William O. Williams

On the thermodynamics of non-simple elastic materials with two temperatures

ZusammenfassungDiese Arbeit behandelt eine thermodynamische Theorie von nichteinfachen, elastischen Stoffen. Es wird gezeigt, dass Substanzen vom Grade höher als eins vorkommen können; vorausgesetzt, dass man das eventuelle Vorhandensein zweier verschiedener Temperaturen in Betracht zieht.

A note on non-simple heat conduction

ZusammenfassungIn einem nicht-einfachen Material kann die Wärmeübertragung durch zwei Temperaturen geregelt werden; die eine regelt die Strahlung, die andere die Leitung. In dieser Arbeit zeigen wir unter Benutzung früherer Ergebnisse, dass bei stationären Bedingungen die Differenz der beiden Temperaturen proportional zur Strahlungsdichte ist.

Geometric Effects in an Elastic Tensegrity Structure

Tensegrity structures are under-constrained, 3-dimensional, self-stressing structural systems. They demonstrate an infinitesimal flex and when loaded they display a nonlinear geometric stiffening. In earlier work many examples of the resulting force–displacement relationship have been demonstrated numerically, and some aspects of the force–displacement relationship have been derived analytically. In this article an energy formulation is presented for the case of a simple but representative tensegrity structure, yielding an exact solution for the force–displacement relationship. The solution makes understandable the different appearance of the force–displacement relationship when comparing a system at zero prestress to one at high prestress, or when comparing a system with almost-inextensible members to one with highly extensible members. The exact solution also is offered as a benchmark against which numerical solutions should be tested. Furthermore, the formulation and the solution reveal conditions of asymmetry of response that have not been noted previously.

Vibration of an elastic tensegrity structure

The dynamic behavior of a simple elastic tensegrity structure is examined, in order to validate observations that the natural damping of the elastic elements in such a structure is poorly mobilized, due to the natural flexibility of the equilibrium position of the structure. It is confirmed, analytically and numerically, that the energy decay of such a system is slower than that of a linearly-damped system.

Shells with thickness distension

Abstract A deductive approach to shell theory is presented, within which a shell is regarded as a constrained three-dimensional continuum with special body structure: more precisely, admissible deformations are given a restricted form, presumed to be consistent with the special shape and partitionability of a shell-like body. In addition to the standard balance equations of forces and torques, an extra balance equation is derived, allowing for a description of dilatation or contraction in the thickness dimension. As an illustrative application, the free oscillations of linearly elastic plates – in particular, thickness–distension waves – are studied.

On rate-type constitutive equations and the energy of viscoelastic and viscoplastic materials

Abstract We discuss the qualitative behavior of the constitutive relation σ = E(ϵ, σ)ϵ + G(ϵ, σ). We show, for example, that this relation exhibits hypoelastic behavior under retardations of the time scale and rate-independent plastic behavior under accelerations of the time scale. We prove further that for a viscoelastic material governed by such a constitutive relation there exists a unique free energy. For a viscoplastic material, however, there are an uncountable infinity of free energies.

On the Viscosities of Liquid Mixtures

There have been many empirical formulae proposed which would give the viscosity of a binary mixture of liquids in terms of the viscosities of the liquids and of the composition. Several years ago kinetic theory arguments were used to rederive an old formula due to Dolezalek and Schulze, which has the virtue of simplicity and, apparently, good accuracy. The parameters of this formula are the two fluid viscosities and a mutual viscosity. In this note we observe that such formulae fit exactly into the continuum theory of mixtures recently developed by us and others. We use Dolezaleks and Schulzes formula and thermodynamic restrictions, elsewhere derived, to present a generalized Navier-Stokes system for the flow of mixtures of liquids.ZusammenfassungEs sind viele empirische Formeln vorgeschlagen worden, die die Viskosität einer Mischung von zwei Flüssigkeiten als Funktion der Viskositäten der beiden Flüssigkeiten ausdrückt. Vor mehreren Jahren wurden Argumente aus der kinetischen Theorie benützt, um eine alte Formel von Dolezalek und Schulze wieder herzuleiten, die den Vorzug der Einfachheit hat und anscheinend ziemlich genau ist. In dieser Arbeit wird bemerkt, dass solche Formeln exakt in die Kontinuumstheorie von Mischungen passen, die vor kurzem von den Autoren und von andern entwickelt wurde. Wir benützten die Formel von Dolezalek und Schulze und thermodynamische Einschränkungen, die anderswo abgeleitet wurden, um eine verallgemeinertes Navier-Stokes-System für die Strömung von Mischungen von Flüssigkeiten anzugeben.

ON FLUIDS OF GRADE n

SommarioI fluidi elastici sono generalmente definiti attraverso il loro gruppo di simmetria; è quindi possibile mostrare la dipendenza della funzione di risposta dalla sola densità. In questa nota diamo una analoga definizione di fluido elastico di grado n. Mostriamo poi che le proprietà di un tale fluido sono determinate dai primi n−1 gradienti spaziali della sua densità.SummaryElastic fluids are generally defined through their symmetry group; it is then possible to show the dependance of the response function of the density alone. In this note we give an analogous definition of elastic fluid of grade n. We then show that the properties of this fluid are determined by the first n−1 spatial gradients of its density.

A generalized stefan condition

SummaryIn this paper we apply the theory of surface thermodynamics to find the jump condition on heat flux across a melting-surface. We distinguish between the effects of surface tension and of surface energy and describe relations between the two which may ensure enhancement or diminution of the curvature of the surface.ZussamenfassungIn dieser Arbeit wenden wir die Theorie der Oberflächen-Thermodynamik auf die Bestimmung der Sprungbedingung für den Wärmefluss an einer Schmelzfläche an. Wir trennen die Oberflächenspannungs-Effekte von den Oberflächenenergie-Effekten und beschreiben Beziehungen zwischen den beiden, die eine Vergrösserung oder Verkleinerung der Oberflächenkrümmung bedingen können.

On Stresses in Mixtures

The first formulation of the theory of mixtures in the terminology of modern continuum mechanics was by TRUESDELL [1], who traced the general ideas to FICK [2] and STEFAN [3]. Since then the theory has been further developed and extended by a number of authors. 1 However, it seems to me that at the fundamental level of formulation of boundary conditions and, more generally, of interpretation of the fields in the theory there remains a great deal of confusion and contradiction. For example, while it is indisputable that the partial stress tensor represents a force exerted per unit area, it is not always so clear exactly upon what the force is exerted and by what. In fact I claim that the usual interpretation of the partial stress tensor is misleading; in this paper I present some arguments against it and present an alternative point of view. In the first section I give an outline of a formal structure for mixture theory and in the second present axioms which formalize the (usually implicit) assumptions which underlie the classical interpretation of the equations of balance of forces for mixtures. I think that this formalization reveals clearly the weaknesses inherent in this interpretation of the theory. 2 In the third section of the paper I present a set of axioms, based on work done by GURTIN, DE LA PENHA, OLIVER, SAMPAIO and me [6-9], which are free of the peculiar defect of the other approach, and I show how they may yield the classical equations. I also present arguments as to why this interpretation of these equations may be more useful when one seeks to create manageable constitutive equations for the theory. In this paper I give no proofs; all results may be established easily by simple translation of the proofs in [7] or [8].