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**Elasticity of transversely isotropic materials.**
*(English)*
Zbl 1101.74001

Solid Mechanics and its Application 126. Dordrecht: Springer (ISBN 1-4020-4033-4/hbk). xii, 435 p. (2006).

With ever greater demand of mathematical analysis of problems of transversely isotropic elastic materials, the utility and interest in the subject play a major role in applied mathematics and engineering. In spite of the fact that elasticity is a classical subject, theory and applications of subject are still predominant in advanced study and research in applied mathematics and engineering. Keeping these in mind, the authors’ main goal is to provide an introduction to the theory and applications of mechanics of transversely isotropic elastic materials.

This book has ten chapters with a long reference and an index. The first chapter deals with a brief presentation of basic equations of anisotropic elasticity, thermoelasticity, boundary and initial conditions. In chapter 2 several methods including the displacement method, stress method and mixed method are developed in solving transversely isotropic elasticity problems. Chapters 3 and 4 deal with problems of infinite elastic solids and with solution of stresses and displacements in a half-space or a layered solid with transverse isotropy using the state-space and Fowler transform methods. Equilibrium of bodies of revolution including circular and annular plates, solid and hollow cylinders, solid and hollow spheres, and solid and hollow cones is discussed in some detail. Chapter 6 is devoted to simplified models of the thermoelastic deformation of transversely isotropic and spherically isotropic elastic solids. However, this is a very brief chapter compared to the vast subject of thermoelasticity. Frictional contact, bending, vibration and stability of plates are the main topics of chapters 7 and 8. Chapter 9 is concerned with the analysis of free vibrations of transversely isotropic cylinders and cylindrical shells of transversely isotropic materials. With some emphasis on engineering applications, three simple vibrational modes including the axisymmetric torsional vibration, breaking vibration modes and thickness shear vibration and asymmetric vibration are discussed. The last chapter 10 deals with the three-dimensional analysis of free vibrations of spherically isotropic spherical shells including single-layered and multi-layer ones. The state-space formulation in spherical coordinates is presented to find the free vibration solution for laminated spherical shells. Additional notes and bibliography to chapters, special functions and nomenclature are included in three appendices.

Some more special comments are in order. First, this book is written to meet the needs of modern topics on mechanics of transversely isotropic elastic solids. However, this is neither a traditional research monograph nor a standard textbook as there are no exercises and worked problems. Second, the book cannot be recommended as a suitable text for a course in the U.S. universities in solid mechanics for applied mathematics, science and engineering as it does not meet the basic requirements of the standard syllabus of a course in solid or continuum mechanics. However, the book is well writtten and it does not contain any wrong or misleading information. It seems to be a useful reference book on the subject. Finally, the first two authors published a large number of good papers on the subject matter of the book. In addition to these research papers, the book can be considered as an important contribution to the engineering literature.

This book has ten chapters with a long reference and an index. The first chapter deals with a brief presentation of basic equations of anisotropic elasticity, thermoelasticity, boundary and initial conditions. In chapter 2 several methods including the displacement method, stress method and mixed method are developed in solving transversely isotropic elasticity problems. Chapters 3 and 4 deal with problems of infinite elastic solids and with solution of stresses and displacements in a half-space or a layered solid with transverse isotropy using the state-space and Fowler transform methods. Equilibrium of bodies of revolution including circular and annular plates, solid and hollow cylinders, solid and hollow spheres, and solid and hollow cones is discussed in some detail. Chapter 6 is devoted to simplified models of the thermoelastic deformation of transversely isotropic and spherically isotropic elastic solids. However, this is a very brief chapter compared to the vast subject of thermoelasticity. Frictional contact, bending, vibration and stability of plates are the main topics of chapters 7 and 8. Chapter 9 is concerned with the analysis of free vibrations of transversely isotropic cylinders and cylindrical shells of transversely isotropic materials. With some emphasis on engineering applications, three simple vibrational modes including the axisymmetric torsional vibration, breaking vibration modes and thickness shear vibration and asymmetric vibration are discussed. The last chapter 10 deals with the three-dimensional analysis of free vibrations of spherically isotropic spherical shells including single-layered and multi-layer ones. The state-space formulation in spherical coordinates is presented to find the free vibration solution for laminated spherical shells. Additional notes and bibliography to chapters, special functions and nomenclature are included in three appendices.

Some more special comments are in order. First, this book is written to meet the needs of modern topics on mechanics of transversely isotropic elastic solids. However, this is neither a traditional research monograph nor a standard textbook as there are no exercises and worked problems. Second, the book cannot be recommended as a suitable text for a course in the U.S. universities in solid mechanics for applied mathematics, science and engineering as it does not meet the basic requirements of the standard syllabus of a course in solid or continuum mechanics. However, the book is well writtten and it does not contain any wrong or misleading information. It seems to be a useful reference book on the subject. Finally, the first two authors published a large number of good papers on the subject matter of the book. In addition to these research papers, the book can be considered as an important contribution to the engineering literature.

Reviewer: Lokenath Debnath (Edinburg)

### MSC:

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

74B05 | Classical linear elasticity |

74E10 | Anisotropy in solid mechanics |

74F05 | Thermal effects in solid mechanics |