- 0/3y3 + b~y~. Our point of departure is Thm. 8.8-1, amended as indicated in Thm. 10.1-1. Before continuing the induction proper (part (v)), we take a closer look at the cancellation of the coefficients of -1 and e ~ (parts (i)to (iv)):
478
[Ch. 10
Nonlinearly elastic flezural shells
(i) When Vg.A,'to(w) r .[0} at each r e 2r
the cancellation of
the coefficient of I leads to the following supplementary conclusions: First, Ei~13 = 0 i n f / . Second, the coefficients u roll3 (~ of e ~ in the formal ezpansions of the functions umll~(e) are given by (the functions (i~l~ are ede ned i n ~ / .
•(0)1113 =
-(1 +
aa2 0
0
Ia112)I3111q-
aa2 0
0
iafll~3112'
~(0) aal 0 0 aal 0 0 2113 = - ( 1 + I,~111)I3112+ I,a112~3111, u(o) o a~l o o Third, the term of order one in the formal ezpansion of u(e) is of the form u1= r
r
= (r r
z3r
_
with r E W(w) and r
E W(w),
i, given t,y a
0
a
0
:= bl(a
_a2J-O ~ 0 _ a a 2 + (1 + a qa112)~3111
_alJ.O
r 0 :-- b2 Ca + (1 + a r _ aala~2 r := _ o r
0 0
_ aal
-
0
0
0
0
0
0
~c,111~3112, ~all2i3111,
Finally, ~0 E .A,"I.F(w) := {r/E W2'4(w); r / - Our/= 0 on 70, Gag~(~/) = 0 in w). To prove that E~I 3 - 0 in ~ simply recall that we showed in part(ii) of the proof of Thin. 8.8-1 and in Thin. 10.1-1 that E~ s - 0 and )~a~E~ ~ + ()~ + 2/~)E~ a - 0 in f~, and that the assumptions 2r C ~/to(W) imply that E~
~ {0} and qFCJ~4o(w) ~ {0) at all
- Ga~(~ ~ = 0 in w, hence in f~.
Sect. 10.1]
A two.dimensional problem satisfied by the leading term
479
The equations 2E~ll3 -- 0 take the form of a nonlinear system with respect to the functions u l ~ , viz., u(o) 0 -4- a ~176 lu (~ 0., (0) II13 -I-- ~'3111 ~I13 --I-- ~'3111=3113 -" O, u(o) o + a ~ o . . u(O) o ~ (o) 2113-4-~3112 all2 /3113 -t-~3112`.3113 - - 0 , (0) aa/3u(O),(o) -4- (o) ( o ) _ O, u3113 -ta113-/311s ual13U3113
where r Ii~ - a~r ~ _ r ~~ r _ bozio i
u(O)
~nd
all3 - 03ua + bai ~ and
i~llZ - a ~
+ b~: ~
(o) = 03u~ "3113 ,
0 3 = ~0illl3' which follow from Osu ~ - 0 in gt have and the relations Uilll been used. As we shall see in part (ii), this nonlinear system can be ezplicitly solved, thanks in particular to the relations E all~ ~ = 0 More (0) satisfies a quadratic equation, specifically, one first shows that "3113 viz.,
(
~(0)~ 2 ( ..al3r0 aala/32 0 0 )2 1 + `.3113] -- 1 + ~ ~alll3 + (~alli~ll2- ~all2<~lll)
Hence either u(o) ~a~o aala~2 ~0 0 ~lll) 3113 = ~ 5~11~-4( ~1t1~112 - ~112 ' or
~o)ll3 _-- - 2 - (a~162176 z + a~176
- r
o
o
(o) S i n c e `.3113
_ 03u~ and .,a13 to -- aa~val 3(~~ and since 03u~ - 0 and `* Salll3 7 ~ ( r ~ - 0 in the linear case (step (v) in Sect. 3.4), the linearization trick leads us to only retain the first solution. One then solves the first two equations of the nonlinear system, (o) a n d , (`.2113" ~ As which constitute a linear system with respect to =1113 shown in (ii), this gives u(0) - - ( l + a a2 0 0 aa2 0 0 1113 -~a112)~3111 -4~a111~3112' u(O) _ aal 0 )~0112+aal~o ~0 2113 - ( 1 + ~alli all2 3111" -
-
Nonlinearly elastic flezural shells
480
[Ch. 10
Using these expressions for the functions %113' - (o) we then find that
03u
u(O) all3---r
- - b a ~ +o
o
03u - (~ - - r ~3113
and
~
where ,r
J.O ,,-0 :_ b?~O + (1 + aa2~Ojl2)~O[[1 - - a _a2 %alJ1%[12'
r ~ : : b~r176+ (1 + a all'Oil 1)(~Oll 2 r
-
-
hal ~.all2~.3ljl ~ 0 0
: : _aa[J~O[[/3 -- a a l a / 3 2 (~.011i~i12 _ ~O[[2~ iii )"
Since U I E W ( ~ ) by assumption and the three functions r independent of x3, we conclude that u I is of the form
u l __ ~1 __ X3r
with ~1 E W(0g) and r
are
E W(w),
the last relation implying that the functions ~o should be in the space W 2, 4(w) in order that the products Oa~~ ~ found in the functions r be themselves in the space WZ,4(w) (recall that this space is an algebra; see, e.g., Vol. I, Thm. 6.1-4). We likewise infer that the derivatives Oa~~ should vanish on 7o in order that the functions r vanish on 70 (we already know that ~o = 0 on V0); hence 0v~/~ - 0 on Vo- We have thus shown that
~0 E .MF(W) -- {~ C W2'4(0;); ~ -- Ov~ -- 0
Ga#(~l)
o n "Y0, -
-
0 in w}.
(ii) Solution of the nonlinear system (found in part (i)) satisfied by the functions.~i113' (o) viz.,
(o) o ~ (o) ~ (o) + ~Oll1 + aa~Olll~ll3 + ~s111~3113 - 0, ~1113 u(o)
o
o
~(o)
+.o
(o)
2113 + ~3112 + aa~ ~'a112"~!13 %!12u3113 - 0, (0) a,~u(O) u(O) ~ (o)~ (o) u3113 + all3 ,6113 + "3113"3ll3 - 0 .
Sect. 10.1]
A two-dimensional problem satisfied by the leading term
481
Recall t h a t the explicit solution of this system, formed by the equations 2E/~13 - 0, was crucially needed in p a r t (i). Let
= :=
lull13 3~ku2ll | (0)
1
,1)
~ (0) z3 :--~3113'
Z ::
,2 i + a ~~
M :--
1
aal~~ 2
(a11 air
A :--
a 21
a22}
( ~'0lll ~'0112~ ~,~Ot[1 r0
..a2 ,*0 \ ct ~aii 1 aa2/.O ~ , I - - ~ - ~ai]2/
d := det M ,
and assume that the matrix M is invertible at all points in ~ (in fact, we shall see later that this assumption is not indispensable) as is the case if, e.g., the norms hence the norms are smaU enough. W i t h these notations, the nonlinear system becomes
I[(~
(l+z3)z+M~-O, (I + z3)2 + ~TAa~ -- 1, so t h a t letting z - - ( 1 + x 3 ) M - l z
(I
in the last equation gives I.
+ z3)2(1 A - z T M - T A M - I z ) -
The rest of the argument essentially consists in computing the number (1 + z T M - T A M - I z ) , which turns out to be a simple expression in terms of d : det M . To this end, we firstremark that MA-~M (since M
- I + ZTA
2E 0II
-
T = A -~ + Z T + Z + Z T A Z and A - A T) and that the relations
0
+
0
+
_~rT,,0
0
0
+ 311~
- o
established in Thin. 10.1-1 can be equivalently w r i t t e n as Z-4- Z T d- Z T A Z ~ - Z Z T -- O, so t h a t we also have MA-1M T - A -i -
zz T : ( I - z z T A ) A -1.
Nonlinearly elastic flezural shells
482
[Ch. 10
Consequently, d 2 -- det M M T -- det A det M A - 1 M T = det A det(I - z z T A ) A -1 - d e t ( I - z z T A ) =
1 -
tr z z T A
-
1 -
;gTA;g
(for any m a t r i x of order two, det(I + B) - 1 + tr B + det B; besides, if B = z z T A , then d e t B - 0 since the m a t r i x z z T is singular); in particular then, 1 - z T A z # 0 since M is a s s u m e d to be invertible. F r o m the above expression of M A - 1 M T, we also consecutively deduce t h a t MA-1MTAz
- (I- zzTA)z
Az - (MA-1MT)-IMA-1MTAz
-- (1 - z T A z ) z ,
- (1- zTAz)M-TAM-lz,
zTAz 1 + zTM-TAM-lz
1 = 1 -- z T A z
- 1 -4- 1 - z T A z
Hence the relation (1 + x3)2(1 + z T M - T A M - l z ) becomes d 2 - (1 + x3)2,
1 = --~" d -- 1 simply
or~ in eztenso~
(
-
(
1+
o r176
+
.
As shown in part (i), the linearization trick leads to only retain the solution (1 + x3) - d. We are thus left with z - -(1 + x3)M-lz
- -(det M)M-lz
- -(Cof
MT)z,
or, in extenso, ~(0)
aa2 0
1113 -
-(1 +
2113 -
-(1 +
~(0)
0
ia112)i3111+
aa2 0
0
iallxi3112,
aal 0 0 aal 0 0 ffalll)~sll2 + ~I12ff3111"
It is then easily verified that the relations d 2 a~ -- - ( C o f M T ) z are still valid if M is singular.
( i + x3) 2 and
(iii) When T r ~ {0} at each ~ E .h/t0(w), the cancellation of the coefficient of 6~ leads to the supplementary conclusion that the functions E/~I3 satisfy ~.,, a1~1571
Ealtl 3 - 0 and ,~,
=all~ + (A + 2#)E~113 - 0 in f~.
Sect. 10.1]
A two.dimensional problem satisfied by the leading term
483
Since all the functions E ~ vanish in ~2 (the relations E ~ I - 0 and E~l 3 - 0 have been respectively established in Thm. 10.1-1 and in part (i) of the present proof) and since L ~ - 0 (the functions fi, 0 and h i'1 vanish; cf. Thin. 10.1-1), the cancellation of the coefficient of e ~ in the variational equations of problem :P(e; ~2) reduces to the equations 1 (v)v~dx = 0 for all v e W ( ~ ) . AiJkl(O)E~lllF -i111
Observing that the cancellation of the coefficient of e-1 in the variational equations of problem T'(e; ~2) led in part (ii) of the proof of Thm. 8.8-1 to analogous equations, only with E kill ~ in lieu of E~IIg we conclude in the same manner that
+ 2#Ea1131(aar + aa,,.a~ru~lta ) - 0
in ~2
(0)
(AaaflEll]ft + (A + 2/.t)E1113)(1 + '~3113)
+2pa a~176
= 0 in ft.
One obvious solution to this system is E~I 13 - 0 a n d X a "'a f i ~ l~all~ + ()~ + 2/~)E 1113 - 0 i n ~ . Denoting by [..-]lin the linear part with respect to (any component of) u ~ u 1, or u 2 in the expression [.-. ] and using the notations of the linear case (Sect. 3.4), we have ~1 ]lin [~llS~
-
[AaafSE~llfI + ()~ + 2#)EsllLs]tin
=
e~ils , +
+
by definition of the functions Elill3 and elliS.1 Since it was found in the linear case (see step (v) in Sect. 3.4) that el_ll 3,~ - 0 a n d e ~ l lso -
+ 2#
aa~eall~
in ~2 ,
the linearization trick suggests that we only retain the "obvious" solution found above.
484
[Ch. 10
N o n l i n e a r l y elastic f l e z u r a l shells
(iv) Equipped with the information that the term of order one is of the f o r m u 1 = r _ X3~0 with ~z E W(w) and ~k~ E W(w) (part (i)), we now show that the functions E~II# (the coefficients of e in the formal expansion of Eall#(e; u(e)), whose explicit expressions will be needed in part (v) for pursuing the induction) are given by ~p, 1 (the functions a '~9" ga/3,1, and ~'i/~ are defined in Thm. 8.5-1):
3E^ 0 II ,
0 -
-
where FOal
1
amn{ o
:=
-
[vP, 1
1
o'.r "
I arnn
0
~ ~ ~ (j.o.{{a(j.,.{{,
f ~ p , 1 kO
lap, 1 0
0
An immediate computation, simply based on the definition of the functions Ealll#, shows that (recall that gi3,1 _ 0)" 2E~I
I~
_
~
(1) _~_~ (1)
~lla
_rnnr,,0
~all~ -t-a
U(1) + / - 0
U (1)
1 0
0
%~mll~ nll~ ~nll~ mlla} + x s g ~ ' ~lla~ll~'
~ma are defined in Thin. 8.5-1)" where (the functions vP'~ ~p,O
1
_~p, 1 0
the notation u(21l~,, which by definition designates the coefficient of e in the formal expansion of Umlla(e), emphasizing that the functions u (1) do not depend solely on u 1 - (up1) talk, Since u 1 - ~1 _ x3r by (i), _
9- m a s p ]
- - X3
mlla - ZZ~rnlla -
--
r~P, 0 ~/,0 ~
--
_
r~P, 1
X3zrna
on the one hand; by definition, F 0
_mnr,,o
2 all,(r/) = flail~ -4- r/~lla q- a
o
~t~rnllar/nll~+ ~nll~r/mlla},
A two-dimensional problem satisfied by the leading term
Sect. 10.1]
with ~Tmlla -- 0ar/m - I'd~ E11 ]~ _
485
on the other. Hence
Fcxli~ 0 (~1 --
x3r
0) -I- ~x3g 1 ,.0 err' 1,.0 "~rllc~"~'llf3
Xamn f pp, 1 0
-~(r~ + 2
~p, 1,,0
0
o~-,1~ + '.~ ~11o))~,"
(v) When v c . ~ o ( ~ ) # {o} ~t e~ch r e .~,to(~,), the c~nceUation o] the coefficient of e leads to the following relations: fi,1 _ 0 in ft and h i,2 - 0 on F+ U r _ ,
r e Vr
:= {rt e w(oJ); F~ ~1
__
t1~ -
0 in w},
^0
-x3E~ll~'
where the functions F ailfj(rl) ~ and ~oaJif3 are those found in (iv). Since fi, o _ 0 and h i'1 - 0 when T r E 2~0 (w) by Thm. 10.1-1, our next try is
r
{0} at each
f i ( e ) -- e f i'l a n d hi(e) - e2h i'2.
The cancellation of the coefficient of e in the variational equations of problem :P(e; f~) then leads to the equations (two terms are needed here from the expansions of the functions AiJm(e)v/g(ei; cf. Thm. 8.5-2):
a
F-l(v)}v/-adx
+ f a BiJm' ~E~'[]l F ~]lJ - l ( v ) dx -- L l ( v ) for aU v C W ( ~ ) . In particular then, we must have
for all v E W ( f t ) that are independent of ~ , ~inc. F,~j1(~) = 0 fo~ such functions v; or equivalently, after the usual identification has been performed,
f a A~jl, l (O)EklltFiiii(~l)v/'a ~ o dx - L (~1) for all r / e W(w) = { r / e Wl'4(w); 17 - 0 on 70}.
Nonlinearly elastic flezural shells
486
[Ch. 10
Using the expressions of the functions A~/kz(0) (Thm. 8.5-2) and the relations satisfied by the functions E~i3 (part (iii)), we are left with (an analogous computation is detailed in part (iii) of the proof of Thm. 8.8-1):
/12 AiJkl(O) Ekll,Filli 1 O (~}) ~ dx --
-
L
(,7)
for all r/E W(w), where
a a ~ r _-
4A# aa~a~r + 2p(aa~ a~ r + aar a/3~)" A+2#
Since
~.o E JVt0(w) -- {W E W(w);
Ga~(W)
-
0 in w}
(Thm. 10.1-1) and since the tangent space T c 0 ~ 0 ( w ) - { r / e W(w); F~
- 0 in w} C W(w)
to the manifold 2vt0(w) at ~0 (recall that E~ ~ - Ga/3(~ ~ and that Fa i i , ( n ) = ' )~}; cf. Thm. 9.1-1) contains nonzero functions by assumption, using such functions in the last variational equations shows that we must let f~, 1 = 0 and h i'2 - 0 in order to fulfill the first requirement (the linearization trick could be also called upon here since the same conclusion was reached in the linear case; cf. part (vi) in Sect. 3.4). The functions Ealll~ being of the form (part (iv)) E~
o
1
^o
where the functions F all,6 ~ (~1) and/~o all~ are independent of the transverse variable x3, we are left with the equations
~_ajga"rr~O
1 0
-0
for all ~/E W(w). Letting r/- (~i in these equations (this is licitsince r E W ( w ) by (i)) shows that (the scaled two-dimensional elasticity tensor of a shell is positive definite;cf. Thin. 3.3-2):
.~"aO[l~(r1) = O.
Hence (~I E 'l['(:o,Ad,o(w) and EaI l l ~-- - x 3 ./~o ,xll~"
Sect. 10.1]
487
A two-dimensional problem satisfied by the leading term
(vi) When Tr ~ {0} at each ~ ~ .h4o(w), the cancellation of the coefficient of s2 shows that the field ~o, which belongs to the space A t e ( w ) by (i), satisfies the following variational equations:
1~~a~, ~ rT~. - ~O l i ~ l l^0 ,(~)~d~ /or all rI -- (Yi) ~ Vr Vr
- L2(~)
where -- {Vl ~ W2'4(w); Vl -- 0~,V/-- 0 on "/0;
~~ Iif~(~7) - 0 in ~}. The functions/~~ are those found in (iv) and the functions POallf~(r/) are given by (see (iv) for some of the notations employed infra): 1 ,,~.
+ (r:~ ~ + - l a ~ { r ~ ( ~ 2
l amn
+ ~
0
{
o
~ + r~Xr176
~p, 1 0
0
-~- --ma
(r
---q, 1 .,0 ]"nf} {~q
'
where ~~ (~) _ (~o (~)) E W ( w ) is defined by ~0(T~) :-- b?T]c~ -~-~3111
v o ( o ) ._ bi',. + ,~112 _
~.0
0
~0(~) ._ - ~ ' . ~ t l ,
Since /i,1 _ 0 and h i'2 - 0 when T r # {0} at each C 2~40(w) by (v), our next try (and final one as it will turn out) is f i ( ~ ) __
~2fi.2 and hi(e)
- ~3hi'3.
The cancellation of the coefficient of s2 in the variational equations of problem ? ( s ; ~) then leads to the equations (no less than
Nonlinearly elastic flezural shells
488
[Ch. 10
three terms are needed here from the expansions of the functions
of. Tam.
ff2 AiJlct(O){EklltFillj(v) 1 1 2 0 3 --1 + EklliFillj(v) + EklltFill j (v)}v/~dz "k
2 -l(v)}da BiJkl'l{E~lll.Fi~[i(v)--k Eklll~l[i
~
+ f a Bqk42~1 ~-,-1 ( " ) d ~ - 1.2 (,,) ~'klltqll~ for all v E W(f~). Note that the highest order term in the formal asymptotic expansion of u(e) appearing in the functions Ei~lj is u 4 (each term involving u s vanishes because it contains some derivative as a factor); this is why this expansion must be "at least" of the form u(e) : E q =4O ~q "uq -t- " " " 9 In particular then, these variational equations must be satisfied for all v E W ( f ~ ) t h a t are independent of z3 since F/]~1(v ) = 0 for such functions v. Equivalently, after the usual identification has been performed, we must have
Osu~
/f AiJlcl(O){EklltFillj 1 1 (rl) + EklliFillj 2 0 (~l) }v/-a dx + f a B i j k l ' 1 ~1
~.,0
L2
for all ~ e W ( w ) - {~/ e WI'4(W); ~ = 0 o n ")'0}, on the one hand. On the other, the cancellation of the coefficient of s shows that we also have
fnAijkt (0){Ekllt F llj( o
2 F-t(v)}v/-~dx + Ekll illJ
+ faBiJk4t~t~klll~.illj ]~-i (v) dx
- 0
for aU v E W ( f l ) , since it was proved in (v) that L 1 - 0. As we shaU see later by means of an independent proof (Thin. 10.1-3), it is remarkable that, given any function r/ in the tangent space Tr to the manifold .A,tF(w), there exists a function v(~/) E W ( f l ) such that
FlJ(") :
,llJ
A two-dimensional problem satisfied by the leading term
Sect. 10.1]
489
Subtracting the last variational equations with v = v(rl) from the penultimate ones and noting that Tr C W(w), we are thus left with the variational equations
/a AiJkZ(O)E~ll,{F~lj(y) - Fi~lj(v(y))}~dx -
L2(.)
for all y C T~O~F(W), which in turn reduce to 1 fn _af3~r ~1
1
-
F0
-
L 2 (.)
for all y E T~o.h4F(W), in view of the expressions of the functions AiJkt(0) and of the relations satisfied by the functions E~l 3 (see Thm. 8.5-2 and part (iii) of this proof). As we shall likewise see later by means of yet another independent proof (Thm. 10.1-4), it is no less remarkable that, given any function Vl e Tr the differences {F~llf3(~)- F ~ appearing in the last variational equations are of the form F 1
0
^1
^0
al]fl ( ~l ) -- Fal]fl ( v ( ~l ) ) - Fallfl ( rl ) - x 3 Fai]fl ( . ) ,
^1 where the functions Fallf3(~) are
independent of x3 and the functions indicated supra, are also independent
__P~ , which are defined as of x3. Since Eal I]~1 -- --x3Eal]/3, ^0
where the functions/~0allf~ are again independent of x3 (part (iv)), we are thus eventually left with the variational equations 0 ^0 5l ~ _ a ~f ~ a r ~,~llrFaljf3(~) as was to be proved.
v/ady
--L2 (~) for all ~ e T ~ o ~ F ( w ) , m
It remains to establish the two properties stated without proof in part (vi) above. Their demonstrations, which are exceedingly delicate and lengthy, were for these reasons postponed in order not to disrupt the otherwise familiar course of the proof of Thm. 10.1-2. The next two theorems are due to Lods ~ Miara [1998, Lemmas 4 and 5].
490
eq
[Ch. 10
Nonlinearly elastic flezural shells
T h e o r e m 10.1-3. Let F i11./ q (v) ' q > - 1 , denote the coefficient of -in the formal asymptotic ezpansion F~,,#(~; ~,(e), v ) -
l(v) ;1 v -~11~
+ Fi~lj(v) + eF~lli(V) + " .
