R. S. Rivlin

Stress-Deformation Relations for Isotropic Materials

It has been shown (see C. Truesdell (1952) for a comprehensive review of this subject) that the mechanics of a homogeneous isotropic ideally elastic material may be developed on the basis of a description of the relevant elastic properties of the material in terms of a strain-energy function W which is a single-valued function of three scalar invariants of the deformation, I 1, I 2 and I 3.

Large Elastic Deformations

In a previous chapter, the foundations of the classical theory of elasticity have been presented. It is concerned with the description and explanation in terms of a unified theory of the relations which are observed between load and deformation for elastic solids of various shapes, sizes, and compostions. The elastic character of the materials to which the theory is applicable may be loosely described as follows: If a body of elastic material is subjected to a load, it will be deformed and on the removal of the load will regain its initial dimensions and shape. In the concept of the elastic solid is also contained the assumption that the mechanical system constituted by an elastic solid and a system of forces applied to it is a system in which energy is conserved. The work done by the applied forces during the iso thermal deformation of the body is balanced by potential energy stored in the elastically deformed body and the kinetic energy of the various parts of the body and the members through which the deforming forces are applied.

Large Elastic Deformations of Isotropic Materials

The theory of the large elastic deformation of incompressible isotropic materials is applied to problems involving thin shells. The inflation of a circular diaphragm of such a material is studied in detail. It is found that the manner in which the extension ratios and curvatures vary in the immediate neighbourhood of the pole of the inflated diaphragm can be determined analytically. However, in order to determine their variation throughout the inflated diaphragm a method of numerical integration has to be employed. Although this is, in principle, valid for any form of the stored-energy function, the calculations are carried through only for the Mooney form.

The Mechanics of Non-Linear Materials with Memory

The mechanics of non-linear materials with memory has been discussed in previous papers [1, 2] and an earlier report [3]*. In these papers, it was assumed that, in a fixed coordinate system, the stress at a point of the material at any instant of time is determined by the deformation gradients at that instant and at previous instants. The limitations imposed on the form of the constitutive equation by the consideration that it is unaltered by a simultaneous rotation of the body and reference system are determined. In the present paper, we assume initially a constitutive equation in the form of implicit relations between the stress, its time derivatives and gradients of displacement, velocity, acceleration, etc. at a number of instants of time in some time interval. The limitations on the form of the constitutive equation resulting from the fact that it is unaltered by the imposition on the body of an additional arbitrary angular velocity, acceleration, etc. are discussed.

The Hydrodynamics of Non-Newtonian Fluids. I

The classical theory of the hydrodynamics of viscous fluids depends on the assumption of a particular law governing the relations between the components of stress in a fluid and those of the strain-velocity. This assumption limits its applicability to Newtonian fluids. Here, the most general possible relations between the stress and strain-velocity components, which can be obeyed by an incompressible, visco-inelastic fluid, are derived. These relations also apply to an incompressible, visco-elastic fluid in a steady state of laminar flow. It is shown how equations of motion and boundary conditions can be obtained if these relations are known. Two problems involving laminar flow are then discussed in some detail. These are: (i) the torsional motion of a cylindrical mass of fluid, produced by means of forces applied to its plane ends, and (ii) the laminar flow of a mass of fluid contained between two coaxial cylinders rotating with different angular velocities. It is found in case (i) that, in general, normal tractions must be applied to the plane surfaces of the fluid, in addition to the azimuthal tractions expected from the classical theory, in order to produce the specified motion. Analogous results are obtained in case (ii). These results apply even when centrifugal forces are neglected and so imply a qualitative difference between the behaviour of fluids in general and those for which the special case of classical hydrodynamics is valid.

The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua

In this paper we show that a symmetric isotropic matrix polynomial in any number of symmetric 3 × 3 matrices can be expressed as a symmetric isotropic matrix polynomial, in which each of the matrix products is formed from at most six matrices and has one of a certain number of forms which are explicitly given. The significance of these results in the mechanics of isotropic continua is indicated.

On Cauchy’s Equations of Motion

ZusammenfassungCauchys Bewegungs- und Momentengleichungen der klassischen Kontinuumsmechanik werden, mit Hilfe von Invarianz-Bedingungen gegenüber starren Zusatzbewegungen, aus dem Energietheorem hergeleitet.

Torsion of a Rubber Cylinder

It has been predicted theoretically that, in general, a right‐circular cylinder of incompressible, highly elastic material, which is isotropic in its undeformed state, cannot be held in a state of pure torsional deformation by means of a torsional couple alone. In addition, normal surface tractions must be exerted over the plane ends of the cylinder. These normal surface tractions depend on the amount of torsion and on position on the plane ends of the cylinder. Experiments are reported here in which this phenomenon is observed in a right‐circular cylinder of pure gum compound. The dependence of the surface traction on amount of torsion and its distribution over the surface of the cylinder is studied by measuring the bulging of the rubber into small holes in a metal plate on one end of the cylinder.

Flow of a Newtonian fluid between eccentric rotating cylinders: Inertial effects

A detailed analysis is carried out of the flow of a Newtonian fluid in the annular region between two infinitely long circular cylinders with parallel axes, resulting from the uniform rotation of one, or both, of the cylinders about their axes. No restriction is placed on the geometry of the system and results are obtained both with the neglect of inertial effects and for the linearized inertial approximation. In both cases, the resultant of the forces exerted by the fluid on the cylinders and the distribution of their normal and tangential components over the cylinders are calculated, and the stream-line patterns are analyzed in some detail. A number of conditions, under which stagnation points, separation points and eddies can exist, are established.

Mechanics of rate-independent materials

Es werden Stoffgleichungen fur endliche Deformationen von geschwindigkeitsunabhangigen Materialien mit Gedachtnis diskutiert. Die Piola-Spannung wird als Funktional des Verformungsweges im Verzerrungsraum angesetzt, unabhangig von der Geschwindigkeit, mit der dieser Weg durchlaufen wird. Ferner werden die Beschrankungen hervorgehoben, die sich fur dieses Funktional aus der ursprunglichen Isotropie ergeben. Es werden differentielle und Integral-Verfahren betrachtet, und es wird eine allgemeine Theorie elastisch-plastischer Materialien, das heisst geschwindigkeitsunabhangiger Stoffe mit einem elastischen Bereich, formuliert.