In this chapter, we develop a theory of shells that we believe is simple yet comprehensive and free of unnecessary assumptions. Thus, as with the special case of beamshells and axishells discussed in Chapters IV and V, we first derive exact equations of motion merely by specializing to shell-like bodies the integral-impulse form of the equations of motion of a three-dimensional continuum. This reduction, which involves no kinematic hypotheses (such as those associated with the names of Kirchhoff or Cosserat) and introduces no expansions through the thickness, leads to two-dimensional stress results and couples, a deformed position vector, and a rotational momentum vector, all expressed as certain weighted integrals with respect to a thicknesslike coordinate. In doing so, we introduce neither a midsurface nor so-called shell coordinates. Exact, conjugate strain measures fall out naturally and automatically from a mechanical work identity. As we emphasize throughout this book, the approximate nature of nonlinear shell theory enters only through the constitutive relations which—as in three-dimensional elasticity—ultimately rest on experimental data. It is at this point (and not before) that we introduce the famous Kirchhoff Hypothesis, now free of the contradictions it engenders if stated as a strict kinematic hypothesis. After a thorough discussion of the field equations and boundary conditions implied by this hypothesis, we end with a classification and discussion of various additional approximations introduced into shell theory to simplify the governing equations, including those leading to quasi-shallow shell theory and (shell-) membrane theory, the subject of the previous chapter.

In this chapter we encounter our first shells—shells that act like beams. The single greatest difference from the chapter on birods is that curvature effects now infest, at least implicitly, nearly every equation we set down. Consequent new features that we shall discuss at some length are: (1) necessary and sufficient conditions for a load potential P to exist if there are distributed as well as edge loads; (2) definitions of different extensional, shear, and bending strains;, (3) restrictions on the strain-energy density imposed by invariance requirements and implied by thinness; and (4) structural stability (buckling) including thermodynamic considerations.

Geometry of the Undeformed Shell and Planar Motion

An infinite, right cylindrical shell is a body whose initial shape is generated by translating a plane region normal to itself. See Fig. 4.1a. Mathematically, the plane region is the image of a rectangular region; the images of the top and bottom of the rectangular region generate the faces of the shell whereas the images of the ends of the rectangular region generate the edges of the shell. Although our intuitive picture is that of a thin-walled panel of, possibly, variable thickness, the notion of thinness shall not enter explicitly until we come to constitutive relations in Section N.

For a quantitative description, consider a fixed, right-handed Cartesian reference frame Oxyz. Let {ex, ey, ez} denote the associated set of orthonormal base vectors with ez perpendicular to the plane region that generates the shell.

The study of waves is important to virtually every branch of science and engineering. Indeed, waves are also important to everyday life. Sound waves allow us to hear, and electromagnetic waves allow us to see. In this book, we restrict our study to one important class of waves often called stress waves. These waves propagate through gases, liquids, and solids. Stress waves in gases and liquids are usually called pressure waves. This nomenclature is derived from the fact that internal forces in solid bodies are represented by a stress tensor, and internal forces in inviscid liquids and gases are represented by a special form of stress called pressure. In Chapter 1, we shall learn the distinction between stress and pressure.

Sound is the most commonly experienced pressure wave. Indeed, anyone standing in a thunderstorm knows that lightening is seen before the report of its thunder is heard. This simple observation reveals the single, most important feature of any wave – its finite velocity. We know there is always a lapse of time between the cause of a sound wave and when we hear it.

A membrane, the two-dimensional analog of a string, is a thin-walled body that can only carry internal forces tangent to its deformed shape. Thus, a membrane is either a shell so thin that it is, essentially, incapable of supporting stress couples or else it is a shell with sensible bending stiffness but which is subject to a combination of external loads and boundary/initial conditions which produce negligible bending. If we wish to distinguish between these two models, we shall refer to the former as a true membrane and to the latter as a (solutiondependent) shell-membrane. True membranes model certain biological tissues, inflatables, soap bubbles, shells subject to very high pressure and very large strains, and the like; shell-membranes model regions of shells that cannot undergo (nearly) inextensional bending and which are not too close to geometric or material discontinuities such as crowns of toroidal shells, cracks, or shell boundaries. Moreover, true membranes cannot support compressive stresses and thus may wrinkle whereas shell-membranes can resist this type of instability.

In various parts of Section V.S we looked at the membrane limit of the simplified, nonlinear, small-strain, axisymmetric Reissner equations. In Section V.T, we developed the general theory of aximembranes undergoing large strains and gave many examples and references. In this chapter, we develop the general theory of membranes, including the subtheory of wrinkling, a topic both mathematically challenging and of growing applicability to natural and manufactured objects.