Given any n = (~i) ~ V~o.A,4.F(w), where ~'~O.~F(0))
--
{n ~ W2'4(~); n
--
Ogvn -- 0 Oil 70,
F~
-
0 in w},
t~t ~ ~ = ( C ( n ) ) e w(..,) - {.7 e w~.~(.,); n - o o.~ "to} be defined by
0
+a al (IOi!1713112+ ~0112~a1[1--
and let
~(n)
"=
0
~'0111~a112
--
0 Ca1127~3,tl),
~,o(n).
Then ~(n) e w ( a )
- {n e w ~ , ~ ( a ) ; ~ - o o= ro},
Fi?lj(. ) - F i ~ ( v ( . ) ) , and C ( n ) - r 1 6 2 1 7 6 fo~ ~11 n c Yr
"wheT'e ~ ( n ) = (~i(n)) E W l ' 4 ( w ) i$ defined ~0~ any n C W2'4(w) by
r
_ a a2 :': b~Ta + (1 + aa2~all2)~311t ~Tallt~Tal12, a al := b~ya + (i + aai~Talli)~73112 ~7aI12~7311i,
r ~3('0) :-- -aa~3rlall~ - aata~32(r/alltrh3112 - r/all2r/1311t)" _
Sect. 10.1]
A two-dimensional problem satisfied by the leading term
491
Proof. The following notations will be used throughout the proof:
r/c, llf~ := cOf~r/c, - I'~7/~ - baf3r/3, r/311fJ "-- Of~r/3 -I- ba~/~,
/~O'[lh __ ~r176
~Z:--
\r/31[2 M(r/)
( all
=(r
(1%- aat~Tall1 1 + a~ :-aal~all2 1 + a al ~~ t
M "--
aallOl[2
] ' d(,)"-aet M(.7).
r
( ~~
~~
\r12111
L
:: Z ( , ) T A -
:=
-dzTM
.21 g22/ ~
aa2(~~II1) = M((:~ d "-- det M -- d(~'~ 1 + aa2(:all2
Z(~) : - (~/1111 '1~112/ , z - -
N(,)
A :=
a12~
- z(r176
aal ' \.~l~"ll ~ a'~2r/alll) aa2r/al[2
, N(v/)
-TA
"--
Some of these notations were introduced in part (ii) of the proof of the preceding theorem; there, it was shown in particular that they satisfy o) \ u1113/ (o)I u2113/
t(
MA-1MTA - I-
-
_
-dM
zzTA, 1
tz,
1
M - T A M - t z - ~ Az,
14-
~ (0) _ ~3113
d,
+ zTM-TAM-Iz
z TAz
1
-- - -
d2 ,
= 1 -- d 2.
These relations will be repeatedly used in the sequel. Since F -1(v) - 0 for all v 6 W(f~) and F~ - 0 for all r/ E T~0,~F(w), it therefore suffices to show that the relations Fi~(v(r/) ) - Fi~13(r/) are satisfied for all ~/C T~o,~4F(w).
492
[Ch. 10
Nonlinearly elastic flezural shells
(i) Let v = (vi) e W(f~) and 17 -- (Wi) e essary and sufficient condition that
W(w) be given. A nec-
is that v be affine with respect to x3 and that
( 03vl
/ (o), [Ull[3~
(b?n~~
L |03v2 i k~SV3
L ( b ~ )~
-
(~73[[1
(o) l + (1 4- .(o) "3113)t ' 3~12
+N(v/)/
/u(~ l
9
3ll3/
\
To prove this relation, we express in matrix form the column vector fields with components F/l~(v ) E L2(n) and F/~j3(v/) E L2(w). First, let v E W(f~) be given. Then the definition of the functions F/[~ (v) shows that 1
2F~a(v )
--
(1 -F
aal 0
~alll)O3v1-{-
aS2 0 0 (0~111~3V2-~- r
(1 +
al1303V2 +
=3ll3/03v3,
. (o)
where the functions ,~i113 are given by (el. part (ii) of the proof of Thm. 10.1-2): _a2,,0 ~ 0 aS2 0 0 u(0) ~lt3 = - ( 1 + a ~112Jr + QI1~3112, U(0)
u(o)
0 aS1 0 0 ~c~111)~3112-~~c~112~3111,
_c~1~0
2113 = -(1 § a 3113 -- d -
1.
Hence these relations together imply that
2F21~(v )
- L /03v2
.
Next, let r/E W(w) be given. Since ~/may be identified in the usual way with a field (still denoted) ~/C W(f~), the relation ~3~/- 0 in f~ shows that Fil~ (~/) - 0. The functions Fill3(~/) , which in this case
Sect. 10.1]
A two-dimensional problem satisfied by the leading term
493
thus become the leading terms in the formal asymptotic expansions of the functions FilIj(e; u(s), r/), are then given by --
--
+~(o~
F3~13(r/)
-
3
o -- ba~,~3o)brrD, ;~
o" 0
+ b ~ ) ( o ~ , 7 . - r~,7~ - b~.,73)
1 a 0 a~ ( O3uo. -t- bo.~a ) b~rrl,6 .
-
Thanks to the relations ~oll ~ _ o ~ o
_ r ~ ~0 _ b~:r
and
i~
cr ~0 - 0 ~ ~ + bai
~3113 the relations giving the functions F~I 3 (r/) can be rewritten in vector form as "1113
~2113 -t-
~31J1 +"3113)
r/ 12
9
"3[[3
for all r/ ~ W(w). Hence the announced assertion is established. (ii) Given any field n - ( ~ i ) E V~o,~F(w) C W(w), r~ - (r~ c w ( w ) be defined by rl0 (r/) := b~rla + ~1 ((1 + aa2"~
l e t the field
~73"II1 -aa2"~ 1 a 0 + ~ a ~311~0zlla,
1 ( o aaZO ) T0(r/) : : b~ya + ~ -aaZ~all2r/311z + (1 + ~allz)r/3112 1 a 0 + ~ a ~'311~r/211a,
~o(~)
1 a
:= ~
~r 0
Then
~(~) := ~ 0 ( ~ ) e w(~) ~nd F,t~(~(,))= F,~I~(,) It is easily verified that r~ C W(w) since W1,4(w) is an algebra; consequently, x3r~ C W(f~).
494
[Ch. 10
Nonlinearly elastic flezura! 8helb
A straightforward computation, based in particular on the relations MA-1MTA = I- zzTA
1 -{- zTM -TAM - l z - 1
and
d2
shows that if the matrix M is invertible, then so is the matrix L with an inverse given by M - I ( I - zzTA) zTA
L_I=
_ dM-lz ) d "
To establish that Fil~(v(W)): Fi~13(W), it suffices by (i) to establish that
,0,,
u1113~ +L-IN(y)
~o(~)
(o) l ~(O))L-1 ~2[]3/ -I" (1 -t" ,~3[13 (0) l
(~ ~73111 1 3~ j ?7 12
,
u3113]
since o3v(.)
= r~
Thanks to the expressions of the matrices L -I and N ( W ) and to the relations
u1113 (o) (o) t u2113
_
_dM-lz,
it is first found that / (0) {Ulll3 L _ I N ( ~ ) I (0)
(o) _ d, 1 + '~3113
M A - 1 M TA - I - zz TA,
(
~3111
( 0 ) l / ; 12 1u2113 + (1 + ~,~3113)L~7 /
(0)
\u3113
z(.)})
- d [ A-1MTA{-z(")TAM-lz + ~ -zTAZ(~I) r A M - l z + z T A z ( y )
"
It thus remains to properly incorporate into the right-hand side of the above vector equation the relations Fo I 1 am. o o I~(~) = ~(~7,~11~+ ~7~11,~+ {imll,~Tnll~+ inll~?mlla} ) -- 0
A two-dimensional problem satisfied by the leading term
Sect. 10.1]
495
satisfied by the function r/ 6 ~'r Another straightforward computation shows that these may be equivalently written in matrix form as MZ(r/) + Z(r/)TM T + zz(rl) T + z(rl)z T -- O. Together with the relations M _ T A M _ 1z - ~-~Az 1 and z T A z - 1 - d 2, this matrix equation shows that
_zT AZ(rl)T AM-lz _
2d1 2 z T A { M Z ( r / ) + Z ( r l ) T M T } A z
--
1 zTA{z(zl)zT_4_ZZ(,t.I)T}Az
2d 2 =
l zT A.z(rl)zT Az -- ( 1
-
1
) zT
hence that
d( - z T AZ(.) TAM-lz + z T Az(r/)) - lzTA.z(y), which takes care of the third component in the vector equation. The relation M A - t M T A = I - z z T A shows that the vector formed by the first two components in the same vector equation is of the form
a(A-1MrA{-Z(n)rAM-lz + z(n)}) = d(M-IzzTAZ(rl)TAM-tz -d(-
- M-tzzTAz(rl))
M-tZ(r/)TAM-tz
+ M-tz(r/)).
Let us transform this expression. The relation (which has just been established)
z T A Z ( r / ) T A M - t z --
1 ) zTAz(r/) 1 - d-5
implies that
M-tzzTAZ(~)TAM-tz
- M-lzzTAz(rl)
= - d - ~1M - t z z W A z ( r l ) - - - ~ M - 1
1z z
(rl)TAz
Nonlinearly elastic flezural shells
496
[Ch. 10
on the one hand; the matrix equation expressing that F ~ll~('~) ~ = o 1 together with the relations M - T A M - l z - ~-~Az and zTAz = 1 - d 2 imply that 1 d---~M-1Z(~)TM TAz
- M -1Z(~) T A M - l z - -
1 1 = d--~Z(.)Az + ~1 M _ l z ( ~ ) z T A z + M - l z z ( ~ ) T A z 1 ( 1 ) 1 = d-sZ(T/)Az + ~ - 1 M - i z ( ~ / ) + -~M-izz(~l)TAz on the other. These relations thus imply that
d(M-lzzTAZ(.)TAM-lz - M-lzzTAz(o)) -d M-1Z(n)TAM-lz + M-lz(.)
~Z(n)Az + ~
-
z(n).
In this fashion, we have established that
/
b~~
u11[3 ~
+ L-1N(y)
U2ll3|(~
~/3[ I +
(1
~(~
-
2
(o),
~o
u3113]
\"# ( ~ ) /"
where (TIO(r/)~
=
(b~'r/a~
+
1
1M-1
z(~)A~
+ d
z(n)'
~3(n) = ~1 zTAz(~/)
It then suffices to combine the definitions of the vector fields z and z(w) and of the matrix fields A and Z(w) with the expressions M_ 1 _
1 (1 + aa2(~
2
_a2/-O
--u,
SailI
\
~'alll/ }
1 + a al 0
Sect. 10.1]
A two-dimensional problem satisfied by the leading term
497
in order to obtain the announced expressions, viz., 1 (
an20
7"0(~1) = b~71a + ~-~ (1 +
_an20
iall2)r/311t
)
Ia111713112
1 act o +~a ~311,orhlla,
1 (_aaZ o _al,,.o ) TO(r/) -- b~'7/a + ~-~ ffallzr/311z + (1 + a ~a111)~/3112 la~o -t-~a ~311/30211a, 1
0
(iii) Let the mapping r - (r for all f I - (~i) E W2'4(03) by
W2'4(60) -+ W1'4(C0) be defined
r
"-- b?rla -4- (1 + aa2rlall2)rl3111
r
:= b~r/a + (1 + aatr/allt)~73112 - aalr/all2r/311t,
r9
:-- -aa/3r/all~3
-
-
-
a~
aatd32(r/alltr//3112
- r/all2r/.Ollt),
and let the mapping ~ - (qoi) W2'4(69) x W2'4(6g) ~ WI'4(W) be all r - (ii) ~ WZ'4(w) and O - (Oi) ~ W z ' 4 ( w ) by "
-f-a'a2(~all2r/3111 4- ~311zr/,all2- ~3112r/alll - ~'c~lltr/3112), Ot
qo2(~, r/) := b2 r/a + r/3112 +aal(~alltr/3112 -4- ~3ll2r/alll -
~311xr/all2
-
~al12r/3111),
qo3(~, r/) :-- -hair/all/3 -aaXa~2(~allxr//3112 + ~ll2r/all 1 - ~all2rh311x - ~111'/a112)" Then
r162
- ~(r rl) for all ~ E 2ere(w), rl e T e a r ( w ) .
In addition, the field r ~
-r~
-r176
r~
- ~~
(if(r/)) e W(w) defined in (ii) satisfies all , / e Tr
Consequently,
where ~ ~
~ ( ~ o r/).
498
Nonlinearly elastic flezural shells
[Ch. 10
The proof of the above assertions is given for ~ - ~0. It is easy to check that it likewise holds for any r E dMy(w). The definition of the function Ca(Y) shows that r
= I - d(y).
To compute the Ggteaux derivative d'(~~ we do not use the definition d(y) = det M(rl); we use instead the relation d(y) 2 - 1 - z ( y ) r A z ( y ) , also satisfied by d(q). This gives
d( ~O)d,(~o)y
=
_z( ~O)TAz(rl );
hence r
_ _d,(~O)rl _
dzTAz(rl) _ TO(y).
While the above computation is valid for any q E W2'4(w), the next one makes a crucial use of the relations F~ ) - 0 in w satisfied by any y C ~(o.h4y(w) C W2'4(w); incidentally, this explains why the preceding computation is substantially simpler to carry out than the next one. The definition of the functions r shows that
+/b(.), where ~b(Vl) - (~ba(v/)):= d(v/)M-t(~/)z(v/). To compute the G~teaux derivative ~b'(~~ we first use the formula for the G~teaux derivative of the inverse matrix mapping, viz. (see, e.g., Vol. I, Sect. 1.2), ( M - t ( ~ 0 ) ) ' Vl where
M'(r
-
_ M - t ( ~ o) ( M ' ( ~ ~ 1 7 6
aal
\ l'7~a[12 ll
a a2 rla a a2 TIa
- z(u) TA.
12
Sect. 10.1]
A two.dimensional problem satisfied by the leading term
499
Combining this formula with that giving the derivative of a trilinear mapping and with the expression found supra for d'(~~ computing r176 this is why we began our computation with the third component), we obtain ~'(~~
(d'(r176176
_ _
~
-d(~~
-1 (~~ Z (v/)T A M -1 (~~ (~~
+d(~~
-1 (~~ (r/)
i (Z(~l)TAz)M_iz d
_ dM_IZ(~I)TAM_iz
+dM-tz(r/). Incorporating the relations F~ matrix formulation
) - 0 under their equivalent
Z(v/) T - _MZ(v/)M -T - z ( ~ I ) z T M - T -- z z ( ~ I ) T M - T , and noting that 1 M - T A M - 1 z - d-5Az
1 + z TM -TAM-lz
and
= 1 d2 ,
we infer that -M-iZ(v/) TAM-lz
-
1
Z(v/)Az
( 1 ) 1 1 + ~-~ 1 M-lz(v/)-4- ~-~M-
zz(o)~Az.
Consequently,
~) , (~ \r i(r176 -/b (r
1
~ z(.IA~
_ (,~
1 lz (.1 + ~M_ (br,o
This relation shows that r 1 7 6 T0(V/). We have thus established that r176 r~ Let us next directly compute the G~teaux derivatives r176 The functions r " W2'4(w) ~ Wl'4(w) being sums of linear and quadratic mappings, this easy task produces the announced relations
r162
C(~).
500
[Ch. 10
Nonlinearly elastic flezural shells
Finally, it must be verified that, when r / E q~o.h4F(W), the field ~o~ belongs to the space W(w). To this end, it suffices to note that the relations r/E W2'4(~) and r/-- 0~,~ = 0 on 70 imply that
o~o- r;~v~- ba~/3
- ~11~ E wX'4(w) and ~11~ - 0
on 3'0,
0ar/3 + b~/~ - r/311a E W 1' 4(w) and ~311a - 0 on 70, and that Wl'4(w) is an algebra. Hence ~o~ E W(w) and consequently v(r/) = za~~ E W(f~) when r/E T~o3ctF(w). II We conclude this analysis by establishing the second property stated without proof in part (vi) of the proof of Thm. 10.1-2. The expressions of the functions F^1ilk(r/) are not provided below simply because they are nowhere needed. T h e o r e m 10.1-4. Let the notations be as in Thm. 10.1-3. Given any 0 - (71i) E Tr let the functions ~k(r/) - (~ki(r/)) E W(w) and ~ ~ (~v~ E W(w) be defined as in ibid. Then
e~.
o (~(.)) _ P21 ( " ) - ~P~
(").
where the functions/~211O(r/) E LZ(w) are independent of z3 and the ^o functions FallO(r/) E L z (w), which ave also independent of z3, are defined by
po**:(.) := Folt:(~o(n))_ ~g o o 1 or..~ {~.o.lla~7.rll/3 q_ ~.r[[/3,qo.lla } I a m n f r , P, 1 #0
pp, 1 0
and the functions ~b~ are given by
~o = ~i(r where ~(r/) E W 1' 4(0)) i8 defined for any rl E W 2' 4(w) by
r
"-- b~r/a + (1 + aa2r/all2)r/311x - aa2r/alllr/3112,
~2(r/) "-- b~'7/~ + (1 + aal~Talll)r/3112 - aalr/al127/3111, ~3(z/) "-- -aa~r/all/3 - aala139(r/alllr/~l12 - r/all2r//3111).
A two.dimensional problem satisfied by the leading term
Sect. 10.1]
501
Proof. By definition, the functions F allf~ 1 (v) are for each v E W ( f / ) the coefficients of e in the formal asymptotic expansions induced by t h a t of the scaled unknown u(s) = u ~ + s u 1 + - . . in the functions 1
+ -~lgmn(s){Umlla(S)VnllO(e) + UnllO(e)Vmlla(e)} where (the functions T~p, ..ma,0 ~p, ..ma,1 a m n and gmn, 1 are defined in T h m . S.5-1):
Vmll~(e) - v m l l ~ -
p, 1 ex3rm~vp +...,
u.~ll~(~)-
6(0~-
0
u~l,o +
~,0 ~
r~.u b -
~p,O~,
O
p,~ 0 ~3r~o%) +..., 0 _~P,O
O
g m n ( e ) -- a mn + e x 3 g m n ' l . + _ . . . .
A simple computation, which in particular takes into account t h a t u 0 _ ~o and u 1 - ~1 _ x3r with ~.o _ (r
E ,&4F(w), ~1 _ ( ~ ) E W ( w ) , and r
_ (r
E W(w)
(see part (i) of the proof of T h m . 10.1-2), then shows that, for any field W E Vr C W ( w ) identified with a field ~ E W ( f / ) ,
F~ I
1 _mnf,,l _l_ x 3
1
(rr,1
0
( ,,I X'3 a m n {
2
0
1.~mnfv,p,1 o 0
(r
rp, 1 o
pp, I tO
0
F~
0
+-m~p)~-l[~+ (r
on the one hand; the definition v(W) - x3~~ (,,(,7))- ~ 3 F ~
)
+ rq~b~q)~Tml[ ~)
shows t h a t
~,o
on the other hand. Subtracting these expressions thus gives F1
o
(~(,7)) _
^~
^o
502
Nonlinearly elastic flezural shells
[Ch. 10
where the functions
fi,11
l a mn
1
1
L2
^0 are indeed independent of z3 and the functions Fall,(r/) E L 2(w) have the announced expressions and are thus also independent of ~. II The proof of Thm. 10.1-2 is thus complete, but the analysis is far from being terminated: There remains to express the variational equations satisfied by the leading term in more "decent" terms (Sect. 10.3). But first, it is time to formulate an essential definition. 10.2.
NONLINEARLY ELASTIC FLEXURAL DEFINITION~ EXAMPLES, AND ASSUMPTIONS ON THE DATA
SHELLS:
The way the variational problem found in Thm. 10.1-2 is formulated clearly shows that the "interesting" cases are those where, first, the manifold .h,'tF(w) contains nonzero elements and, second, ~ t F ( w ) possesses nonzero tangent vectors at each one of its points. This observation is the basis for the following definition: Let w be a domain in IR2 with boundary 7 and let 0 E C2(~; mR3) be an injective mapping such that the two vectors Oa8 are linearly independent at all points y of ~. A nonlinearly elastic shell with middle surface S = O(~) is called a n o n l i n e a r l y elastic f l e x u r a l shell if the following three conditions are simultaneously satisfied: (i) The shell is subjected to a boundary condition of place along a portion of its lateral face with 0(70) as its middle curve, where the subset 70 C 7 satisfies length 70 > 0. (ii) Define the manifold: tFC
)
"- {.
-
e
=
a~(r/) - a ~
=
o
= 0 in w},
Sect. 1 0 . 2 ]
Definition,ezamples, and assumptions on the data
503
where aaf~(r/) denote the covariant components of the metric tensor of the surface (0 + rliai)(-~). Then # {o).
In other words, there exist nonzero displacement fields rIia i of the middle surface S with covariant components r/i in the space W2,4(w) that are i n e x t e n s i o n a l d i s p l a c e m e n t s of S, in the sense that they preserve the metric of the surface S (this is what the relations aaf3(r/) -aaf3 - 0 in w express), and that are also a d m i s s i b l e , in the sense that they satisfy boundary conditions "of strong clamping" along the curve 0(70), expressed here by means of the boundary conditions r / = 0 v r / = 0 on 70 on the associated field r / = (7/i) (these boundary conditions are interpreted later; cf. Sect. 10.5). (iii) At each ~ E ~ F ( w ) , the tangent space to the manifold J~tF(w) contains nonzero functions, i.e.,
Vr
# {0} at each ~ E .h4F(w), where
72~j~4F(W) "-- {r/ E W2'4(w); r / = 0vr/-- 0 on 3'0, - o
and (Thm. 9.1-1) 1 a~e(n) = ~(a~,(n)
-
a~e)
1 = ~(:r/all~ + ~7~lla -4- amn~lmllarlnll~),
,
1
amn{
Examples of nonlinearly elastic flexural shells are given in Figs. 10.2-1 to 10.2-3 and justified in Exs. 10.1 to 10.3. A w o r d of c a u t i o n . Like that of a nonlinearly elastic membrane shell, the definition of a nonlinearly elastic flexural shell depends only on the subset of the lateral face where a boundary condition of place is imposed (via the set 70) and on the geometry of the middle surface of the shell (via its second fundamental form and Christoffel symbols, which appear in the functions ~7ill~); cf, Ex. 10.4. i
504
Nonlinearly elastic flezural shells
[Ch. 10
\
/
Fig. 10.2-1: Nonlinearly elastic flezural shells. A shell whose middle surface is a portion S = 0(~) of a cylinder and which is subjected to a boundary condition of place (i.e., of vanishing displacement field) along a portion (darkened on the figure) of its lateral face whose middle curve 0(70) is contained in one or two generatrices of S provides an instance of a nonlineaxly elastic flexural shell, i.e., one for which the manifold ~ v t F ( ~ ) = { ~ ~ w ~ , ' ( ~ ) ; ~ = o ~ n = o o~ 70, ~ ( n )
- ~
= 0 m
~}
contains nonzero ftmetions r / a n d the tangent space at each one of its elements contains nonzero fimctions; of. Ex. 10.1. The "two-dimensional boundary conditions of strong damping" 17 = igor/= 0 on 70 are so named because they express that the points and the tangent vectors to the coordinate lines of the deformed and undeformed middle surfaces coincide along the set e(70) (as suggested on the "two-dimensional" figures); cf. Sect. 10.5. This example and that of Fig. 9.1-1 illustrate the crucial r61e played by the set 8(7o) for determining the type of a shell, like in the linear theory (Figs. 5.8-2 and 6.1-1).
