Finite Element Analysis: Thermomechanics of Solids

Explore a Unified Treatment of the Finite Element Method The finite element method has matured to the point that it can accurately and reliably be used, by a careful analyst, for an amazingly wide range of applications. With expanded coverage and an increase in fully solved examples, the second edition of Finite Element Analysis: Thermomechanics of Solids presents a unified treatment of the finite element method in theremomechanics, from the basics to advanced concepts. An Integrated Presentation of Critical Technology As in the first edition, the author presents and explicates topics in a way that demonstrates the highly unified structure of the finite element method. The presentation integrates continuum mechanics and relevant mathematics with persistent reliance on variational and incremental-variational foundations. The author exploits matrix-vector formalisms and Kronecker product algebra to provide transparent and consistent notation throughout the text. Nearly twice as long as the first edition, this second edition features: § Greater integration and balance between introductory and advanced material § Increased number of fully solved examples § Selected developments in numerical methods, detailing accelerating computations in eigenstructure extraction, time integration, and stiffness matrix triangularization § More extensive coverage of the arc length method for nonlinear problems § Expanded and enhanced treatment of rotating bodies and buckling Provides Sophisticated Understanding of Capabilities and Limitations This new edition of a popular text includes significant illustrative examples and applications, modeling strategies, and explores a range of computational issues. Written by a professor with years of practical engineering and instructional experience, the book provides a strong foundation for those requiring a sophisticated understanding of the method’s capabilities and limitations.

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