Elasticity: theory, applications, and numerics

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Elasticity: Theory, Applications, and Numerics
Copyright
Preface
	Contents summary
	The subject
	Exercises and web support
	Feedback
Acknowledgments
About the Author
Part 1: Foundations and elementary applications
1 -
Mathematical preliminaries
	1.1 Scalar, vector, matrix, and tensor definitions
	1.2 Index notation
	1.3 Kronecker delta and alternating symbol
	1.4 Coordinate transformations
	1.5 Cartesian tensors
	1.6 Principal values and directions for symmetric second-order tensors
	1.7 Vector, matrix, and tensor algebra
	1.8 Calculus of Cartesian tensors
		1.8.1 Divergence or Gauss theorem
		1.8.2 Stokes theorem
		1.8.3 Green's theorem in the plane
		1.8.4 Zero-value or localization theorem
	1.9 Orthogonal curvilinear coordinates
	References
	1 . . Exercises
2 -
Deformation: displacements and strains
	2.1 General deformations
	2.2 Geometric construction of small deformation theory
	2.3 Strain transformation
	2.4 Principal strains
	2.5 Spherical and deviatoric strains
	2.6 Strain compatibility
	2.7 Curvilinear cylindrical and spherical coordinates
	References
	2 
Exercises
3 -
Stress and equilibrium
	3.1 Body and surface forces
	3.2 Traction vector and stress tensor
	3.3 Stress transformation
	3.4 Principal stresses
	3.5 Spherical, deviatoric, octahedral, and von Mises stresses
	3.6 Stress distributions and contour lines
	3.7 Equilibrium equations
	3.8 Relations in curvilinear cylindrical and spherical coordinates
	References
	3 
Exercises
4 -
Material behavior—linear elastic solids
	4.1 Material characterization
	4.2 Linear elastic materials—Hooke's law
	4.3 Physical meaning of elastic moduli
		4.3.1 Simple tension
		4.3.2 Pure shear
		4.3.3 Hydrostatic compression (or tension)
	4.4 Thermoelastic constitutive relations
	References
	4 
Exercises
5 -
Formulation and solution strategies
	5.1 Review of field equations
	5.2 Boundary conditions and fundamental problem classifications
	5.3 Stress formulation
	5.4 Displacement formulation
	5.5 Principle of superposition
	5.6 Saint–Venant's principle
	5.7 General solution strategies
		5.7.1 Direct method
		5.7.2 Inverse method
		5.7.3 Semi-inverse method
		5.7.4 Analytical solution procedures
			Power series method
			Fourier method
			Integral transform method
			Complex variable method
		5.7.5 Approximate solution procedures
			Ritz method
		5.7.6 Numerical solution procedures
			Finite difference method
			Finite element method
			Boundary element method
	5.8 Singular elasticity solutions
	References
	5 
Exercises
6 -
Strain energy and related principles
	6.1 Strain energy
	6.2 Uniqueness of the elasticity boundary-value problem
	6.3 Bounds on the elastic constants
		6.3.1 Uniaxial tension
		6.3.2 Simple shear
		6.3.3 Hydrostatic compression
	6.4 Related integral theorems
		6.4.1 Clapeyron's theorem
		6.4.2 Betti/Rayleigh reciprocal theorem
		6.4.3 Integral formulation of elasticity—Somigliana's identity
	6.5 Principle of virtual work
	6.6 Principles of minimum potential and complementary energy
	6.7 Rayleigh–Ritz method
	References
	6 
Exercises
7 -
Two-dimensional formulation
	7.1 Plane strain
	7.2 Plane stress
	7.3 Generalized plane stress
	7.4 Antiplane strain
	7.5 Airy stress function
	7.6 Polar coordinate formulation
	References
	7 
Exercises
8 -
Two-dimensional problem solution
	8.1 Cartesian coordinate solutions using polynomials
	8.2 Cartesian coordinate solutions using Fourier methods
		8.2.1 Applications involving Fourier series
	8.3 General solutions in polar coordinates
		8.3.1 General Michell solution
		8.3.2 Axisymmetric solution
	8.4 Example polar coordinate solutions
		8.4.1 Pressurized hole in an infinite medium
		8.4.2 Stress-free hole in an infinite medium under equal biaxial loading at infinity
		8.4.3 Biaxial and shear loading cases
		8.4.4 Quarter-plane example
		8.4.5 Half-space examples
		8.4.6 Half-space under uniform normal stress over x≤0
		8.4.7 Half-space under concentrated surface force system (Flamant problem)
		8.4.8 Half-space under a surface concentrated moment
		8.4.9 Half-space under uniform normal loading over −a≥x≥a
		8.4.10 Notch and crack problems
		8.4.11 Pure bending example
		8.4.12 Curved cantilever under end loading
	8.5 Simple plane contact problems
	References
	8 
Exercises
9 -
Extension, torsion, and flexure of elastic cylinders
	9.1 General formulation
	9.2 Extension formulation
	9.3 Torsion formulation
		9.3.1 Stress–stress function formulation
		9.3.2 Displacement formulation
		9.3.3 Multiply connected cross-sections
		9.3.4 Membrane analogy
	9.4 Torsion solutions derived from boundary equation
	9.5 Torsion solutions using Fourier methods
