Walter Noll

The thermodynamics of elastic materials with heat conduction and viscosity

The basic physical concepts of classical continuum mechanics are body, configuration of a body, and force system acting on a body. In a formal rational development of the subject, one first tries to state precisely what mathematical entities represent these physical concepts: a body is regarded to be a smooth manifold whose elements are the material points; a configuration is defined as a mapping of the body into a three-dimensional Euclidean space, and a force system is defined to be a vector-valued function defined for pairs of bodies1. Once these concepts are made precise one can proceed to the statement of general principles, such as the principle of objectivity or the law of balance of linear momentum, and to the statement of specific constitutive assumptions, such as the assertion that a force system can be resolved into body forces with a mass density and contact forces with a surface density, or the assertion that the contact forces at a material point depend on certain local properties of the configuration at the point. While the general principles are the same for all work in classical continuum mechanics, the constitutive assumptions vary with the application in mind and serve to define the material under consideration.

An approximation theorem for functionals, with applications in continuum mechanics

We start with a brief discussion of the physical motivation behind the mathematical considerations to be presented in Part I of this paper.

A Mathematical Theory of the Mechanical Behavior of Continuous Media

Until not long ago continuum mechanics meant to most people the theories of inviscid and linearly viscous fluids and of linearly elastic solids. However, the behavior of only few real materials can be described adequately by these classical theories. Experimental scientists, who had to deal with real materials, developed a science of non-classical materials called rheology. But they did not succeed in fitting their experimental results into a general mathematical framework. Most of the rheological theories are either one-dimensional, and hence appropriate at best for particular experimental situations, or are confined to infinitesimal deformations, in which case they are only of limited use, because large deformations occur easily in the materials these theories are intended to describe.

Foundations of Linear Viscoelasticity

The classical linear theory of viscoelasticity was apparently first formulated by Boltzmann1 in 1874. His original presentation covered the three-dimensional case, but was restricted to isotropic materials. The extension of the theory to anisotropic materials is, however, almost immediately evident on reading Boltzmann’s paper, and the basic hypotheses of the theory have not changed since 1874. Since that date, much work has been done on the following aspects of linear viscoelasticity: solution of special boundary value problems,2a reformulation3,4 of the one-dimensional version of the theory in terms of new material functions (such as “creep functions” and frequency-dependent complex “impedances”) which appear to be directly accessible to measurement, experimental determination2b of the material functions for those materials for which the theory appears useful, prediction of the form of the material functions from molecular models, and, recently, axiomatization5,6 of the theory.

A New Mathematical Theory of Simple Materials

The original theory of simple materials was formulated by me in 1958 in reference [N2]. I proposed this theory in an attempt to unify and clarify the confusing variety of theories of mechanical behavior of materials that had been proposed in the literature up to that time. One can perhaps say that the attempt was moderately successful, considering that this first theory of simple materials has served as a foundation for a large part of the research in continuum physics since 1958, and considering that the concepts and even the notations of [N2] are now used routinely and without reference in textbooks on continuum mechanics.

Lectures on the Foundations of Continuum Mechanics and Thermodynamics

This is an outline of a series of lectures I delivered at the Technion in Haifa, Israel, in the summer of 1972. This outline gives my view of what the basic concepts of modern continuum mechanics and thermodynamics are and how they should be presented to graduate students of mathematics and theoretical mechanics. This view evolved gradually over the past eight years. A first version was given in a series of lectures I delivered at the Summer Session in Bressanone, Italy, in 1965 (reference [1]). The material was reworked several times for an introductory graduate course repeatedly given at Carnegie-Mellon University and for lecture series given at the University of Karlsruhe, Germany, in 1968 and in Jablonna, Poland, in 1970. This paper is the latest version, and it renders reference [1] obsolete.

RECENT RESULTS IN THE CONTINUUM THEORY OF VISCOELASTIC FLUIDS

This article is concerned with phenomenological aspects of the mechanical behavior of viscoelastic? fluids. The discussion is limited to those special circumstances in which all nonmechanical influences can be neglected; that is, no consideration is given to thermal, chemical, and electromagnetic phenomena. After reviewing in Section 1 some kinematical prerequisites, we give in Section 2 a mathematical definition of the concept of a simple fluid. This definition is sufficiently general to include perfect fluids, Newtonian fluids, and viscoelastic fluids as special cases. In Section 3 we discuss the behavior of a simple fluid in steady simple shearing flow. It turns out that for incompressible fluids several steady flow problems can he solved by direct appeal to the definition. Some properties of these solutions that may he of interest to experimenters are discussed in Section 4. Recent results on the behavior of simple fluids in periodic motions arc given in Section 5. In Section 6 we formulate a general “smoothness assumption,” which makes it possible to prove a t,heorem that seems to justify the intuitive notion that for most fluids the theory of the Newtonian fluid should be a first-order correction to the theory of perfect fluids in the limit of “slow motions.” A rigorous procedure for determining the form of higher-order corrections is sketched in Section 7. It is not our intention to supply here formal mathematical proofs of all our assertions, for this article is not intended primarily for mathematicians working in continuum mechanics. Our goal is to summarize in an article of reasonable length certain recent results in the mechanics of fluids that may be of interest to pctlymer physicists. The omitted proofs not already published elsewhere will appear shortly in the Archive for Rational Mechanics and Analgsis. * The work reported in this paper was supported in part by grants from the Air Force Office of Scientific Research, Washington, D.C., under Contract AF 49(638)541 with the Mellon Institute, Pittsburgh, Pa., and from the National Science Foundation, Washington, D.C., under Grant NSF-G 5250 to Carnegie Institute of Technology, Pittsburgh, Pa. t The term viscoelastic does not, have a precise meaning except in linearized theorics. Here we use the word to indicate that we are dealing with materials that do not obey the classical laws of fluid mechanics or elasticity. Our procedure is as follows.

On certain steady flows of general fluids

In a recent article on the mechanical behavior of continuous media [1], the physical concept of a fluid was given a mathematical definition. We believe that this definition covers almost all real fluids (whenever thermal and other non-mechanical effects can be disregarded) and is more general than any proposed previously because it accounts for all hereditary effects including stress relaxation. The general fluids covered by this definition may even exhibit, in some situations, physical phenomena which are usually attributed to solids.

Material Symmetry and Thermostatic Inequalities in Finite Elastic Deformations

This paper is concerned with elastic materials; these are substances for which the present stress S on a material point depends on only the present local configuration1 M of that point: