Review of Fundamental Solutions in Elastodynamics, by Eduardo Kausel: Cambridge Univ. Press, Cambridge, U.K.; 2006; ISBN 13 978-0-521-855709-9; 251 pp. Price: US$75.00.

This book presents a comprehensive collection of fundamental solutions (or Green’s functions) for classic elastodynamic problems. This book is the first one of its kind and provides a handbook of fundamental solutions for numerical analyses of, for example, wave-propagation problems in elasticity, soil dynamics, and earthquake engineering. Researchers who are working on the boundary element method (BEM) or the boundary integral method (BIM) will find the book particularly useful.

Although the solutions included in the book can be found from the literature, they were presented by using different conventions and symbols and in different coordinate systems. One of the useful features of the book is that it shows all the formulas in a common format, including using a consistent notation and sign convention. Both two- and three-dimensional problems are considered, and appropriate formulas are always given for the three most commonly used coordinate systems, i.e., Cartesian, cylindrical, and spherical systems.

All the formulas are presented in the book without proof, whereas references are given to published papers or books for most of them. Thus, this book cannot serve as a textbook or a research monograph for researchers who want to study fundamental solutions and their applications. Sound knowledge of differential equations, calculus, vectors, and matrices is required to read the book.

The book has 10 main chapters that are divided into five sections. Section 6 contains three appendixes, including a MATLAB programs chapter. An index could have been given for the benefit of the reader.

The first section comprises two chapters that deal with fundamental equations in relation to elastodynamics and the differentiation technique used in deriving fundamental solutions for dipoles.

Chapter 1, entitled “Fundamentals,” provides a list of notations and symbols that are used throughout the book. It also provides an overview of some of the most useful differential operators, strain and stress components and the elastic wave equation. All the formulas are presented explicitly in terms of the Cartesian, cylindrical, and spherical coordinate systems, so this chapter is useful not only for dealing with fundamental solutions but also for solutions of differential equations and two- or three-dimensional elasticity.

Chapter 2 deals with the differentiation technique for obtaining fundamental solutions when the sources are couples or tensile crack sources. The solutions also include dipoles formed by line sources, couples, line, and point blast loads. The chapter provides a general procedure for constructing new fundamental solutions on the basis of those obtained from simple cases.

Sections 2 and 3, respectively, list the fundamental solutions for full-space (Chapters 3 and 4) and half-space (Chapters 5 and 6) problems. The formulas are presented for both two-dimensional and three-dimensional solutions for various sources, such as point loads, line loads, tension cracks, seismic moments, and torques.

Section 4 has only one chapter (Chapter 7), which discusses two-dimensional problems (subjected to in-plane loads) in homogeneous plates and strata. The author devotes significant space to the formulas for plates with mixed boundary conditions. It is unfortunate that plates subjected to transverse loads (bending of plates) are excluded from this section.

Section 5 is a rather independent part, where, in addition to providing a summary of the general solutions to the Helmholtz and wave equations, full derivations to these equations and solutions are given in all the three major coordinate systems. There are three chapters in this section.

Chapter 8 summarizes the equations and their solutions of these classic problems. They are presented in parallel in terms of the three coordinate systems and are reasonably easy to follow, assuming that readers have good knowledge of the subject areas. In most cases, the solutions are expressed in compact matrix form that simplifies applications to more-complex problems, such as layered media and numerical calculations using computers.

Chapter 9 briefly introduces the integral transform method. That method is one of the most popular methods for solving elastodynamic problems in unbounded continua. Although the concept of integral transform is nicely described in this chapter, like the previous chapters, this chapter cannot be used for a study of the method. Two well-explained examples are given in the chapter to demonstrate how the integral transform method can be used to solve practical problems.

The final chapter of this section is a nice addition to this book. It introduces the stiffness matrix method that is primarily related to the numerical analysis of structures. The chapter extends the applications of the closed-form solutions to complex problems that require numerical treatment, for example, problems with layered media. More examples that can help readers are found in this chapter. Problems solved include waves in a layer of a half-space, There is a summary of the method and a nice comparison of the method with the propagator matrix method at the beginning of the chapter. These are conceptually important and are often ignored in textbooks. The author also provides step-by-step instructions for using the stiffness matrix method, which is very useful for readers who are not familiar with the method.

The book has three appendixes that are grouped in Section 6 into three separate chapters. The first gives basic properties of Bessel functions and Legendre polynomials. The second contains a listing of integral transforms, including Fourier transforms, Hankel transforms and spherical Hankel transforms. The third chapter contains sixteen MATLAB programs that produce solutions to problems including SH line load in half or full space, SVP line load in half or full space, blast line source in half or full space, two- or three-dimensional cavity in full space, point source in half or full space, and torsional point source in half or full space, SH line source in homogeneous plate or stratum, SVP line source inplate with mixed boundary conditions, and spheroidal and torsional modes of homogeneous sphere.

Overall, the book is well-produced, and the contents are well structured. I am sure that this book will be welcomed by researchers and will be a nice addition to academic libraries.