Body Tensor Fields in Continuum Mechanics

Body Tensor Fields in Continuum Mechanics

With Applications to Polymer Rheology

1st Edition - January 1, 1974

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  • Author: Arthur S. Lodge
  • eBook ISBN: 9781483262994

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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.

Table of Contents

  • Preface


    List of Notation

    1 Some Mathematical Prerequisites

    1.1 Primitive Concepts

    1.2 Linear Spaces

    1.3 Matrices and Determinants

    1.4 Functionals

    2 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 Values

    3 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 Flow

    4 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 System

    5 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 History

    6 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 Hydrodynamics

    7 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-Freeflow

    8 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 Problems

    9 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 Flow

    10 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 N2

    11 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 Note

    Appendix A Equations in Cylindrical Polar Coordinates

    Appendix B Equations in Spherical Polar Coordinate Systems

    Appendix C Equations in Orthogonal Coordinate Systems

    Appendix D Summary of Definitions for Unidirectional Shear Flow

    Appendix E Summary of T Operations for Covariant Strain Tensors

    Appendix F Calculations for Viscoelastic Liquids


    Solutions to Problems

    Author Index

    Subject Index

Product details

  • No. of pages: 336
  • Language: English
  • Copyright: © Academic Press 1974
  • Published: January 1, 1974
  • Imprint: Academic Press
  • eBook ISBN: 9781483262994

About the Author

Arthur S. Lodge

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