In this second edition, we have added a chapter (VII) on general membrane theory (including wrinkling) and a chapter (VIII) on general shell theory. Thus, the subtitle “One Spatial Dimension” of the first edition has been dropped. In addition, we have updated many parts of Chapters I-VI of the first edition, adding references, correcting typographical errors, and rewriting several sections to improve both their style and substance.

Originally, we planned to write an entire second volume on general shell and membrane theory and to include an extensive discussion of variational, buckling, and thermoelastic theories, just as we had done in the first edition for shells of one spatial dimension. We also intended to discuss comparisons with three-dimensional theory as well as various approximate theories and their analytical solutions. This agenda proved to be too ambitious, so we have settled for the more reasonable goal of what we hope is a thorough (but readable and useful) treatment of the foundations of our subject.

In surveying the vast literature on shell and membrane theory and in attempting to distill its essence, we have come to five broad conclusions:

1. There are far too many papers in the literature on shells that make little or no attempt to study (much less cite) earlier work.

2. There is a plethora of so-called “higher-order” or “refined” theories that do not recognize that the formulation of proper refined boundary conditions requires the consideration of threedimensional effects in all but the simplest cases (e.g., simple support), although there exists a rather extensive literature on this question extending back to the early 1940s.

3. […]

Elastic Material

It is a common observation that individual materials react differently to applied loads.

We often calculate nonlinear wave phenomena with complex numerical computer codes. With such powerful capabilities you may be tempted to simply characterize the energy state of a material with large tables of data that yield values of equilibrium stress, temperature, stiffnesses, specifics heats, and thermal expansion coefficients for given values of F and η. Accurate computations of some very common materials rely heavily upon this method. The computation of nonlinear pressure waves in water is an important example. However, even with this capability, questions arise about the constitutive descriptions of these materials. For example, if water is heated by a shock wave, it often spontaneously “flashes” to steam when the pressure drops again in response to a release wave.

Index Notation

We begin by introducing a compact and convenient way to describe quantities in three spatial dimensions with a Cartesian coordinate system. We also show how to transform quantities between different Cartesian coordinate systems.

Cartesian Coordinates and Vectors

A Cartesian coordinate system has three straight coordinates that are mutually perpendicular to each other (see Figure 1.1). The coordinates are named x1, x2, and x3. The subscripts 1, 2, and 3 are called indices. When we let k represent any of these indices, we can refer to the coordinates by the compact notation xk. Each coordinate xk also has a shaded arrow that represents a unit coordinate vectorik. We use a boldface symbol to denote a vector. The length of each ik is equal to unity.

Now consider another vector v that is drawn with arbitrary angles to the three coordinates. We draw a box with sides parallel to ik and with v as a diagonal.

Why must we study thermodynamics? In Chapter 2 we studied a wide range of wave phenomena using the mechanical concepts of force and motion. Using only the constitutive model for an elastic material, we developed detailed analyses of the nonlinear structured wave, the transition of a compression wave into a shock wave, and the decay of a shock wave as it is overtaken by a rarefaction wave. We have analyzed wave motion in solids, liquids, and adiabatic gases.

Guide to Notation

Shell theory uses many symbols. Some have become standard (e.g., N's for stress resultants, M's for stress couples, Q's for transverse shear stress resultants), but others vary from writer to writer. Below, we give a few guidelines to our scheme of notation and list some of the more important global symbols. We have not attempted to list local symbols, i.e., symbols peculiar to a section or subsection. However, we warn the reader that some global symbols have a different meaning locally. In part, this was forced on us because of the limited number of alphabets and fonts available on our laser printer and we sometimes had to use other than optimal symbols.

General scheme of notation

Bold letters (e.g., F) denote vectors.

Script letters (e.g, F) usually denote lines, regions, matrices, column vectors, functionals, and operators.

A comma followed by a subscript (as in F,θ) denotes differentiation with respect to the subscript.

A prime (as in F') denotes differentiation with respect to undeformed arc length σ.

A dot [as in or] denotes differentiation with respect to time t.

An overbar (as in) denotes a quantity associated with the deformed configuration. The same quantity, without the overbar, is associated with the undeformed configuration.

A hat (as in) denotes a prescribed quantity.

This book is a greatly expanded version of parts of our monograph, “Nonlinear Elastic Shell Theory,” that appeared on pages 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 principles, 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 original. Space, time, and our proclivities have prevented us from discussing numerical solutions of shell equations, an important area that needs to be surveyed, summarized, and streamlined. The state of the art may be 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 presentation 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 three-dimensional continuum mechanics.

During July 1994 while I was writing this book, fragments of the comet Shoemaker–Levy 9 crashed into Jupiter. As each fragment entered the Jovian atmosphere, it formed a shock wave, similar to the bow wave in front of a boat. Heat generated within these waves ignited the atmosphere of Jupiter creating fireballs the size of the earth.