Table Of ContentThe Nonlinear Theory of Elastic Shells
One Spatial Dimension
A. Libai
Department of Aeronautical Engineering
Technion, Israel Institute of Technology
Haifa, Israel
J. G. Simmonds
Department of Applied Mathematics
University of Virginia
Charlottesville, Virginia
ACADEMIC PRESS, INC.
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Library of Congress Cataloging-in-Publication Data
Libai, A. (Avinoam), 1929-
The nonlinear theory of elastic shells of one
spatial dimension.
Includes bibliographies and index.
1. Shells (Engineering) 2. Elasticity.
3. Nonlinear theories. I. Simmonds, James G.
II. Title.
TA660.S5L457 1988 624.1'7762 87-33497
ISBN 0-12-447940-5
88 89 90 91 9 8 7 6 5 4 3 2 1
Printed in the United States of America
To our beloved families
Preface
This book is a greatly expanded version of parts of our monograph, "Non
linear Elastic Shell Theory," that appeared on pp. 271-371 of Volume 23 of
Advances in Applied Mechanics (J. W. Hutchinson & T. Y. Wu, eds.),
Academic Press, 1983. We have added several new chapters and rewritten or
supplemented others extensively. Discussions of stability and variational princi
ples, which were omitted from the monograph for lack of space, are included
here. Furthermore, some material on load potentials, nonlinear constitutive
laws, thermodynamics (as applied to shells), and boundary conditions is origi
nal. Space, time, and our proclivities have prevented us from discussing numer
ical solutions of shell equations, an important area that needs to be surveyed,
summarized, and str1eamlined. The state2 of the art may 3b e gleaned from books
by Hinton & Owen, Hughes & Hinton, and Bushnell.
This book has two main goals: to lay a foundation for the nonlinear theory
of thermoelastic shells undergoing large rotations and large strains and to
present, early on, relatively simple equations for practical application. We have
tried to write for those who know some continuum mechanics but little shell
theory, but we think that experts will find here much that is novel—in presenta
tion if not in content. Thus, after an introductory chapter that describes what we
mean by a nonlinear elastic shell and spells out our approach, we expose, in the
next chapter, the bedrock of 3-dimensional continuum mechanics. On this foun
dation we build an ascending stairway of four chapters. Our hope is to raise
gradually the reader's understanding of shell theory by developing, for rods and
special shells, concrete results that are useful in themselves. There are some
topics, however, that we think are best treated as special cases of, or as approxi
mations to, the general shell equations. It is our intention to include these in a
second volume—the descending portion of the stairway.
To aid the reader, we have posed and solved, in various places, simple
problems to illustrate general principles, have used an asterisk (*) to indicate
those sections that can be omitted without loss of continuity, and have
lM
2Finite Element Software for Plates and Shells," Pineridge Press, Swansea, U. K., 1984.
"Finite Element Methods for Plate and Shell Structures," vols. 1 and 2, Pineridge Press,
Swansea, U. K., 1986.
'"Computerized Buckling Analysis of Shells," Nijhoff, Dordrecht, The Netherlands, 1985.
xiii
Preface
summarized our notational scheme and listed the important symbols we use in
an appendix at the end of the book.
The chapters on "birods", "beamshells","axishells", and "unishells" also
illustrate, in situations of increasing complexity, our mixed approach to shell
theory, namely, to use those equations of 3-dimensional continuum mechanics
that are independent of material properties to derive corresponding rod or shell
equations, but to postulate the form of those rod or shell equations that depend
unavoidably on material properties.
4
We might take as an epigraph for this book TruesdelFs dictum (1983):
In mathematical practice today it is, unfortunately, often forgotten that to
derive basic equations is even so much a mathematician's duty as to study
their properties
5
to which we would add Koiter's observation (1969) that
Flexible bodies like thin shells require a flexible approach.
In obedience to these mentors, we have tried to derive the equations of shell
theory with care and to cast them in many different forms, knowing from
experience that a set of dependent variables or a reference frame that works for
one class of problems may be unsuitable for another. We also acknowledge the
profound influence of Eric Reissner, as our many citations to his papers attest.
