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E-raamat: Continuum Mechanics and Linear Elasticity: An Applied Mathematics Introduction

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Continuum Mechanics and Linear Elasticity: An Applied Mathematics IntroductionSuurem pilt
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This is an intermediate book for beginning postgraduate students and junior researchers, and offers up-to-date content on both continuum mechanics and elasticity. The material is self-contained and should provide readers sufficient working knowledge in both areas. Though the focus is primarily on vector and tensor calculus (the so-called coordinate-free approach), the more traditional index notation is used whenever it is deemed more sensible.

With the increasing demand for continuum modeling in such diverse areas as mathematical biology and geology, it is imperative to have various approaches to continuum mechanics and elasticity. This book presents these subjects from an applied mathematics perspective. In particular, it extensively uses linear algebra and vector calculus to develop the fundamentals of both subjects in a way that requires minimal use of coordinates (so that beginning graduate students and junior researchers come to appreciate the power of the tensor notation). 

Arvustused

My opinion about the book is really positive. I advise everybody interested in teaching linear elasticity to have a look at it. (Giuseppe Saccomandi, Mathematical Reviews, June, 2020)

Part I Elements of Continuum Mechanics
1 Vector, Tensors, and Related Matters
3(118)
1.1 Matrices
3(5)
1.2 Vector Spaces
8(3)
1.3 Euclidean Vector Spaces
11(7)
1.4 Euclidean Point Spaces
18(2)
1.5 Second-Order Tensors: Fundamentals
20(5)
1.6 Examples of Tensors: Elementary Projections
25(1)
1.7 Basic Properties of Tensors
26(6)
1.8 Linear Mappings as Geometric Transformations
32(3)
1.9 Transformation Rules for a Change of Basis
35(3)
1.10 Higher Order Tensors
38(2)
1.11 A Special Property: Isotropy
40(1)
1.12 Pseudo-scalars/vectors/tensors
41(2)
1.13 Other Uses of Vector and Tensor Products
43(7)
1.14 Fourth-Order Tensors
50(6)
1.15 Finite Rotations
56(3)
1.16 On the Relationship Between Skw and
59(2)
1.17 More General Bases
61(6)
1.17.1 Cylindrical Polar Coordinates
63(2)
1.17.2 Spherical Polar Coordinates
65(1)
1.17.3 Physical Components for Vectors and Tensors
65(2)
1.18 Invariants
67(4)
1.19 Eigenvalues
71(5)
1.20 Some Important Theorems
76(5)
1.21 Projections in Lin
81(1)
1.22 Vector and Tensor Calculus
82(23)
1.22.1 Differential Calculus in Finite-Dimensional Euclidean Vector Spaces
85(6)
1.22.2 Differential Operators: DIV, GRAD, CURL
91(5)
1.22.3 Vector and Tensor Identities
96(3)
1.22.4 Non-Cartesian Coordinates
99(3)
1.22.5 Some Important Integral Theorems
102(3)
1.23 Differentiation of Tensor Functions
105(7)
1.24 Exercises
112(9)
Bibliography
120(1)
2 Kinematics
121(40)
2.1 Defoimable Bodies: Definition and Generalities
121(2)
2.2 Examples of Deformations/Motions
123(4)
2.3 Velocity and Acceleration Fields. Material Time Derivatives
127(5)
2.4 The Deformation Gradient
132(4)
2.5 Changes in Area and Volume
136(3)
2.6 Strain Tensors
139(4)
2.7 Examples of Particular Strain Tensors
143(4)
2.8 The Interpretation of the Polar Decomposition Theorem
147(1)
2.9 The Spatial Gradient of Velocity
147(4)
2.10 Transport Formulae
151(3)
2.11 Exercises
154(7)
Bibliography
159(2)
3 Balance Laws
161(36)
3.1 The Principle of Mass Conservation
162(4)
3.2 Body and Surface Forces
166(4)
3.3 Global Form
170(1)
3.4 Local Form
171(2)
3.5 Some Technical Proofs
173(5)
3.6 Further Properties of the Cauchy Stress Tensor
178(6)
3.7 Particular States of Stress
184(2)
3.8 The Piola--Kirchhoff Stress Tensors
186(3)
3.9 Exercises
189(8)
Bibliography
196(1)
4 Constitutive Relationships
197(46)
4.1 The Principle of Material Frame-Indifference
198(8)
4.2 Other Important Constitutive Principles
206(3)
4.3 Cauchy-Elastic Materials
209(1)
4.4 Material Symmetry
210(7)
