Nonlinear Problems of Elasticity

Mathematical statements such as formulas, theorems, figures, and exercises are numbered consecutively in each section. Thus formula (III.4.11) and Theorem III.4.12 are the eleventh and twelfth numbered statements in Sec. 4 of Chap. III. Within Chap. III, these statements are designated simply by (4.11) and Theorem 4.12.

The main purpose of this chapter is to give a derivation, which is mathematically precise, physically natural, and conceptually simple, of the quasilinear system of partial differential equations governing the large motion of nonlinearly elastic and viscoelastic strings. A part of this treatment is a careful study the Principle of Virtual Power and the equivalent Impulse-Momentum Law, which are physically and mathematically important generalizations of the governing equations. These formulations form the foundation of the treatment of concrete problems, which is begun in this chapter for very simple problems and is pursued in greater depth in Chaps. III and VI. These formulations also serve as models for those for more complicated bodies, such as rods, shells, and three-dimensional bodies, which we study later.

The development of both continuum mechanics and mathematics during the eighteenth century was profoundly influenced by the successful treatment of conceptually simple, but technically difficult problems. (Indeed, one can argue that the dominant philosophical attitudes of the Age of Reason were founded on an awareness of these scientific triumphs, if not on their understanding.) Among the most notable of these classical problems are those of determining the equilibrium states of inextensible strings hung between two points and subjected to various systems of loads. (An inextensible string is one for which the stretch v is constrained to equal 1, no matter what force system is applied to the string.) The problem of the catenary is to determine the equilibrium states of such a string when the applied force is the weight of the string. The problem of the suspension bridge is to determine these states when the applied force is a vertical load of constant intensity per horizontal distance. (The string does not correspond to the bridge, but to the wires from which it is suspended.) The problem of the velaria is to determine these states when the applied force is a normal pressure of constant intensity. In the related problem of the lintearia, the applied force is a normal pressure varying linearly with depth. (This problem describes the deformation of a cylindrical membrane holding a liquid, the string representing a typical section of the membrane.) In a fifth problem, the applied force is the attraction to a fixed point. In Sec. 8 we outline the progression from haphazard conjecture to elegant solution of these problems in the seventeenth and eighteenth centuries.

A theory of rods is the characterization of the motion of slender solid bodies by a finite number of equations in which there is but one independent spatial variable. (The theory of strings, formulated in Chap. II, is thus an example of a theory of rods.) In this chapter we formulate and analyze equilibrium problems for the planar deformation of elastic rods. The intrinsically one-dimensional theory that we employ, which may be called the special Cosserat theory of rods, has several virtues: It is exact in the same sense as the theory of strings of Chap. II is exact, namely, it is not based upon ad hoc geometrical approximations or mechanical assumptions. It is much more general than the standard theories used in structural mechanics. Many important concrete problems for the theory admit detailed global analyses, some of which are presented below.

If a naturally straight thin rod, such as a plastic or metal ruler, is subjected to a small compressive thrust applied to its ends, it remains straight. If the thrust is slowly increased beyond a certain critical value, called the buckling load, the rod assumes a configuration, called a buckled state, that is not straight. See Fig. 1.1. This process is called buckling. Depending on the precise mode of loading and the nature of the rod, the transition to a buckled state can be very rapid. If the thrust is further increased, the deflection of the rod from its straight state is likewise increased. If this entire process is repeated, the rod may well buckle into another configuration such as the reflection of the first through a plane of symmetry. The performance of a whole series of such experiments on different rods would lead to the observation that the buckling loads and the nature of buckled states depend upon the material and shape of the rod and upon the manner in which it is supported at its ends. It can also be observed that the results of experiments are highly sensitive to slight deviations of the rod from perfect straightness or of the thrust from perfect symmetry. The study of buckling for different bodies is one of the richest sources of important problems in nonlinear solid mechanics.

In this chapter we apply the theory of Chap. V to specific bifurcation problems for strings and rods. Here we confront certain singular boundary-value problems (like those described in Ex. V.5.6). We concentrate on effects caused by the nature of constitutive response. We develop techniques by which we can convert given boundary-value problems into mathematical forms suitable for global analysis, and techniques by which we can extract very detailed information about specific classes of problems. We begin by studying the multiplicity of steady rotating states of elastic strings.

In this chapter we describe a few simple applications of the calculus of variations to one-dimensional problems of nonlinear elasticity, with the purpose of showing how some very useful physical insights can be easily extracted from the basic theory. The reader should review Secs. II.10 and III.3. We require a mathematical setting somewhat different from that used in bifurcation theory; the technical aspects of this setting are given below in small type.