Sect. 1 0 . 2 ]
Definition, ezamples, and assumptions on the data
505
Fig. 10.2-2: Another ezample of a nonlinearly elastic flezural shell. A shell whose middle surface S is a portion of a cone (excluding its vertex) and which is subjected to a boundary condition of place along a portion (darkened on the figure) of its lateral face whose middle line is contained in one generatrix of S provides another example of a nonlinearly elastic flexttral shell; cf. Ex. 10.2.
A n o t h e r w o r d o f c a u t i o n . A l t h o u g h the above definition is strongly suggested by the a s y m p t o t i c analysis ( T h m . 10.1-2), it is certainly perfectible, t h r o u g h an ad hoc weakening of the required regularity on the covariant components of the admissible inextensional displacements. Be t h a t as it may, these should be "at least" in W 2 ' p ( w ) for some p >_ 1, if only to insure t h a t the traces 0vrl = 0 on ")'0 make sense. A n ideal classification of nonlinearly elastic shells would be one that leaves no gap between the " m e m b r a n e " and "flezural" ones (in this respect, see a similar "another word of caution" in Sect. 9.1), like in the linear case! m R e m a r k s . (1) In the linear case (Sect. 6.1), only one condition was seemingly needed, viz., V F ( w ) ~ {0}, where VF(
)
:=
-
e
•
qi = 0 ~ s = 0
• on
70, 7at3(rl) = 0 in w}.
506
Nonlinearly elastic flezural shells
[Ch. 10
r
Fig. 10.2-3: Another ezarnple of a nonlinearly elastic flezural shell. A plate subjected to a boundary condition of place on a portion of its lateral face whose middle line 70 is a segment provides an instance of a nonlinearly elastic flexural shell; cs Ex. 10.3. By contrast, if 70 is the union of two parallel segments, the plate becomes a nonlinearly elastic membrane shell! cs Fig. 9.1-2. But this condition is likewise one (albeit "in disguise") on the tangent spaces to V y ( w ) , since T c V y ( w ) = V y ( w ) at all ~ e V y ( w ) in the linear case. (2) We shall see later (Sect. 10.6) that the proper framework for the existence theory rests on an ad hoc "enlargement" of the manifold to
{ r / e H2(w); r / - - 0vr/-- 0 on 70, aa/3(17) -- aa/3 -- 0 in w}, a sign t h a t the definition of nonlinearly elastic flexural shells proposed here is still perfectible (as hinted at above). (3) By adding condition (iii) in the above definition, I depart here from the definition given in Lods & Miara [1998]. II The conclusions reached in T h m . 10.1-2 then naturally lead to the following a s s u m p t i o n s o n t h e d a t a f o r a f a m i l y of nonlinearly elastic flexural shells, with each having the s a m e middle surface S = 0(~) and with each subjected to a b o u n d a r y condition of place along a portion of its lateral face having the s a m e set 0(7o) as its middle curve, as their thickness 2e approaches zero:
Sect. 10.3]
The t w o - d i m e n s i o n a l equations as a variational problem
507
We require that the Lamd constants and the applied body and surface force densities be such that )~ = )~
and
#e =/~,
_ ~2/~.2(.)
fo~ all
.~ - . ~ .
~
h*.~(. ~) - ~3h~.3(.)
for all
.r - . ~
C r ~ u r~_.
/r
~r
where the constants )~ > 0 and Iz > 0 and the functions fi,2 E L2(fl) and h i'3 C L2(r+ u r_) are independent of e (Fig. 3.2-1 recapitulates the definitions of the sets fit, fl, r~_, r + , r ~ , and r _ ) . Note that the same limit two-dimensional equations are evidently obtained if the following more general a s s u m p t i o n s on t h e d a t a are made:
and
~ue = r
= ~2+.1,.2(.)
for an
.~ = ~ .
~
h ,, ~(.~) _ ~ + , h ,, 3 ( . )
for an
.~ - ~ .
~ r ~ u r ~_
A e = eta
/,.~(.~)
~.
where the constants )~ > 0 and # > 0 and the functions fi,2 E L2(fl) and h i, 3 E L2(r+ Ur_) are again independent of e and t is an arbitrary real number. Besides the analysis that led to Thm. 10.1-2 shows that these assumptions on the data are the only ones possible for nonlinearly elastic flexural shells. 10.3.
THE TWO-DIMENSIONAL VARIATIONAL PROBLEM
EQUATIONS
AS A
It was shown in Thin. 10.1-2 that, when ~t0(w) ~ {0} and ~Fr contains nonzero functions at each ~ E 2~0(w), the leading term of the formal asymptotic expansion of the scaled unknown u(e) is independent of the transverse variable x~ and may thus be identified with a function r of two variables; it was further shown in ibid. that ~0 belongs to the manifold ~ t F ( w ) C M0(w) and that r satisfies ad hoc variational equations for all rl E ~r C ~r The definition of a nonlinearly elastic flexural shell given in the previous section precisely ensures that this variational problem makes sense for such a shell. It remains, however, to shed a brighter light on the somewhat mysterious functions/~0all~ and/~0a[[~(rl) appearing in the variational
Nonlinearly elastic flezural shells
508
[Ch. 10
equations of Thm. 10.1-2: In the same manner that the functions E~ IIf3 and Fallf3(Vl) 0 occurring in the variational equations of a nonlinearly elastic membrane shell are simply equal to G ~ ( r ~ and a ,z~ ( r 0 )rl (Thin. 9.1-1), where G~(,7)
1
- a~)
- ~(a~('7)
are the covariant components of the change of metric tensor associated with a displacement field ~?iai of the middle surface, the functions /~~IIf3 and Fallf3(17) ^0 have definitely more illuminating expressions and interpretations, notably in terms of the covariant components R.,(o)
= bo,(n) - b.,
of the associated change of curvature tensor (see notably Thms. 10.3-1 and 10.4-1). The following result takes care of the functions E~ As stated here, it extends Lods & Miara [1998, Thm. 2], in that I prove that it holds not only for the leading term ~0, but in fact for any element 17 in the manifold 2vtF(w). This extension is particularly convenient for interpreting the two-dimensional equations as a minimization problem (Sect. 10.4). Note that the intricate expressions given in the next theorem for the functions R~f3(17) can be considerably simplified; cf. Thin. 10.3-2. 10.3-1. Let w be a domain in I~2, and let 0 E C3(~; ~3) be an injective mapping such that the two vectors aa - OaO are final Aa2 early independent at all points of-~, let a3 -- la I A a21' and let the Theorem
vectors a i be defined by a i . a j - J~. Given a displacement field ~Tiai of the surface 0(-~) with smooth enough covariant components yi "-~ ~ IR and such that the two vectors aa(rl) := Oa(O + Oia i) are linearly independent at all points of w, let the covariant components of the c h a n g e of c u r v a t u r e t e n s o r associated with this displacement field be defined by : =
-
where bao and ba~ (0) are the covariant components of the curvature tensors of the surfaces 0(-~) and (0 + ~Tiai)(-w) (Fig. 2.3-1).
Sect. 10.3]
The t w o - d i m e n s i o n a l equations as a variational problem
509
Then the vectors aa(~l) are linearly independent, and consequently the functions ba#(~l) and Ra#(~7) are well defined, for all ~1 E .A4F(W). Let the functions R~# (~7) E L2 (w) be d e ~ a ~o~ ~ y o e W ~' ~(~) by 1
{TT
p, 1 +(r:~ ~ + ~1 a m n {r~oo.ll, + r p,. , 1~11o})~,
where ~11~ "= 013r/~ - r-a[3~l~- ba[3~13and ~7311~:- 0[3~73A-b~l~, the functions Fall,(r/, r E L2(w) are defined for any 17 E W2'4(w) and any r E wl'4(og) by
1 "-- ~(r
Fall/3(r/, r
and finally, the field r E W 2 ' 4 ( O g ) by
-f- r
-f- amn{71mllaCnll# + ~Tnll#r (~bi(n)) E wl'4(6g) i8 defined for any
r := b~r/a + (1 + aa2rlall2)rlzllI - aa2r/alll~3ll2, ~/32(Ir/) :'-- b'~rla + (1 + aalr/alll)r/31l 2 - aalr/all2r/3ll 1, r : - -aa~rlall~ - aata~2 (rlalll rl~ll2 - r/all2r/~lll). Then
R~(~)-
R ~ ( ~ ) for a11 ~ E , ~ F ( w ) .
Let the functions ~oall~ E L 2(w) be those appearing in the variational problem satisfied by the leading term ~o E ~4F(w) (el. Thin. 10.1-2; their definitions are given in part (iv) of its proof), viz., /~0
.
0
1 err
0
+(r:b~ + ~1 a m n flap, , ' ~ -1/'0 ,,,
0
+ r~
o
510
[Ch. 10
Nonlinearly elastic flezural shells
where 1
F~ r
_mnr~O
:= ~(r/alll3 + r/~l[a -f- a := (r with r ._ r
0"
(~m[[a~ln[[[3+ ~nli/3~lm[la}),
Then
~~ ~ - R~,(r o) = R~,(r176
^o ~ - Ra~(~ b Proof. That E,~II ~ immediately follows from the defi-
nitions of the functions R~13(~/) and/~0
It thus remains to show
that ba13(r/) is well defined and that R~(~/) - ba~3(~/)- bal3 for all n ~ A4F(~). (i) For any '1 E A 4 y ( w ) , the vectors aa(~l) are linearly independent at all points in-~. Since the set ~ is compact, the assumption that the vectors aa E C2 (~; I~3) are linearly independent implies that there exist constants Cl and c2 such that 0
< 1
and
lal" a2l < cllallla2l in ~,
0 < c2 < 1
and
c2 < la,~l < c~-~ in ~,
on the one hand. By definition of the manifold .A4y(w),
lal(n)l- v/~ii(n)- ~
= la~l i= ~,
1~2(n)l = v/~2~(ni = ~
= ia21 in ~,
a l (~/)" a2(lr/) = a l 2 ( U ) -- a12 in ~,
on the other. Hence
I~l(n)" a2(,n)l ~ ~llo-.,(n)ll~2(n)l
~2 _< la~(w)l ~ ~
in ~,
i= ~.
Since Cl < 1, these relations mean that the vectors al(~/) and a2(~/) are likewise linearly independent in ~ (they are even "uniformly" linearly independent in w, a crucial property of isometric
Sect. 10.3]
511
The two-dimensional equations as a variational problem
surfaces" t h a t will be put to another essential use in the existence theory of Sect. 10.6).
ax(n)
(ii) For a n y ~1 e . ~$ F ( W ) , the vector a3(r/) -- lab(r/) A a2(r/) I ' w h i c h is well defined since the vectors a a ( r l ) are t h e n linearly i n d e p e n d e n t by (i), is also g i v e n by
a3(,1) x (n)a -
where
Xl(~)
:"-
- ( 1 -f- aa2~7all2)r/31l 1 -4- aa2rlalllrl3ll2,
X2(r/) := aal~7,~l12r/3111 - (1 + aalr/,~lll)r/3112, x3(r/) "-- 1 + a'~13r/all/3 + aala~2(r/alllr//3112 - r/,~l12r//3111). The formulas of Gauf~ and Weingarten (Thm. 2.3-1) imply that al(r/) A a2(r/) -- ( a l + 01(~liai)) A (a2 + 02(r//ai)) :
a l A a2 -4- ~lallla a A a2 -4- r//3ll2al A a 13
+r/3llla3 A a2 -4- r/3ll2al A a 3 A- ~Tal11~l[3112aa A a [3 -f-~Talll~73112aa A a 3 + r/3111r/13ll2a3 A a 13.
Incorporating the formulas (the first ones were established in parts (i) and (ii) of the proof of T h m . 2.5-1; the others are immediately proved): al A a2-
-v/-aa 3
a l A a 3 - - - v / a a 2,
a a A a2 -- aa[3a~ A a2 -- ~
a a l a 3,
a 3 A a2
-- ~
a1
a l A a [3 -- vraa/32a 3,
a a A a 3 -- a a ~ a ~ A a 3 -- ~/~ ( a a 2 a 1 - a a l a 2 ) , a a A a [3 -- ~
( a a l a [32 - a a 2 a ~ l ) a 3,
where a - det(aal3), into the expression of a l ( r / ) A a2(r/) then gives
(,7) A
-
x (,7)a
where the functions Xi(~l) are of the announced forms. It remains to observe that, by definition of the manifold 2~4F(W), A
- ~/det aal3(r/) - v/det aal3 - v/~.
512
[Ch. 10
Nonlinearly elastic fle=ural shells
(iii) For any rI E .h~tF(w), the .functions ba~(~7) Oaa~(n)'a3(~l), which are well defined since the vector a3(r/) is then well defined by (i), are also given by -
1
b.,(~)
-
.~ (~,(~) + ~o(~))x~(n),
where the functions xp(rl) are defined in (ii) and
~(n)
:= b~ + 0~{~sil~} + b~,ll~.
The definition of the functions ba13(r/) = b#a(rl), the expression found in (ii) for the vector a3(r/), and the formulas of Gaufl and Weingarten together give b ~ , ( n ) = o ~ a ~ ( n ) . a3(n)
: ( 0 ~ { ~ + ~ t l ~ ~ + ~311~}). (x,(n)~ ~) 1 where the functions ~P~(r/) are of the announced forms. (iv) For any n e .t~4F(w), the functions R~f3(~?) as defined in the statement of the theorem satisfy
where X(~7) =: (Xi(17)) and the functions Xi(~7) are defined in (ii).
The definitions of the fields r (see the statement of the theorem) and X(r/) immediately reveal that
-r
= x ( n ) + ~c(n),
where K:(n ) = (K:i(n)), the functions K:i(n) being defined by Ka(y) := -b,~r/,,. and Ea(Y):= -1.
The two-dimensional equations as a variational problem
Sect. 10.3]
513
Consequently, the functions R~o(r/) may be also written as
R~(n) - -F~jl~(n, x ( n ) ) - F~ll~(n, ~C(n)) 1 ~r'r
-~g
' lr/~llar/rll~ lamn
+(r~:~ + ~
p, 1
p, 1
{r~o~,l ~ + r.~ ~,lo})~.
There thus remains to compute the functions 1
+ ~lamn (r/mlla{K:n(rl) } Ilfl + ~Tnll~{K:m(r/)} IIa) " The definitions of the functions K:i(r/), of the covariant derivatives ba[ ~ (Thm. 2 . 5-1), and of the functions vP'I ~aft (Thin. 8.5-1) together show t h a t
O"
Or
O"
= ba~ - b~l~r/~ - b~r/~ll~ -
7"
b~b~rla
~,p, 1
= ba~ + t a~ yp - b~r/~ll~,
{~:~(n)}ll~ - 0 ~ { ~ ( n ) }
+ b;~(n)
- -b~b;~. -
l~r 1
_~ ~.
Consequently, ~p, 1 l azr
p, 1
1 aZ r (bza + Fp, 1
+~
_,~a
1
)~
r/p - b~r/.ll a)r/rll ~
1pr, 1
-
b ~ - ~g
, lr/~ll~r/~ll~
p, 1 p, 1 + ( r ~ ~ + ~l a m n {r~o~,,~ + r.~ ~.llo})~,
since
a~rb~r -- 89
(Thin. 8.5-1). Therefore,
~ , ( n ) = -F~ll,(n, x(n)) - ~ .
514
[Ch. 10
Nonlinearly elastic flezural shells
~o11~(., x ( . ) ) = - ~ ( . ) . C o n s e q u e n t l y , f o r a n y rl ~ . h / t ~ ( w ) ,
A straightforward computation, based on the definitions of the functions Xi(r/), shows that these functions satisfy X~(~/) + am"~/mll~X.(~/) - 0 for all ~/~
wl'4(0g).
Then the relations ~o (X~(~) + "mn~mil'Xn(~)) = 0, together with the relations
O a a m3 -
0
(recall that
O a a a r -- O a a ~ . a r + a ~ 9O a a r --
a m3 -
~m3) and
_rX~a/S~ _ rX/sa~/s,
imply that -amnOa{rlmll~}Xn(rl)
-
Oa(xO(~l)} t a'~r
--(a
s
+ amnrlmlloOaXn(~l) a#rpa
+-
--att)r/allf~XT(r/)
Incorporating these relations into the expressions of the functions ~Pf~(r/) defined in (iii) then gives
~.(n)x~(n) - rxzx~(n) + b.zx~(n) + =
(a ar
,~ ror~ll~ +
-{xz(o)}ll~
am"O,~{~?mllZ}X.(~l)
~
ba~/311f3)x~(r/) +
r
b,~71rlt~x3(rl)
- a""{xm(n)}ll~n.llz,
so that the expressions found in (iii) for the functions b~f3(r/) when r/E .h4F(w) become 1
= _1 ({x~(.)}llo + {xo(.)}ll~) 2
1 am n
= -F~l,~(~, x(~)). The proof is thus complete.
II
Sect. 10.3]
The two-dimensional equations as a variational problem
515
Remark. The proof shows that, in fact, R~f3(rl)- Raf3(l?) for all W2'4(CO) that satisfy aa~(y) - aaf3 -- 0 in w, i.e., the boundary conditions satisfied by the fields y in the manifold .h4F(w) play no C
rhle here.
[]
We next show that the functions R af3 ~ (y) have considerably simpler expressions in some realistic cases; this observation is due to Roquefort [2000]. m T h e o r e m 10.3-2. Let w be a domain in I~2, let 0 E C3(~; I~3) be an injective mapping such that the two vectors aa - OaO are linearly independent at all points in w, and let the ]unctions R~af~(17) E L 2 (w) be defined for each ,? E W2'4(w) as in Thm. 10.3-1. Then one also has 1
for all ~ e W2'4(w) such that det(aaf3(y)) > 0 in where ha(y) "- Oa(e + ~?ia'). These relations imply in particular that
= b.,(n)
-
= R.,(,)
.
e
as was already established in Thm. 10.3-1. Proof. The proof follows the same lines as that of Thm. 10.3-1. If, instead of the relations haft(y) -aaf3 = 0 in w, a vector field rl E W 2' 4(w) satisfies the less restrictive relation
a(rl) := det(aafj(Vl) ) > 0 in w, then the vector a3(~) and the functions ba~(~7) are now given by (compare with parts (ii) and (iii) of the proof of Thm. 10.3-1, whose notations are employed here): i a3(~7)-
aa(~7)Xi(~)ai,
516
[Ch. 10
Nonlinearly elastic flezural shells
so that (compare with part (v) of the proof in ibid.) a
II
for such a vector field, and the assertion is established.
The next result, due to Lods ~ Miaxa [1998, Thin. 3], takes care of the functions F_ ~^0 iif~(~7). ..
Let the functions R ~ ( y ) e L 2 ( w ) b e def i ~ e d f07" a ~ y ~7 ~ W2'4(~g) as i~, Thin. 10.3-1 and let the functions ~o11~(,~) ~ L~(~) b~ tho,~ a ~ ~ i , ~ i,~ th~ ~a~atio,~t ~q~atio,~, ~tisfied for all ~1 ~ T~0c~e(w) by the leading term ~o ~ .~te(w) (cf. Thin. 10.1-2; their definitions are given in part (vi) of its proof), viz., Theorem
10.3-3.
1
(p,1
I am n y~p, 1/,0
I am n
0
O'7"
~ pp, 1 0
FP, 1 [,O
~q, 1 0
where F2I
l
i~('~) "= ~(~ii, + ,7~11~+
a mn
0
{i.,il~'7.1i~ + r
r/ailf3 :-- OfP7a -- r aaf3~70- - ba/3~73,
~73iif3:-- 0f3~73 ~- b~r/0-
and the fields ~0(~/) _ (~o(17)) e W(w) and r given by:
v0(,)
0
e W(w) are
= (r
.= b?,o + ,~ll~ +~"~(r
+ i~
- i3~
~ - i~
+aal (~~
-I- (:~
- (:~ lvlail2 - ~all2~Taill),
o
~~ (~7) :---aaf3~Tallf3 -aala~2(~'a~
2 + ~'f~li2~Talll- ~~
- ~f~lll~Tal12),
Sect. 10.3]
The two-dimensional equations as a variational problem
r
:= r (~o) _ b~.o + (1
+ aa2(~~176
r
: : r162
+ aal(~Ojll)(~30]]2
= b~r + (1
r 0 :--- r
----aCZ/~iOi]/~
aal -
-
517
1 -a-a2 v.o~aj]l~-3]02,0 -
aalr
-
0 0 0 0 a/~2(~aJ]l~/~ll2 ~ai]2~/~]}l)" -
-
Then
#~
-
(R~) , (~0)y for all y e Vr ~
where (R~f~)'(~"~ designates the Frdchet derivative at r of each mapping R~/~ " W2'4(w) -+ L2(w).
e .h4v(w)
Pro@ For details about the notions of differential calculus used in~a, see, e.g., Vol. I, Sect. 1.2. The functions R~af3(v$) are of the form
R~(,~) - a~,,~(,~)+ H~,,~(~), where
a~it~(n ) .- F.it~(., r 1
ar
i
+ ( r ~ ~+ ~
a m n.f~p, 1
p, 1
~-~o,-11~+ r.~ ,~t,o}),~.
The mappings (~7, ~ ) " W2'4(w) • wl'4(w) -+ Fall/3(~, ~) E L2(w) are affine with respect to 17, linear with respect to ~, and continuous; hence they are differentiable. As a sum of continuous linear and quadratic mappings, the mapping ~ 9W2,4(w) -+ w l , 4 ( w ) is likewise differentiable. The expressions of the G gteau.x derivatives for such functions together with the chain rule then give
a'lt~(r fo~ an r
-
Fo,t~(r r162
~ w2,4(~). Be~ide~ (Thin. 10.V3), r162
C W2'4(0~),
_ ~o0(.) for a l l . E ~r
so that
r~ll~(r ~ r162176 for a11 ~ E qF~o~F(w).
