Elements of Continuum Mechanics

  • ID: 2170334
  • Book
  • 346 Pages
  • John Wiley and Sons Ltd
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As the ever–growing capabilities of numerical tools are being used to simulate how fluids and solids respond to high tempuratures and severe loads, a fundamental understanding of the theory of large coupled thermo=mechanical deformations will play an ever–increasing role. This requires a sound understanding of the basic principles of nonlinear continuum mechanics such as those given in this textbook. This book differs from several others in the market in that it provides numerous example problems from every day life which should help develop a better understanding of nonlinear theory. The book is oriented more toward engineers then mathematicians.

After briefly reviewing the mathematical preliminaries in Chapter 2, the theory of large deformations of a continuum is developed in subsequent chapters. Special emphasis is placed on constitutive relations. Theories developed in earlier chapters are illustrated through solutions of several nonlinear and linear problems in latter chapters.

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Chapter 1 Introduction.

1.1 What is Mechanics?.

1.2 Continuum Mechanics.

1.3 An Example of an Ad–Hoc Approach.

Chapter 2 Mathematical Preliminaries.

2.1 Summation Convention, Dummy Indices.

2.2 Free Indices.

2.3 Kronecker Delta.

2.4 Index Notation.

2.5 Permutation Symbol.

2.6 Manipulations with the Indicial Notations.

2.7 Translation and Rotation of Coordinate Axes.

2.8 Tensors.

2.9 The Divergence Theorem.

2.10 Differentiation of Tensor Fields.



Chapter 3 Kinematics.

3.1 Description of motion and continuum.

3.2 Referential and Special Descriptions.

3.3 Displacement Vector.

3.4 Restrictions on Continuous Deformation of a Deformable Body.

3.5 Material Derivative.

3.6 Finding Acceleration of a Particle from a Given Velocity Field.

3.7 Deformation Gradient.

3.8 Strain Tensors.

3.9 Principle Strains.

3.10 Deformation of Areas and Volumes.

3.11 Mass Density, Equation and Continuity.

3.12 Rate of Deformation, Strain–Rate Tensor, Spin.

3.13Polar Decomposition.

3.14 Infinitesimal Deformations.

3.15 Infinitesimal Deformations Superimposed Upon Finite.

3.16 Volumetric and Deviatoric Strains.

3.17 Transformation of Tensors Under a Change of Bases.

3.18 Plain Strain Deformation.

Appendix A: Solution of a Cubic Equation.



Chapter 4 The Balance Laws, Stress Tensors.

4.1 Kinetics of a Continuous Media.

4.2 Traction Boundary Conditions.

4.3 The Nominal Stress Tensor.

4.4 Transformation of Stress Tensors Under the Rotation of Axes.

4.5 Principle Stresses; Maximum Shear Stress.

4.6 Relations Among Stress Tensors for Infinitesimal Deformations.

4.7 Plane Stress.

4.8 Deviatoric Stress, von–Mises Stress.

4.9 Balance of Energy.

4.10 Entropy Inequality, The Clausius–Duhem Inequality.

4.11 Summary of Equations Governing Deformations of a Body.

4.12 Nonuniquences of Solutions for Static Problems.

Appendix B: The Transport Theorem.


Chapter 5 Constitutive Relations.

5.1 Introductory Remarks.

5.2 Thermoelastic Material.

5.3 Principle of Material Objectivity.

5.4 Linear Constitutive Relations for Finite Deformations of a Thermoelastic Body.

5.5 Isotropic Thermoelastic Materials.

5.6 Comparison of Results from Four Linear Constitutive Relations in Isotropic Finite Elasticity.

5.7 Transversely Isotropic Thermoelastic Materials.

5.8 Orthotropic Thermoelastic Materials.

5.9 Coincidence of Principle Axes of Stress and Strain in Isotropic Elastic Materials.

5.10 Coincidence of Principle Axes of Stress and Strain in  Transversely Isotropic Elastic Materials.

5.11 Incompressible Elastic Materials.

5.12 Comparison of Results fr5om Constitutive Relations.

5.13 Constitutive Relations fro Infinitesimal Deformations of Elastic Materials.

5.14 Constitutive Relations for Special Isotropic Nonlinear Elastic Materials.

5.15 Infinitesimal Deformations Superimposed Upon Finite Deformations of an Isotropic Elastic Body.

5.16 Constitutive Relations for Plane Deformations of a Thermoelastic Body.

5.17 Thermoviscoelastic Materials.

5.18 Summary.



Chapter 6 Torsion of a Circular Cylinder.

6.1 Torsion of  a Linear Elastic Circular Cylinder.

6.2 Torsion of a Second Order Elastic Circular Cylinder.

6.3 Infinitesimal Twist of a Finitely Stretched Circular Cylinder.

6.4 Finite Torsion of a Circular Cylinder.

Appendix C: A Uniqueness Theorem.



Chapter 7 Fluid Flow.

7.1 Steady Flow Between Two Parallel Plates.

7.2 Steady Isothermal Flow of an Incompressible Fluid Down an Inclined Plane.

7.3 Steady Flow of an Incompressible Fluid in a Horizontal Circular Pipe.


Chapter 8 Bending of Beams.

8.1 Bending of a Rectangular Beam.

8.2 Bending of a Nonlinear Elastic Rectangular Beam.

8.3 Air Stress Function for Bending of a Beam.


Chapter 9 Wave Propagation.

9.1 Singular Surface.

9.2 Kinematics of a Singular Surface.

9.3 Acceleration Waves in Linear Elasticity.

9.4 Progressive Waves.

9.5 Incompressible Linear Elastic Materials.

9.6 Acceleration Waves in Nonlinear Elastic Bodies.

9.7 Infinitesimal Deformations Superimposed Upon Finite Deformations.


Chapter 10 Spherical and Cylindrical Pressure Vessels.

10.1 Radial Expansion of a Spherical Pressure Vessel.

10.2 Radial Expansion of an Incompressible Hookean Sphere.

10.3 Radial Expansion of a Cylindrical Pressure Vessel.

10.4 Radial Expansion of an Inhomogeneous and Incompressible Hookean Cylinder.

10.5 Finite Radial Expansion of a NeoHookean Cylinder.


Supporting Materials.

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Romesh Batra
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