ELASTICITY AND FRACTURE MECHANICS. Vijay G. Ukadgaonker

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1 THEORY OF ELASTICITY AND FRACTURE MECHANICS y x Vijay G. Ukadgaonker

2 Theory of Elasticity and Fracture Mechanics VIJAY G. UKADGAONKER Former Professor Indian Institute of Technology Bombay Delhi

3 THEORY OF ELASTICITY AND FRACTURE MECHANICS Vijay G. Ukadgaonker 2015 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN The export rights of this book are vested solely with the publisher. Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi and Printed by Syndicate Binders, A-20, Hosiery Complex, Noida, Phase-II Extension, Noida (N.C.R. Delhi).

4 To My Teachers, Students and my Family

5

6 Contents Preface xiii 1. Analysis of Stress and Strain Introduction Importance of Theory of Elasticity Solutions Analysis of Stresses Definition of Stress Notations Force Equilibrium Equations in Two-dimensional Cartesian Co-ordinates Equilibrium Equations in 3D Cartesian Co-ordinates Equilibrium Equations in Polar Co-ordinates Equilibrium Equations in Cylindrical Co-ordinates Equilibrium Equations in Spherical Co-ordinates Stress at a Point: 2D Boundary Condition Principal Stresses and Mohr s Circles: 2D Cases Stress Ellipsoid and Stress Director Surface Analysis of Strain Definition of Strain Strain Displacement Relations in 2D Cartesian Co-ordinates Strain Displacement Relations in 3D Cartesian Co-ordinates Strain Displacement Relations in Polar Co-ordinates Strain Displacement Relations in Cylindrical Co-ordinates Strain Displacement Relations in Spherical Co-ordinates Strain at a Point (2D Case) The Rigid Body Rotation Cubical Dilation Plane Stress and Plane Strain Compatibility Conditions Stress Strain Relations Properties Definitions Tension Test Generalized Hooke s Law Proof of Principal Stress Directions and Principal Strain Directions are Same Correlation between the Elastic Constants 44 v

7 vi Contents 1.6 Formulation of Elastic Problems Different Types of Fundamental Boundary Value Problems Boundary Value Problem in Elasticity Elastic Equations Using Index Notations General Theories in Elasticity Principle of Superposition Theorem of Minimum Strain Energy Principle of Virtual Work Castigliano s Theorem Saint Venant s Principle 55 Exercises Finite Element Method Introduction Six Steps of FEM in the Solution of General Continuum Problems Six Steps in FEM for Solid Mechanics Figures for FEM Element Stiffness Matrix (K) for Triangular Continuum Element Assembly of Element Stiffness Matrices or Global or Overall Stiffness Matrix Application Basic Seven Steps in the Derivation of the Element Stiffness Characteristics Theory of Elasticity Solution to the Beam Problem Rectangular Element with Four Nodes Rectangular Finite Element for Plate Flexure Example Axisymmetric Shells of Revolution Example for Circular Cylindrical Thin Shell Thick Shells of Revolution 101 Exercises Two-dimensional Elasticity Problems Introduction Biharmonic Equation in 2D Problems in Rectangular Co-ordinates Biharmonic Equation in Polar Co-ordinates Solution of 2D Elasticity Problems in Rectangular Co-ordinates Stress Function in the Form of a Polynomial Bending of a Cantilever Loaded at its End Simply Supported Beam with Uniformly Distributed Load Solution of 2D Problem in the Form of a Fourier Series D Problems in Polar Co-ordinates Axisymmetric Stresses Pure Bending of Curved Bars 125