FoiI~(r 0, ~,~
-
Foll~(v,o(,7))
518
[Oh. 10
N o n l i n e a r l y elastic f l e z u r a l shells
As sums of continuous linear and quadratic mappings~ the mappings Hall3 " W2'4(w) --+ L2(w) are differentiable and their G~teaux derivatives are given for any ~, ~/E W2'4(w) by 1 (rr
lamn
+ (r:~~ + ~
p, 1
{r~or
1 _mn r~p, 1
+ r~.~~r
l~p, 1
Combining the above relations thus yields
R~e(r
_ a.lle(r
+ H. le(r
_ p0
for all rl E T~;o.h4F(w) and the proof is complete.
I
We are now in a position to recast the variational problem of Thm. 10.1-2 in a more suitable form, as an immediate corollary to Thms. 10.3-1 and 10.3-3: T h e o r e m 10.8-4. Consider a family of nonlinearly elastic flezural shells according to the definition of Sect. 10.2, with thickness 2~ > O, with each having the same middle surface S = O(~) and with each satisfying a boundary condition of place along a portion of its lateral face having the same set {?(To) as its middle curve, and let the assumptions on the data be as in Sect. 10.2. Finally, assume that 0 ~ Ca (~; I~3). Then the leading term u ~ of the formal asymptotic ezpansion of the scaled unknown u(e) is independent of the transverse variable x a and ~o "-._ !2 f l 1 uO dx3 satisfies the following sealed twod i m e n s i o n a l v a r i a t i o n a l p r o b l e m T'F(w) of a n o n l i n e a r l y elastic flexural shell: ~0 C .M.F(W)"-- {~ C W2'4(W); ~ -- ~ , ~ -- 0 on "Yo,
o , ~ ( n ) = o i~
1L
a3~r b
L
'
"
for all r / - (r/i) E T r Vr
:= {n c w~,4(~o); n
~},
where
- o ~ n = o oll "to,
G'/3(r176
0 in w},
Sect. 10.4]
The two.dimensional equations as a m i n i m i z a t i o n problem
519
1 where the functions Oa~ (0) -- ~ (aa# (~1) - aa#) and R~a#(~1) are defined in Thins. 9.1-1 and 10.3-1, and finally, aaf3~rr := ~ 4Xl.t + 2p a~a~rr + 2p(aa~ra~r + aara~r), 9 := pZ,2
i , 3 -4- h i_,3 a n d h ~ 3 " - h z'"3 (., -4-1). f z ,92 d x 3 + h.,%
1
E
10.4.
T H E T W O - D I M E N S I O N A L E Q U A T I O N S AS A MINIMIZATION PROBLEM
A crucial property has yet to be exploited: When a field ~1 = (~}i) belongs to the manifold .h4F(w), the functions R~f3(17) become equal to the covariant components of the change of curvature tensor associated with the displacement field 7}iai of the middle surface S (Thm. 10.3-1). This goal is achieved in two stages: First, by recognizing as in Lods & Miara [1998, Thm. 3] that the variational equations of problem 7~F(w) (Thm. 10.3-4) express that the Gdteaux derivatives (j~F)'(~~ of an ad hoc functional J~F vanish for all the fields ~7 in the tangent space to the manifold .A/tf(w) at ~0; second, by noting that, over this manifold, J~F becomes equal to a functional jF whose integrand is (apart from its linear terms accounting for the applied forces) a quadratic and positive definite expression, via the scaled twodimensional elasticity tensor of the shell, in terms of the change of curvature tensor introduced in Thm. 10.3-1. These decisive observations in turn allow to recast the two-dimensional variational problem :PF(w) as a minimization problem, which now exhibits a remarkable simplicity. In this minimization problem, which will be studied for its own sake in Sect. 10.6, lies the apex of the application of the method formal asymptotic expansions to nonlinearly elastic flexural shells. T h e o r e m 10.4-1. Given a nonlinearly elastic flezural shell according to the definition given in Sect. 10.2, let the manifold ~&4F(w)
520
[Ch. 10
Nonlinearly elastic flezural shells
and the functional j~ " W2,4(w) --~ I~ be defined by ~v(~)
"- { n = ( ~ ) e w 2 , 4 ( ~ ) ;
n - o~
- o on ~0,
a.z(n) -~,
j~(17) "-- ~
a
R~r(~7)R~f3(.)v/ady-
= 0 i=
~),
p"27Ii~dy
for all O ~ W~,4(w),
where the functions aaf3(17) and Rba[3(n) are defined in Thins. 9.1-1 and 10.3-1 and the functions a a/3ar and pi,2 are defined in Thm. 10.3-3. Then the functional j~ is differentiable over the space W2'4(w) and ~o C .&4F(w) is a solution to the variational problem PF(w) of Thin. 10.3-4 if and only if it is a stationary point of the functional j~ over the manifold ,A,4F(w), i.e., it satisfies (j~),(~o)~/ _ 0 for all in the tangent space ~go,A,4F(w) to the manifold ,h/tF(w) at ~o. Since the fianctions bar3(U) are well defined for all ~ G .A/tF (w) and R~,(n)
= b.~(u)-
b.~ Io~ an ,7 e ~ v ( ~ ) (Tam. 10.3-2), w~tic~t~
solutions to problem 7OF(w) are obtained by solving the minimization problem: Find ~ such that
E ~It~(w) and j F ( ~ ) =
inf iF(v/), where neAav(~)
1 f~ aaf3~r - ~pi,2~Tivfady.
Proof. The differentiability of the functional j~ over the space W2,4(w) is a simple consequence of the definition of the functions R~(v/) (Thm. 10.3-1). An argument similar to that found in the proof of Thm. 8.2-3 then shows that the G~teaux derivatives (j~)'(~)Vl are given by
(j~)'(~)v/- ~
a
Rar(~)((Rba/3)'(~)Vl)v~dy-
p"2rliv~dy
for all ~, ~/E W2'4(w). Hence an inspection of the variational equations satisfied by ~o E A4F(w) for all vl 6 ~ o ~ t F ( w ) (rhm. 10.3-4)
Sect. 10.5]
The two.dimensional equations derived by a formal analysis
521
immediately reveals that they coincide with the equations (jbF)'(r176 : 0 for an ~ e Vr Conversely, if r E .~vtF(w) is a stationary point of the functional J~F over the manifold 2vtF(w), it also satisfies the variational problem 7>F(w) of Thm. 10.3-4, since the argument that led in Thm. 10.3-3 to the relations _
po
all~(~7) for all ~ C Tr
holds in fact if ~0 is replaced by any ~ in the manifold 2vt~(w) (this observation relies in particular on part (iii) of the proof of Thm. 10.1-3). Particular solutions to problem T'F(w) may thus be obtained by solving the minimization problem: Find r such that C ~'tF(W) and j ~ ( ~ ) -
inf
j~(rl).
Since the functionals j~ and jF coincide over the manifold ~ F ( W ) , this minimization problem may be equivalently defined in terms of the functional iF. I
The functional jF : ,A/tF(W) --+ R is called the scaled two-dim e n s i o n a l e n e r g y of a n o n l i n e a r l y elastic flexural shell. We shall see in Sect. 10.6 that the minimization problem in terms of the functional jp is in fact well-posed over a manifold larger than .A/tF(w), in the definition of which the space H2(w) replaces the space W 2' 4 (w). 10.5.
THE TWO-DIMENSIONAL EQUATIONS OF A NONLINEARLY ELASTIC FLEXURAL SHELL DERIVED
BY MEANS
ASYMPTOTIC
OF A FORMAL
ANALYSIS; COMMENTARY
In order to get physically meaningful formulas, it remains to descale the components ~o of the vector field ~0 that satisfies the scaled two-dimensional problems found in Thms. 10.3-4 and 10.4-1. In view of the scalings ui(~)(x)
- u ~ ( x e) for all x e - ~r~x e ~e
522
[Ch. 10
Nonlinearly elastic flezural shells
made on the covariant components of the displacement field (Sect. 8.4), we are naturally led to defining for each s > 0 the covaria n t c o m p o n e n t s ~ 9W --+ IR of the l i m i t d i s p l a c e m e n t field ~e 9 _w --~ i~3 of the middle surface S of the shell by letting (the vectors a i form the contravariant basis at each point of S)"
Cf ._ r
and ~ := era'.
A w o r d of c a u t i o n . As always, the fields ~ := ( ~ ) and ~ - ~f a i must be carefully distinguished! The former is essentially a convenient mathematical "intermediary", but only the latter has physical significance. II Recall that fi,~ E L2(n e) and h i'e E L2(r~_ u r ~ ) represent the contravariant components of the applied body and surface forces actually acting on the shell and that A* and #* denote the actual Lain6 constants of its constituting material. We then have the following immediate corollary to Thins. 10.3-4 and 10.4-1: T h e o r e m 10.5-1. Let the assumptions and notations be as in Thin. 10.3-4. Then the vector field ~ := ( ~ ) formed by the covariant components of the limit displacement field ~ a i of the middle surface S satisfies the following t w o - d i m e n s i o n a l v a r i a t i o n a l p r o b l e m P~(w) of a n o n l i n e a r l y elastic f l e x u r a l shell:
~s E ,/q~F(69)"-- {~ -- (TIi) E W2'4(w); i t / - Ov~ - 0 on 70,
for all v/E ~Fr
:= {~/E
W2'4(w);
n -
0vn -
0 o n "fo, o in
,,,),
Sect. 10.5]
The two-dimensional equations derived by a formal analysis
523
where a.,(n)
1
.-
-
4)~~#~ a,~a~, - + 2/.t~(aa,, a f~r + a,~,a~,~ ),
aC,f~rr, ~ :=
A~ + 2p ~
pi, S ._
f
'~
9
9
.
.
f"~ dx i + h; ~ + h i'~_ and h~ ~ : : h ''~(., +e),
and the functions R~(17) are defined in Thm. 10.3-1. it is a stationary point over the manifold ,h,4F(w) of the functional j~ defined by e3 ~ aa~ar, eR~r(~l)R~(U)V/_~ dy - ~ pi, e71iv/~dy
~07" all ~ -- (~i) e
W2'4(W).
Particular solutions to problem 7~(w) are thus obtained by solving the minimization problem: Find ~e such that (the functions ba~(~/) are the covariant components of the curvature tensor of the surface (0 + ~Tiai)(-~); cf. Thm. 10.3-1)
~6 E .A4F(w) and j~(~e) _
inf
j~(~/), where
~3 f aa~r' (b~r(~l) - bar)(ba~(~l) - ba~)v/-ady
J~(~l) "- -~ J~
- ~ Pi'~?iv~dY"
m
Any one of these problems satisfied by ~e constitutes one version of the t w o - d i m e n s i o n a l e q u a t i o n s of a n o n l i n e a r l y e l a s t i c f l e x u r a l s h e l l (the specific meaning conveyed by "flexural" is given below). The functions 1 Gaff(r/) - ~(aal3(r/) - aa~) and Raf~(r/) - (baf3(r/) - baf~)
Nonlineavly elastic flezural shells
524
[Ch. 10
are respectively the covariant components of the c h a n g e of m e t r i c t e n s o r and c h a n g e of c u r v a t u r e t e n s o r associated with a displacement field yia i of the middle surface S (Thms. 9.1-1 and 10.3-1) and the functions a a ~ ' e are the contravariant components of the t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of t h e shell. The functional j~ 9.A4F(w) --~ I~ is the t w o - d i m e n s i o n a l energy, and the functional
~3 ~ aa~ar'~(b,r'r(W) _ b,rr)(ba~(rl) _ ba~)v~dy is the two-dimensional strain energy~ of a nonlinearly elastic flexural shell. Finally~ the boundary conditions Y -- O~y -- 0 on 7o express that the shell is strongly c l a m p e d along 7o (the geometric interpretation of these boundary conditions is given infra). A major conclusion is thus that~ without any recourse to any a priori assumption of a geometrical or mechanical nature, the method
of formal asymptotic expansions provides a justification of the twodimensional equations of a nonlinearly elastic flexural shell, in the
forms found in Thm. 10.5-1. This justification by Lods & Miara [1998] constitutes a generalization to shells of the formal analysis of Fox, Raoult & Simo [1993] in the planar case, i.e., when the mapping 0 is of the form 0(yl, Y2) = (Yl, Y2, 0) for all y -- (Yl, Y2) E w (see also Vol. II, Sect. 4.12). Further important conclusions and comments are in order about the present justification: First and foremost, the resulting shell theory is a n o n l i n e a r flexu r a l t h e o r y in the sense that the s t o r e d e n e r g y f u n c t i o n of a n o n l i n e a r l y elastic f l e x u r a l shell, defined by:
,-
e
-
-
b.,)
is a quadratic and positive definite expression (via the two-dimensional elasticity tensor of the shell) in terms of the change of curvature tensor, i.e., of the exact difference between the curvature tensor of the deformed middle surface and that of the unde]ormed one, while the associated energy is to be minimized over displacement fields for which the change of metric tensor, i.e., the exact difference between
Sect. 10.5]
The two-dimensional equations derived by a formal analysis
525
the metric tensor of the deformed middle surface and that of the undeformed one, vanishes. The existence theory for this minimization problem is treated in the next section, where further relevant comments are also to be found. Note in passing the truly remarkable simplicity of this minimization problem! Second, the resulting theory is f r a m e - i n d i f f e r e n t in the sense that the value of the above stored energy function is unaltered if, given any r/ = (~7i) E J ~ F ( W ) , ~]~ : = 0 + ~7iai is replaced by Q r where Q is any orthogonal matrix of order three (Ex. 10.5). It is also a l a r g e d i s p l a c e m e n t , or equivalently a l a r g e d e f o r m a t i o n , t h e o r y , in the sense that the de-scaling produces a displacement field that is 0(1) with respect to ~. We now interpret the boundary conditions r/ = 0v r/ = 0 on 3'0. To this end, let rli ai be a displacement field of the middle surface S = 0(~) with smooth enough, but otherwise arbitrary, covariant components TIi : ~ ~ I~. The tangent plane at an arbitrary point O(y) + ~Ti(y)ai(y), y E ~, of the deformed surface (0 + ~Tiai)(-~) is thus spanned by the vectors o
(o +
-
+
+
if these are linearly independent. Since r / = 0v r / = 0 on 3'0 =r ~/i = 0 a r ] i -- 0 on 3'0, it follows that
O(y) + yi(y)ai(y) - O(y) and Oa(O + yiai)(y) - aa(y) for all y e 7o. These relations thus show that not only the points and the tangent spaces (as when only the weaker "boundary conditions of clamping" ~i -- 0u~3 - 0 are imposed on 3'0; cf. Sect. 6.3), but also the tangent vectors to the coordinate lines, of the de]ormed and undeformed middle surfaces coincide along the curve 0('/0). Such t w o - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s of s t r o n g c l a m p i n g are suggested in Figs. 10.2-1 to 10.2.3. The formal linearization of the equations found in Thm. 10.5-1 produces the equations of a linearly elastic flexural shell (Chap. 6). This is perhaps best seen on the minimization problem: In the linearization process, the covariant components Raf3(r/) --- baf3(r/)- bar3
Nonlineavly elastic flezural shells
526
[Ch. 10
of the change of curvature tensor entering the energy become by definition the covariant components paf3(r/) of the linearized change of curvature tensor (Sect. 2.5); likewise, the covariant components Ga~(~l) - 89 - aa~) of the change of metric tensor entering the definition of the manifold J~F(w) become by definition the covariant components 7afj(v/) of the linearized change of metric tensor (Sect. 2.4), so that the energy becomes that of the linear flexural theory (Sect. 6.3) and the manifold .h4F(w) likewise becomes the vector space
vF(
) := { . -
e HI( ) • HI( ) • r/i = 0v7/3 = 0 on 9'0, 7af3(v/) = 0 in w}
of the linear flexural theory (Sect. 6.3)
Remarks. (1) The function spaces and the boundary conditions, which become in the linearization process those of the space VF(W), have to be modified so that the two-dimensional equations of the linear flexural shell theory become well-posed (Thm. 6.3-1). (2) As observed by Lods & Miara [1998], the equations of a linearly elastic flezural shell are also recovered by considering forces of higher order; cf. Ex. 10.6. m Finally, note that Collard & Miara [1999] have shown that the formal analysis of Lods & Miara [1998] described supra also leads to the explicit computation of the limit stresses in a nonlinearly elastic flexural shell. 10.6.
E X I S T E N C E OF S O L U T I O N S T O T H E MINIMIZATION PROBLEM
In this section, which is based on Ciarlet & Coutand [1998], we show that the minimization problem for a nonlinearly elastic flexural shell found in Thm. 10.5-1 has at least one solution, once the manifold ~4F(w) is replaced by a larger manifold, where the space H2(w) replaces the space W2,4(w). For notational convenience, we keep the same notation ,h4F(W) for the enlarged manifold and we suppress in this section all the ezponents "s" appearing in the formulation of this problem. Our point of departure is thus the following:
Sect. 10.6]
Ezistence of solutions to the minimization problem
527
The unknowns are the three covariant components ~i " w -+ of the limit displacement field ~iai of the middle surface S of the shell. This means t h a t ii(y)ai(y) is the displacement of the point O(y) of the middle surface for each y E ~. The unknown vector field :-- (r " w --+ i~3 should then satisfy the following minimization
problem: r ~ A4~(~)
- {,I - (v~) ~ I~2(~); ,7 - o~,i - o o ~ "~o,
aaf~(~/) - aaf3 - 0 in w},
jF(C,)-
inf
n~A4p(w)
jF(rl), where
~3
iF(n)- -~ / aaf3~ (b,;r(rl) - b,Tr)(baf3(rl) - baf3)v/-ddy - L p i r l i ~ dy,
where 70 is a portion of the b o u n d a r y 7 satisfying
length 70 > 0, 0v denotes the outer normal derivative operator along 7, the functions aaf~(~/)" ~ -+ I~ and baf3(Vl)" ~ -+ I~ are the covariant components of the metric and curvature tensors of the deformed surface (O+~liai)(-~) corresponding to an arbitrary field v$ - (v/i) E JV~F(w), aaf }~r :_
4A# aaf3aO.r + 2/z(aa~af3 r + aar a~a ) A+2#
designate the contravariant components of the two-dimensional elasticity tensor of the shell, and finally, the given functions pi E L2(w) account for the applied forces. Two immediate comments are in order about this minimization problem: First, it is clear that the "interesting" situations covered by the present theory are those where the manifold ~/t1~(w) contains other fields ~} than ~} = O. Observe in this respect t h a t such an assumption, which in effect depends only upon the geometry of S a n d u p o n the subset Vo of V, is one of the assumptions made in the definition of a nonlinearly elastic flexural shell (Sect. 10.2).
Nonlinearly elastic flezural shells
528
[Ch. 10
As already noted, this assumption means that there exist nonzero displacement fields 71iai of the middle surface S that are inextensional, in the sense that they preserve the metric of the surface S, and that are also admissible, in the sense that they satisfy the two-dimensional boundary conditions of strong clamping r/ = Our/ = 0 on 7o. This is the case for instance if the middle surface S is either a portion of a cone (excluding its vertex) or a portion of a non-planar cylinder and the shell is subjected to a boundary condition of place along a portion of its lateral face whose middle curve 0(70) is a subset of one of its generatrices (if S is a non-planar cylinder, 0(70) may consist of subsets of two generatrices); cf. Figs. 10.1-1 and 10.1-2 and Exs. 10.1 and 10.2. These developable surfaces constitute canonical examples, but more "exotic" examples can be found; cf. Ex. 10.7. Second, we must verify that the functional jF is well defined on the manifold .A~tF(w). To this end, we will show (part (ii) of the proof of Thin. 10.6-1) that the vectors Oa(O+yia i) are linearly independent if r / - - (7/i) E dg4F(w) and that the functions bat3(r/), which are then well defined, belong to L2(w). It turns out that the existence theory is substantially easier to carry out if, instead of ~ = (~i) and ~1 = (71i), the vector fields
~o "= 0 + ~ia i and '0 := 0 + ~i ai are taken as the new unknown and new "trial functions". In other words, the new unknown is the l i m i t d e f o r m a t i o n field of the middle surface of the shell: This means that ~o(y) - O(y)+ r is the deformation of the point O(y) of the middle surface for each yEW. A w o r d of c a u t i o n . The minimization problem is thus still expressed in terms of curvilinear coordinates (the coordinates ya of the points y E ~), but the unknowns are no longer covariant components (the functions ~i :w --+ I~) over the contravariant bases along the surface S. Instead, the unknowns are now the Cartesian components, i.e., over a fixed Cartesian frame, of the unknown deformation field 3. m Using in particular the relations r/j - (71iai).aj, we remark that the equivalence -
H2(
)
0 +
- r
H2(
)
Sect. 10.6]
Ezistence of solutions to the minimization problem
529
holds, since the assumption 0 E C3(~; I~3) made throughout this chapter (see, e.g., Thm. 10.3-4) implies that ai C C2(~; R 3) and a ~ ~ C2 (~; R ~). If r E H2(w), we define a.e. in w (here and subsequently, the abbreviation "a.e." stands for "almost everywhere") the vector fields
and the functions
~(r
:= . ~ ( r
a,(r
= .~.(r
If r E H2(w) is such that the two vectors a a ( r are linearly independent a.e. in w, we also define a.e. in w the vector field
~(r
~(r "- I~(r
A
~2(r A ~2(r
the functions
and finally, the vector fields ai(r
~(r
by means of the relations
aj(r
i - ~j.
The vectors aa(r and a ~ ( r thus form respectively the covariant and contravariant bases of the tangent plane to the deformed surface r at the point r for almost all y E w.