	9.6 Torsion of cylinders with hollow sections
	9.7 Torsion of circular shafts of variable diameter
	9.8 Flexure formulation
	9.9 Flexure problems without twist
	References
	9 
Exercises
Part 2: Advanced applications
10 -
Complex variable methods
	10.1 Review of complex variable theory
	10.2 Complex formulation of the plane elasticity problem
	10.3 Resultant boundary conditions
	10.4 General structure of the complex potentials
		10.4.1 Finite simply connected domains
		10.4.2 Finite multiply connected domains
		10.4.3 Infinite domains
	10.5 Circular domain examples
	10.6 Plane and half-plane problems
	10.7 Applications using the method of conformal mapping
	10.8 Applications to fracture mechanics
	10.9 Westergaard method for crack analysis
	References
	10 -
Exercises
11 -
Anisotropic elasticity
	11.1 Basic concepts
	11.2 Material symmetry
		11.2.1 Plane of symmetry (monoclinic material)
		11.2.2 Three perpendicular planes of symmetry (orthotropic material)
		11.2.3 Axis of symmetry (transversely isotropic material)
		11.2.4 Cubic symmetry
		11.2.5 Complete symmetry (isotropic material)
	11.3 Restrictions on elastic moduli
	11.4 Torsion of a solid possessing a plane of material symmetry
		11.4.1 Stress formulation
		11.4.2 Displacement formulation
		11.4.3 General solution to the governing equation
	11.5 Plane deformation problems
		11.5.1 Uniform pressure loading case
	11.6 Applications to fracture mechanics
	11.7 Curvilinear anisotropic problems
		11.7.1 Two-dimensional polar-orthotropic problem
		11.7.2 Three-dimensional spherical-orthotropic problem
	References
	11 
Exercises
12 -
Thermoelasticity
	12.1 Heat conduction and the energy equation
	12.2 General uncoupled formulation
	12.3 Two-dimensional formulation
		12.3.1 Plane strain
		12.3.2 Plane stress
	12.4 Displacement potential solution
	12.5 Stress function formulation
	12.6 Polar coordinate formulation
	12.7 Radially symmetric problems
	12.8 Complex variable methods for plane problems
	References
	12 
Exercises
13 -
Displacement potentials and stress functions: applications to three-dimensional problems
	13.1 Helmholtz displacement vector representation
	13.2 Lamé's strain potential
	13.3 Galerkin vector representation
	13.4 Papkovich–Neuber representation
	13.5 Spherical coordinate formulations
	13.6 Stress functions
		13.6.1 Maxwell stress function representation
		13.6.2 Morera stress function representation
	References
	13 
Exercises
14 -
Nonhomogeneous elasticity
	14.1 Basic concepts
	14.2 Plane problem of a hollow cylindrical domain under uniform pressure
	14.3 Rotating disk problem
	14.4 Point force on the free surface of a half-space
	14.5 Antiplane strain problems
	14.6 Torsion problem
	References
	14 
Exercises
15 -
Micromechanics applications
	15.1 Dislocation modeling
	15.2 Singular stress states
	15.3 Elasticity theory with distributed cracks
	15.4 Micropolar/couple-stress elasticity
		15.4.1 Two-dimensional couple-stress theory
	15.5 Elasticity theory with voids
	15.6 Doublet mechanics
	15.7 Higher gradient elasticity theories
	References
	15 
Exercises
16 -
Numerical finite and boundary element methods
	16.1 Basics of the finite element method
	16.2 Approximating functions for two-dimensional linear triangular elements
	16.3 Virtual work formulation for plane elasticity
	16.4 FEM problem application
	16.5 FEM code applications
	16.6 Boundary element formulation
	References
	16 
Exercises
A -
Basic field equations in Cartesian, cylindrical, and spherical coordinates
	Strain–displacement relations
		Cartesian coordinates
		Cylindrical coordinates
		Spherical coordinates
	Equilibrium equations
		Cartesian coordinates
		Cylindrical coordinates
		Spherical coordinates
	Hooke's law
		Cartesian coordinates
		Cylindrical coordinates
		Spherical coordinates
	Equilibrium equations in terms of displacements (Navier's equations)
		Cartesian coordinates
		Cylindrical coordinates
		Spherical coordinates
B -
Transformation of field variables between Cartesian, cylindrical, and spherical components
	Cylindrical components from Cartesian
		Displacement transformation
		Stress transformation
	Spherical components from cylindrical
		Displacement transformation
		Stress transformation
	Spherical components from Cartesian
		Displacement transformation
		Stress transformation
C -
MATLAB® Primer
	C.1 Getting started
	C.2 Examples
	References
D -
Review of mechanics of materials
	D.1 Extensional deformation of rods and beams
	D.2 Torsion of circular rods
	D.3 Bending deformation of beams under moments and shear forces
	D.4 Curved beams
	D.5 Thin-walled cylindrical pressure vessels
Index
	A
	B
	C
	D
	E
	F
	G
	H
	I
	K
	L
	M
	N
	O
	P
	R
	S
	T
	U
	V
	W
	Y
	Z
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