Advanced Mechanics of Continua
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Book Description
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 contravariant 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 selfstudy.
Table of Contents
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 ContraVariant Bases
Jacobian of Deformation
Change of Description, Co and ContraVariant 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 HypoElastic Solids
Rate Constitutive Theory of Order `N'
Rate Constitutive Theory of Order Two
Rate Constitutive Theory of Order One
Compressible Generalized HypoElastic Solids of Order One
Incompressible Ordered HypoElastic Solids
Incompressible Generalized HypoElastic 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
EulerLagrange Equation: Lagrangian Description
EulerLagrange Equation: Eulerian Description
Summary and Remarks
Appendices
Author(s)
Biography
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.
Reviews
"… this comprehensive and selfsufficient monograph, written on a very strict mathemat
ical level and clear physical background, contains a detailed and wellstructured 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
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