Remark. I hope, but am not entirely convinced, that I will be forgiven for the lack of consistency observed between, e.g., the notations ai(~7) used so far and ai(r used in this section. At least, these inconsistencies spared me the burden of introducing yet further notations! I We now establish an existence result ]or the minimization problem of a nonlinearly elastic flexural shell, reformulated in terms of the new unknown ~o and trial functions r Note that, for simplicity, we impose only one boundary condition on ~'0, viz., ~o -- ~o0 on 70 (this boundary condition is thus more general than its special case ~o -- 8 on 3'0 considered until now), as the following existence theory holds
530
[Ch. 10
Nonlinearly elastic flezural shells
in this case of a shell that is s i m p l y s u p p o r t e d , as well as in the case (considered so far) of a shell that is strongly clamped. Note also that this existence result holds under a substantially weaker regularity assumption on the mapping 0 than that made so far, viz., 0 6 C3(~; IR3). This assumption was nevertheless essential for carrying out the asymptotic analysis to its term (Thm. 10.3-4). The following ezistence result is due to Ciarlet & Coutand [1998]. It extends to shells the existence theorem established by Coutand [1997a] (see also Vol. II, Thm. 4.12-3) for the minimization problem justified by Fox, Raoult & Simo [1993] in the planar case. The proof is, however, substantially different. The uniqueness or multiplicity of solutions to this problem are studied in Coutand [1999a, 1999f]. T h e o r e m 10.6-1. Let there be given a mapping 0 E W2'p(w; IR3) with p > 2, such that the two vectors a a -- OaO are linearly independent at all points of-~. Let there be given a subset 70 C 7 such that length 70 > 0 and let there be given a mapping 7~o :70 --~ IR3 such that the m a n i f o l d of a d m i s s i b l e i n e x t e n s i o n a l d e f o r m a t i o n s
9
:= {r
e
r
=
on
-
-
0 in
is not empty. Then if r E q~y(W), the vectors a a ( r = 0 a r are linearly independent a.e. in w and the ]unctions ba~ (r are in L2 (w). Given a continuous linear form L on H2(w), define the twod i m e n s i o n a l e n e r g y IF : q~y(w) -~ R of a n o n l i n e a r l y elastic f l e x u r a l shell by
~3 ~ aa~r(b~r(r
- b~r)(ba~(r
- ba~)v~dy - L(r
for all r 6 q~F(w). Then there is at least one 7~ such that
7~ C q~y(w) and I y ( ~ ) =
inf
IF(C).
Sect. 10.6]
Ezistence of solutions to the minimization problem
531
Proof. For the sake of clarity, the proof is broken into a series of seven parts, numbered (i) to (vii). In the first five parts, we establish various properties of the manifold ~F(w), while in the remaining two, we establish properties of the functional IF over this manifold, the combination of which eventually leads to the existence theorem. Note in passing that the properties established in parts (i) to (v) may also be viewed as properties of isometric surfaces. The norms in the spaces L2(w) and Hm(w), rn ~ 1, are denoted as usual I 910,~ and ][. [[m,~ and those in the spaces L~176 and Wl,~176 are denoted I- ]0,oo,~ and I[" I[1,oo,~. The same notations are used for the norms in the corresponding spaces of vector fields, such spaces being then denoted by boldface letters. Strong and weak convergences are denoted by -4 and --~, respectively (a review of all the properties relevant here about weak convergence and lower semicontinuity is found, e.g., in Vol. I, Sects. 7.1 and 7.2). The proof follows a pattern familiar in the calculus of variations: After showing that the manifold ~F(w) is sequentially weakly closed (part (i)), we establish that the functional IF is sequentially
(p ts and (vii)), all these properties holding with respect to the topology of the space H2(w). The existence of a minimizer of Iv over ~F(w) then classically follows from these properties; see, e.g., Dacorogna [1989, Chap. 1] or Struwe [1990, Chap. 1]. Note that the coerciveness of the functional hinges on the crucial property that the manifold 'I'F(W) lies in a bounded subset of W 1, oo(w) (part (iii)).
(i)
of It2( ), the manifold ~F(w) is sequentially
weakly closed, i.e., r
E 'I'F(W), I ~_ 1, and r
___~r in H2(w) =~ r E ~F(w).
Let e l E 'I'F(W), I > 1, be such that e l __~ r in H2(w). Since the trace operator tr from H2(w) into L2(70) is continuous with respect to the strong topologies of both spaces, it remains so with respect to the weak topologies of both spaces (see, e.g., Brezis [1983, Tam. III.9]). Hence tr e l __~ tr r in L2(70) and thus tr r - ~0 on 70 since tr e l _ ~0 on 70 for all I > 1.
Nonlinearly elastic flezural shells
532
[Ch. 10
By the Rellich-Kondra~ov imbedding theorem (see, e.g., Vol. I, Thin. 6.1-5), Ct __, r in Hi(w); hence
~(r
~o(r
-
~(r
-~ ~(r
~(r
- ~o~(r
in Z~(~).
Since a ~ ( r t) - aa~ a.e. in w for all l, we conclude that a ~ ( r - a~ a.e. in w; hence ~ E ~I'y(w) as was to be proved. Should the manifold ,~,(w) include a second boundary condition of the form 0~k : ~01 on 70 (recall that the manifold 2r comprises a boundary condition of the form 0~r/ = 0 on 70), a similar argument shows that such a manifold is again sequentially weakly closed. (ii) There ezists C1 such that, for all vector fields r satisfying aa~ (r - aa/3 a.e. in w,
E H2(w)
0 < c~ ~ la~(~) A a 2 ( ~ ) l ~.e. in ~, C~-~ _< la~(~)l _< C~ a.e. in ~.
Consequently, the vectors ai(~) and ai(~) associated with such vector fields ~ are well defined and "uniformly" linearly independent a.e. in w, the corresponding functions ba~(r are in L2(w), and the functional IF is well defined over the manifold ~ y ( w ) (that the functions ba#(r are indeed well-defined when ~ belongs to the manifold 9 F(w) was already observed in Thin. 10.3-1). Since the set ~ is compact, the vectors aa - OaO are "uniformly" linearly independent in ~ (they belong to the space WI,p(w; R3), which is continuously imbedded into the space Co (~; ~3) since p > 2), in the sense that there exist Cl and c2 such that 0 < cl < 1 and lal 9a21 < ~la~lla21 in ~,
0 < c2 < 1 and c2 < la~l ___c; x in ~. Furthermore, a1(~3),
a2(~3) -
I~x(~)l 2
a12(~3) -
-- a 1 1 ( ~ 3 ) -
la2(~)l 2 --
a22(r
a12 -
all -lal[
al.a
2 a.e. i n w ,
2 a , e , i n w,
-- a22 - - i a 2 l 2 a . e . i n w;
consequently, lal(r
a2(~)l ~
cllal(~)lla2(~)l
a.e. in co.
Sect. 10.6]
Ezistence of solutions to the minimization problem
533
This inequality shows that the vectors a 1 ( r and a2(~])) are likewise "uniformly" linearly independent a.e. in w, since cl < 1. Hence there exists c3 > 0 such that
c31a1(r A a2(r
>~ l a l ( r 1 6 2
lallla2[ a.e. in w,
and thus there exists a constant C1 such that the two announced inequalities hold. The vector
a3(1]~) -- a 3 ( r
-- lal(~))A a2(r
is thus well defined a.e. in w. Consequently, ba[3(~b) - Oa[3~b"
a3(r
C L2(W),
since la3(r : 1 a.e. in w. The vectors a a ( r are likewise well defined and "uniformly" linearly independent a.e. in w. (iii) Let r e H2(w) be such that aa[3(~b) - aa/3 a.e. in w. Then r C wl'~176 and there exists C2 such that < C2 for all r E I-I2(w).
[0~r
In addition, there exists C3 such that
IIr
<__ c3 for all r E ~F(W).
Let r - (r E H2(w) be such that aa~(r - aa~ a.e. in w. We already noticed that there exists c2 > 0 independent of such fields r such that, for almost all y C w,
IO~r
2
-la~(r
2 = i ~ ( y ) i 2 <__~ 2 .
This shows that 0 a r C L~(w), hence that @ C Wl'~176 since in addition @ e H2(w) r C~ I~3); it also shows that there exists C2 independent of such fields r such that I0ar cr < C2. Let q > 2 be fixed. By assumption, length 3'0 > 0; hence, by the generalized Poincard inequality (seer e.g.~ Vol. I~ Thm. 6.1-8), there exists c4 - c4(q, 3'0) such that, for all r E w l ' a ( w ) ,
a
0
Nonlinearly elastic flezural shells
534
[Ch. 10
Let r ~ ,~F (w). Then
A ~ I~1~ ,Zu~2c~A dyandIX ~ c~
0
o
Since the field ~oo 9Vo --->I~3 is continuous on Vo (as the trace on Vo of a function in H2(w); the set ~I'F(w) is not empty by assumption), there exists c5 - c~(c4, C2, ~Oo) such that, for all ~b ~ ~I'F(w),
q(~)
1r + Z IOar
dy < c5.
The Sobolev imbedding w~,q(w) ~ C~ then implies the existence of cs - c6(c5) such that, for all r ~ ~v(w),
and the second assertion is proved. If ~ E ,I,~,(w), the components c9a~r a ~ ( ~ ) of the vector fields c9a/3~ - cDaa~(~) E L~(w) over the vectors a ~ ( ~ ) of the contravariant basis of the tangent plane to the deformed surface ~ ( ~ ) are in L2(w), since a ~ ( ~ ) - c9~ E L~176 by (iii). We next show that these components remain in a bounded subset of the space L2(w) when varies in the set 'I~F(w): (iv) There exists C4 such that
[Oa~" a~(~)10,w _~ C4 for aU ~ E ~F(w). By assumption, 0 E W2'P(w; R 3) with p > 2; as a consequence, Oa~O. 0,,0 E LP(w) C L2(w). Differentiating the relations o~r
o~r = a~(r
- ~
- 0 ~ o . 0~o
in the sense of distributions (which is licit, as is immediately verified) then shows that there exists cz such that, for all ~ C cI'F(w),
1012~" cO2~10,.o_ c7,
Sect. 10.6]
535
Ezistence of solutions to the minimization problem
The relations 011~]J" 0 2 r = (011r
0 2 r + 012r
022r
0 1 r -[- 012r
01~/~ -- (022r
02r
-- 012r
01r
-- 012r
02~]~,
then imply that [011~-~" 02~b[0, w ~ 2C7,
[022r 01r
w ~ 2C7.
Thanks to parts (ii) to (iv), a lower bound for the norms Ib~(r when r C 'I'F(w) can now be established. This lower bound will be essential for proving in part (vii) the coerciveness of the functional IF over the manifold '~F(w). (v)
There exists C5 such that Ib~(r
2O,w >~ 11r 2 W + Cs fo= a11 'r e "I'v(w)
a,/3
Let r e ~F(W). For almost all y e w, the vectors ai(r are linearly independent by (ii), so that the vectors ai(~b)(y) are well defined by the relations ai(r aj(r - ~} for almost all y e w. We can then expand Oa~r as Oa/3~b -- {Oa[3~b. aa(~]J)}a~r(r
+ {Oa/3~b. a3(~.~)}a3(~]J) a.e. in w.
Since bad(C) - 0 a ~ r a3(r la3(r we have, for almost all y E w,
IO~r
2 -I{o~r
a~(r162
1, a n d a 3 ( r
9a a ( ~ b ) -
2 § lb~(r
O,
2
Since the vector fields a ~ ( r lie in a bounded subset of Z~176 when r varies in the set 'I'F(W) (parts (ii) and (iii)) and since the functions Oa~r162 lie in a bounded subset of L 2 (w) when r varies in 'I'v(w) (part (iv)), there exists c8 such that [~0~,
a~(~)}a~(~)[0,~ ~ cs for all ~ E ~I'v(w),
Consequently, there exists c9 such that, for all ~ e 'I'F(w),
a,/3
a,/3
Nonlinearly elastic flezural shells
536
~_ a~0 for all ~p E ~F(w) (see
Since there exists al0 such that llr part (iii)), we finally have w
[Ch. 10
--
,w
w
1,w
c 9 - c20 for all @ E 'I'F(w).
-> ]1r
We now turn our attention to the functional IF. (vi) The functional IF is sequentially weakly lower semi-continuous over the manifold @F(w), i.e.,
r
E ~F(w), l :> 1, and r
___~r E ~F(w) in H2(w)
implies that
Iv(C) _< l i1--+oo minflF(r The weak convergence r
~ r in H2(w) clearly implies that
0af3r '---~ 0af3r in L2(w) and aa(r l) --+ a a ( r
in L2(w),
the last convergences being consequences of the Rellich-Kondra~ov imbedding theorem. We first show that it also implies that a3(r
~ a3(~P) in L2(w).
To this end, we observe that [a3(~bl)] - 1 a.e. in w and that there exists a subsequence ( r m oo of (r I oo such that
since a a ( r
-+ a a ( r
in L2(w). The definition
thus shows that ~(r
A ~2(r
= ~(r
~s m -+ o~
for almost all y E w (by (ii), the vectors aa(r m > 1, and a a ( ~ ) are well defined and "uniformly" linearly independent a.e.
Sect. 10.6]
Ezistence of solutions to the minimizationproblem
537
in w). Therefore, a3(~pm) --4 az(r in L2(w) by Lebesgue's dominated convergence theorem. Since the limit a3(~) is unique, the whole sequence (as(~bt))~l strongly converges in L2(w) to this limit. Using these properties, we next show that
b ~ ( r ~) - ~ r
~3(r
~ a~r
~3(r
- b~(r
in L2(~).
To this end, fix a and ~, let ft E L2(w) denote one component of 0 ~ r t (the same for all l ___ 1), let gt E L~176 denote the same component of a3(~l), and finally, let f E L2(w) and g E L~(w) likewise denote the corresponding components of (gaf~ and a3(~)). In thi~ fashion, the t~o sequ~nce~ ( / ' ) ~ a~d (g~)~=~ ~ti~fy: fl ___xf in L2(w), g! -+ g in L2(w) and Ig'10,oo,~ _< 1 for all l.
It then follows that yg E LZ(w) and
y~9 ~ - / g
i~ LZ(w).
Although these implications are standard, we provide a proof for completeness. For any qo E 79(w), the bilinear form (f, g) E L2(w) • L2(w) -+ f~ .fgqady is strongly continuous; hence f'--~ f in L2(w) and gl _.~ g in L2(w) =~ ~ flgl~ody -4 ~ fg~o dy. Let (/mgm)~= 1 be an arbitrary subsequence of (flgl)~ 1. Since
[fmg~lo,~ < [fm[0,~ and the weakly convergent sequence (.fro)m~176is bounded in L2(w), there is a subsequence (].g.)Oo n--1 of (fmgm)Oo m--1 that weakly converges in Lz(w) to some h E LZ(w). Therefore,
f fngn~d~1-4~hqadY=ffg~ad~tforall~Eg(w), and thus h - fg. Since the limit fg of the subsequence (fngn)n~176 is unique, the whole sequence (y'9~) l~176 = l weakly converges in LZ(w) to this limit. In particular then, we have established that ba#(r t) ~ ba~(~) in L2 (w).
Nonlinearly elastic flezural shells
538
[Ch. 10
We are now in a position to establish the sequential weak lower semi-continuity of IF over '~F(W). Let L2(w) denote the space of all fields of symmetric matrices of order two with components in L2(w). The symmetric bilinear form B" L2(w) x L2(w) --+ R defined by
for all (S, T) - ((Sa~), (ta~)) e L2(w) x L2(w) is strongly continuous and positive definite since there exist constants Cll and cl2 such that 0 < cll and a '~;3'~r(Y)t~rta~ >_ cll
~ Iraqi2 a,~
for all y e ~ and all symmetric matrices (ta~) (Thm. 3.3-2) and -1 0 < cl2 < 1 and 0 < cl2 < %/a(y) < cl2
for all y 6 ~. Being strongly continuous and (strictly) convex, the mapping S 6 L2(w) ~ B(S, S) is thus weakly lower semi-continuous. Let s ta~ "- ba~ (r
_ ba~ and Sa~ "- ba~ ( r
- ba~ .
Then since ba~(r l) ~
ba~(~b)in L~(w), and
thus
B ( S , S) < liminfB(S l, Sl). l--+oo
This shows that the functional IF, which is defined by g3
IF(r
=
%-
for all r C ~F(W) is sequentially weakly lower semi-continuous on 'I'F(w) (recall that L is by assumption a continuous linear form on H2(w)). (vii) There ezist constants C6 and C~ such that 6'6 > 0 and IF(C) > C6[[r
+ C7 for all r c ~ v ( w ) .
Consequently, the functional I v is coercive on the manifold ~V(W).
Ezercises
539
By definition of the functional IF, we have (the constants Cll and c12 appeared in the proof of part (vi))
~3 - c1311r a,/3
where
C13
denotes the norm of the continuous linear form L. Since 1
2
and since, by (v),
ib (r
,~ >__l]r
-4-C5 for a11 r e ~F(w),
the assertion follows, and the existence of a minimizer of the functional IF over the manifold '~F(w) is thus established. II Remarks. (1) The manifold ~F(w) is not convex (save if it contains only one element); cf. Ex. 10.8. (2) A property analogous to that proved in (iii) also holds for isometric three-dimensional manifolds in ]~3; for a precise statement of this property, which is due to Luc Tartar, see, e.g., Vol. I, Ex. 1.16. II
An interesting question consists in identifying the Lagrange multiplier associated with the "constraints" a a ~ ( r aal3 = 0 in w. In this direction, see the formal approach of Fox, Raoult & Simo [1993, Sect. 5.4] in the planar case. EXERCISES
10.1. Let if - (fa) e C3([0, 1]; 1~2) be an injective mapping such that f ' ( t ) ~ 0 for all t e [0, 1] and 3r 1]) is not a segment. Let S - 0(~), where w =]0, 1[• 1[ and O(t, z) = fa(t)e a A- z e 3 for (t, z) E ~. The surface S is thus a portion of a cylinder orthogonal to, and passing through, the planar curve f([0, 1]); cf. Fig. 10.2-1. (1) Assume that 7 0 C { ( 0 , z) C R 2 ; 0 < z < l } U { ( 1 ,
z) E i ~ 2 ; 0 < z < l } .
Nonlinearly elastic flezural shells
540
[Ch. 10
Show that the manifold ~F(W)
= { n = (~i) E W 2 ' 4 ( W ) ; n = O~'n -- 0 on ")'o;
aa/3(r/) -- aao = 0 in w} contains nonzero functions r/. (2) Show that, at each ~ E ,A,tF(w), the tangent space T r to Ji,'tF(w) contains nonzero functions (such a tangent space is defined, e.g., in Sect. 10.2). 10.2. Let :f - (fa) E e3([0, 1];R 2) be an injective mapping such
that f'(~) ~ o fo~ ~II ~ e [0, I]. Let S - 8(~), where w -]0, 1[• 1[ for some 0 < so < 1 and 8(t, s) = s f a ( t ) e a + (1 - s)e 3 for (t, s) C ~. The surface S is thus a portion of a cone with vertex e 3 and passing through the planar curve :f([0, 1]); cf. Fig. 10.2-2. (1) Assume that 7 o - - {(0, s) E R2;so ~_ s <__ 1}. Show that the manifold ~F(W)
-- { n -- (~i) E W 2 ' 4 ( W ) ; n -- ~ u n -- 0 o n 70;
aa/3(r/) --aa/3 -- 0 in w) contains nonzero functions r/. (2) Show that, at each ~ E .A4F(w), the tangent space Tr to .Ad.F(w) contains nonzero functions. (3) Assume that
"/o = {(0, s) E
I~2; so <_ s _< 1}
L.J {(1, s) E
I~2; so <_ s _< 1}.
Are the conclusions of (1) and (2) still valid? 10.3. Let w C I~2 be a domain and let 70 C O w be a segment; cf. Fig. 10.2-3. (1) Show that the manifold
.A4.F(w) --
{ ~ / = (7/i) E W 2 ' 4 ( w ) ; U - O v a / - 0 on 70;
contains nonzero functions r/. (2) Show that, at each ~ 6 ,s the tangent space Tr to .A4E(w) contains nonzero functions.
Ezercises
541
10.4. The notations used in this exercise should be self-explanatory. Let the manifold .h4y(w) and its tangent spaces_Tr be defined as in Sect. 10.2. Let a surface S = O(~) = 0 ( { ~ } - ) be equipped with two systems of curvilinear coordinates (Ya) C -~ and (Ya) e {5~}- and assume that 0 ( 7 0 ) - 0('~0). (1) Show that ~ 4 e ( w ) ~ {0} implies that ~4~,(~) ~ {0}. (2) Assume that, at each ~ C ~ t F ( w ) , the tangent space qI'r to ~,'tF(w) contains nonzero functions. Then show that, at each E 2~4F(~), the tangent space qr~MF(~) contains nonT,ero functions. In other words, the definition of a nonlinearly elastic flexural shell (Sect. 10.2) is independent of the system of curvilinear coordinates employed for representing the surface S. 10.5. Show that the stored energy function of a nonlinearly elastic flexural shell:
~3 ~l - (~i) e .]~F(W) ~ -~aal~rr'e(bcrr(~) - bcrr)(bal~(~l) - bai~) is frame-indifferent, in the following sense: Given any r / C ~/tF(w), its value is unaltered if r "- 0 + yia i is replaced by Qr where Q is any orthogonal matrix of order three. 10.{i. This exercise shows that, if higher-order forces are considered, the leading term of the formal expansion (then also of a higher order) satisfies the equations of a linearly elastic flexural shell, as found in the linear theory (Thm. 3.4-3). This observation is due to Lods & Miara [1998, p. 367]. Let the assumptions be as in Thm. 10.3-4, save that the applied body and surface forces are now such that fi, e(Xe ) -- ~2+r~i, 2+r(X )
for all
xe=rexEf~e,
hi,~ (x~) _ sa+, h i, a+r (x)
for all
x ~ -7r~x C P~_ U P~_
where r is an arbitrary integer > 1 and the functions fi, 2+r C L2(~) and h i, 3+r E L 2 ( r + U P _ ) are independent of s. (1) Show that the leading term of the formal asymptotic expansion of the scaled unknown u(s) is u r (in particular then, u ~ vanishes).
