For the first time, the Micropolar Theory of Elasticity is applied to solving a wide variety of problems connected to the specifics of nanomaterials. Namely, their unique physical-mechanical characteristics and behaviors under various stress-induced conditions.
These theories have been constructed based on the equations of the classical theory of elasticity as well as other equations that have till now remained untouched in their application to molecular theories of solid deformable media. The book also introduces a new applied micropolar theory of thin shells which is based on Cosserat's pseudo-continuum. It explores the theory’s application to a category of nanomaterial shells and plates previously neglected from classical theories due to their unconventional size and structure. Theoretical results are accompanied by solutions of certain problems, essential for various applications.
The book consists of six chapters. The first chapter is a review of the essential data on the non-symmetric theory of elasticity. The second and third chapters are devoted to various theories of plate bending and solutions to some basic problems. Chapter four refers to membrane or, so-called, momentary shell theory. Chapter five deals with the theory of very shallow shells. Finally, chapter six presents the geometry of the nonlinear theory of plates and the theory of very shallow shells.
The book is intended for researchers, postgraduate students, and engineers, interested in the design of structures from nanomaterials and in the problems of mechanics of deformable bodies, theories of shells and plates, and their applications in micromechanics.
Author(s): Sergey A. Ambartsumian
Series: Foundations of Engineering Mechanics
Publisher: Springer
Year: 2021
Language: English
Pages: 188
City: Cham
Preface
Preface to the Second Edition
References
Introduction
References
Contents
1 Main Equations and Relations of the Theory of Non-symmetric Elasticity
1.1 Preliminary Remarks
1.2 Equations of Motion
1.3 Defining Equations or the Law of Small Elastic Deformations
1.4 Compatibility Equations
1.5 Orthogonal Curvilinear System of Coordinates
1.6 Main Equations and Correlations in the Case of the Geometrically Nonlinear Problem
References
2 Bending of Plates
2.1 Remarks
2.2 Initial Assumptions and Hypothesis
2.3 Stresses, Internal Forces, and Momenta
2.4 Averaged Equations of Motion and Boundary Conditions
2.5 Another Example of a Bending Equation of a Pate at X pm = 0,Y pm = 0
2.6 Theory of Bending of a Plate at α= 0
2.7 Axisymmetric Bending of a Plate
2.8 Few Remarks on the New Generalizing Elasticity Constant
References
3 Some Trivial Problems of Plates
3.1 Remarks
3.2 Bending of a Plate Over a Cylindrical Surface
3.3 A Sample Problem of Bending of a Hinged Rectangular Plate Under Normally Applied Sinusoidal Load
3.4 Bending of a Hinged Rectangular Plate Under Arbitrary, Normally Applied Load
3.5 Bending of a Semi-infinite Plate by a Load Distributed Along the Edge
3.6 On the Problem of Axisymmetric Bending of a Circular Plate
3.7 Yet Another Problem of the Statical Stability of a Plate
3.8 Free Vibrations of a Plate: One-Dimensional Problem
3.9 Free Vibrations of a Rectangular Plate Hinged Along the Entire Contour
References
4 Membrane Theory of Plates
4.1 Remarks
4.2 Initial Statements and Hypothesis
4.3 Stresses, Internal Forces, and Momenta
4.4 Averaged Equations of Motion and Boundary Conditions
4.5 Cylindrical Shell
4.6 Some Problems of a Circular Cylindrical Shell
4.7 Axisymmetrically Loaded Shells of Revolution
4.8 Membrane Theory of Very Shallow Shells
References
5 Theory of Very Shallow Shells
5.1 Remarks
5.2 Initial Statement and Hypothesis
5.3 Stresses, Internal Forces and Momenta
5.4 Averaged Equations of Motions and Boundary Conditions
5.5 Circular Cylindrical Shell
5.6 Axisymmetrically Loaded Shells of Revolution
5.7 Axisymmetric Vibrations of a Circular Cylindrical Shell
References
6 Other Aspects of the Theory of Plates and Shells
6.1 Remarks
6.2 Geometrically Nonlinear Theory of Rectangular Plates
6.3 Rectangular Plate Hinged Along the Entire Contour. Nonlinear Problem
6.4 Theory of Very Shallow Shells. Full Account of the Transverse Shifts
6.5 Bending of a Plate Along a Cylindrical Surface
References
