Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates.

*(English)*Zbl 0645.73037
Studies in Applied Mechanics, 10. Amsterdam etc.: Elsevier. VIII, 281 p.; $ 66.75; Dfl. 180.00 (1985).

This book is devoted to the application of boundary integral equation (BIE) methods to eigenvalue problems of elastodynamics and thin plates. Its contents is that of the doctoral dissertation of the author which was submitted to Kyoto University in March 1984.

This book is divided into two parts which are themselves divided into several chapters. Let us indicate briefly now the contents of each of them.

Part 1: Applications of BIE methods to eigenvalue problems of elastodynamics. Firstly in Chapter 1, the basis of BIE method is summarized for elastodynamics. A general method for obtaining fundamental solutions is carefully described. Next by considering the analogy between the potential theory and the elasticity theory, the associated simple or double layer potentials are introduced. Moreover Green’s displacement and traction formulae in steady-state elastodynamics are summarized. - Chapter 2 contains and discusses the formulation of BIE by the use of layer potentials (“indirect method”) and Green’s formula (direct method) for the three kinds of boundary value problems (displacement, traction or mixed conditions) of steady-state elastodynamics. These BIE are summarized in very convenient tables. - Chapter 3 is concerned with the formulation of eigenvalue probems by the BIE. The three types of boundary value problems are considered and the corresponding BIE are again summarized in convenient tables. - Chapter 4 is devoted to the analytical treatment of previous integral equations for the circular geometry (circular and annular domains). - In order to evaluate these previous integral equations for arbitrary-shaped domains, Chapter 5 proposes some numerical procedures. - Chapter 6 examines the numerical aspects of the BIE method to obtain eigenvalues, eigendensities and eigenmodes for the antiplane shear problem. This is applied to the analysis of resonance phenomena of an inhomogeneous protusion on a stratum. - Finally in Chapter 7, the BIE methods are applied to the numerical analysis of eigenvalue problems of in-plane elastodynamics. Examples including discs, annular domains and dam-type structures are given and discussed.

Part 2: Applications of BIE methods to eigenvalue problems of thin plates. The first chapter briefly summarizes the fundamentals of the boundary integral equation method for plate problems, in particular the eigenvalue problems including the buckling problem. Four kinds of boundary value problems are considered: clamped, simply supported, free and mixed problems. - Then Chapter 2 formulates the BIE for thin plates and eigenvalue problems in terms of not only layer potentials but also Green’s formula. Finally the Chapter 3 reports the numerical analysis of eigenvalue problems for three types of plate problems (plate vibration, plate vibration subjected to uniform in-plane force, and buckling). The accuracy of eigenfrequencies, buckling loads, eigendensities and eigenmodes is discussed. In addition, the BIE methods are applied to eigenvalue problems for an arbitrary shaped plate with a mixed boundary condition.

In conclusion, this is a very nice book, clearly written, very well illustrated and documented. Although I am not working directly on boundary integral equations, after reading this book I would wish to extend, more or less, the second part of the book to the study of eigenvalue problems of thin shells.

This book is divided into two parts which are themselves divided into several chapters. Let us indicate briefly now the contents of each of them.

Part 1: Applications of BIE methods to eigenvalue problems of elastodynamics. Firstly in Chapter 1, the basis of BIE method is summarized for elastodynamics. A general method for obtaining fundamental solutions is carefully described. Next by considering the analogy between the potential theory and the elasticity theory, the associated simple or double layer potentials are introduced. Moreover Green’s displacement and traction formulae in steady-state elastodynamics are summarized. - Chapter 2 contains and discusses the formulation of BIE by the use of layer potentials (“indirect method”) and Green’s formula (direct method) for the three kinds of boundary value problems (displacement, traction or mixed conditions) of steady-state elastodynamics. These BIE are summarized in very convenient tables. - Chapter 3 is concerned with the formulation of eigenvalue probems by the BIE. The three types of boundary value problems are considered and the corresponding BIE are again summarized in convenient tables. - Chapter 4 is devoted to the analytical treatment of previous integral equations for the circular geometry (circular and annular domains). - In order to evaluate these previous integral equations for arbitrary-shaped domains, Chapter 5 proposes some numerical procedures. - Chapter 6 examines the numerical aspects of the BIE method to obtain eigenvalues, eigendensities and eigenmodes for the antiplane shear problem. This is applied to the analysis of resonance phenomena of an inhomogeneous protusion on a stratum. - Finally in Chapter 7, the BIE methods are applied to the numerical analysis of eigenvalue problems of in-plane elastodynamics. Examples including discs, annular domains and dam-type structures are given and discussed.

Part 2: Applications of BIE methods to eigenvalue problems of thin plates. The first chapter briefly summarizes the fundamentals of the boundary integral equation method for plate problems, in particular the eigenvalue problems including the buckling problem. Four kinds of boundary value problems are considered: clamped, simply supported, free and mixed problems. - Then Chapter 2 formulates the BIE for thin plates and eigenvalue problems in terms of not only layer potentials but also Green’s formula. Finally the Chapter 3 reports the numerical analysis of eigenvalue problems for three types of plate problems (plate vibration, plate vibration subjected to uniform in-plane force, and buckling). The accuracy of eigenfrequencies, buckling loads, eigendensities and eigenmodes is discussed. In addition, the BIE methods are applied to eigenvalue problems for an arbitrary shaped plate with a mixed boundary condition.

In conclusion, this is a very nice book, clearly written, very well illustrated and documented. Although I am not working directly on boundary integral equations, after reading this book I would wish to extend, more or less, the second part of the book to the study of eigenvalue problems of thin shells.

Reviewer: M.Bernadou

##### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74H45 | Vibrations in dynamical problems in solid mechanics |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74K20 | Plates |

65R20 | Numerical methods for integral equations |

74E30 | Composite and mixture properties |

74B99 | Elastic materials |

74H99 | Dynamical problems in solid mechanics |