Table of Contents

Introduction To The Finite Element Method

Introduction

Overview of the Finite Element Method

Mesh Development

Mathematical Foundations: Vectors and Matrices

Introduction

Vectors

Matrices

Eigenvalues and Eigenvectors

Coordinate Transformations

Orthogonal Curvilinear Coordinates

Gradient Operator in Orthogonal Coordinates

Divergence and Curl of Vectors in Orthogonal Coordinates

Appendix: Divergence and Curl of Vectors in Orthogonal Curvilinear Coordinates

Mathematical Foundations: Tensors

Tensors

Divergence of a Tensor

Invariants

Positive Definiteness

Polar Decomposition Theorem

Kronecker Products of Tensors

Examples

Introduction to Variational Methods

Introductory Notions

Properties of the Variational Operator

Example: Variational Equation for a Cantilevered Elastic Rod

Higher Order Variations

Examples

Fundamental Notions of Linear Solid Mechanics

The Displacement Vector

The Linear Strain and Rotation Tensors

Examples of Linear Strain and Rotation Tensors

Traction and Stress

Equilibrium

Stress and Strain Transformations

Principal Stresses and Strains

Stress Strain Relations

Principle of Virtual Work in Linear Elasticity

Thermal and Thermomechanical Response

Balance of Energy and Production of Entropy

Classical Coupled Linear Thermoelasticity

Thermal and Thermomechanical Analogs of the Principle of Virtual Work and Associated Finite Element Equations

One-Dimensional Elastic Elements

Interpolation Models for One Dimensional Elements

Strain-Displacement Relations in One Dimensional Elements

Stress-Strain Relations in One Dimensional Elements

Element Mass and Stiffness Matrices from the Principle of Virtual Work

Integral Evaluation by Gaussian Quadrature: Natural Coordinates

Unconstrained Rod Elements

Unconstrainted Elements for Beams and Beam-Columns

Assemblage and Imposition of Constraints

Damping in Rods and Beams

General Discussion of Assemblage

General Discussion of the Imposition of Constraints

Inverse Variational Method

Two- and Three-Dimensional Elements in Linear Elasticity and Linear Conductive Heat Transfer

Two Dimensions

Interpolation Models in Three Dimensions

Strain Displacement Relations and Thermal Analogs

Stress-Strain Relations

Stiffness and Mass Matrices and Their Thermal Analogs

Thermal Counterpart of the Principle of Virtual Work

Conversion to Natural Coordinates in Two and Three Dimensions

Assembly of Two and Three Dimensional Elements

Solution Methods for Linear Problems - I

Numerical Methods in FEA

Time Integration: Stability and Accuracy

Properties of the Trapezoidal Rule

Integral Evaluation by Gaussian Quadrature

Modal Analysis by FEA

Solution Methods for Linear Problems -II

Introduction

Solution Method for an Inverse Problem

Accelerated Eigenstructure Computation in FEA

Fourth Order Time Integration

Additional Topics in Linear Thermoelastic Systems

Transient Conductive Heat Transfer in Linear Media

Coupled Linear Thermoelasticity

Incompressible Elastic Media

Torsion of Prismatic Bars

Buckling of Elastic Beams and Plates

Introduction to Contact Problems

Rotating and Unrestrained Elastic Bodies

Finite Elements in Rotation

Critical Speeds in Shaft-Rotor Shaft

Finite Element Analysis for Unconstrained Elastic Bodies

Appendix: Angular Velocity Vector in Spherical Coordinates

Aspects on Nonlinear Continuum Thermomechanics

Introduction

Nonlinear Kinematics of Deformation

Mechanical Equilibrium and the Principle of Virtual Work

Principle of Virtual Work Under Large Deformation

Nonlinear Stress-Strain-Temperature Relations: The Isothermal Tangent Modulus Tensor

Introduction to Nonlinear FEA

Introduction

Types of Nonlinearlity

Newton Iteration

Combined Incremental and Iterative Methods: A Simple Example

Finite Stretching of a Rubber Rod Under Gravity

Newton Iteration Near a Critical Point

Introduction to the Arc Length Method

Incremental Principle of Virtual Work

Incremental Kinematics

Stress Increments

Incremental Equation of Balance of Linear Momentum

Incremental Principle of Virtual Work

Incremental Finite Element Equation

Contributions From Nonlinear Boundary Conditions

Effect of Variable Contact

Interpretation as Newton Iteration

Buckling

Tangent Modulus Tensors for Thermomechanical Response of Elastomers

Introduction

Compressible Elastomers

Incompressible and Near-Incompressible Elastomers

Stretch-Ration Based Models: Isothermal Conditions

Extension to Thermohyperelastic Materials

Thermomechanics of Damped Elastomers

Constitutive Model in Thermoviscohyperelasticity

Variational Principles and Finite Element Equations for A Thermoviscohyperelastic Material

Tangent Modulus Tensors for Inelastic and Thermoinelastic Materials

Plasticity

Tangent Modulus Tensor in Small Strain Isothermal Plasticity

Plasticity Under Finite Strain

Thermoplasticity

Tangent Modulus Tensor in Viscoplasticity

Continuum Damage Mechanics

Selected Advanced Numerical Methods in FEA

Iterative Triangularization of Perturbed Matrices

Stiff Arc Length Constraint in Nonlinear FEA

Non-Iterative Solution of Finite Element Equations in Incompressible Solids

References

Index