542
[Ch. 10
Nonlinearly elastic flezural shells
(2) Show that u r 9 ~ --+ R 3 is independent of the transverse variable and that, once identified with a function (:~ 9~ ~ IR3 it satisfies the variational problem ~r e W F ( W ) --
{~- (TIi) e
W l ' 4 ( w ) x W1,4(w) x W2,4(w);
r/i = 0vr
g1 L aa/3ZrPz,(: )pa[3(~l)x/~dy:
: 0 on 3'0, 7af3(v/) -- 0 in w},
/-
p"2+e~Iix/~dy
for all ~ - (~1/) e W F ( w ) , where the functions 7a~(r/) and Pa~(Y) are the covariant components of the linearized change of metric and change of curvature tensors and
" 3+, and h~3+r - h i ' 3 + ' ( ., + 1 ) . p,," 2+,, = /~ f,," 2+,, dz3 + h~S+" + h'2 1
10.7. The following problem provides an "exotic" example of a
manifold 2~4F(W), as used in the definition of a nortlinearly elastic flexural shell (Sect. 10.2). Given arty function f E C~176 1]) that satisfies f(t) - 1 if 0 < t < 89and f(t) = 0 if 2 < t < 1, let the mapping 0 E C~176 ~2) be defined by O(yl, y2) - (yl, Y2, f(V/y 2 + yg)) for all (Yl, Y2)E ~, where w = {(yl, y2)C I~2; y2 + y2 < 1}. (1) Show that the manifold /~tF(W) 9 = {~ "- (TIi) e W2'4(W); 71 -- 0v~ -" 0 Oil 7,
aa/3(~/) -an/3 = 0 in w} contains the two distinct elements r / - 0 and ~ / - - 2 0 . a 3 , the second one thus corresponding to art everted state. (2) Are there other elements in the manifold ,/k4F(w)? (3) Are the conclusions of (1) and (2) modified if the space H2(w) replaces the space W2,4(w) in the definition of the manifold Jk4F(w)? Remark. This example can be generalized in various ways, for instance by letting 0(yl, y 2 ) - (yl, y2, fn(V/y2 + y22)) for some n > 1, where the functions fn E C~176 1]), n _ 1, are defined by
fo = f and
fn(g)_
fn--l(f;) _~_fr~--l(3nt) if 0 ~__ f;
/"-~(t)
if
=
_<_ t _< 1, n >__ 1.
1 3--~ ,
l~'zercises
543
10.8. Let 70 C "7 be such that length 70 > 0, let the manifold 9 f(w) be defined as in T h m . 10.6-1, viz., 9 r ( ~ ) - { r c H~(~); r - ~0 o~ 70, ~ , ( r
~
-0
i~ o,},
and let r and r be two distinct elements in @F(w). (1) Show t h a t there exist an index i0 C {1, 2, 3} and a subset cr of w with area o > 0 such that
(2) Let r
fw lOior
tr
+ (1 - t ) r 2. Show that
2 dy < f~ [OioO[2 dy for all 0 < t < 1,
and conclude t h a t the manifold @e(w) is not convez. Remark. This elegant proof of the non-convexity of to D. Coutand.
@e(w) is due
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C H A P T E R 11 KOITER'S EQUATIONS AND OTHER TWO-DIMENSIONAL NONLINEAR SHELL THEORIES
INTRODUCTION The objective of this short chapter is to provide a succinct repertoire of, and ad hoc references to, "other" two-dimensional nonlinear shell theories, i.e., that are neither the membrane nor the flexural theories justified in the previous chapters. No exhaustiveness is to be expected, however, even within the few topics selected for this chapter. Among these, Koiter's nonlinear shell theory (Sect. 11.1) is particularly appealing: Its corresponding strain energy is simply the sum of the strain energy of a nonlinearly elastic membrane shell and that of a nonlinearly elastic flexural shell, justified respectively in Chaps. 9 and 10. Finally, we briefly review in Sects. 11.2 and 11.3 other two-dimensional equations that have been proposed for modeling nonlinearlg elastic shells, notably Naghdi's nonlinear shell theory, which is based on the notion of one-director Cosserat surface, and those theories that are deemed appropriate for nonlinearlg elastic "shallow" shells. 11.1.
THE TWO-DIMENSIONAL KOITER EQUATIONS FOR A NONLINEARLY SHELL
ELASTIC
In a landmark paper, Koiter [1966] has proposed nonlinear equations 1 which may well constitute, possibly after some amendments, an "all-purpose nonlinear shell theory", in the same way that Koiter's linear equations constitute an "all-purpose linear shell theory" in the sense that they are valid for all kinds of geometries of the middle surface and boundary conditions (Sect. 7.2). 1A n e x c e r p t from this p a p e r is r e p r o d u c e d on page lxi.
546
Koiter's equations and other nonlinear shell theories
[Ch. 11
Another major advance in nonlinear shell theory is due to John [1965, 1971], who established (by means of exceedingly delicate computations) that, "away from the lateral face", the state of stress inside a shell is approximately planar and the stresses parallel to the middle surface vary approximately linearly across the thickness if the thickness is small enough. This means that, given any point ~ E S - O(w) sufficiently far from the "boundary" 8(y) of S and a local Cartesian frame with origin 9 and basis vectors ea in the tangent plane to S at Y, the components ~ij, e of the stress tensor are approximately of the form &aft, ~ _ &a~3,0 + fz~aa~3,t and &i3,e _ 0 in a sufficiently small three-dimensional neighborhood of Y, where $~ denotes the Cartesian coordinates along the basis vector e3 (normal to S at 9) and the functions &a~3,0 and &at3,t are independent of $~. In Koiter's approach, these approximations constitute an a priori assumption of a mechanical nature, which is associated with another a priori assumption, of a geometrical nature. This second assumption is the K i r c h h o f f - L o v e a s s u m p t i o n , so named after Kirchhoff [1876] and Love [1934]; it asserts that any point situated on a normal to the middle surface in the reference configuration remains on the normal to the deformed middle surface after the deformation has taken place and that, in addition, the distance between such a point and the middle surface remains constant (complements about this assumption are found in Ex. 7.1). Combining these two a priori assumptions, W.T. Koiter concludes that the displacement field across the thickness of the shell can be completely determined from the sole knowledge of the displacement field of the middle surface and he identifies the two-dimensional problem (described infra) that this field should satisfy. More specifically, let w be a domain in I~2 with boundary 7, let 0 E C3(~; I~3) be an injective mapping such that the two vectors aa = OaO are linearly independent at all points of ~, and let 70 be a portion of 7 that satisfies length 70 > 0. Consider a nonlinearly elastic shell with middle surface S = 0 ( ~ ) and thickness 2e > 0, i.e., a nonlinearly elastic body whose reference configuration is the set O(~e), where
.= o(y,
0(y) +
(y,
e
The two-dimensional Koiter equations
Sect. 11.1]
547
The material constituting the shell is homogeneous and isotropic and the reference configuration is a natural state; hence the material is characterized by its two Lamd constants Ae > 0 and #e > 0. The shell is subjected to a boundary condition of place along a portion of its lateral face with 8(70) as its middle curve, i.e., the three-dimensional displacement vanishes on | where = • Finally, the shell is subjected to applied body forces in its interior O(f~ e) and to applied surface forces on its "upper" and "lower" faces O(r~_) and O(re_), given by the contravariant components fi,~ E L2(f~ e) and h i'e C L2(r u V ) of their respective densities, where F~= -- w • {+e}. Let
aaf3o.r, ~
:=
4Ae# ~ aa~a '~" + 2#~(aa'~a~ + aara~'~), A e + 2# e .
9
pZ, e ._
.
.
.
' dx i + h ; e + h z'e and h~: e "- h~'e( ., +e),
the functions a af3~r' e thus denoting as usual the contravariant components of the two-dimensional elasticity tensor of the shell. Then the t w o - d i m e n s i o n a l K o i t e r e q u a t i o n s for a n o n l i n e a r l y e l a s t i c shell take the following form when they are expressed as a minimization problem (Koiter [1966, eqs. (4.2) and (8.3)]): The unknown ~ : - (~:,i), where the functions i~:,i" ~ --+ R are the covariant components of the displacement field ~ , i a~ of the middle surface S (this means that ~i~,g(y)ai(y) is the displacement of the point O(y)) should minimize the energy jeg defined by
e f~ aa~ar, e(a~(~l ) _ a~r)(aaf3(r/) - aaf3)v/-ady
e3 ~w a a~r' e (b~(~l) - ba~)(ba,(~7) - baf3)x/ady + -~ -
f~ pi'erli~/~ dy
over an appropriate manifold of vector fields v / - (yi) satisfying ad hoc boundary conditions, for instance the boundary conditions of strong
548
Koiter's equations and other nonlinear shell theories
[Ch. 11
clamping ~ -- Ourl -- 0 on 7o found in the two-dimensional equations of a nonlinearly elastic flexural shell (Sect. 10.5). The functions 89 and (ba/3(y)-ba/3), already encountered and denoted Ga/3(rl) and Ra/3(r/) in Chaps. 9 and 10 (see, e.g., Thms. 9.4-1 and 10.5-1), are the covariant components of the change of metric and change of curvature tensors associated with a displacement field ~?iai of the middle surface S. We recall that aat3(r/) and ba/3(rl) respectively denote the covariant components of the metric and curvature tensors of the deformed surface (0 + yiai)(-~). In other words, the s t r a i n e n e r g y r/--+ ~g f~ aal3crr'e (aar(r/) - a~r)(aa/3(rl) - aa/3)v/a dy g3
+-( found in the energy jeg is the sum of the strain energy of a nonlinearly elastic membrane shell (Sect. 9.4) and of the strain energy of a nonlinearly elastic flexural shell (Sect. 10.5). Note in passing that this expression of the strain energy immediately shows that Koiter's equations for a linearly elastic shell (Sect. 7.1) are indeed obtained by linearizing the nonlinear ones. But the functions bat3(r/) are not defined at those points o f f where the two vectors aa(rl) := Oa(O+~lia i) are linearly dependent. For this reason, it is not clear to decide over which manifold the energy j ~ should be minimized. One way to circumvent this difficulty may consist in replacing in the strain energy the functions Ra/3(r/) = (ba/3(rl)- baD) by the functions R~/3(r/) introduced in Thm. 10.3-1, which are well defined for all smooth enough fields rl = (yi), irrespective of whether or not the vectors aa(y) are linearly dependent in a subset of ~. We recall that these functions, which are also given by (Thm. 10.3-2) 1 when det aa/3(y) > 0, coincide with the functions (ba~(y)-ba/3) when r/ corresponds to an inextensional displacement yia i of the middle surface, i.e., when aat3(rl) - aa/3 - 0 in w (Thm. 10.3-1). It is interesting to note that the same functions R~/3(rl) as above have also been proposed by W.T. Koiter (cf. Koiter [1966, eq. (4.11)]), most likely out of different considerations!
Sect. 11.2]
11.2.
Other nonlinear shell theories
OTHER NONLINEAR
549
SHELL T H E O R I E S
Nonlinear "shallow" shell theories are treated separately (see Sect. 11.3.
Naghdi~s nonlinear shell t h e o r y w i t h directors and ext e n s i o n s . While Koiter's theory relies in particular on the KirchhoffLove assumption, Naghdi [1963] has proposed another nonlinear shell theory, in which the a priori assumption on the stresses is the same as in Koiter's, but the a priori assumption of a geometrical nature affords more freedom on the displacements inside the shell; more specifically, the points situated on a line normal to the undeformed middle surface again stay on a line after the deformation has taken place and the distances are unmodified along this line, but this line need no longer remain normal to the deformed middle surface. Unlike Koiter's, Naghdi's theory may thus accommodate shear inside the shell. In this approach, the displacement of any point 0 ( y ) + x~a3(y) inside the undeformed shell is of the form ~e(y) + x~de(y), where ~e (y) _ ~ ( y ) a i ( y ) is the unknown displacement vector of the point O(y) of the middle surface S - 0(~) and de(y) is another unknown vector, called the director at 0(y), that measures the "rotation" of the normal vector after the deformation has taken place. In this fashion, the shell is modeled as a one-director Cosserat surface, whose deformed configuration is specified not only by the displacement field ~ a i 9~ -+ IR3 of the points of the middle surface, but also by a director field d ~ : ~ --+ R 3. This notion is due to Cosserat & Cosserat [1909].
Remark. An example of a one-director Cosserat surface is provided by Naghdi's linear shell theory, with d e - r ae a a as the director field; cf. Sect. 7.4. A planar example is provided by the ReissnerMindlin theory for linearly elastic plates (Vol. II, Sect. 1.9). m After its foundations were properly laid (see notably Naghdi [1963, 1972, 1982], Green, Naghdi & Wainwright [1965], Green & Naghdi [1974], and Reissner [1974]), Naghdi's theory has undergone significant developments, often under the appellations of multi-director shell theories or geometrically ezact shell theories.
550
Koiter's equations and other nonlinear shell theories
[Ch. 11
In these directions, see in particular the illuminating introduction to such theories given in Antman [1995, Chap. 14]. See also Basar [1987], Basar & Kr/itzig [1988], Antman [1989], Simo & Fox [1989], Simo, Fox & Rifai [1989, 1990a, 19905], Fox & Simo [1992], Antman [1997], Valid [1995]; Ge, Kruse & Marsden [1996], Kirchg/issner & Djurdjevic [1997], Djurdjevic [1999] for time-dependent nonlinearly elastic shells; and Basar, Ding & Schultz [1993], Kriitzig [1993], Sasar gr Ding [1995] for the modeling of multi-layered, or laminated, nonlinearly elastic shells by multi-director theories. " F i r s t - o r d e r " a n d " h i g h e r - o r d e r " shell theories. Shell theories, linear and nonlinear, have often been elaborated by assuming that the ratio of 6 to the smallest absolute value of the radii of curvature of the middle surface is also a "small" parameter (for instance the linear Novozhilov and nonlinear Donnell-Mushtari-Vlasov models for shallow shells can be derived in this manner; cf. Sects. 7.6 and 11.3). A shell theory obtained in this fashion is deemed "of the first order" if all terms of order _ 2 with respect to ?7 are neglected in the various formal asymptotic expansions considered and "of a higher order" otherwise. This approach, which leads to theories that are often scarcely distinguishable from the one-director or multi-director theories mentioned supra, is frequently used for modeling multi-layered, or laminated, shells, whose constituting elastic material is thus anisotropic. See notably Librescu [1975], Kriitzig [1974, 1976], Basar & Kriitzig [1985], Ambartsumian [1991], Simmonds [1992], Pomp [1996]. N o n l i n e a r shell t h e o r i e s b a s e d on t h e m e t h o d of internal c o n s t r a i n t s . Another approach advocated by P.M. Naghdi (see notably Green, Laws & Naghdi [1968] and Green & Naghdi [1970]) for generating nonlinear shell theories consists in directly incorporating ad hoc internal constraints in the three-dimensional equations. For instance, the KirchhoJf-Love assumption can play the r61e of such an internal constraint, imposed a priori on the admissible displacements (in its linearized version, this is the approach of Podio-Guidugli [1990]; cf. Sect. 7.5). Special care must be exercised, however, as serious inconsistencies arise if the Lagrange multipliers associated with such internal constraints (in the sense of optimization theory) are not properly
Sect. 11.2]
Other nonlinear shell theories
551
interpreted. Such inconsistencies have been neatly clarified by Antman & Marlow [1991], who showed that the Lagrange multipliers are "reactive" stresses that maintain the constraints, while ad hoc "active" stresses must be added to them so as to make up the total stresses (which are no longer given by constitutive equations); see also Antman [1995, Chap. 14].
Time-dependent nonlinear shell theories can likewise be developed by the method of internal constraints; see Antman [1995, Chap. 14] and Lembo [1996].
A nonlinear Budiansky-Sanders t h e o r y . Destuynder [1982, 1983] has proposed a nonlinear Budiansky-Sanders theory, where the B S (Y) of the "modified linearized change of covariant components Pa~ curvature tensor" are the same as in the linear Budiansky-Sanders theory (Sect. 7.5) and the covariant components
1 a a~ (17) - -~( aa~ ( ~ ) -- aaf3 ) 1 (aZr - 'Ta,o ('r/) + ~ r/o.lla'r/.,-II~ § 'r/311ar/311~) of the change of metric tensor (see, e.g., Thm. 9.1-1) are replaced by the shorter functions
Nonlinear terms thus only appear in the "membrane" part of the strain energy. P. Destuynder then establishes an existence theorem for the associated minimization problem, provided the Christoffel symbols of the middle surface are small enough or an ad hoc geometric assumption is satisfied. His proof makes an essential use of the inequality of Korn's type on a general surface (Thm. 2.6-4). " I n t r i n s i c " n o n l i n e a r shell t h e o r y . The intrinsic approach of Delfour & Zol~sio [1995], which is based on a "tangential differential calculus" and on the "oriented distance function" (Sect. 7.4), has been recently extended to the modeling of nonlinearly elastic shells by Delfour & Zhao [1999].
552 11.3.
Koiter's equations and other nonlinear shell theories NONLINEAR
SHALLOW
[Ch. 11
SHELL THEORIES
Nonlinear shallow shell theories in Cartesian coordinates are already treated in Vol. II, Sects. 4.14 and 5.12. Accordingly, the additional commentary and bibliographical notes found in this section concern mostly nonlinear shallow shell theories expressed in curvilinear coordinates. We recall that, according to the definition proposed and justified by a formal asymptotic method by Ciarlet & Paumier [1986], an elastic shell is deemed "shallow" if, in its reference configuration, the deviation of its middle surface from a plane is of the order of the thickness. More specifically, there exists a smooth enough function 0 :-~ --~ I~ independent of ~ such that
S ~ = 0e(~), where
0e(Yl, Y2) = (Yl, Y2, sO(y1, Y2)) for all (Yl, Y2) C ~.
A w o r d of c a u t i o n . As in the linear case, this specification of how the middle surface should "vary with e" thus also constitutes an assumption on the data, specific to shallow shell theory. II The two-dimensional equations of nonlinearly elastic shallow shells in curvilinear coordinates have been justified by Busse [1995, 1997, 1999] and Andreoiu-Banica [1998, 1999], Andreoiu [1999b] by means of the method of formal asymptotic expansions. S. Busse has considered the case of a clamped shallow shell, while G. Andreoiu has identified Marguerre-von Kdrmdn equations in curvilinear coordinates for a specific class of boundary conditions (introduced by Ciarlet [1980] in the plane case). Since the asymptotic justification and the existence theory for such equations essentially resemble those for the equations of nonlinearly shallow shells in Cartesian coordinates (Vol. II, Sects. 4.14 and 5.12), they are not incorporated into this volume. To give a flavor of these theories, we simply record the twodimensional equations of a nonlinearly elastic clamped shallow shell in curvilinear coordinates, expressed here as a minimization problem:
Sect. 11.3]
553
Nonlinear shallow shell theories
Let
bal3~r, e :__
4Ae#~
Ae + 2# e
F
+
fi, e dx~ + h~ e + h i'e_
p$~ e ~ ._
q~,~ := f~ Sh' ~
+
x~3]o,,~dxe3 + e(h~ 'e
_
1
1
and let a z, ~ designate the vectors of the contravariant bases along the middle surface S e. Then the unknown ~ - (i~), where the functions ~ 9~ -~ I~ are the covariant components of the displacement field ~ a z'e of the middle surface S e, minimizes the energy j sh, e defined by o
{ eba/3~r, eEsh, ~,
, ,~sh, e
} dy
+_~e3ba~crr, e 0err ~]30a~ T~3
over the space (the same space as for Koiter's linear equations; cf. Sect. 7.1)VK(W) := { ~ 7 - (Yi) e H l ( w ) •
Y i - 0~?~3 --0 on 70}-
An inspection of this minimization problem immediately reveals a strong resemblance with the minimization problem found for the "same" nonlinearly elastic shallow shell, but where the unknowns are the Cartesian components of the unknown displacement field (Vol. II, Sect. 4.14). As in the linear case (Sect. 7.6), the resulting theory is more reminiscent of a plate theory than of a shell theory! Using a different definition of "shallowness", requiring in particular that the absolute value of the radii of curvature be everywhere on the middle surface "sufficiently large", Koiter [1966, eqs. (11.43) to (11.50)] also obtains a nonlinear shallow shell theory in curvilinear coordinates, which takes the following form when it is expressed as a
554
Koiter's equations and other nonlinear shell theories
[Ch. 11
minimization problem: The unknown ~e = (~), where the functions ~ : ~ -+ R are the covariant components of the displacement field of e the middle surface, should minimize the energy J.sh, g defined by
g fwaa~ar'e
sh
sh
~3 ~ a a/3~r' +--6en31~rrl3ta~3 V~ dY
- ~pi,"rli~dy , where
sh
9-
+
1 0a~730~73 and
9-
0
,73
-
over the space (again the same as for Koiter's linear equations): VK(W) "= {y = (r/i) e Ht(w)•215
r l i - 0vrl3 = 0 on 70)-
Using the theory of pseudo-monotone operators developed by Lions [1969], Bernadou & Oden [1981] have established the existence of a solution to the associated variational problem if the tangential components of the applied forces and the curvature of the middle surface are "sufficiently small". Their approach makes an essential use of the inequality of Korn's type on a general surface (Thm. 2.6-4). Introducing an Airy stress function in curvilinear coordinates and using the same method as in Ciarlet & Rabier [1980] for establishing the existence of a solution to the yon K~rms equations (see also Vol. II, Sect. 5.8; this method is itself an elaboration over Berger [1967, 1977]), Alexandrescu-Iosifescu [1995a] has also shown that the same nonlinear shallow shell model of W.T. Koiter has at least one solution when the applied tangential components of the applied forces vanish. In addition, Alexandrescu-Iosifescu [1995b] has studied the behavior of the solutions to this model when the shell "becomes a plate". The analysis of the eversion problem for shallow shells modeled by the above Koiter equations has been carried out by Geymonat, Rosati & Valente [1989], Podio-Guidugli, Rosati, Schiaffino & Valente [1989], Geymonat & L~ger [1994]. In this direction, see also Paumier [1978a, 1978b], Paumier & Rao [1989].
Sect. 11.3]
Nonlinear shallow shell theories
555
The same nonlinear shallow shell model is also called the DonneUMushtari-Vlasov model, so named by Sanders [1963] after Donnell [1933], Vlasov [1944], and Mushtari-Galimov [1961]. Under this name, it has been justified by Figueiredo [1990a] by means of a formal asymptotic method with two "small" parameters, the thickness 2e and the ratio of e to the smallest absolute value of the radii of curvature of the middle surface. Figueiredo [1990b] has also established the existence of a solution, by means of the implicit function theorem. Other "classical" equations have been likewise deemed appropriate for modeling nonlinearly elastic shallow shells. A thorough treatment of their mathematical properties, together with a detailed discussion of the various a priori assumptions they are based upon, is found in the recent book by Vorovich [1999], which contains in addition an extensive list of references from the Russian literature on shell theory. The modeling of nonlinearly elastic shallow shells lying on an obstacle leads to interesting "unilateral eigenvalue problems", which can be solved by means of pseudo-monotone operator theory as shown by Gratie [1998, 1999a, 1999b]. In this direction, see also Goeleven, Nguyen & Th~ra [1993a, 1993b], Goeleven [1996], Gratie & Pascali [1997, 1999], Pauchard, Pomeau & Rica [1997], Le & Schmitt [1997], Goeleven & Motreanu [1998].
