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Skip to main content# Body Tensor Fields in Continuum Mechanics

## With Applications to Polymer Rheology

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- 1st Edition - January 1, 1974
- Author: Arthur S. Lodge
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 6 2 9 9 - 4

Body Tensor Fields in Continuum Mechanics: With Applications to Polymer Rheology aims to define body tensor fields and to show how they can be used to advantage in continuum… Read more

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Body Tensor Fields in Continuum Mechanics: With Applications to Polymer Rheology aims to define body tensor fields and to show how they can be used to advantage in continuum mechanics, which has hitherto been treated with space tensor fields. General tensor analysis is developed from first principles, using a novel approach that also lays the foundations for other applications, e.g., to differential geometry and relativity theory. The applications given lie in the field of polymer rheology, treated on the macroscopic level, in which relations between stress and finite-strain histories are of central interest. The book begins with a review of mathematical prerequisites, namely primitive concepts, linear spaces, matrices and determinants, and functionals. This is followed by separate chapters on body tensor and general space tensor fields; the kinematics of shear flow and shear-free flow; Cartesian vector and tensor fields; and relative tensors, field transfer, and the body stress tensor field. Subsequent chapters deal with constitutive equations for viscoelastic materials; reduced constitutive equations for shear flow and shear-free flow; covariant differentiation and the stress equations of motion; and stress measurements in unidirectional shear flow.

PrefaceAcknowledgmentsList of Notation1 Some Mathematical Prerequisites 1.1 Primitive Concepts 1.2 Linear Spaces 1.3 Matrices and Determinants 1.4 Functionals2 The Body Metric Tensor Field 2.1 Introduction 2.2 Body and Space Manifolds and Coordinate Systems 2.3 The Definition of Contravariant Vector Fields 2.4 The Definition of Covariant Vector Fields 2.5 The Definition of Second-Rank Tensor Fields 2.6 Metric Tensor Fields for the Body and Space Manifolds 2.7 Magnitudes and Angles 2.8 Principal Axes and Principal Values3 The Kinematics of Shear Flow and Shear-Free Flow 3.1 Introduction 3.2 Shear Flow 3.3 Base Vectors and Strain Tensors for Shear Flow 3.4 Torsional Flow between Circular Parallel Plates in Relative Rotation about a Common Axis 3.5 Torsional Flow between a Cone and a Touching Plate in Relative Rotation about a Common Axis 3.6 Helical Flow between Coaxial Right Circular Cylinders 3.7 Orthogonal Rheometer Flow 3.8 Balance Rheometer Flow 3.9 Shear-Free Flow4 Cartesian Vector and Tensor Fields 4.1 Rectangular Cartesian Coordinate Systems 4.2 The Definition of Cartesian Vector Fields 4.3 The Definition of Cartesian Second-Rank Tensor Fields 4.4 Relations between Cartesian and General Tensor Fields 4.5 Cartesian Base Vectors for a Curvilinear Coordinate System5 Relative Tensors, Field Transfer, and the Body Stress Tensor Field 5.1 Relative Tensors 5.2 Tensors of Third and Higher Rank 5.3 Quotient Theorems 5.4 Correspondence between Body and Space Fields at Time t 5.5 Volume and Surface Elements 5.6 The Body Stress Tensor Field 5.7 Isotropic Functions and Orthogonal Tensors 5.8 Constant Stretch History6 Constitutive Equations for Viscoelastic Materials 6.1 General Forms for Constitutive Equations 6.2 Constitutive Equations from Molecular Theories 6.3 Perfectly Elastic Solids 6.4 Integral Constitutive Equations 6.5 Differential Constitutive Equations 6.6 Alternative Forms for Constitutive Equations 6.7 Memory-Integral Expansions 6.8 Boltzmann’s Viscoelasticity Theory: Small Displacements 6.9 Classical Elasticity and Hydrodynamics7 Reduced Constitutive Equations for Shear Flow and Shear-Free Flow 7.1 Incompressible Viscoelastic Liquids in Unidirectional Shear Flow 7.2 Oscillatory and Steady Shear Flow: Low-Frequency Relations 7.3 Orthogonal Rheometer: Small-Strain Limit 7.4 Shear-Freeflow8 Covariant Differentiation and the Stress Equations of Motion 8.1 Divergence and Curl 8.2 Covariant Differentiation in a Euclidean Manifold 8.3 Covariant Derivatives of Body Tensor Fields 8.4 Curvature of Surfaces 8.5 Stress Equations of Motion 8.6 Covariant Differentiation in An Affinely Connected Manifold 8.7 The Affine Connection for a Riemannian Manifold 8.8 Compatibility Conditions 8.9 Boundary Conditions 8.10 Simultaneous Equations for Isothermal Flow Problems9 Stress Measurements in Unidirectional Shear Flow: Theory 9.1 Stress Equations of Motion for Unidirectional Shear Flow 9.2 The Importance of N1 and N2 9.3 Torsional Flow, Parallel Plates 9.4 Torsional Flow, Cone and Plate 9.5 Steady Helical Flow10 Constitutive Predictions and Experimental Data 10.1 Shear and Elongation of Low-Density Polyethylene 10.2 Fast-Strain Tests of the Guassian Network Hypothesis 10.3 Polystyrene/Aroclors Data and Carreau's Model B 10.4 Measurements of Ν1 and N211 Relations between Body- and Space-Tensor Formalisms 11.1 Convected Components 11.2 Embedded Vectors 11.3 Objectivity Condition for Space-Tensor Constitutive Equations 11.4 Formulation of Constitutive Equations: Historical NoteAppendix A Equations in Cylindrical Polar CoordinatesAppendix B Equations in Spherical Polar Coordinate SystemsAppendix C Equations in Orthogonal Coordinate SystemsAppendix D Summary of Definitions for Unidirectional Shear FlowAppendix E Summary of T Operations for Covariant Strain TensorsAppendix F Calculations for Viscoelastic LiquidsReferencesSolutions to ProblemsAuthor IndexSubject Index

- No. of pages: 336
- Language: English
- Edition: 1
- Published: January 1, 1974
- Imprint: Academic Press
- eBook ISBN: 9781483262994

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