This book examines in detail the Theory of Elasticity which is a branch of the mechanics of a deformable solid. Special emphasis is placed on the investigation of the process of deformation within the framework of the generally accepted model of a medium which, in this case, is an elastic body. A comprehensive list of Appendices is included providing a wealth of references for more in depth coverage. The work will provide both a stimulus for future research in this field as well as useful reference material for many years to come.

The classical theory of elasticity maintains a place of honour in the science ofthe behaviour ofsolids. Its basic definitions are general for all branches of this science, whilst the methods forstating and solving these problems serve as examples of its application. The theories of plasticity, creep, viscoelas­ ticity, and failure of solids do not adequately encompass the significance of the methods of the theory of elasticity for substantiating approaches for the calculation of stresses in structures and machines. These approaches constitute essential contributions in the sciences of material resistance and structural mechanics. The first two chapters form Part I of this book and are devoted to the basic definitions ofcontinuum mechanics; namely stress tensors (Chapter 1) and strain tensors (Chapter 2). The necessity to distinguish between initial and actual states in the nonlinear theory does not allow one to be content with considering a single strain measure. For this reason, it is expedient to introduce more rigorous tensors to describe the stress-strain state. These are considered in Section 1.3 for which the study of Sections 2.3-2.5 should precede. The mastering of the content of these sections can be postponed until the nonlinear theory is studied in Chapters 8 and 9.

According to established tradition, courses on analytical mechanics include general equations of motion of holonomic and non-holonomic systems, vari­ ational principles, theory of canonical transformations, canonical equations and theory of their integration (the Hamilton-Jacobi theorem), integral in­ variants, theory of last multiplier and others. The fundamental laws of mechanics are taken for granted and are not subject to discussion. The present book is concerned with those issues of the above listed sub­ jects which, in the author's opinion, are most closely related to engineering problems. Application of the methods of analytical mechanics to non-trivial prob­ lems at the very stage of constructing the equations requires detailed knowl­ edge of the issues that are normally only briefly touched upon. With this perspective considerable attention is paid to ways of introducing the gener­ alised coordinates, the theory of finite rotation, methods of calculating the kinetic energy, the energy of accelerations, the potential energy of forces of various nature, and the resisting forces. These introductory chapters, which have to some extent independent significance, are followed by those on methods of constructing differential equations of motion for holonomic and non-holonomic systems in various forms. In these chapters the issues of their interrelations, determination of the constraint forces and some prob­ lems of analytical statics are discussed as well. It is thought useful to include geometric considerations of the motion of a material system as motion of the representative point in Riemannian space.

According to established tradition, courses on analytical mechanics include general equations of motion of holonomic and non-holonomic systems, vari­ ational principles, theory of canonical transformations, canonical equations and theory of their integration (the Hamilton-Jacobi theorem), integral in­ variants, theory of last multiplier and others. The fundamental laws of mechanics are taken for granted and are not subject to discussion. The present book is concerned with those issues of the above listed sub­ jects which, in the author''s opinion, are most closely related to engineering problems. Application of the methods of analytical mechanics to non-trivial prob­ lems at the very stage of constructing the equations requires detailed knowl­ edge of the issues that are normally only briefly touched upon. With this perspective considerable attention is paid to ways of introducing the gener­ alised coordinates, the theory of finite rotation, methods of calculating the kinetic energy, the energy of accelerations, the potential energy of forces of various nature, and the resisting forces. These introductory chapters, which have to some extent independent significance, are followed by those on methods of constructing differential equations of motion for holonomic and non-holonomic systems in various forms. In these chapters the issues of their interrelations, determination of the constraint forces and some prob­ lems of analytical statics are discussed as well. It is thought useful to include geometric considerations of the motion of a material system as motion of the representative point in Riemannian space.

This book examines in detail the Theory of Elasticity which is a branch of the mechanics of a deformable solid. Special emphasis is placed on the investigation of the process of deformation within the framework of the generally accepted model of a medium which, in this case, is an elastic body. A comprehensive list of Appendices is included providing a wealth of references for more in depth coverage. The work will provide both a stimulus for future research in this field as well as useful reference material for many years to come.

The classical theory of elasticity maintains a place of honour in the science ofthe behaviour ofsolids. Its basic definitions are general for all branches of this science, whilst the methods forstating and solving these problems serve as examples of its application. The theories of plasticity, creep, viscoelas­ ticity, and failure of solids do not adequately encompass the significance of the methods of the theory of elasticity for substantiating approaches for the calculation of stresses in structures and machines. These approaches constitute essential contributions in the sciences of material resistance and structural mechanics. The first two chapters form Part I of this book and are devoted to the basic definitions ofcontinuum mechanics; namely stress tensors (Chapter 1) and strain tensors (Chapter 2). The necessity to distinguish between initial and actual states in the nonlinear theory does not allow one to be content with considering a single strain measure. For this reason, it is expedient to introduce more rigorous tensors to describe the stress-strain state. These are considered in Section 1.3 for which the study of Sections 2.3-2.5 should precede. The mastering of the content of these sections can be postponed until the nonlinear theory is studied in Chapters 8 and 9.

According to established tradition, courses on analytical mechanics include general equations of motion of holonomic and non-holonomic systems, vari­ ational principles, theory of canonical transformations, canonical equations and theory of their integration (the Hamilton-Jacobi theorem), integral in­ variants, theory of last multiplier and others. The fundamental laws of mechanics are taken for granted and are not subject to discussion. The present book is concerned with those issues of the above listed sub­ jects which, in the author's opinion, are most closely related to engineering problems. Application of the methods of analytical mechanics to non-trivial prob­ lems at the very stage of constructing the equations requires detailed knowl­ edge of the issues that are normally only briefly touched upon. With this perspective considerable attention is paid to ways of introducing the gener­ alised coordinates, the theory of finite rotation, methods of calculating the kinetic energy, the energy of accelerations, the potential energy of forces of various nature, and the resisting forces. These introductory chapters, which have to some extent independent significance, are followed by those on methods of constructing differential equations of motion for holonomic and non-holonomic systems in various forms. In these chapters the issues of their interrelations, determination of the constraint forces and some prob­ lems of analytical statics are discussed as well. It is thought useful to include geometric considerations of the motion of a material system as motion of the representative point in Riemannian space.

According to established tradition, courses on analytical mechanics include general equations of motion of holonomic and non-holonomic systems, vari­ ational principles, theory of canonical transformations, canonical equations and theory of their integration (the Hamilton-Jacobi theorem), integral in­ variants, theory of last multiplier and others. The fundamental laws of mechanics are taken for granted and are not subject to discussion. The present book is concerned with those issues of the above listed sub­ jects which, in the author''s opinion, are most closely related to engineering problems. Application of the methods of analytical mechanics to non-trivial prob­ lems at the very stage of constructing the equations requires detailed knowl­ edge of the issues that are normally only briefly touched upon. With this perspective considerable attention is paid to ways of introducing the gener­ alised coordinates, the theory of finite rotation, methods of calculating the kinetic energy, the energy of accelerations, the potential energy of forces of various nature, and the resisting forces. These introductory chapters, which have to some extent independent significance, are followed by those on methods of constructing differential equations of motion for holonomic and non-holonomic systems in various forms. In these chapters the issues of their interrelations, determination of the constraint forces and some prob­ lems of analytical statics are discussed as well. It is thought useful to include geometric considerations of the motion of a material system as motion of the representative point in Riemannian space.