1R10. Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. Applications of the Bubnov-Galerkin and Finite Difference Numerical Methods. - J Awrejcewicz (Dept of Automatics and Biomechanics, Tech Univ of Lodz, 1/15 Stefanowskiego St, Lodz, 90-924, Poland) and VA Krys’ko (Dept of Math, Saratov State Tech Univ, 77 Polyteshnycheskaya St, Saratov, 41005, Russia). Springer-Verlag, Berlin. 2003. 428 pp. ISBN 3-540-43880-7. $89.95.
Reviewed by MV Shitikova (Dept of Struct Mech, Voronezh State Univ of Architec and Civil Eng, ul Kirova 3-75, Voronezh, 394018, Russia).
Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells is one of the latest titles from the Scientific Computation series published by Springer-Verlag. This monograph describes some approaches to the linear and nonlinear dynamic theory of thermoelastic plates and shells. By nonclassical problems the authors mean the problems described by two and three-dimensional differential equations of hyperbolic and parabolic type, in so doing the interaction between the strain and temperature fields is taken into account. When solving nonlinear problems, both physical and geometrical nonlinearities are considered.
The book involves 9 chapters, 247 references, a subject index, 21 tables, and 222 figures. Chapter 1 presents a brief review of the literature devoted to the dynamic coupled thermoelastic problems, in so doing the emphasis is made to Eastern references published in Polish and Russian, which can be of great interest to a Western reader. Since there exists a very large number of books, papers and commercial software packages devoted to application of different numerical methods for the analysis of plates and shells, the authors restrict their efforts to the Ritz method and the Bubnov-Galerkin method, resulting only in mathematical and numerical investigations of some specific problems related to the coupled theory of elastic and elasto-plastic plates and shells.
Chapter 2 presents the basics of the coupled linear thermoelasticity of shallow shells modeled by both the Timoshenko and Kirchhoff-Love theories. The existence and uniqueness of a general solution are discussed. The authors have not introduced any limitations on the temperature distribution through the shell thickness; therefore, the temperature depends on the three spatial coordinates and the time, while the components of the displacement vector of a point of the middle surface of the shell depend on the two spatial coordinates and the time, resulting in a set of equations of different types (hyperbolic and parabolic ones) with different dimensions. The second part of Chapter 2 is focused on the dynamics of an elastic infinite cylindrical panel within a transonic gas flow. Free and forced panel vibrations are analyzed using the finite-difference method. The stability loss of the panel within a transonic flow is investigated.
Chapter 3 discusses estimation of the errors of the Bubnov-Galerkin method applied for solving a system of linear differential equations, which corresponds to coupled thermoelastic problems for plates and shallow shells with variable thickness.
Numerical investigations of the errors of the Bubnov-Galerkin method are given in Chapter 4 for the following problems: vibration of a transversely loaded simply supported square plate without a thermal load, the vibrations of a simply supported plate with a given initial distribution of deflection or of its velocity with neither a mechanical nor a thermal load.
Fundamental assumptions and relations for coupled nonlinear thermoelastic problems and the equations describing vibration of a Timoshenko-type shell are formulated in Chapter 5. The existence and uniqueness of a solution of some initial-boundary value problems of coupled thermoelasticity as well as the convergence of the Bubnov-Galerkin method are discussed.
Theory of Kirchhoff-Love type shallow shells with physical nonlinearities in the form of small elasto-plastic deformations and coupling of temperature and strain fields is developed in Chapter 6.
Chapters 7 and 8 are devoted to the numerical solution of some nonlinear problems described by the hybrid form of the differential equations obtained in Chapters 5 and 6. Numerical procedures based on the finite-difference method and the relaxation method are presented for the analysis of the vibration of an isolated shell subjected to an impulse load which is constant in time and uniformly distributed, for the dynamic stability of shells under thermal shock, as well as for the analysis of the regular and chaotic behavior of the plate. Vibrations and stability of elasto-plastic shells subjected to cyclic loading are discussed within the theory of small elastic-plastic deformations.
Some nonlinear dynamic problems of thermoelastic shells to be solved by the authors in the nearest future are outlined in Chapter 9.
In this book, despite of traditional approach, when the heat transfer equation is reduced to the two-dimensional form by integrating over the shell thickness, the three-dimensional heat transfer equation is used. The utility of using the two-dimensional equations of shell motion and the three-dimensional heat transfer equation taking the coupling between the train and temperature fields is not apparent to the reviewer’s opinion, since in the expansions of shell displacements in terms of the shell thickness the terms of the zero and first orders are held, thereafter the temperature terms of the orders larger than the first entering into the equations of motion should be discarded, since they have the order higher than the other terms of equations.
The book is well written and involves a large amount of good quality figures illustrating numerical examples. The reviewer could recommend this monograph for using in graduate student courses devoted to the shell theory and for purchase by the university libraries. It also could be of interest for graduate students, researchers and practical engineers dealing with numerical investigation of thermomechanical behavior of plates and shallow shells.