He was the teacher of one of us (JGS), and his lessons have not been forgotten.
In assembling material for this book we have encountered many gaps. In
most cases, we have resisted the temptation—with its potential for distraction
and frustration—to try to fill them. Here and there, therefore, the reader will
find open questions and problems.
Parts of this book were written while Professor Libai was visiting the
Department of Applied Mathematics at the University of Virginia, on sabbatical
leave from the Department of Aeronautical Engineering, the Technion, Israel
Institute of Technology. We thank the Center for Advanced Studies at the
University of Virginia for its partial support of Professor Libai*s visit. We also
thank the Fund for the Promotion of Research at the Technion and the National
Science Foundation for supporting much of our research. Our colleagues, Ernst
Axelrad and Hubertus Weinitschke, and our students, Dawn Fisher, Jim Fulton,
and Dror Rubinstein, have made useful suggestions in various chapters. Rohn
England, another student, has read the entire manuscript with an eye to both
syntax and semantics. Tom Tartaglino and Judith Trott prepared all of the fig
ures, except for 5.10-5.12, which Jim Fulton prepared using the troff preproces
sors grap and pic. Thanks to all!
4
The influence of elasticity on analysis: the classical heritage. Bull. Amer. Math. Soc. 9,
293-310.
foundations and basic equations of shell theory: A survey of Recent Progress. In "Theory of
Thin Shells," Proc. I.U.T.A.M. Sympos., Copenhagen, 1967 (F. I. Niordson, ed.), pp. 93-105.
Springer-Verlag, Berlin, etc.
xiv
Preface
We typeset the manuscript ourselves using the ditrofftext processing sys
tem with 4.2 BSD UNIX on a VAX 11/780, run jointly by the Departments of
Computer Science and Applied Mathematics at the University of Virginia.
Thus, all errors—substantial or typographical—rest squarely on us.
XV
Chapter I
Introduction
A. What Is a Shell?
A shell is a curved, thin-walled structure. A quantitative definition will be
given in later chapters. Two important degenerate classes of shells are plates
(shells which are flat when undeformed) and membranes (shells whose walls
offer no resistance to bending). Shells may be made of a single inhomogeneous
or anisotropic material or may be made of layers of different materials. The pri
mary function of a shell may be to transfer loads from one of its edges to
another, to support a surface load, to provide a covering, to contain a fluid, to
please the eye or ear, or a combination of these. Shells occur in nature and as
artifacts. Aortic valves, automobile hoods, balloons, beer cans, bellows, bells,
bladders, bowls, contact lenses, crab carapaces, domes, ducts, egg coverings,
footballs, funnels, inner tubes, light bulb casings, loudspeaker cones, manhole
covers, parachutes, peanut hulls, Ping-Pong balls, panels, pipes, pressure
vessels, silos, skulls, straws, tents, tires, trumpets, umbrellas, vaults, wine
glasses, and woks are all shells. The aim of shell theory is to describe the static
or dynamic behavior of structures like the above by equations that involve no
more than one or two spatial variables.
B. Elastic Shells and Nonlinear Behavior
If a shell is prevented from moving as a rigid body, then it is elastic if,
upon application and removal of a sufficiently small load, it tends to return to its
initial shape.
The behavior of an elastic shell is said to be nonlinear if, under static con
ditions, the deflection of any point of the shell is not proportional to the magni
tude of an applied load. Two sources of nonlinearity are often distinguished:
geometric and material. One speaks of a geometrically nonlinear shell theory if
the strain-displacement relations are nonlinear but the stress-strain relations
linear. Most papers on nonlinear shell theory make this assumption and for
good reason: traditional engineering materials such as steel and aluminum
remain elastic only if the principal strains are relatively small, say less than half
l
2 Libai & Simmonds
of a percent. While there have been attempts to formulate shell theories in
which the strain-displacement relations are linear but the stress-strain relations
nonlinear (e.g., Zerna, 1960), the consistency of this assumption, at least in 3-
dimensional elasticity, has been challenged by Bharatha & Levinson (1977).
