# Continuum Mechanics

695

Author: Chandramouli P N

ISBN: 9789380381398

Copy Right Year: 2014

Pages:  854

Binding: Soft Cover

Publisher:  Yes Dee Publishing

SKU: 9789380381398 Category:

## Description

The fundamental concepts of continuum mechanics have been widely used at various stages in the study of solid mechanics, fluid mechanics, geomechanics and material engineering. The basic concepts of continuum mechanics are used in their simpler form for initial analysis before preliminary design of engineering structures and the advanced principles are used for complex analysis before final design. Many of the undergraduate and graduate courses employ the concepts of continuum mechanics as the basis for further development of engineering principles. Therefore, it is necessary as a first step to understand clearly the basic principles and corresponding mathematical expressions involved in continuum mechanics. Since the solutions to most of the engineering problems are obtained by solving the governing equations, it is imperative that one should learn the origin of such equations and the basis on which they have been derived. For this reason, attention has been given in this book to present the derivations of various fundamental equations in an extensive manner.

Weight 1.2 kg 23 × 18 × 4 cm

## Table of Content

PREFACE

ACKNOWLEDGEMENTS

CONTENTS AND COVERAGE

CHAPTER 1 INTRODUCTION TO CONTINUUM MECHANICS

1.1 Overview of continuum mechanics

1.2 Background of continuum mechanics

1.3 Overview of theory of elasticity

1.4 Differences between elementary theory and

Theory of elasticity

1.5 Procedure to be followed in theory of elasticity

1.6 Conditions applied in theory of elasticity

1.7 Assumptions made in elementary theory and

Theory of elasticity

1.8 Classification of materials

1.9 Applications of elasticity

1.10 Overview of fluid dynamics

Review questions

CHAPTER 2 THE STRESS FIELD

2.1 Introduction

2.2 Types of forces

2.3 Concept of three-dimensional stress

2.4 General state of stress on an element

2.4.1 Sign conventions

2.4.2 Index notation

2.4.3 Two-dimensional state of stress

2.5 Differential equation of equilibrium in a general

Three-dimensional stress system

2.6 Stress on a general plane

2.6.1 Direction cosines

2.6.2 Axis transformation

2.6.3 Stress on oblique plane through a point

(Cauchy’s Formula)

