Nonlinear Theory of Elasticity

The following sections are included:

• Preface

• Contents

Problems involving deformation of soft biological tissues are among the most challenging in applied mechanics. In addition to undergoing large strains, soft tissues exhibit nonlinear time-dependent behavior similar to that of viscoelastic materials (Fung, 1993). Moreover, biological tissues are not simply passive materials like those used in traditional engineering applications. Rather, they actively contract, grow, and remodel. Further complicating matters, problems in soft tissue biomechanics often involve complex three-dimensional geometry, dynamic loading conditions, and fluid-solid interaction. These problems are highly nonlinear in general…

The following sections are included:

• Reference Frames and Coordinate Systems

• Vectors
  • Base Vectors
  • Reciprocal Base Vectors

• Dyadics and Tensors
  • Dyadics
  • Tensors
  • Matrix Representation of Tensors
  • Some Properties of Tensors

• Coordinate Transformation
  • Transformation of Base Vectors
  • Transformation of Vector Components
  • Transformation of Tensor Components

• Tensor Invariants
  • Trace
  • Determinant
  • Eigenvalues and Eigenvectors

• Special Tensors
  • Metric Tensor
  • Permutation Tensor
  • Orthogonal Tensors

• Physical Components
  • Physical Components of Vectors
  • Physical Components of Tensors

• Vector and Tensor Calculus
  • Differentiation with Respect to Scalars
  • Differentiation with Respect to Vectors and Tensors
  • Gradient Operator
  • Integral Relations

• Problems

The following sections are included:

• Deformation in One Dimension

• Coordinate Systems and Base Vectors

• Deformation Gradient Tensor

• Deformation and Strain Tensors
  • Deformation Tensors
  • Strain Tensors
  • Physical Components of Strain

• Strain-Displacement Relations

• Geometric Measures of Deformation
  • Stretch
  • Shear
  • Volume Change
  • Area Change

• Principal Strains
  • The Eigenvalue Problem
  • Strain Invariants

• Stretch and Rotation Tensors
  • Polar Decomposition
  • Principal Stretch Ratios

• Approximations
  • Small Displacement
  • Small Deformation
  • Small Deformation and Moderate Rotation
  • Small Rotation

• Deformation Rates
  • Time Rates of Change
  • Rate-of-Deformation and Spin Tensors

• Compatibility Conditions

• Problems

The following sections are included:

• Body and Contact Forces

• Stress Tensors
  • Cauchy Stress Tensor
  • Pseudostress Tensors

• Relations Between Stress Components

• Physical Interpretation of Stress Components

• Principal Stresses

• Equations of Motion
  • Principle of Linear Momentum
  • Principle of Angular Momentum
  • Lagrangian Form of Equation of Motion
  • Component Forms of Equations of Motion

• Problems

The following sections are included:

• Thermodynamics of Deformation
  • First Law of Thermodynamics
  • Second Law of Thermodynamics

• Fundamental Constitutive Principles
  • Principle of Coordinate Invariance
  • Principle of Determinism
  • Principle of Local Action
  • Principle of Equipresence
  • Principle of Material Frame Indifference
  • Principle of Physical Admissibility

• Strain-Energy Density Function
  • Orthotropic Material
  • Transversely Isotropic Material
  • Isotropic Material

• Stress-Strain Relations
  • Isotropic Material
  • Transversely Isotropic Material
  • Orthotropic Material
  • Curvilinear Anisotropy

• Incompressibility

• Linear Elastic Material
  • Isotropic Material
  • Transversely Isotropic Material
  • Orthotropic Material

• Determining W for Soft Tissues
  • Microstructural and Phenomenological Approaches
  • Experimental Considerations
  • Some Types of Experiments

• Problems

The following sections are included:

• Boundary Value Problems

• Extension and Compression of Soft Tissue
  • Governing Equations
  • Biaxial Stretching of a Membrane
  • Uniaxial Extension of a Bar

• Simple Shear of Soft Tissue
  • Governing Equations
  • Solution

• Extension and Torsion of a Papillary Muscle
  • Governing Equations
  • Solution

• Extension, Inflation, and Torsion of an Artery with Residual Stress
  • Artery Model
  • Governing Equations
  • Solution
  • Geometric and Material Properties
  • Residual Stress in Unloaded Artery
  • Loaded Artery

• Passive Filling of the Left Ventricle
  • Model for the Left Ventricle
  • Analysis
  • Geometric and Material Properties
  • Results

• Blastula with Internal Pressure
  • Governing Equations
  • Solution
  • Geometric and Material Properties
  • Results

• Bending of the Embryonic Heart
  • Governing Equations
  • Solution
  • Geometric and Material Properties
  • Results

• Problems

The following sections are included:

• Appendix A Linear Theory of Elasticity
  • Mathematical Preliminaries
    • Base Vectors
    • Vectors, Dyadics, and Tensors
    • Coordinate Transformation
    • Vector and Tensor Calculus
  • Analysis of Deformation
    • Strain
    • Rotation
    • Principal Strains
    • Compatibility Conditions
  • Analysis of Stress
    • Body and Contact Forces
    • Stress Tensor
    • Principal Stresses
    • Equations of Motion
  • Constitutive Relations
    • Thermodynamics of Deformation
    • Strain-Energy Density Function
    • Linear Elastic Material
  • Boundary Value Problems
    • Torsion of a Circular Cylinder
    • Torsion of a Noncircular Cylinder

• Appendix B Special Coordinate Systems
  • Cylindrical Polar Coordinates
  • Spherical Polar Coordinates
  • Toroidal Coordinates

• Bibliography

• Index