4.5 Hyperelastic Solids
217(6)
4.6 Constitutive Representations for Isotropic Hyperelastic Solids
223(4)
4.7 Internal Constraints
227(4)
4.8 Particular Forms of the Strain-Energy Function
231(1)
4.9 Simple Fluids
232(5)
4.10 Exercises
237(6)
Bibliography
240(3)
Part II Topics in Linear Elasticity
5 Linear Elasticity: General Considerations and Boundary-Value Problems
243(38)
5.1 Introduction
243(1)
5.2 Linearised Kinematics
244(1)
5.3 Distortional and Spherical Strain
245(1)
5.4 Linearised Constitutive Behaviour
246(5)
5.5 Linearised Field Equations
251(4)
5.6 Restrictions on the Elastic Constants
255(3)
5.7 The Navier--Lame Equations
258(2)
5.8 Principle of Superposition
260(2)
5.9 Saint-Venant's Principle
262(1)
5.10 Worked Examples
263(8)
5.11 Standard Simplifications
271(4)
5.11.1 Plane Strain
272(1)
5.11.2 Plane Stress
273(1)
5.11.3 Antiplane Strain/Stress
274(1)
5.12 Exercises
275(6)
Bibliography
280(1)
6 Compatibility of the Infinitesimal Deformation Tensor
281(38)
6.1 Introduction
281(2)
6.2 Simply and Multiply Connected Domains
283(2)
6.3 The incompatibility' Operator
285(3)
6.4 Conservative Fields
288(1)
6.5 Cesaro--Vol terra Formula
289(7)
6.6 Alternative Forms of the Compatibility Equation
296(3)
6.7 Beltrami--Michell Equations
299(2)
6.8 Explicit Calculations and Examples
301(6)
6.9 Weingarten-Volterra Dislocations
307(4)
6.10 Exercises
311(8)
Bibliography
318(1)
7 Torsion
319(52)
7.1 Introduction
319(2)
7.2 Some Auxiliary Notation
321(2)
7.3 Governing Equations
323(1)
7.4 Circular Cylinder
324(3)
7.5 Non-circular Cylinder
327(7)
7.6 A Closer Look at the Torsional Rigidity
334(5)
7.7 Prandtl Stress Function
339(3)
7.8 Modified Stress Function
342(3)
7.9 Multiply Connected Domains
345(2)
7.10 The Shear Stress
347(3)
7.11 Complex Variables Formulation
350(4)
7.12 Worked Examples
354(11)
7.13 Exercises
365(6)
Bibliography
369(2)
8 Two-Dimensional Approximations
371(48)
8.1 Introduction
371(2)
8.2 Plane Strain
373(2)
8.3 Plane Stress
375(1)
8.4 Generalised Plane Stress
376(3)
8.5 The Airy Stress Function
379(6)
8.5.1 Main Definitions
379(1)
8.5.2 The Governing Equation for Φ
379(2)
8.5.3 Physical Interpretation of the Boundary Conditions
381(3)
8.5.4 The Displacement Field
384(1)
8.6 Worked Examples
385(19)
8.7 Volterra Distortions
404(4)
8.8 Exercises
408(11)
Bibliography
418(1)
9 Special Two-Dimensional Problems: Unbounded Domains
419(40)
9.1 The Bi-harmonic Equation via Fourier Transforms
419(10)
9.2 Remarks on the Displacement Field
429(1)
9.3 The Direct Approach
429(5)
9.4 A Modification of the Method
434(3)
9.5 The Elastic Quarter-Plane
437(5)
9.6 Displacement Boundary-Value Problems
442(10)
9.6.1 The Papkovitch--Neuber Representation
443(3)
9.6.2 The Stress Tensor in Terms of Ψ and B
446(1)
9.6.3 Particular Case: Plane Elasticity
447(5)
9.7 Exercises
452(7)
Bibliography
458(1)
Appendix A Vector and Tensor Identities 459(2)
Appendix B Cylindrical and Spherical Coordinates 461(6)
Appendix C Geometry of Areas 467(8)
Appendix D Fourier Transforms 475(16)
Appendix E The Bi-harmonic Equation 491(22)
References 513(2)
Index 515
Ciprian D. Coman is currently a Senior Lecturer at the School of Computing and Engineering at the University of Huddersfield in the UK. A seasoned applied mathematician with over two decades of professional experience, Dr. Coman has held academic positions at various universities in the UK (Leicester, Glasgow, Nottingham), as well as in several industrial organisations (Institute of Solid Mechanics of the Romanian Academy, Schlumberger Ltd, and National Physical Laboratory/UK). His main research interests are in the areas of theoretical and applied mechanics, with a special focus on mathematical modelling involving differential equations and singular perturbation techniques, elastic stability problems, and the micromechanics of particulate solids. He has published more than 50 peer-reviewed journal articles and has co-authored over 25 technical reports.
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