5R2. The Finite Element Analysis of Shells: Fundamentals. Computational Fluid and Solid Mechanics. - D Chapelle (INRIA-Rocquencourt, Le Chesnay, France) and K-J Bath (MIT, Cambridge MA). Springer-Verlag, Berlin. 2003. 330 pp. ISBN 3-540-41339-1. \$79.95.

Reviewed by C Meyer (Dept of Civil Eng, Columbia Univ, 500 W 120th St, MC 4709, New York NY 10027-6699).

The modern theory of shells, which dates back to Love, more than one hundred years ago, has ever since been a major topic for applied mathematicians and engineers who don’t mind solving eighth-order partial differential equations. The introduction of the finite element method in the 1960s seemed to eliminate this need for mathematical astuteness, as practitioners could get the impression that a standard finite element software system would provide correct answers to just about any practical problem, without the need for a fundamental understanding of shell behavior on the user’s part.

Nothing could be further from the truth. Each “shell element” in a commercially available software system is based on one particular shell theory, and it so happens that different shell theories tend to give different results for the same problem and exhibit widely varying convergence characteristics with mesh refinement. Each shell theory is characterized by its specific modeling assumptions of the kinematic constraints “through the thickness” of the shell. This applied primarily for thin-shell theories, because for thick shells, a discretization with solid elements will typically eliminate most potential problems. The primary service that this book by Chapelle and Bathe renders to the profession is that it uses a rigorous mathematical approach to organize these various theories. It assumes that the reader is intimately familiar with tensor notation and the mathematical theory of shells. Readers without such background are unlikely to obtain an understanding of thin shell theory from this book.

After a very short introduction, Chapter 2 introduces geometric preliminaries of vectors and tensors in three-dimensional curvilinear coordinates, which are then used to define the shell geometry. Chapter 3 presents the elements of functional and numerical analysis, that is, Sobolev spaces and their associated norms as well as variational formulations and finite element approximations. With Chapter 4, the book starts getting interesting. It is here where the various kinematic shell models are analyzed with mathematical rigor. Likewise, the presentation of asymptotic behaviors of different shell models, subject of Chapter 5, contains intriguing results of practical significance. Also, the derivation of displacement-based shell finite elements in Chapter 6 proves very elegantly that the popular facet-shell, ie, flat; elements do not satisfy the mathematical criteria for convergence, whereas more efficient displacement-based elements based on general shell theory are available. Chapter 7 presents an in-depth analysis of the influence of the shell thickness on the various theories, and Chapter 8 derives the formulation of effective general shell elements.

Up to this point, the book is limited to small deformations and linear elastic material behavior, characterized by two material constants, Young’s modulus and Poisson’s ratio. This is a serious limitation. Although the authors claim, in Chapter 9 on the nonlinear analysis of shells, that almost their entire theory is applicable to the nonlinear domain as well, this reviewer is not convinced. Many problems in engineering practice are nonlinear. For metal shells, buckling behavior often controls the design. For concrete thin-shell structures, nonlinear constitutive relations and creep behavior can play significant roles. The theory presented in this book may form a solid foundation, but taking the significant step to nonlinear applications requires more than the few cursory, though valid comments advanced in the very short final chapter.

As already mentioned, the target audience for The Finite Element of Analysis of Shells-Fundamentals is the mathematically astute, relatively small group of experts who may be called upon for the development of better software codes, ie, better finite element formulations. The vast majority of practitioners are users of such software and not expected to be experts in mathematical shell theory. It would have been useful to provide them with some guidelines on how to evaluate the relative accuracy and convergence characteristics of different finite element formulations. But this was obviously beyond the purpose and scope of this fine and compact monograph on mathematical shell theory.