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SOKOLOWSKI, J.; ZOL]~SIO, J.P. [1992]: Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer-Verlag, Heidelberg. SPIVAK, M. [1975]: A Comprehensive Introduction to Differential Geometry, Volumes I to V, Publish or Perish, Boston. SrtUBSHCHIK, L.S. [1968]: Nonstit~ess of a nonshallow spherical dome, J. Appl. Math. Mech. 32, 435-445. SB.UBSHCHIK,L.S. [1972]: On the problem of non-stit~ess in the nonlinear theory of shallow shells, Izv. Akad. Nauk SSSR, .qer. Math., 86, 890-909 (in Russian). SRUBSHCHIK, L.S. [1980]: Precritical equilibrium of a thin shallow shell of revolution and its stability, J. Appl. Math. Mech. 44, 229-235. STEIN, E. [1970]: Singular Integrals and Differentiability Properties of Functions, Princeton University Press. STOKe.R, J.J. [1968]: Nonlinear Elasticity, Gordon and Breach, New York. STOKe.R, J.J. [1969]: Differential Geometry, John Wiley, New York. STOLARSKI, H.; BSr.YrSCHKO, T. [1982]: Membrane locking and reduced integration for curved elements, J. Appl. Mech. 49, 172-176. STOLARSKI, H.; BELYTSCHKO,T.; L~.~., S.H. [1995]: A Review of Shell Finite Elements and Gorotational Theories, in Computational Mechanics Advances, Vol. 2, 125-212. STRUIK, D.J. [1961]: Lectures on Classical Differential Geometry, Second Edition, Addison-Wesley, Reading. STRUW~., M. [1990]: Variational Methods, Springer-Verlag, Berlin. SuRI, M. [1997]: A reduced constraint hp finite element method for shell problems, Math. Gomp. 6 6, 15-29. SuP.x, M.; BABUgKA, I.; SCHWAB, C. [1995]: Locking effects in the finite element approximation of plate models, Math. Gomp. 64, 461-482. SZABO, B.A.; SAHRMANN, G.J. [1988]: Hierarchic plate and shell models based on p-extension, Internat. Y. Numer. Methods Bngrg. 2tl, 1855-1881. SzP.RI, A.J. [1990]: On the everted states of spherical and cylindxical shells, Quart. Appl. Math. 47, 49-58. TARTAR, L. [1978]: Topics in Nonlinear Analysis, Publications Math6matiques d'Orsay No. 78.13, Universit6 de Paris-Sud, Orsay. TP.r.EGA, J.J.; LBWIgrSKI, T. [1998a]: Homogenization of linear elastic shells: Pconvergence and duality. Part I. Formulation of the problem and the effective model, Bull. Polish Acad. Sci., Technical Sci., 46, 1-9. T~.LSGA, J.J.; L~.WIr~SKI, T. [1998b]: Homogenization of linear elastic shells: F-convergence and duality. Part II. Dual homogenization, Bull. Polish Acad. Sci., Technical Sci., 46, 11-21. THOMAS, T.Y. [1934]: Systems of total differential equations defined over simply connected domains, Annals Math. $5, 730-734. TIMOSHENKO, S.P. [1951]: Theory of Elasticity, McGraw-Hill, New York. TIMOSH~.NKO, S.P.; WOINOWSKY-KRI~.G~.R, S. [1970]: Theory of Plates and Shells, McGraw-Hill, New York. TRABUCHO, L.; VxAgro, J.M. [1987]: Derivation of generalized models for linear elastic beams by asymptotic expansion methods, in Applications of Multiple Scalings in Mechanics (P.G. CIARI,ST & E. SANCH~,Z-PAr.~.NCIA,Editors), pp. 302-315, Masson, Paris. TRABUCHO, L.; VIA~O, J.M. [1996]: Mathematical modelling of rods, in Handbook of Numerical Analysis, Vol. I V (P.G. CIARI,ST & J.L. LIONS, Editors), pp. 487-974, North Holland, Amsterdam.
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INDEX
a p r i o r i assumption of" a geometrical or mechanical nature : 336, 363, 546 admissible: applied force : 265, 292, 293, 298, 351 m displacement : 303 inextensional deformation : 530 inextensional displacement: 439, 503 linearized inextensional displacement : 303 analytic functions: 121, 125 anisotropic elastic material : 162, 178, 234, 326, 360, 366, 550 Ansatz of the m e t h o d of formal asymptotic expansions : 164, 413 applied b o d y force : 7, 25, 386 assumptions o n - s : 152, 162,164, 179, 180, 199, 229, 232, 233, 247, 265, 305, 322, 326, 414, 442,443, 458, 507 applied force ( a d m i s s i b l e - - ) : 265, 292, 293, 298, 351 applied surface force : 7, 26, 386 assumptions on m s : 152, 162, 164, 179, 180, 199, 229, 247, 265, 305, 322, 414, 442, 443, 507 area element : 19 area element on a surface : 70 assumptions on the data : See "applied body force", "applied surface force", "Lam~ constants", "shallow shell" justification of m : 162, 425, 443, 476, 507 asymptotic expansions (method of f o r m a l - - ) : 164, 413 asymptotic line : 83, 295 b e n c h m a r k problem : 361 bending: badly-inhibited ~ : 292 not i n h i b i t e d - : 325 w e l l - i n h i b i t e d - : 232 bending in shells : See "flexuxal-dominated behavior", "flexuxal theory", "linearly elastic flexuxal shell", "nonlineafly elastic flexuxal shell" bending m o m e n t : 344 b o d y force : See "applied b o d y force" boundary condition: of place : 8, 32, 387 of pressure : 455 of traction : 388 c o m p l e m e n t i n g - : 123, 236, 237, 373 t w o - d i m e n s i o n a l - s of clamping : 101, 116, 304, 305, 306, 322, 344 two-dimensional m s of simple support : 115, 134, 339, 439, 451, 530 two-dimensional ~ s of strong clamping : 503, 504, 524, 525, 530, 548 t w o - d i m e n s i o n a l - s of weak simple support : 234
Indez
584
b o u n d a r y layer : 351, 362,368, 371, 374
BUDIANSKY-SANDERS linearized change of c u r v a t u r e tensor : 367 BUDIANSKY-SANDERS shell t h e o r y : l i n e a r - - - : 367 nonlinear m : 551 canonical e x t e n s i o n : 144
CARTESIAN c o m p o n e n t s : 6 of the displacement field : 7, 387, 459, 463
CARTESIAN coordinates : 6, 67 GREEN-ST VENANT strain tensor i n : 388 KORN'S inequality i n : See " K o r n ' s inequality" linearized change of metric tensor in m : 9 linearized elasticity i n : 10 linearized strain tensor i n : 9 linearized strains i n : 9, 36 nonlinear elasticity in ~ : 391 thxee-dimensional elasticity tensor in ~ : 11 CARTESIAN frame: 6 CAUCHY-GREEN strain tensor (right m ) : 9 change of c u r v a t u r e tensor on a surface : 508, 519, 524, 548 change of c u r v a t u r e tensor on a surface (linearized ~ ) 308, 322, 338, 344 BUDIANSKY-SANDERS ~ : 367 NAGHDI'S ~ : 366
: 93, 174, 181,
change of m e t r i c tensor in a t h r e e - d i m e n s i o n a l d o m a i n : in CARTESIAN coordinates : 388 in curvilinear coordinates : 398 change of m e t r i c tensor in a three-dimensional d o m a i n (linearized m ) : ill CARTESIAN coordinates : 9 in curvilinear coordinates : 36 change of m e t r i c tensor on a surface : 437, 450, 524, 548 change of m e t r i c tensor on a surface ( l i n e a r i z e d - - ) : 91, 170, 181,210, 228, 308, 321,338, 344, 366
CHRISTOFFEL symbols : 29, 33 of the first kind : 55, 60 of t h e second kind : 55, 60
CHRISTOFFEL symbols on a surface : 88, 136 of the first kind : 130 of the second kind : 130 clamping : two-dimensional b o u n d a r y conditions o f - - : 101,116, 304, 305, 306, 322, 344 two-dimensional b o u n d a r y conditions of strong ~ : 503, 504, 524, 525, 530, 548 CODAZZI-MAINARDI i d e n t i t i e s : 131, 133 coerciveness
of
a
f u n c t i o n a l : 431, 452, 466, 538
c o m p l e m e n t i n g b o u n d a r y condition : 123, 236, 237, 373 c o m p o s i t e m a t e r i a l (shell m a d e with a ~ ) cone : 296, 305, 329, 505, 540
: 368, 371
Indez
585
constitutive equation : three-dimensional: 389 two-dimensional: 229, 451 c o n t i n u a t i o n t h e o r e m ( u n i q u e m ) : 125, 126, 294 c o n t r a v a r i a n t basis : 21 c o n t r a v a r i a n t basis on a surface : 86 of t h e t a n g e n t p l a n e : 72 contravariant - - of of of of of of of w of of of of - -
- -
- -
components 9 a v e c t o r : 56 t h e a p p l i e d b o d y force d e n s i t y : 25 t h e a p p l i e d surface force d e n s i t y : 26 t h e first PIOLA-KIRCHHOFF stress t e n s o r : 403 t h e l i n e a r i z e d stress t e n s o r : 38 t h e m e t r i c t e n s o r : 20, 57 t h e m e t r i c t e n s o r of a surface : 72 t h e r e s u l t a n t stress t e n s o r : 224 t h e s e c o n d PIOLA-KIRCHHOFF stress t e n s o r : 402 t h e t h r e e - d i m e n s i o n a l e l a s t i c i t y t e n s o r : 32 t h e t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of a shell : 101
c o n t r o l l a b i l i t y of shells : 234, 3 6 1 , 3 7 1 convergence : of d i s p l a c e m e n t s : 183, 209, 231, 233, 268, 291, 308, 324, 370, 461 of stresses : 234, 237, 326, 331 F-: 233, 326, 459 c o o r d i n a t e line : 17 on a surface : 68 c o o r d i n a t e s (see also "CArtTSSXAN c o o r d i n a t e s " a n d " c u r v i l i n e a r c o o r d i n a t e s " ) : cylindrical--: 15 s p h e r i c a l u : 15, 67 s t e r e o g r a p h i c ~ : 67 COSSErtAT surface (one d i r e c t o r - - )
: 363, 549
c o v a r i a n t b a s i s : 17 of t h e t a n g e n t p l a n e : 69 on a surface : 86 - -
covariant components : of a v e c t o r : 25, 56 of t h e c h a n g e of m e t r i c t e n s o r : 398 - - of t h e c u r v a t u r e t e n s o r : 81 - - o f t h e d i s p l a c e m e n t f i e l d : 25 of t h e d i s p l a c e m e n t field of a surface : 86 - - of t h e GRSEN-ST VBNANT s t r a i n t e n s o r : 397 of t h e l i n e a r i z e d c h a n g e of c u r v a t u r e t e n s o r : 93 of t h e l i n e a r i z e d c h a n g e of m e t r i c t e n s o r : 36 of t h e l i n e a r i z e d c h a n g e of m e t r i c t e n s o r on a surface : 91 of t h e l i n e a r i z e d s t r a i n t e n s o r : 36 of t h e m e t r i c t e n s o r : 18, 57 of t h e m e t r i c t e n s o r of a surface : 69 - -
covariant derivative: a t e n s o r field : 38, 94, 133, 224, 402 - - of a v e c t o r field : 30, 33, 451 - - of a v e c t o r field on a surface : 88 s e c o n d order ~ : 94, 345 -
-
o
f
Indez
586
c u r v a t u r e (see also " c u r v a t u r e t e n s o r on a surface") : center o f - - : 7'4, ?7 - - o f a p l a n a r curve : ?5, 76 G A U S S I A N - : 82, 83, 84, 121, 133 line o f : 82 mean: 82 p r i n c i p a l u : 82 principal radius of: 82 r a d i u s o f - - : 74 total: 82 cuxvatuxe t e n s o r on a surface : c o v a r i a n t c o m p o n e n t s of t h e : 81 m i x e d c o m p o n e n t s of t h e : 81 RIEMANN u : 136 cuxvilinear c o o r d i n a t e s : 15 GREEN-ST VENANT s t r a i n t e n s o r i n : 397 KORN's i n e q u a l i t y i n : 44, 48, 53, 59 l i n e a r i z e d change of m e t r i c t e n s o r i n : 36 l i n e a r i z e d elasticity i n : 3? l i n e a r i z e d s t r a i n t e n s o r i n - - : 36 l i n e a r i z e d strains i n : 36 n a t u r a l - - for a shell : 14, 145 n o n l i n e a r elasticity i n : 392 t h r e e - d i m e n s i o n a l e l a s t i c i t y t e n s o r in - - : 32 cuxvilinear c o o r d i n a t e s on a surface : 65 c y l i n d e r : 296, 328, 440, 465, 504, 539 circular ~ : 66, 191 c y l i n d r i c a l c o o r d i n a t e s : 15 d e f o r m a t i o n : 9 , 3 8 ? , 456 a d m i s s i b l e i n e x t e n s i o n a l - : 530 large - - n o n l i n e a r flexuxal shell t h e o r y : 525 large - - n o n l i n e a r m e m b r a n e shell t h e o r y : 452, 463 s c a l e d - - : 458 d e f o r m a t i o n g r a d i e n t : 10 d e f o r m e d c o n f i g u r a t i o n : 9, 287 d i r e c t o r field on a surface : 363 d e v e l o p a b l e surface : 83 d~velopp~e : 75 d i r e c t o r shell t h e o r y : m u l t i - - - : 549 o n e - - - : 363, 549 displacement : admissible-
: 303, 439, 503
CARTESIAN c o m p o n e n t s of t h e -
: ?, 387, 459 convergence of t h e s : 183, 209, 231, 233, 268, 291, 308, 324, 370, 461 c o v a r i a n t c o m p o n e n t s of t h e : 25 c o v a r i a n t c o m p o n e n t s of t h e - - of a surface : 86 field in t h r e e - d i m e n s i o n a l e l a s t i c i t y : ? m field of a surface : 86 error e s t i m a t e s f o r - s : 233, 350, 351, 358, 360 i n e x t e n s i o n a l - - : 439, 503
Indez
587
limit ~ s : 226, 232, 289, 319, 447, 462, 522 linearized inextensional ~ : 162 n o r m a l u of the middle surface : 232, 325 scaled m s : 152, 199, 246, 303, 408, 458 t a n g e n t i a l m of the middle surface : 232, 325 d i s p l a c e m e n t gradient : 10, 389 d i s p l a c e m e n t - t r a c t i o n p r o b l e m : 390 l i n e a r i z e d - - : 10 d o m a i n in R n : 6, 40 DONNELL-MUSHTARI-VLASOV nonlinear shallow shell t h e o r y : 555 e d g e : See "fold" eigenvalue p r o b l e m s for shells : 234, 326, 358, 371 u n i l a t e r a l m : 555 elastic m a t e r i a l : anisotropic m : 162, 178, 234, 326, 360, 366 homogeneous: 6, 388 isotropic u : 6, 388 linearly m : See "linearized elasticity" nonhomogeneous: 162, 178, 234, 326, 360, 366 n o n l i n e a r l y - : 389, 392, 454 soft ~ : 464 ST VENANT-KIRCHHOFF ~ : 389, 392, 430, 455 elasticity tensor : t h r e e - d i m e n s i o n a l - in CARTESIAN coordinates : 11, 392 t h r e e - d i m e n s i o n a l ~ in curvilinear coordinates : 32, 57 t w o - d i m e n s i o n a l R of a s h e l l : 101, 117, 228, 291,322, 338, 344, 365, 450, 527 e l a s t o d y n a m i c s for a shell : See " t i m e - d e p e n d e n t equations" e l l i p s o i d : 121, 197, 297 elliptic m e m b r a n e shell (linearly elastic R ) : See "linearly elastic elliptic m e m b r a n e shell" elliptic m i d d l e surface (shells with ~ )
: 120, 196, 293
elliptic p o i n t : 83 elliptic s u r f a c e : 120, 122, 126, 130, 196 elliptic s y s t e m : s t r o n g l y ~ : 236, 373 u n i f o r m l y ~ : 123, 236, 237, 373 energy : See also " t h r e e - d i m e n s i o n a l energy", "two-dimensional energy", " t w o - d i m e n s i o n a l strain energy '~ existence of a m i n i m u m : 224, 228, 290, 321, 339, 340, 3 9 1 , 4 6 2 , 530, 551 m i n i m i z e r of an ~ : See " m i n i m i z a t i o n of a functional" stored ~ f u n c t i o n : 391, 392 strain ~ : See "two-dimensional strain energy" e q u a t i o n s of e q u i l i b r i u m : in CARTESIAN coordinates : 387 in curvilinear coordinates : 403 linearized m in CARTESIAN coordinates : 12 linearized ~ in curvilinear coordinates : 40 t w o - d i m e n s i o n a l - - : 229, 451 error e s t i m a t e s for displacements : 233, 350, 351, 358, 360
Indez
588 EUCLIDEAN space : three-dimensional ~ e three-dimensional- : 5
: 5
EUL~.R characteristic of a surface : 83 eversion of shells : 442, 542, 554 existence of solutions : 52, 101, 111, 115, 126, 223, 317, 338, 340, 366, 367, 376, 392, 452, 453, 529, 551, 554 face of a shell : lateral - - : 146 lower - - : 148 u p p e r m : 148 f l e x u r a l - d o m i n a t e d behavior : 361 flexural shell (linearly elastic ~ )
: See "linearly elastic flexural shell"
flexural shell (nonlinearly elastic - - ) : See "nonlinearly elastic flexural shell" flexural t h e o r y : See "shell t h e o r y " f o l d : 76, 295, 296, 297 force : See "applied b o d y force", "applied force", "applied surface force" formal a s y m p t o t i c e x p a n s i o n : 164, 414 leading t e r m in a - - : 164, 414 m e t h o d of m s : 164, 413 t e r m of order q in a : 141 frame-indifferent flexural t h e o r y : 525, 541 frame-indifferent m e m b r a n e t h e o r y : 452, 463, 466 functional (see also "energy" a n d " m i n i m i z a t i o n of a functional") : coerciveness of a : 431, 452,466, 538 as a F-limit : 460 s t a t i o n a r y point of a : 399, 445, 448, 520 weak lower semi-continuity of a : 431, 452, 536 f u n d a m e n t a l form of a surface : f i r s t - : 69, 72 s e c o n d - : 81 third: 99 I~-convergence : 233, 326, 459 F - l i m i t : 460
GAuss:
formula o f - : 88 T h e o r e m a egregium o f - - : 131, 136
GAUSS-BONNET t h e o r e m : 83, 133
GAUSSIAN c u r v a t u r e : 82, 83, 84, 121, 133 generalized m e m b r a n e shell (linearly elastic - - ) : See "linearly elastic generalized m e m b r a n e shell" genus of a surface : 83 g e o m e t r i c a l l y exact shell t h e o r y : 549
GREEN-ST VENANT strain tensor : - - i n CARTESIAN coordinates : 388 --- in curvilinear coordinates : 397 hierarchic shell t h e o r y : 368 HILBERT uniqueness m e t h o d (HUM) : 361
589
Indez
HOLMGI%EN'S uniqueness t h e o r e m : 124 h o m o g e n e o u s elastic m a t e r i a l : 6, 3 8 8 h o m o g e n i z a t i o n of shell equations : 234, 360 h y p e r b o l i c middle surface (shells w i t h - - )
: 295, 362
h y p e r b o l i c p a r a b o l o i d : 132 h y p e r b o l i c point : 83 h y p e r b o l o i d of revolution : 294 hyperelastic m a t e r i a l : 391, 392, 454 implicit function t h e o r e m : 144, 392, 453, 555 inextensional d e f o r m a t i o n : 530 inextensional displacement : 439, 503 l i n e a r i z e d - : 162, 303 infinitesimal rigid displacement : 46 on a surface : 109, 115 infinitesimal rigid displacement l e m m a : 49 o n a general surface : 109, 366, 377 on a general surface with little regularity : 114 N on an elliptic surface : 122, 295 infinitesimal rigidity of a surface : 297 infinitesimal r o t a t i o n field : 133 interior layer : 362 internal constraints ( m e t h o d o f - - ) : 368, 550 intrinsic shell t h e o r y : l i n e a r - : 368 n o n l i n e a r - : 551 isometric s u r f a c e s : 85, 439, 441, 531 isotropic elastic m a t e r i a l : 6, 3 8 8 j u n c t i o n s b e t w e e n shells : 360, 454 KII%CHHOFF-LOVE a s s u m p t i o n : 336, 372, 546, 550 l i n e a r i z e d - : 336, 360, 368, 372 KIB.CHHOFF-LOVE plate t h e o r y : 227, 320, 328 KOITER energy : 344, 548 KOITER s t r a i n energy 9 344, 548 KOIT~.I~'s linear shell equations : 100, 111, 115, 337, 339, 341, 344, 360, 373, 375 admissible applied forces for - - : 351 eigenvalue p r o b l e m for - - : 358 for shells whose middle surface has little regularity : 115, 339, 362 t i m e - d e p e n d e n t - : 347, 361 KOITER'S nonlinear shallow shell equations : 553 KOITER'S nonlinear shell equations : 547 KORN's inequality : in CARTESIAN coordinates : 49, 113, 207, 217, 253, 257 in curvilinear coordinates : 44, 48, 53, 59
590
Indez
KORN'S type (inequality o f - - ) : on a general surface : 103, 111, 118, 134, 318, 339, 356, 377, 551, 554 w on a general surface with little regularity : 115, 340 on an elliptic surface : 126, 197, 208, 216, 225, 261, 349 t h r e e - d i m e n s i o n a l - for a family of linearly elastic elliptic membrane shells : 205, 212, 220, 221 three-dimensional m for a family of linearly elastic shells : 259, 269,
271, 310 LAGRANGE multiplier: 319, 539, 550 LAMI~ constants : 7, 389 assumptions on the ~ : 152, 162, 199, 229, 247, 303, 322, 442, 443, 507 laminated shell : 368, 550 large deformation, or large displacement, flexural theory : 525 large deformation, or large displacement, membrane theory : 452, 463 lateral face of a shell : 146