In this chapter we study spatial deformations for nonlinearly elastic rods. We collect here the governing partial differential equations for transversely isotropic rods from the preceding chapter. Equations (VIII.2.3) and (VIII.2.4) yield

We now give a deeper treatment of the material introduced in Secs. I.4 and IV.1. Much of our exposition consists of assertions of standard results, the proofs of which are given in the references cited at the end of Sec. 2.

In this chapter we present a formulation of continuum mechanics directed toward the treatment of the behavior of solids. It is based on the material (or Lagrangian) description of motion. Virtually all the problems we treat will be cast in this formulation. At the conclusion of the chapter, we describe the modifications needed for the spatial (or Eulerian) formulation, which is essential for fluid mechanics and which has been used in solid mechanics. References for standard material of this chapter expressed in a related style are Chadwick (1976), Gurtin (1981a), and Truesdell & Noll (1965). The last work has extensive historical notes. References for specialized topics in this chapter are given in the individual sections.

We now collect all the results of Chap. XII that are pertinent to elasticity. p(z,t) is the position of material point z of body B at time t. We define The classical form of the balance of linear momentum is where T is the first Piola-Kirchhoff stress tensor, f is the prescribed body force intensity per unit reference volume, and ρ is the mass density per unit reference volume.

In Chap. N we defined a theory of rods to be the characterization of the motion of slender solid bodies by a finite number of equations in which there is but one independent spatial variable, which we denote by s. There are several kinds of rod theories, reflecting different ways to construct them. Perhaps the most elegant are the intrinsic(ally one-dimensional) theories (Cosserat Theories), the simplest example of which is that presented in Chap. N. In intrinsic theories, the configuration of a rod is defined as a geometric entity, equations of motion are laid down, and constitutive equations relating mechanical variables to geometrical variables are prescribed. But, as we saw in Chap. VIII, there are parts of the theory that are best developed under the inspiration of the three-dimensional theory. In the special Cosserat theory of Chap. VIII, the classical equations of motion, namely, the balances of linear and angular momentum, suffice to produce a complete theory. They are inadequate for more refined intrinsic theories. In our treatment of refined intrinsic theories in Sec. 7, we discuss the construction of the requisite additional equations of motion.

In this chapter we discuss a general class of materials with memory, the plastic materials, which are useful in describing the behavior of metals. Our purpose is to present the basic theory, in which some concepts of Chap. XII are further developed and illustrated, and in which the theory of elasticity plays a central role, in as simple a context as is compatible with the underlying physics. The exposition is simpler than that of most treatments because we consistently use internal variables in the material formulation, which obviates the need for a complicated treatment of frame-indifference, as is necessary in the spatial treatment of theories involving stress rates. To illustrate the nature of such theories, we treat a model with a lot of physical structure in Secs. 2 and 3. In Sec. 4 we show how to formulate a natural numerical approach for the solution of a particular dynamical problem. In Sec. 5 we give a general formulation of antiplane problems, whose degeneracies illuminate subtle difficulties with the concept of permanent plastic deformation. We conclude this chapter with a brief discussion of discrete models, which are used to motivate various theories.

In this chapter we treat a collection of elementary but illuminating dynamical problems for elastic and viscoelastic bodies. This material merely serves as an entrée to some parts of the rich and fascinating modern research on the quasilinear hyperbolic and parabolic systems applicable to elasticity. We develop the theory only in the context of concrete problems, so that we can concentrate on the effects of constitutive hypotheses.

A Banach space is a vector space with very attractive convergence properties. For our purposes, the most important Banach spaces are spaces of functions. Formally, a Banach space is a complete, normed, vector space. Let us now define each of these terms. By a scalar we mean a real or complex number.

In many of the problems treated in this text, we have detailed information about special solutions, especially those termed trivial. We can often determine solutions in a neighborhood of the special solutions or determine solutions of problems with nearby data by using methods relying on versions of the Implicit Function Theorem. In this chapter we develop these methods of local nonlinear analysis. Each of our results is a consequence of the Contraction Mapping Principle, which we now state and prove.

Throughout this section, we take Ω to be a bounded open subset of ℝ ⁿ and take cl Ω ∋ × → f(x) ∈ ℝ ⁿ to be continuous. We wish to estimate the number of solutions x in cl Ω of the equation We shall often be content with demonstrating that there is (at least) one solution. We denote the collection of solutions of (1.1) lying in c152 by f⁻¹ ({0}).