2.6.4 Stress transformation

2.7 Principal stresses and planes

2.8 Boundary conditions

Summary

Review questions

CHAPTER 3 THE DISPLACEMENT FIELD AND STRAIN FIELD

3.1 Introduction

3.2 Elementary concept of strain

3.3 Strain displacement relation

3.4 Strain at a point

3.5 Strain components at a given point in any direction

3.6 Principle strains and their directions

3.7 Strain rosettes

3.7.1 Rectangular strain rosette

3.7.2 Delta rosette

3.8 Mohr’s circle of strain

Summary

Review questions

CHAPTER 4 CONSTITUTIVE RELATIONS

4.1 Introduction

4.2 Response model

4.3 1-D Hooke’s law

4.4 Generalized Hooke’s law (Anisotropic Form)

4.5 Non-isotropic linear elastic behavior

4.6 Stress-strain relation for isotropic material

4.7 Stress-strain relation for orthotropic material

4.8 Stress-strain relation for transverse isotropic material

Review questions

CHAPTER 5 TWO-DIMENSIONAL PROBLEMS OF ELASTICITY

5.1 Introduction

5.1.1 Two-dimensional state of stress

5.1.2 Two-dimensional state of strain

5.2 Plane stress problems

5.3 Plane strain problems

5.4 Equation of compatibility

5.5 Mathematical conditions of compatibility

Summary

Review questions

CHAPTER 6 TWO-DIMENSIONAL PROBLEMS IN CARTESIAN

COORDINATE SYSTEM

6.1 Introduction

6.2 Airy’s stress functions

6.3 Saint-Venaunts’ principle

6.4 Two-dimensional problems in Cartesian coordinate

6.4.1 Airy’s stress function method

6.4.2 Solution by polynomials

6.5 Solution for bending of a cantilever loaded at the

free end using stress function as a polynomial

6.6 Bending of a beam by uniform load using

the stress function as a polynomial

Summary

Review questions

CHAPTER 7 TWO-DIMENSIONAL PROBLEMS IN POLAR

COORDINATES SYSTEM

7.1 Introduction

7.2 Two-dimensional differential equation of equilibrium

in polar coordinates

7.3 Derivations of Airy’s stress function in

polar coordinates

7.4 Stress-strain relationship in polar coordinates

7.5 Strain displacement relations

7.6 Compatibility equation

7.7 Stresses due to concentrated loads

7.8 Bending of a curved bar by a force at the end

7.9 Semi-infinite medium loaded with a concentrated

force at the boundary

Summary

Review questions

CHAPTER 8  AXI-SYMMETRIC STRESS DISTRIBUTION

8.1 Introduction

8.2 Plane stress and plane strain

8.3 Compatibility equation for axi-symmetric case

8.4 Rotating circular disc 237

8.5 Thick cylinder subjected to internal and external

8.6 Pure bending of curved bars

Summary

Review questions

CHAPTER 9 TORSION ON PRISMATIC BARS

9.1 Introduction

9.2 Saint-Venants’ theory

9.3 Torsion of elliptical cross-section

9.4 Torsion of equilateral triangle cross-section bar

Summary

Review questions

CHAPTER 10 THEOREMS OF ELASTICITY

10.1 Introduction

10.2 Uniqueness theorem

10.3 Principle of superposition

10.4 Method of virtual work and minimum potential

energy principle of elasticity

10.5 Complimentary strain energy

10.6 The Crotti-Engesser theorem

10.7 Castigliano’s theorem

10.8 Maxwell reciprocal theorem

10.9 Clapeyron’s theorem in linear elastic theory

Review questions

CHAPTER 11 STRESS CONCENTRATION

11.1 Introduction

11.2 Stresses concentration around circular hole

Review questions

CHAPTER 12 STRESSES DUE TO ROTATION

12.1 Introduction

12.2 Rotational stresses in thin cylinder or rotating ring

12.3 Expression for stresses in a rotating thin disc

12.3.1 Expression for radial and circumferential

stresses in a solid disc

12.3.2 Expression for radial and circumferential

stresses for disc with central hole

12.4 Disc of uniform strength

12.5 Long cylinders

Summary

Review questions

CHAPTER 13 CURVED BEAMS

13.1 Introduction

13.2 Assumptions made in the derivation

of stresses in a curved bar

13.3 Expression for stresses in a curved bar

13.3.1 To fi nd the position of neutral axis

13.4 Determination of factor ‘k’ for various sections

13.4.1 Rectangular section

13.4.2 Triangular section

13.4.3 Trapezoidal section

13.4.4 Circular section

13.4.5 T-section

13.4.6 I-section

13.5 Resultant stress in a curved bar subjected to

direct stresses and bending stresses

13.5.1 Resultant stress in a hook

13.5.2 Stresses in circular ring

13.5.3 Stresses in a chain link

Summary

Review questions

CHAPTER 14 SHEAR CENTRE

14.1 Introduction

14.2 Shear flow

14.3 Principle involved in finding the shear centre

14.3.1 Shear centre for a channel section

14.3.2 Shear centre for unequal I-section

14.3.3 Shear centre for an angle section

14.3.4 T-Section

14.3.5 I-Section

14.3.6 Half thin walled cylindrical section

Summary

Review questions

CHAPTER 15 UNSYMMETRICAL BENDING

15.1 Introduction

15.2 Product of inertia for an area

15.3 Parallel – axis theorem

15.4 Moment of inertia of an area about inclined axes

15.5 Principal moments of inertia

15.6 Shear centre (unsymmetrical sections)

15.7 Unsymmetrical bending

15.8 Determination of bending stress through product

of inertia