LAX-MILGRAM lemma : 54, I01, I17, 192, 225, 278, 285, 288, 319, 339 layer : b o u n d a r y w : 351, 362, 368, 371, 374 i n t e r i o r - : 362 length element : 20 length element on a surface : 70 limit displacements : 226, 232, 289, 319, 447, 462, 522 limit stresses : 234, 237, 326, 331,452, 546 line of curvature : 82 linear shell theory : See "shell theory" linearization trick : 415 linearized change of curvature tensor : 93, 174, 181, 308, 322, 338, 344, 526 BUDIANSKY-SANDERS - - : 367 NAGHDI'S ~ : 366 linearized change of metric tensor in a three-dimensional domain : in CARTESIAN coordinates : 9, 36 m in curvilinear coordinates : 36 linearized change of metric tensor on a surface : 91, 170, 181, 210, 228, 308, 321, 338, 344, 366, 452, 526 linearized elasticity (three-dimensional--) : in CARTESIAN coordinates: 8, 10, 455 in curvilineax coordinates : 32, 38, 52 linearized inextensional displacement : 162, 303 linearized KIRCHHOFF-LOVE assumption : 336, 360, 368, 372 linearized rotation field : 364, 376 linearized strain (see also "change of metric") 9 in CARTESIAN coordinates : 9 in curvilinear coordinates : 36 M tensor : 9 s c a l e d - - : 153
Indez
591
linearized stress in CARTESIAN c o o r d i n a t e s : 11 in curvilinear coordinates : 38 linearized t r a n s v e r s e shear s t r a i n tensor : 366 linearly elastic elliptic m e m b r a n e shell : definition of a m : 196 e x a m p l e of a - - : 197 KOITBR'S e q u a t i o n s for a : 347 t w o - d i m e n s i o n a l constitutive equations for a : 229 t w o - d i m e n s i o n a l e n e r g y of a : 224, 229 t w o - d i m e n s i o n a l equations of a ~ : 117, 126, 210, 223, 227, 359 t w o - d i m e n s i o n a l equilibrium equations for a : 229 t w o - d i m e n s i o n a l s t r a i n energy of a ~ : 229 linearly elastic flexural shell : 111, 541 definition of a : 199, 302 e x a m p l e of a m : 303, 304, 305, 328, 329, 330 KOITER'S equations for a u : 355 t w o - d i m e n s i o n a l e n e r g y of a : 322 t w o - d i m e n s i o n a l equations of a : 101, 309, 318, 320, 359 t w o - d i m e n s i o n a l s t r a i n energy of a ~ : 322 linearly elastic generalized m e m b r a n e shell : admissible applied forces for a ~ : 265, 292, 293, 298 definition of a ~ : 245 e x a m p l e of a - - : 293, 294, 295, 296, 297 of t h e first k i n d : 262, 268, 293 of t h e second k i n d : 262, 283, 293 KOITER'S e q u a t i o n s for a ~ : 351 t w o - d i m e n s i o n a l energy of a ~ : 290 t w o - d i m e n s i o n a l equations of a : 268, 283, 359 t w o - d i m e n s i o n a l s t r a i n energy of a ~ : 290 linearly elastic m a t e r i a l : See "linearized elasticity" linearly elastic m e m b r a n e shell : 199, 246, 466 linearly elastic shallow shell : definition of a ~ : 369, 371 t w o - d i m e n s i o n a l e n e r g y of a -
: 370
LIONS ( l e m m a of J.L. ~ ) : 42, 105, 119, 218, 256, 279, 376 locking: p h e n o m e n o n : 362, 366 membrane: 362, 366, 368 lower face of a shell : 148 m a n i f o l d : 439, 472, 502, 520, 526, 529, 542, 543 t a n g e n t space to a : 472, 503, 520 MAB.GUERRE-VON K.~RM~N equations in curvilinear coordinates : 552 m e a n c u r v a t u r e : 82 m e m b r a n e - d o m i n a t e d b e h a v i o r : 361 m e m b r a n e locking : 362, 366 m e m b r a n e shell (linearly elastic ~ ) : See "linearly elastic elliptic m e m b r a n e shell", "linearly elastic generalized m e m b r a n e shelr'~ "shell t h e o r y " m e m b r a n e shell (nonlinearly e l a s t i c - - ) : See "nonlinearly elastic m e m b r a n e shell" m e m b r a n e t h e o r y : See " p l a n a r m e m b r a n e t h e o r y " , "shell t h e o r y "
592
Indez
m e t r i c tensor : 55 contravariant c o m p o n e n t s of t h e : 19 covariant c o m p o n e n t s of the ~ : 18 m i x e d c o m p o n e n t s of the ~ : 57 m e t r i c tensor on a surface : contravariant c o m p o n e n t s of the ~ : 72 covariant c o m p o n e n t s of the ~ : 69 m i d d l e surface of a shell : 143 with folds : 76 with little regularity : 76, 114, 115, 339, 362, 366 with no b o u n d a r y : 143, 234, 292, 297 m i n i m i z a t i o n of a functional : 224, 228, 290, 321, 339, 340, 391, 446) 448, 452, 456, 462, 520, 523, 529, 553 mixed component : of the c u r v a t u r e tensor on a surface : 81 of the metric tensor : 57 multi=dixector shell t h e o r y : 549 m u l t i -l a y e re d shell : 368, 371,454, 550 multiplicity of solutions : 530 NAGHDI'S linear shell equations : 365, 376 for shells whose middle surface has little regularity : 366 NAGHDI's linearized change of curvature tensor : 366 NAGHDI'S nonlinear shell t h e o r y : 549 n a t u r a l curvilinear coordinates for a shell : 14, 145 n a t u r a l s t a t e : 6, 388 n o n h o m o g e n e o u s elastic m a t e r i a l (shell m a d e with a - - ) 326, 360, 366
: 162, 178, 234,
nonlinear elasticity ( t l a r e e = d i m e n s i o n a l - - ) : - - i n CARTESIAN c o o r d i n a t e s : 391 in curvilinear coordinates : 392 nonlinear p l a n a r m e m b r a n e t h e o r y : 451,453, 454, 460 non l i n e a r shell t h e o r y : See "shell theory" nonlineaxly elastic flexural shell : definition of a - - : 502 e x a m p l e of a : 504, 505, 506, 539, 540 two-dimensional energy of a - - - : 524 two-dimensional equations of a : 518, 522, 523 two-dimensional strain energy of a : 524 n o n l i n e a r l y elastic m a t e r i a l : 389, 392, 454 n o n l i n e a r l y elastic m e m b r a n e shell (theory derived by formal analysis) : definition of a : 439 e x a m p l e of a m : 440, 4 4 1 , 4 6 5 two-dimensional constitutive equations for a : 451 two-dimensional energy of a : 445, 446, 450 two-dimensional equations of a --- : 444, 448, 450 two-dimensional equilibrium equations for a : 451 two-dimensional strain energy of a : 450
Indez n o n l i n e a r l y elastic m e m b r a n e shell (theory derived by P-convergence) : stored energy function of a : 463 two-dimensional energy of a : 463 n o n l i n e a r l y elastic shallow shell: definition of a : 552, 553, 555 t w o - d i m e n s i o n a l energy of a : 553, 554 n o n u n i q u e n e s s of solutions : 530 NOVOZHILOV'S shell t h e o r y : 371 n u m e r i c a l a p p r o x i m a t i o n of shell problems : 362, 363, 368 obstacle (shell lying on a n - - ) : 555 one-director COSSBRAT surface : 363, 549 one-director shell t h e o r y : 363, 549 parabolic m i d d l e surface (shells w i t h - - ) : 296 parabolic p o i n t : 83 PIOLA-KIRCHHOFF stress tensor ( ~ s t - - ) : 388, 403 PIOLA-KIRCHHOFF stress tensor ( s e c o n d - - ) : 387 p l a n a r m e m b r a n e t h e o r y ( n o n l i n e a r - - ) : 451,453, 454, 460 p l a n a r p o i n t : 82, 132 p l a t e : 306, 329, 4 6 5 , 5 0 6 shell b e c o m i n g a : 192, 234, 326, 358, 375 p l a t e t h e o r y : 152, 326, 358, 370 KII%CHHOFF-LOVE: 227, 320, 328 KEISSNER-MINDLIN- : 363, 549 p o i n t on a surface : e l l i p t i c - : 83 h y p e r b o l i c - : 83 parabolic: 83 planar: 82, 132 u m b i l i c a l - : 82, 133 POISSON ratio : 389 p o l y c o n v e x stored energy function : 392 pressure l o a d : 455 principal c u r v a t u r e : 82 principal d i r e c t i o n : 82 principal radius of c u r v a t u r e : 82 principle of v i r t u a l work : l i n e a r i z e d - in CARTESIAN c o o r d i n a t e s : 11 l i n e a r i z e d - in curvilinear coordinates : 40 in CARTESIAN c o o r d i n a t e s : 388 - - in curvilinear coordinates : 403 quasiconvex : envelope : 459, 462 function : 459 -
-
reference configuration : 6, 386 of a shell : 12, 143 r e g u l a r i t y of solutions : 60, 122, 235, 341, 373 P~EISSNER-MINDLIN p l a t e t h e o r y : 363, 549 RIEMANN c u r v a t u r e tensor on a surface : 136
593
594
Indez
rigid displacement l e n u n a : See "inFinitesimal rigid displacement", "infinitesimal rigid displacement l e m m a " rigidification of a surface : 297 rigidity of a surface : 442 i n f i n i t e s i m a l - : 297 r o t a t i o n field : infinitesimal u : 133 l i n e a r i z e d - - : 364, 376 scaled s t r a i n : See "strain" scaled t l ~ e e - d i m e n s i o n a l elasticity tensor of a shell : 154, 410 scaled t l ~ e e - d i m e n s i o n a l equations : for a linearly elastic shell : 154 - - for a nonlinearly elastic shell : 410 scaled two-dimensional elasticity tensor of a shell : 158, 170, 181, 210, 308, 430 scaled u n k n o w n : See "scalings of displacements" scaled vector field: 152, 408 scalings of deformations : 458 scalings of displacements : 152, 199, 246, 303, 408, 458 justification o f - - : 162, 415 scalings of stresses : 237 sensitive variational p r o b l e m : 293, 295, 296 shallow shell : a s s u m p t i o n on the middle surface of a - - : 369, 371, 552 linearly e l a s t i c - : 369 n o n l i n e a r l y e l a s t i c - : 552 shape o p t i m i z a t i o n for shells : 361 shear : linearized t r a n s v e r s e - - tensor : 366 shell (see also "shallow shell") : 12, 143 b o u n d a r y layer in a : 351, 362 c l a m p e d u : 101, 116, 146, 339 controllability of m s : 234, 361 eigenvalue p r o b l e m for - - s : 234, 326, 358, 3 7 1 , 5 5 5 eversion o f - - s : 442, 542, 554 h o m o g e n i z a t i o n o f - - e q u a t i o n s : 234, 360 j u n c t i o n s b e t w e e n m s : 360, 454 laminated: 368, 550 lateral face of a - - : 146 linearly elastic elliptic m e m b r a n e : See "linearly elastic elliptic m e m b r a n e shell" linearly elastic f l e x u r a l - - - : See "linearly elastic flexural shell" linearly elastic generalized m e m b r a n e - - : See "linearly elastic generalized m e m b r a n e shell" lower face of a : 148 m i d d l e surface of a m : 143 m u l t i - l a y e r e d - - : 368, 371,454, 550 n a t u r a l curvilinear coordinates for a : 14, 145 n o n l i n e a r l y elastic f l e x u r a l - : See "nonlinearly elastic flexural shell" n o n l i n e a r l y elastic m e m b r a n e : See "nonlinearly elastic m e m b r a n e shell"
Indez
595
n u m e r i c a l a p p r o x i m a t i o n of u p r o b l e m s : 362, 363, 368 shape o p t i m i z a t i o n for u s : 361 b e c o m i n g a p l a t e : 192, 234, 326, 358, 375 m a d e w i t h a composite m a t e r i a l : 368, 371 m a d e w i t h a n o n h o m o g e n e o u s m a t e r i a l : 162, 234, 326, 360, 366 m a d e with a soft m a t e r i a l : 464 m a d e w i t h an anisotropic m a t e r i a l : 162, 234, 326, 360, 366, 550 whose m i d d l e surface has little regularity : 76, 114, 115, 339, 362, 366 whose m i d d l e surface has no b o u n d a r y : 143, 234, 292, 297 w i t h folds : 76, 295, 296, 297 w i t h variable thickness : 234, 238, 326, 330, 360 simply s u p p o r t e d : 115, 134, 339, 439, 451, 530 strongly c l a m p e d : 503, 504, 525, 530, 548 thickness of a : 143 u p p e r face of a ~ : 148 weakly simply s u p p o r t e d u : 234 shell t h e o r y : C O S S B R A T - : 363, 549 DONNELL-MUSHTARI-VLASOV nonlinear s h a l l o w - : 555 first-order ~ : 550 f r a m e - i n d i f f e r e n t - : 452, 463, 466, 525, 541 geometrically exact m : 549 h i e r a r c h i c - : 368 higher-order i : 550 intrinsic linear u : 368 intrinsic n o n l i n e a r - : 551 large deformation, or large d i s p l a c e m e n t , - : 452, 463, 525 linear - - based on the m e t h o d of internal constraints : 368 linear BUDIANSKY-SANDERS ~ : 367 linear f l e x u r a l - : 101, 162, 171, 180, 182, 199, 526 linear K O I T E R ' S - : 100, 115, 336, 346, 548 linear m e m b r a n e u : 117, 126, 162, 170, 178, 182, 196, 199, 232, 246, 452, 466 linear NAGHDI'S u : 363, 365 linear s h a l l o w - : 369 MARGUERRE-VON KARMAN ~ : 552 multi-director ~ : 549 nonlinear B U D I A N S K Y - S A N D E R S : 551 nonlinear flexuzal ~ : 502, 505, 525 nonlinear K O I T E R ' S shallow ~ : 553 nonlinear K O I T E R ' S u : 547 nonlinear m e m b r a n e ~ : 439, 451, 463, 464, 505 nonlinear N A G H D F S ~ w i t h directors : 549 nonlinear shallow i : 552 nonlinear ~ b a s e d o n the m e t h o d of internal constraints : 550 NOVOZHILOV ~ : 371 one-director ~ : 363, 549
of higher-order : 550 of the first order : 550 time-dependent I. 234, 326, 347, 550, 551
simple s u p p o r t : t w o - d i m e n s i o n a l b o u n d a r y conditions of m : 115, 134, 339, 439, 451, 530 t w o - d i m e n s i o n a l b o u n d a r y conditions of weak - - : 234
Indez
596 s i n g u l a r p e r t u r b a t i o n p r o b l e m : 154 SOBOLEV spaces : 41
soft elastic m a t e r i a l : 464 s p h e r e : 67, 133, 135, 190, 441 s p h e r i c a l c o o r d i n a t e s : 15, 67 ST VENANT-KIRCHHOFF m a t e r i a l : 389, 392, 430 s t o r e d e n e r g y f u n c t i o n of a m : 391, 455, 459, 464, 467 s t a t i o n a r y p o i n t of a f u n c t i o n a l : 391, 448, 520, 523 s t e r e o g r a p h i c c o o r d i n a t e s : 67 s t o r e d e n e r g y f u n c t i o n : 391, 392, 4 5 1 , 4 5 5 , 4 6 3 strain : l i n e a r i z e d - s in CARTESIAN c o o r d i n a t e s : 9 linearized m s in curvilinear c o o r d i n a t e s : 36 scaled l i n e a r i z e d - : 153 scaled: 410 S i n CARTESIAN c o o r d i n a t e s : 388 s in curvilinear c o o r d i n a t e s : 398 s t r a i n e n e r g y : See " t w o - d i m e n s i o n a l s t r a i n e n e r g y " strain tensor : CAUCHY-GREEN-
: 9
linearized-
in CARTESIAN c o o r d i n a t e s : 9 G R E E N - S T V E N A N T - in CARTESIAN c o o r d i n a t e s : 388 G R E E N - S T VENANT - - i n c u r v i l i n e a r c o o r d i n a t e s : 397
stress : convergence o f - - e s : 234, 237, 326, 331 limites : 234, 237, 326, 331, 452, 546 l i n e a r i z e d - es in CARTESIAN c o o r d i n a t e s : 11 l i n e a r i z e d - es in curvilinear c o o r d i n a t e s : 38, 237 es in CARTESIAN c o o r d i n a t e s : 388 - - e s in curvilinear c o o r d i n a t e s : 402 stress couple : 344, 345 stress r e s u l t a n t : 224, 229, 344, 345, 450 stress t e n s o r : first P I O L A - K I R C H H O F F - : 388 l i n e a r i z e d - : 38 second P I O L A - K I R C H H O F F - : 387, 402 s t r o n g l y elliptic s y s t e m : 236 s u p p l e m e n t a r y c o n d i t i o n on L : 123, 236, 373 s u r f a c e : 65 d e v e l o p a b l e - : 83 elliptic ~ : 120, 293 hyperbolic: 295 infinitesimal rigidity of a ~ : 297 isometrics : 439, 441, 531 p a r a b o l i c - - : 296 rigidification of a - - : 297 r i g i d i t y of a - - : 442 surface force : See " a p p l i e d surface force"
597
Indez tensor : BUDIANSKY-SANDERS l i n e a r i z e d c h a n g e of c u r v a t u r e CAUCHY-GREEN s t r a i n (right) : 9 c h a n g e of c u r v a t u r e - - : 508 c h a n g e of m e t r i c m : 388, 398 c h a n g e o f m e t r i c - - o n a s u r f a c e : 437 curvature: 81 first PIOLA-KIRCHHOFF s t r e s s : 388, 403 G R E E N - S T VENANT s t r a i n : 388 linearized change of curvature: 93 linearized change of metric: 9, 36 l i n e a r i z e d c h a n g e o f m e t r i c - - o n a s u r f a c e : 91 l i n e a r i z e d s t r a i n ~ : 9, 36 l i n e a r i z e d s t r e s s ~ : 38 linearized transverse shear strain: 366 m e t r i c ~ : 18 m e t r i c ~ o n a s u r f a c e : 69 RIEMANN c u r v a t u r e : 136 s e c o n d PIOLA-KIRCHHOFF s t r e s s ~ : 387, 402 three-dimensional elasticity: 11, 32, 392, 396 t h r e e - d i m e n s i o n a l e l a s t i c i t y ~ of a shell : 148 t w o - d i m e n s i o n a l e l a s t i c i t y - - of a shell : 101, 117 Theorema
e g r e g i u m : 131, 136
thickness: shell with variable o f a shell : 143 three-dimensional
~ e
~
:
234, 238, 326, 330, 360
EUCLIDEAN s p a c e : 5
three-dimensional elasticity tensor : s c a l e d ~ : 154, 410 i n CARTESIAN c o o r d i n a t e s : 11, 59, 392 in c u r v i l i n e a r c o o r d i n a t e s : 32, 50, 59, 182, 396 three-dimensional energy : minimizer of a: 392 i n CARTESIAN c o o r d i n a t e s : 391 i n c u r v i l i n e a r c o o r d i n a t e s : 396 t h r e e - d i m e n s i o n a l e q u a t i o n s for a l i n e a r l y e l a s t i c shell : 146 s c a l e d ~ : 154, 200, 247, 307 three-dimensional scaled ~:
e q u a t i o n s for a n o n l i n e a r l y e l a s t i c shell : 406 410
three-dimensional EUCLIDEAN s p a c e : 5 three-dimensional linearized elasticity : in CARTESIAN c o o r d i n a t e s : 8 in c u r v i l i n e a r c o o r d i n a t e s : 32 three-dimensional nonlinear elasticity: i n CARTESIAN c o o r d i n a t e s : 391 i n c u r v i l i n e a r c o o r d i n a t e s : 401 time-dependent equations : for a l i n e a r l y e l a s t i c s h e l l : 234, 326, 347 for a n o n l i n e a r l y e l a s t i c shell : 550, 551 t o r u s : 66, 133, 292 t o t a l c u r v a t u r e : 82
: 367
598
Indez
transverse variable : 145, 150 average with respect to t h e - : 201 two-dimensional boundary conditions : u of clamping: 101, 116, 304, 305, 306, 322, 344 m of simple support : 115, 134, 439, 451, 530 m of strong clamping : 503, 504, 524, 525, 530, 548 of weak simple support : 234 two-dimensional constitutive equation : 229, 451 two-dimensional elasticity tensor of a shell : 101, 117, 190, 228, 291, 322, 338, 344, 365, 450, 524 s c a l e d - : 158, 170, 181, 210, 308, 430, 519 two-dimensional energy : of a linearly elastic elliptic membrane shell : 229 of a linearly elastic flexural shell : 322 of a linearly elastic generalized membrane shell : 290 of a nonlinearly elastic flexural shell : 524, 530 of a nonlinearly elastic membrane shell (theory derived by formal analysis) : 450 of a nonlinearly elastic membrane shell (theory derived by F-convergence): 463 two-dimensional equations" for a linearly elastic elliptic membrane shell : 117, 126, 210, 223, 227 for a linearly elastic flexural shell : 101, 309, 318, 320, 321 for a linearly elastic generalized membrane shell : 268, 283 for a nonlinearly elastic flexural shell : 518, 530 for a nonlinearly elastic membrane shell (theory derived by formal analysis): 444, 450 for a nonlinearly elastic membrane shell (theory derived by r-convergence): 463 two-dimensional equations of equilibrium : 229, 451 two-dimensional KOIT]~R energy : of a linearly elastic shell : 344 of a nonlinearly elastic shell : 548 two-dimensional KOITER equations : eigenvalue problem for the --- : 358 time-dependent m : 347, 361 for a linearly elastic shell : 100, 111, 115, 337, 339, 341, 344, 360, 373, 375 for a nonlinearly elastic shell : 547 two-dimensional KOITER strain energy : o f a linearly elastic shell : 344 o f a nonlinearly elastic shell : 548 two-dimensional strain energy (see also "two-dimensional KOITER strain energy") : o f a linearly elastic elliptic membrane shell : 229, 344, 361 of a linearly elastic flexural shell : 322, 344, 361 of a linearly elastic generalized membrane shell : 290 of a nonlinearly elastic flexural shell : 524, 548 of a nonlinearly elastic membrane shell (theory derived by formal analysis) : 450, 548 of a nonlinearly elastic membrane shell (theory derived by r-convergence) : 463
Indez
umbilical point : 82, 133 uniformly elliptic system : 123, 236, 237 uniqueness of solutions : 52, 101, 111,115, 122, 124, 126, 223, 317, 338, 340, 366, 367, 376, 530 upper face of a shell : 148 volume element : 18 weak lower semi-continuity of a functional : 431, 452, 536 WEINGARTEN (formula of--) : 88 Y O U N G modulus : 389
599
