Advanced Mechanics of Continua Applied and Computational Mechanics Series

Mathematically demanding, but also rigorous, precise, and written using very clear language, Advanced Mechanics of Continua provides a thorough understanding of continuum mechanics. This book explores the foundation of continuum mechanics and constitutive theories of materials using understandable notations. It does not stick to one specific form, but instead provides a mix of notations that while in many instances are different than those used in current practice, are a natural choice for the information that they represent. The book places special emphasis on both matrix and vector notations, and presents material using these notations whenever possible.

The author explores the development of mathematical descriptions and constitutive theories for deforming solids, fluids, and polymeric fluids?both compressible and incompressible with clear distinction between Lagrangian and Eulerian descriptions as well as co- and contravariant bases. He also establishes the tensorial nature of strain measures and influence of rotation of frames on various measures, illustrates the physical meaning of the components of strains, presents the polar decomposition of deformation, and provides the definitions and measures of stress.

• Einstein?s notation
• Index notations
• Matrix and vector notations
• Basic definitions and concepts
• Mathematical preliminaries
• Tensor calculus and transformations using co- and contra-variant bases
• Differential calculus of tensors
• Development of mathematical descriptions and constitutive theories

Advanced Mechanics of Continua prepares graduate students for fundamental and basic research work in engineering and sciences, provides detailed and consistent derivations with clarity, and can be used for self-study.

Introduction

Concepts and Mathematical Preliminaries

Introduction

Summation Convention

Dummy Index and Dummy Variables

Free Indices

Vector and Matrix Notation

Index Notation and Kronecker Delta

Permutation Tensor

Operations Using Vector, Matrix, and Einstein's Notation

Change of Reference Frame, Transformations, Tensors

Some Useful Relations

Summary

Description of Motion

Lagrangian and Eulerian Descriptions

Material Particle Displacements

Continuous Deformation and Restrictions on the Motion

Material Derivative

Acceleration of a Material Particles

Coordinate Systems and Bases

Covariant Basis

Contravariant Basis

Alternate Way to Visualize Co- and Contra-Variant Bases

Jacobian of Deformation

Change of Description, Co- and Contra-Variant Measures

Notations For Covariant and Contravariant Measures

Deformation, Measures of Length and Change in Length

Covariant and Contravariant Measures of Strain

Changes in Strain Measures Due To Rigid Rotation of Frames

Invariants of Strain Tensors

Expanded Form of Strain Tensors

Physical Meaning of Strains

Polar Decomposition: Rotation and Stretch Tensors

Deformation of Areas and Volumes

Summary

Definitions and Measures of Stresses

Cauchy Stress Tensor

Contravariant and Covariant Stress Tensors

General Remarks

Summary of Stresses and Considerations in Their Derivations

General Considerations

Summary of Stress Measures

Conjugate Strain Measures

Relations between Stress Measures and Useful Relations

Summary

Rate of Deformation

Decomposition of [ L], the Spatial Velocity Gradient Tensor

Interpretation of the Components of [D]

Rate of Change or Material Derivative of Strain Tensors

Physical Meaning of Spin Tensor [ W ]

