Nonlinear Theory of Elasticity

The following sections are included:

• Dedication
• Preface to the Revised Edition
• Preface to the First Edition
• 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.

Physical quantities can be represented by tensors. Zeroth-order tensors (scalars) have only a magnitude (e.g., temperature), first-order tensors (vectors) have a magnitude and direction (e.g., velocity), second-order tensors (dyadics) have a magnitude and two associated directions (e.g., stress), and so on. The literature on nonlinear (finite) elasticity is filled with tensors, which provide a convenient tool for deriving the governing equations. For solving these equations, tensor quantities then can be expressed in terms of components relative to any convenient coordinate system. Because much of the historical literature on finite elasticity deals with general curvilinear coordinates, such generality is a main feature throughout this book.

The mechanics of hard tissues, such as bones, teeth, and horns, can be analyzed using the linear theory of elasticity, in which deformation is assumed to be “small.” Under this condition, the distinction between the geometries of the undeformed and the deformed body can be ignored. Soft tissues, in contrast, often undergo “large” (finite) deformation. In this case, geometric nonlinearity enters the analysis, even if the material properties (stress-strain relations) are linear. This chapter deals only with geometry. Nonlinear stress-strain relations characterize material nonlinearity, as discussed in Chap. 5…

The classical definition of stress is force per unit area. But which area: the undeformed area or some other reference area? For small deformation, the distinction is immaterial, since changes in area are negligible. For large deformation, however, we must distinguish between the undeformed and deformed geometries. In this case, the only physically meaningful definition for stress is force per unit deformed area, which is called true stress. But the deformed geometry of a solid body is not known a priori, often is difficult to measure, and may be time-dependent. Experimentalists, therefore, find it useful to define a Lagrangian stress or engineering stress as the force per unit undeformed (reference) area. In addition, the governing equations sometimes have simpler forms or may be easier to solve when written in terms of yet another type of stress (to be defined later). We call any stress that is not the true stress a pseudostress, since it has no physical significance. Pseudostresses always must be converted into true stress for physical interpretation. This chapter introduces various types of stress tensor and discusses the relations between them.

The analyses of deformation and stress in the previous chapters are valid for any solid body that can be represented as a continuum, regardless of the type of material that comprises the body. Deformation and stress, however, are linked by constitutive relations that must be found experimentally for each type of material. In biomechanics, determining constitutive relations is complicated by material nonlinearity, complex geometry, the composite nature of biological tissues, and the influence of a wide range of environmental variables. In general, even the functional forms of these relations are unknown, but thermodynamic considerations place important restrictions on their form. This chapter considers some of these restrictions and discusses some strategies for determining constitutive relations for soft tissues.

Finite element and other computational methods now make it possible to solve virtually any problem in biomechanics, no matter the complexity of the geometry, deformation, and material properties. Nevertheless, “exact solutions” remain useful for several reasons. First, they provide insight into the fundamental nonlinear behavior of soft tissues. Second, they provide benchmarks for checking the accuracy of numerical solutions. Third, as discussed in the previous chapter, exact solutions can be used in combination with experiments to determine material properties. Such experiments often involve relatively simple geometries and deformations, making the use of exact solutions convenient.

This chapter considers a number of fundamental problems in soft tissue biomechanics. Besides integrating the material presented in previous chapters, we use these problems to illustrate the effects of compressibility and anisotropy. Although soft tissues usually are assumed to be incompressible, extracellular fluid flow can produce significant compressibility effects, and microstructural considerations suggest that many soft tissues can be treated as orthotropic or transversely isotropic materials. Realistic models must take these features into account. For simplicity, gravity and other body forces, as well as inertia effects, are ignored in all problems of this chapter (with the exception of Problem 6.9).

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

The following sections are included:

• Problem Solutions
• Bibliography
• Index