C. Truesdell

Mechanical Basis of Diffusion

Four types of theory of diffusion are distinguished, presented, and compared: kinematical, hydrodynamical, kinetic, thermodynamic. A simple mechanical theory, based upon recognizing the diffusive drags as forces that produce motions, is shown to include and unify all earlier attempts. In this theory a hypothesis of binary drags is formalized and shown to lead to the symmetry relations proposed originally by Stefan. The Onsager relations for pure diffusion are proved to be equivalent to Stefans relations. Known results show that these relations hold as a first approximation in the kinetic theory of monatomic gas mixtures. Whether or not they hold in higher approximation is unknown.

General and Exact Theory of Waves in Finite Elastic Strain

After the classical researches of Christoffel, Hugoniot, Hadamard, and Duhem on waves in elastic materials, it might seem that little remains to be learned. Such is not the case. As for most parts of mechanics, it has been necessary in the last decade to go over the matter again, not only so as to free the conceptual structure from lingering linearizing and to fix it more solidly in the common foundation of modern mechanics, but also so as to derive from it specific predictions satisfying modern needs for contact between theory and rationally conceived experiment. After reading the recent papers by Toupin & Bernstein [1961, 1] and by Hayes & Rivlin [1961, 2], I have seen that more can be learned than is there proved. In the present paper I follow Toupin & Bernstein’s approach to the general theory yet try to achieve the elegant and explicit directness of Ericksen’s earlier treatment of isotropic incompressible materials [1953]. At the same time, all the results of Hayes & Rivlin are obtained in shorter but more general form as immediate corollaries.

On a function which occurs in the theory of the structure of polymers

plays an essential role in his researches on the theory of the structure of polymers. In this context s is a rational number, such as 3, -2 or I and x is a real number between 0 and 1. While the series (1) is rapidly convergent near 0, near 1 it is so slowly convergent as to be quite useless for computation, and Mr. Jacobson asked me to find some other representation for ?(x, s) which is amenable to numerical evaluation when x is only slightly less than 1. In response to his request I discovered the expansions (9) and (10), which it is the purpose of the present paper to derive, along with some other properties of ?(x, s) which seem of possible interest though not useful in M\Ir. Jacobsons problems. I have also computed a table of the values of ?(x, s) when 0 1 does the series remain convergent at the point x = 1; then

Static grounds for inequalities in finite strain of elastic materials

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Static grounds for inequalities in finite strain of elastic materials C. Truesdell, R. Toupin

An Essay toward a Unified Theory of Special Functions

The description for this book, An Essay Toward a Unified Theory of Special Functions. (AM-18), will be forthcoming.

Wave propagation in dissipative materials

The laws of wave propagation cast light upon the nature of material response by analysis. They show how the material reacts, locally and instantaneously, to a small change or impulse in a tiny region. To compose the small effects into a motion of the body as a whole is a much harder problem even in the simplest of theories, a problem generally too hard to solve, yet we gain insight and assurance by taking the preliminary step even though we rarely follow through. The term “small” has two distinct meanings: that the disturbance itself is small, and that it is confined to a small region. There are several different approaches to wave motion, resting upon different concepts of smallness. In the commonest of these, the differential equations of motion are shorn of their non-linear terms so as to yield a linear system which may be visualized as an assembly of harmonic oscillators, whose motions may be described in the terms hallowed by centuries of contemplation of the pendulum: frequency, amplitude, wave length, phase shift. It was to this method that HUGONIOT referred when, in 1885, he wrote: “… hypotheses have been imposed upon the equations of hydrodynamics which are disguised, it is true, by the word of approximation, but which singularly alter the value of such results as can be deduced from them.” Hugoniot himself developed in fairly general terms a different concept of wave propagation in which the disturbance is limited, rigorously, to a region of no volume at all, namely, a surface, but the disturbance itself may be of any amount.

Maria Gaetana Agnesi

M. G. Agnesi (1718-1799), mathematicienne. Sa vie et son oeuvre. Le role des femmes intellectuelles au 18e siecle

Foundations of Continuum Mechanics

Last week Mr. Noll presented a new, compact, and general theory of space-time structure for Euclidean continuum mechanics. In mathematical science organization can be effected at various levels. For example, we may construct the real numbers from the integers, or we may take axioms for the real numbers themselves as our starting point. Mr. Noll’s space-time structure furnishes a foundation for the theory of constitutive equations he formulated some years ago, but it does not change that theory or render it either more or less precise. Along with deepening the foundations, mathematical science strives also to broaden the structure. The great clarity gained from the abstract approach to constitutive equations for purely mechanical phenomena has made it possible to extend the structure of rational mechanics so as to include thermo-energetic effects in comparable generality. Thermostatics, now a century old, was never intended to apply to problems of deformation and motion; the linear “irreversible thermodynamics”, which rests on applying classical thermostatics to volume elements, has fallen so far behind recent views on mechanics as to put it quite out of the running, and the new thermodynamics, developed by Mr. Coleman, makes no use of its concepts or apparatus.

Absolute temperatures as a consequence of Carnot's general axiom

1. Program In our book, Concepts and Logic of Classical Thermodynamics,1 Mr. BHARATHA and I developed classical thermodynamics on the basis of Part I of Carnots General Axiom, namely, the motive power of a Carnot cycle is positive and is determined by its operating temperatures and by the amount of heat it absorbs. For a given body, then, there is a function G such that for any Carnot cycle # L(%) = G(0+,0 ,C + (^j)>0. (1) The domain of G is the set of operating temperatures and heats absorbed that may appertain to Carnot cycles for the body in question. It is part of the definition of a Carnot cycle that 6+>6~ and that C+(^)>0. This definition and 1 C. Truesdell & S. Bharatha, Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech, N.Y. etc., Springer-Verlag, 1977.

History of classical mechanics

Continuing the story begun in Part I, this article sketches the development of rational mechanics in the nineteenth century. Then it was taught to every physicist and, as time went on, to an ever greater proportion of engineers. Not only was it the core of their training in natural science, but also it served as the paradigm of a scientific discipline. The rise of relativity and the theory of quanta left classical mechanics the preserve of engineers for the first half of the 20th century. Recently classical mechanics, particularly in reference to severely deformable bodies, has come to be studied and developed again as a science in its own right, much as it was in the Age of Reason and the Enlightenment.