8 Contents vii The Effect of Circular Hole on Stress Distribution in Infinite Plate (Figure 3.14) Problems of Stress Concentration around Holes Bending of Curved Bar by a Force Applied at its Ends Concentrated Force at a Point on Semi-infinite Plate 136 Exercises Complex Variable Approach Introduction Complex Functions Representation of Biharmonic Function Representation of Stresses Representation of Displacements Representation of Resultant Forces and Moments Acting on the Boundary Transformation of Co-ordinates Translation of Rectangular Co-ordinates Rotation of Rectangular Co-ordinates Polar Co-ordinates Structure of Functions f(z) and c (z) Arbitraryness in Choosing the Stress Functions Finite Domain Infinite Regions Different Methods for Obtaining f(z) and c (z) Use of Equation (4.33) Use of Equation (4.35) Conformal Mapping Integro Differential Equations Problem of Linear Relationship (Hilbert Problem) Schwarz s Alternating Method for Multiply Connected Region Application of the Theory Circular Plate under Arbitrary Edge Thrust Infinite Plate with a Circular Hole Infinite Region Bounded by an Ellipse Infinite Region with Two Circular Holes A Novel Method of Stress Analysis of Infinite Plate with Circular Hole with Uniform Loading at Infinity A Novel Method of Stress Analysis of an Infinite Plate with Elliptical Hole with Uniform Tensile Stress A Novel Method of Stress Analysis of an Infinite Plate with Small Corner Radius Equilateral Triangular Hole A Novel Method of Stress Analysis of an Infinite Plate with a Rectangular Hole of Rounded Corners under Uniform Loading at Infinity 199 Exercises 210

9 viii Contents 5. Interaction Effect on Stresses of Two Unequal Holes in Infinite Plate Schwarz s Alternating Method Convergence of Schwarz s Alternating Method Infinite Plate with Two Unequal Circular Holes Subjected to Uniform Pressures Results Infinite Plate with Two Unequal Circular Holes Subjected to Uniform Shearing Stresses First Approximation Second Approximation Third Approximation Numerical Results Infinite Plate with Two Unequal Circular Holes Subjected to Uniform Tension at Infinity along the Line of Symmetry First Approximation Second Approximation Third Approximation Stress Field Numerical Results Two Unequal Circular Holes Subjected to Uniform Tension at Infinity Perpendicular to the Line of Symmetry First Approximation Second Approximation Third Approximation Stress Field Numerical Results Plate with two Unequal Circular Holes Subjected to Uniform in Plane Shearing Stress at Infinity Two Unequal Arbitrarily Oriented Elliptical Holes or Cracks First Approximation Second Approximation Uniform Shear at Infinity Stress Field Two Unequal Collinear Elliptical Holes Stress Analysis of Door and Window of a Passenger Aircraft 262 Exercises Anisotropic Elasticity Basic Cases of Elastic Symmetry Plane of Elastic Symmetry Three Planes of Elastic Symmetry (Orthotropic Body) Axis of Rotational Symmetry or Transversely Isotropic Body Isotropic Body Curvilinear Anisotropy Cylindrical Anisotropy 278

10 Contents ix 6.8 Spherical Anisotropy Some Anisotropic Elastic Constants for Engineering Materials Fundamental Equations of the Theory of Anisotropic Elasticity Complex Representation of the Stress Functions Boundary Conditions Representation of Resultant Forces and Moments Expressions for the Functions f(z 1 ) and y(z 2 ) for Multiply-connected Region First Fundamental Problem for Infinite Anisotropic Plate with Elliptical Hole An Anisotropic Plate with Elliptical Hole with Uniform Tensile Stresses at Infinity at an Angle Uniform Tension p in x-direction Crack of Length 2a Edge of Elliptical Hole Subjected to Uniform Tangential Stress T Elliptical Hole Subjected to Uniform Pressure p A Novel Method of Stress Analysis of an Infinite Anisotropic Plate with Elliptical Hole or Crack with Uniform Tensile Stress Stress Distribution around Triangular Holes in Anisotropic Plates A General Solution for Stress Resultants and Moments around Holes in Unsymmetric Laminates A General Solution for Moments Around Holes in Symmetric Laminates A General Solution for Stresses Around Holes in Symmetric Laminates under Inplane Loading Stress Analysis for an Orthotropic Plate with an Irregular Shaped Hole for Different In-plane Loading Conditions Part Exercises Introduction to Fracture Mechanics Importance of Fracture Mechanics Modes of Fracture The Griffith Criterion: As Surface Energy A Crack Structure The Stresses at the Crack Tip and Irwin s Stress Intensity Factor Relationship betwee K I and G I Crack Propagation and Paris Law Fracture Control Plans 415 Exercises Stress Analysis of Fracture Westergaard s Stress Function Stresses in a Cracked Body Mode I Crack Problem with Biaxial Stress and Using Wertergaard s Stress Function Crack in Finite Width Plate 423

11 Theory Of Elasticity And Fracture Mechanics 25% OFF Publisher : PHI Learning ISBN : Author : UKADGAONKER, VIJAY G. Type the URL : Get this ebook

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