Vorticity Vector and Vorticity

Material Derivative of Determinant of J

Material Derivative of Volume

Rate of Change of Area: Material Derivative of Area

Stress And Strain Measures for Convected Time Derivatives

Convected Time Derivatives

Conjugate Convected Time Derivatives of Stress And Strain Tensors

Summary

Introduction

Mass Density

Conservation Of Mass: Continuity Equation

Transport Theorem

Conservation Of Mass: Continuity Equation

Balance of Linear Momenta

Kinetics of Continuous Media: Balance of Angular Momenta

First Law of Thermodynamics

Second Law of Thermodynamics

A Summary of Mathematical Models

Summary

Introduction

Mathematical Model for Deforming Matter in Lagrangian Description

Conservation Of Mass: Continuity Equation

Balance of Linear Momenta

Balance of Angular Momenta

First Law of Thermodynamics

Second law of thermodynamics in terms of Φ

Second law of thermodynamics in terms of Ψ

Summary of Mathematical Models

First and Second Laws for Thermoelastic Solids

Summary

General Considerations in the Constitutive Theories

Introduction

Axioms of Constitutive Theory

Objective

Solid Matter

Fluids

Preliminary Considerations in the Constitutive Theories

General Approach of Deriving Constitutive Theories

Summary

Ordered Rate Constitutive Theories for Thermoelastic Solids

Introduction

Entropy inequality in Φ: Lagrangian description

Constitutive Theories for Thermoelastic Solids

Constitutive Theories Using Generators and Invariants

Strain energy density π: Lagrangian description

Stress in terms of Green strain based on π: Lagrangian

Stress in terms of Cauchy strain based on π: Lagrangian

Constitutive Theories for the Heat Vector: Lagrangian

Alternate Derivations: Strain In Terms Of Stress

Alternate Derivations: Heat Vector In Terms Of Stress

Summary

Introduction

Constitutive Theories Using Helmholtz Free Energy Density

Constitutive Theories Using Gibbs Potential

Comparisons of constitutive theories using Φ and Ψ

Thermoviscoelastic Solids with Memory

Introduction

Constitutive Theories Using Helmholtz Free Energy Density

Constitutive Theories Using Gibbs Potential

Comparisons of constitutive theories using Φ and Ψ

Introduction

Second Law of Thermodynamics: Entropy Inequality

Dependent Variables and Their Arguments

Development of Constitutive Theory for Thermo Fluids

Rate Constitutive Theory of Order N

Rate Constitutive Theory of Order Two

Rate Constitutive Theory of Order One

Generalized Newtonian and Newtonian Fluids

Incompressible Ordered Thermo Fluids of Orders N, 2 And 1

Incompressible Generalized Newtonian, Newtonian Fluids

Conjugate Measures, Validity of Rate Constitutive Theories

Summary

Introduction

Second Law of Thermodynamics: Entropy Inequality

Dependent Variables and Their Arguments

Development of Constitutive Theory for Polymers

Rate Constitutive Theory of Orders `M' and `N'

Rate Constitutive Theory of Orders M=1 and N=1

Rate Constitutive Theory of Orders M=1 and N=2

Constitutive Theories for Incompressible Polymers

Numerical Studies Using Giesekus Constitutive Model

Introduction

Second Law of Thermodynamics: Entropy Inequality

Dependent Variables and Their Arguments

Development of Constitutive Theory for Hypo-Elastic Solids

Rate Constitutive Theory of Order `N'

Rate Constitutive Theory of Order Two

Rate Constitutive Theory of Order One

Compressible Generalized Hypo-Elastic Solids of Order One

Incompressible Ordered Hypo-Elastic Solids

Incompressible Generalized Hypo-Elastic Solids: Order One

Summary

Introduction

Thermodynamic Pressure: Compressible Matter

Mechanical Pressure: Incompressible Matter

Specific Internal Energy

Variable Transport Properties or Material Coefficients

Final Form of the Mathematical Models

Summary

Introduction

Hamilton's Principle in Continuum Mechanics

Euler-Lagrange Equation: Lagrangian Description

Euler-Lagrange Equation: Eulerian Description

Summary and Remarks

Appendices

Karan S. Surana attended undergraduate school at Birla Institute of Technology and Science (BITS), Pilani, India and received a B.E. in mechanical engineering in 1965. He then attended the University of Wisconsin, Madison where he obtained M.S. and Ph.D. in mechanical engineering in 1967 and 1970. He joined The University of Kansas, Department of Mechanical Engineering faculty where he is currently serving as Deane E. Ackers University Distinguished Professor of Mechanical Engineering. His areas of interest and expertise are computational mathematics, computational mechanics, and continuum mechanics. He is the author of over 350 research reports, conference papers, and journal papers.