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Explore the Computational Methods and Mathematical Models That Are Possible through Continuum Mechanics Formulations

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.

Comprised of 16 chapters, this text covers:

- 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.

"… this comprehensive and self-sufficient monograph, written on a very strict mathemat-

ical level and clear physical background, contains a detailed and well-structured description of

the advanced mechanics of continua. This book should be very useful to students and teachers

in the fields of engineering and mathematical physics."

*—Zentralblatt MATH,*1326

"The book **Advanced Mechanics of Continua** is a complete reference for both students and researchers in the field of material sciences and applied mechanics. It contains comprehensive and detailed representation for the mechanical behavior of solids and fluids. The constitutive laws for different types of materials are presented separately in this book which helps the readers to understand the concepts without confusion. The choice of gradually developing formulations for different subjects eases the grasping of material for the readers and makes the book an excellent reference for graduate students.

—George Z. Voyiadjis, Louisiana State University, Baton Rouge, USA

"This textbook gives a very rigorous analysis of continuum mechanics covering complex topics in Eulerian and Lagrangian kinematics and stress measures for finite deformation problems. … The book covers all the key topics required to gain a foothold on the complex concepts of continuum mechanics. … The discussion on kinematics provides a rigorous analysis of kinematics in different reference frames. This is often treated briefly in other comparable textbooks. … The author has succeeded in providing an exhaustive resource for advanced graduate students and researchers interest[ed] in mastering continuum mechanics."

—William S. Oates, Florida State University, Tallahassee, USA

**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

Kinematics of Motion, Deformation and Their Measures

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, Strain Rate, and Spin

Tensors

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

Conservation and Balance Laws in Eulerian Description

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

Conservation and Balance Laws In Lagrangian Description

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

Thermoviscoelastic Solids without Memory

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

Ordered Rate Constitutive Theories for Thermofluids

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

Ordered Rate Constitutive Theories for Polymers

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

Ordered Rate Constitutive Theories for Hypoelastic Solids

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

Mathematical Models with Thermodynamic Relations

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

Principle of Virtual Work

Introduction

Hamilton's Principle in Continuum Mechanics

Euler-Lagrange Equation: Lagrangian Description

Euler-Lagrange Equation: Eulerian Description

Summary and Remarks

Appendices

- SCI041000
- SCIENCE / Mechanics / General
- TEC009070
- TECHNOLOGY & ENGINEERING / Mechanical
- TEC063000
- TECHNOLOGY & ENGINEERING / Structural