Our aim in this book is to present a shell theory that, in its general form, is
both geometrically and materially nonlinear. Thus, we intend to present equa
tions that can be applied to biologic or rubber-like shells undergoing large defor
mations as well as to more traditional shells.
C. Approaches to Shell Theory
An exact 2-dimensional theory of shells does not exist: no matter how
thin, a shell remains a 3-dimensional continuum. Indeed, the response of a shell
to external loads—mechanical or thermal—depends critically on its thickness.
Different approaches to shell theory may be classified according to how the
reality of 3-dimensions is handled.
At one extreme lies the direct approach which, from the start, denies that
a shell is 3-dimensional. This approach, pioneered by the Cosserat brothers
(1908, 1909), proposed independently in a long-overlooked paper by Weather-
bum (1925), revived independently by Ericksen & Truesdell (1958) and Giinter
(1958), and brought to perfection by Green, Naghdi, & Wainwright (1965) and
Cohen & DeSilva (1966), starts by modeling a shell as a 2-dimensional
"oriented" continuum, represented by a surface with one or more attached direc
tors (vectors) and endowed with pointwise properties such as mass, bending
stiffness, etc. Next, 2-dimensional laws of conservation of mass; balance of
linear, rotational, and "director" momentum; and thermodynamics are postu
lated. Finally, the addition of a 2-dimensional free-energy density yields a com
plete thermoelastic theory of shells.
A limitation of the direct method is its apparent isolation from 3-
dimensional continuum mechanics. To remedy this, Green, Laws, & Naghdi
(1968) assume for the deformed position vector of a 3-dimensional, shell-like
continuum an infinite series expansion in a thickness coordinate and identify the
directors of the direct approach with the coefficients in this expansion. See the
treatise by Naghdi (1972) for a complete exposition.
At the other end of the spectrum lies the derived approach to shell theory.
Here, one starts from the 3-dimensional equations of thermoelasticity and
attempts, by formal, asymptotic methods, to exploit the special geometry and
loading that characterize those bodies we call shells. The end results are "inte
rior" equations that describe the behavior of shells away from edges and other
geometric or load discontinuities. Contributors to this approach (who confine
themselves to isothermal, linear theory) include Goodier (1938), Goldenveiser
(1945, 1962, 1963, 1966, 1969a, 1969b), Green (1962), Johnson & Reissner
(1958), Reissner (1960, 1963, 1969), Reiss (1960, 1962), Johnson (1963), and
Cicala (1966).
I. Introduction 3
A drawback of the derived approach, aside from its tediousness, is that we
must know at the edges of the shell the distribution of the applied stress or dis
placements over the thickness. As Koiter (1970) has emphasized, we never
know the stress distribution precisely, except at a free edge. Another drawback
of the derived approach is that, because the thickness of the shell is always
incorporated in the expansion parameter, one set of uniformly valid interior (i.e.,
shell) equations does not emerge. Instead, there is one set of equations for a
membrane state, another for an inextensional bending state, another for a "sim
ple" edge-effect, another for a "degenerate" edge-effect, and, if one is dealing,
say, with an infinite cylindrical shell subject to self-equilibrating edge loads, still
another set of equations is needed to recover the "semi-membrane" theory of
Vlasov (see Novozhilov, 1970, Section 49).
In contrast, the approach of John (1965a, 1965b, 1969, 1971) to shell
theory is remarkable. His analysis is nonlinear, his error estimates are rigorous
rather than formal, and in his 1969 and 1971 papers, he obtains uniformly valid
interior shell equations (at the expense, ho1wever, of introducing cubic terms in
the 3-dimensional strain-energy density). John's work has been extended to
shells of variable thickness under distributed surface loads by Berger (1973).
