The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

This paper gives a historical review of the theories that have been developed for the analysis of multilayered structures. Attention has been restricted to the so-called Zig-Zag theories, which describe a piecewise continuous displacement field in the plate thickness direction and fulfill interlaminar continuity of transverse stresses at each layer interface. Basically, plate and shell geometries are addressed, even though beams are also considered in some cases. Models in which the number of displacement variables is kept independent of the number of constitutive layers are discussed to the greatest extent. Attention has been restricted to those plate and shell theories which are based on the so-called method of hypotheses or axiomatic approach in which assumptions are introduced for displacements and/or transverse stresses. Mostly, the work published in the English language is reviewed. However, an account of a few articles originally written in Russian is also given. The historical review conducted has led to the following main conclusions. 1! Lekhnitskii ~1935! was the first to propose a Zig-Zag theory, which was obtained by solving an elasticity problem involving a layered beam. 2! Two other different and independent Zig-Zag theories have been singled out. One was developed by Ambartsumian ~1958!, who extended the well-known Reissner-Mindlin theory to layered, anisotropic plates and shells; the other approach was introduced by Reissner ~1984!, who proposed a variational theorem that permits both displacements and transverse stress assumptions. 3 ! On the basis of historical considerations, which are detailed in the paper, it is proposed to refer to these three theories by using the following three names: Lekhnitskii Multilayered Theory, ~LMT!, Ambartsumian Multilayered Theory ~AMT!, and Reissner Multilayered Theory ~RMT!. As far as subsequent contributions to these three theories are concerned, it can be remarked that: 4! LMT although very promising, has almost been ignored in the open literature. 5! Dozens of papers have instead been presented which consist of direct applications or particular cases of the original AMT. The contents of the original works have very often been ignored, not recognized, or not mentioned in the large number of articles that were published in journals written in the English language. Such historical unfairness is detailed in Section 3.2. 6! RMT seems to be the most natural and powerful method to analyze multilayered structures. Compared to other theories, the RMT approach has allowed from the beginning development of models which retain the fundamental effect related to transverse normal stresses and strains. This review article cites 138 references. @DOI: 10.1115/1.1557614#

Historical review of Zig-Zag theories for multilayered plates and shells

Volume II J. S. R. Chisholm and Rosa M. Morris Amsterdam: North-Holland. 1964 Pp. xviii + 717. Price 72s. This book is a valuable addition to an ever-growing collection of mathematical texts written expressly for the physicist. The authors suggest that the book might also be useful to undergraduates in chemistry and engineering and research workers in other fields, but the content is clearly chosen to meet the needs of modern undergraduate physics.

http://iopscience.iop.org/article/10.1088/0031-9112/16/8/015/pdf

Mathematical Methods in Physics

Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.

Introduction to Matrix Analysis. By Richard Bellman. 2nd Edition. Pp. xxiii, 403. £7·50. (McGraw-Hill.)

Elastic analysis of rotating functionally graded polar orthotropic disks

Elastoplastic behavior of a reinforced layer

Plasticity (strength) condition for a reinforced layer

The problem of rational profiling of thermoelastic three-layered reinforced rotating disks of gas turbines is formulated. The conditions of equal-stressed reinforcement of the midlayer and the condition of equal-strength binder of the reinforced layer or external isotropic layers are used as criteria for rational designing. An iterative method is proposed for solving such problems. The problem of rational profiling of disks with radial and radially circumferential reinforcements, on condition that the fibers of the radial family are equally stressed, is solved analytically. The calculation results for B-Al and B-Ti disks, obtained by using both the criteria of rational design, are given for plane-parallel and 3D reinforcements. It is shown that, at an equal load-carrying capacity, the weight of such disks can be 30-60% smaller than that of reinforced disks of constant thickness, or, at an equal weight, they can have a considerably higher load-carrying capacity. The criterion of equal-stressed reinforcement, as a rule, is more efficient than the condition of equal-strength binder.

Rational Profiling of Reinforced Rotating Disks

Dynamic behavior of rigid-plastic rectangular plates

We propose a general approach to the solution of direct dynamic problems of the numerical analysis of composite rods subjected to thermal and force loads for a large variety of their materials and loading conditions. We take into account the shear effects and interactions between the elements of structures and supporting media. The proposed inhomogeneous rods are regarded as elements of rod systems and characterized by higher parameters of strength and stiffness and lower costs of their production as compared with homogeneous elements. In the equations of motion and physical relations, we introduce four functional characteristics of stiffness and viscosity and three mass functional characteristics equivalently reflecting the dynamic deformation of composite rods with the help of a one-dimensional model. With the help of trigonometric Fourier series, the dynamic loads, displacements, and temperature are represented as products of functions of the coordinate and time. The solution of the homogeneous problem is obtained via the matrizant of the system of equations of the first order. Partial solutions for loads of various types and generalized temperature loads are obtained on the basis of approximation of given quantities and the required displacements by trigonometric series.

Yu. V. Nemirovskii

Papers

Elastoplastic behavior of a reinforced layer

Plasticity (strength) condition for a reinforced layer

Rational Profiling of Reinforced Rotating Disks

Dynamic behavior of rigid-plastic rectangular plates

Dynamic Analysis of Composite Rods Under Thermal and Force Loads