Summary

Review questions

CHAPTER 16 FLUID STATICS

16.1 Fluid flow concepts

16.2 Continuum concept

16.3 Fundamental concepts

16.4 Stress relationships at a point in a fluid

16.5 Pressure at a point

16.6 Pressure variation in an incompressible static fluid

16.7 Pressure variation in a compressible fluid

Summary

Review questions

CHAPTER 17  KINEMATICS

17.1 Introduction

17.2 Relation between the local and individual

time rates

17.3 Acceleration

17.4 Scalar, vector and tensor quantities – fields

17.5 Types of fluid flow

17.5.1 Three-, two- and one-dimensional flow

17.5.3 Uniform and non-uniform flow

17.5.4 Laminar and turbulent flows

17.5.5 Compressible and incompressible flow

17.5.6 Rotational and irrotational flow

17.5.7 Ideal and real fluid flow

17.6 Description of fluid motion

17.6.1 Components of acceleration in other

coordinate systems

17.7 Fundamentals of flow visualization

Summary

Review questions

CHAPTER 18 FLOW EQUATIONS-CONTINUITY EQUATIONS

18.1 System and control volume

18.1.1 Intensive and extensive properties

18.2 Control volume transformation equation

18.3 Continuity equation for a control volume

18.4 Continuity equation for an infinitesimal control

volume

18.5 Mass conservation (or continuity) equation

along a stream tube

18.6 Three dimensional continuity equation in

Cartesian coordinates

18.7 Equation of continuity in the Lagrangian method

18.8 Equivalence of the two forms of the equation

of continuity

18.9 Equation of continuity in polar coordinates

18.10 Continuity equation in cylindrical polar coordinates

18.11 Continuity equation in spherical coordinates

18.12 Conservation of mass in orthogonal

curvilinear coordinates

Summary

Review questions

CHAPTER 19 FLOW EQUATIONS

19.1 Euler’s equation of motion

19.2 Energy equation

19.3 Boundary surface

19.4 Momentum equation

19.5 Control volume momentum equation

19.6 Law of conservation of angular momentum or

law of conservation of momentum of momentum

19.7 Equation of motion under impulsive forces

19.8 Kinetic energy and momentum correction factors

(Coriolis coefficients)

Summary

Review questions

CHAPTER 20 CIRCULATION AND ROTATION

20.1 Circulation

20.2 Energy dissipation, shear deformation and

rotationality

Summary

Review questions

CHAPTER 21 SCALAR FUNCTIONS

21.1 Potential function

21.2 Stream function

21.3 Cauchy Riemann equation

21.4 Relationship between stream function ψ and

the velocity components Vr and V in cylindrical

polar coordinates

21.5 Orthogonality of stream lines and potential lines

Summary

Review questions

CHAPTER 22 POTENTIAL FLOW

22.1 Introduction

22.2 Uniform flow (u or u0)

22.2.1 Uniform flow parallel to x –axis

22.2.2 Uniform flow parallel to y-axis

22.3 Source flow (q or m)

22.4 Sink flow (-q or -m)

22.5 Free vortex flow

22.6 Super-imposed flow

22.6.1 Source–sink pair

22.6.2 Source kept near a wall (Method of image)

22.6.3 A plane source in a uniform flow/ source

placed in a rectilinear flow/flow past

a half body

22.6.4 Doublet

22.6.5 A source and a sink pair in a uniform

flow/flow past a Rankine oval shape

22.6.6 A doublet in a uniform flow

(Flow past a circular cylinder)

22.6.7 Flow past a circular cylinder

with circulation

22.7 Drag and lift

22.8 To find drag and lift in the case of a circular cylinder

without circulation

22.9 To find drag and lift in the case of circular cylinder

with circulation

Summary

Review questions

CHAPTER 23 REAL FLUID FLOW

23.1 Introduction

23.1.1 Classification of viscous flow

23.1.2 Relation between shear and pressure

23.2 Navier-Stokes’ equation

23.3 Exact solutions of Navier-Stokes’ equations

23.3.1 Flow through a circular pipe

(Hagen-Poiseuille theory)

23.3.2 Flow of viscous fluid between two parallel

stationary plates

23.3.3 Flow of viscous fluid between two parallel

plates, if one plate is moving with a

constant velocity

Summary

Review questions

CHAPTER 24 APPLICATION OF COMPLEX VARIABLES TO

TWO-DIMENSIONAL FLUID FLOW

24.1 Introduction

24.2 Complex number

24.3 De Moivre’s theorem

24.4 Conjugate complex number

24.5 The logarithm of a complex number

24.6 Functions of complex variable

24.7 Relation of functions of a complex variable to

irrotational flow

24.8 Conformal transformation

24.9 Simple conformal transformation

24.10 Complex potential for a source

24.11 Complex potential for vortex

24.12 Complex potential for a doublet

24.13 Image in two-dimensions

24.13.1 Image of a source with regard to a line

24.14 Source and sink of equal strength

24.15 Steady flow around a circular cylinder

without circulation

24.16 Circulation about a circular cylinder

24.17 Flow past a circular cylinder with circulation

24.18 Milne Thomson method to determine complex

function

24.19 Blasius theorem

24.20 Theorem of Kutta and Joukowski

Summary

Review questions

OBJECTIVE QUESTIONS

ABBREVIATIONS

BIBLIOGRAPHY

INDEX