D. The Approach of this Book to Shell Theory
Our approach is a mixed one, based on the following dichotomy. The
equations of 3-dimensional continuum mechanics fall into two groups: the gen
eric equations, which are independent of the material properties of a body,
namely conservation of mass, balance of linear and rotational momentum, heat
flow, and the Clausius-Duhem inequality (the Second Law of Thermodynam
ics); and those which are not, namely Conservation of Energy (the First Law of
Thermodynamics) and the constitutive relations. The second group of equations
serve to join the mechanical and thermal variables which never appear together
in the generic equations. The internal energy that appears in the Conservation
of Energy and the constitutive relations must be determined experimentally;
hence it is unavoidable that these relations be approximate.
To derive a nonlinear shell theory, we descend from the generic 3-
dimensional equations, via a weighted integration through the thickness, to
obtain generic 2-dimensional equations of balance of momentum, heat flow, and
an entropy inequality. Conservation of mass is satisfied identically because we
use a material (Lagrangian) formulation. On the other hand, because Conserva
tion of Energy and the constitutive relations in 3-dimensions are not exact, we
do not use them, but instead postulate analogous 2-dimensional relations. That
is, we throw all approximations in shell theory into those field equations which
by their very nature are approximate. The reason for calling our approach
"mixed" is now clear: the exact parts of our shell equations follow from 3-
^oiter & Simmonds (1973) have shown how to obtain John's results using only the standard
quadratic strain-energy density.
4 Libai & Simmonds
dimensional continuum mechanics; the approximate parts are postulated, ab ini
tio.
In more detail, our procedure begins with the 3-dimensional equations of
balance of linear and rotational momentum and the Clausius-Duhem inequality
for a body, written in integral-impulse form and referred to the reference shape
of the body. [Thus, impulsive or concentrated loads and discontinuous
unknowns are incorporated simply and naturally, and the artifice of delta "func
tions" is never needed. See Truesdell & Toupin (1960, p. 232, footnote #4) or
Truesdell (1984a, p. 33-34, footnote #19 and p. 38) for a history and an elabora
tion of this point.] By specializing these equations to a shell-like body, we
obtain analogous, exact 2-dimensional equations for shells, together with defini
tions of stress resultants, stress couples, a deformed position vector, a spin vec
tor, and several thermal variables, all expressed as certain weighted integrals
through the thickness of the 3-dimensional body in its reference shape. We
emphasize that the descent from 3-dimensions to 2 involves no series expan
sions in the thickness coordinate.
Next, assuming sufficient smoothness of various fields in space and time,
we obtain differential equations of motion and a differential entropy ineq2uality.
From the differential equations, we derive a Mechanical Work Identity. In the
process, definitions of 2-dimensional strains fall out automatically.
At this point, if we assume isothermal deformations, the machinery is in
place to develop a complete Mechanical Theory of Shells. In particular, an elas
tic theory may be defined by assuming the existence of a strain-energy density
that depends only on the strains delivered by the Mechanical Work Identity. We
develop this theory extensively in Chapters III-V.
If the temperature is not constant, we must couple the mechanical vari
ables with the 2-dimensional thermal variables; the latter include heat flux, the
entropy resultant, and an entropy flux vector. This is done by postulating a 2-
dimensional Law of Conservation of Energy. A thermoelastic shell is then
defined as one in which the internal energy depends on the present values of the
"state" variables (the strains plus some of the thermal variables) and their spatial
gradients. By arguments in the spirit of Coleman & Noll (1963), we deduce
from the shell version of the Clausius-Duhem inequality suitable forms of, and
restrictions on, the constitutive re3lations. As one application of our results, we
use the ideas of Duhem (1911), Ericksen (1966a,b), Koiter (1969, 1971), and
Gurtin (1973,1975) to cast the problem of stability of an equilibrium configura
tion in a thermodynamic setting.
2
Antman & Osborn (1979) have shown that the closely related Virtual Work Principle can, in
3-dimensions, be obtained directly from the integral form of balance of linear and rotational momen
tum. This is a most satisfying result—to obtain the Virtual Work Principle via differential equations
requires more smoothness than is necessary. A similar observation was made by Carey & Dinh
(1986).3
See Truesdell (1984b, pp. 38-44) for a summary of DuhenTs long-neglected ideas on the sta
bility of deformable, heat conducting bodies.
