Allen C. Pipkin

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.

Note on Ericksen's problem

ZusammenfassungDer Aufsatz beweist die Existenz einer Drei-Parameter-Familie von inhomogenen Deformationen mit konstanten Deformationsinvarianten. Diese Deformationen lassen sich in jedem homogenen, isotropen, inkompressiblen elastischen Material durch Oberflächenkräfte aufrecht erhalten.

Equilibrium of Elastic Nets

A general equilibrium theory for nets constructed from two families of perfectly flexible elastic fibres is presented. The fibres are assumed to be continuously distributed and to offer negligible resistance to shear distortion. Configurations of nets are shown to be minimizers of the potential energy of deformation only if the associated fibre stretches are points of convexity of the fibre strain energy functions and the stresses in the fibres are tensile. These results are used to construct a relaxed energy density that automatically accounts for wrinkling of the network. Universal solutions are obtained. These are the deformations that can be maintained in every elastic net by the application of edge tractions and lateral pressure alone. A detailed study of the differential geometry of nets is included to aid in their interpretation. The equilibrium theory for half-slack (wrinkled) nets is developed and applied to the solution of some representative examples.

Stress Analysis for Fiber-Reinforced Materials

Publisher Summary Stress analysis for highly anisotropic materials is a matter of some current interest because of the growing practical importance of fibre-reinforced composites. Some of the most important problems concerning composites deal with the prediction of their properties from those of their constituents, or stress analysis on the microscopic scale, in which individual fibers are recognized. However, there is also a need for the practical methods of stress analysis at the macroscopic level. For this purpose, several closely related continuum models of the mechanical behavior of fibrous materials have been developed in recent years. These models treat the material as a homogeneous continuum with known properties. Much of the theory of ideal composites is independent of the nature of the shearing stress response of the material. The first theory concerned is the finite elastic shearing response; the main development of the theory came in the context of rigid-plastic response. The chapter also discusses the theory of infinitesimal plane strain of ideal composites. The theory is also applicable to the plane stress of fabrics. Displacement boundary value problems are trivial in this theory, but traction problems lead to a system of differential–difference equations of unusual form.

Asymmetric relaxation and compliance matrices in linear viscoelasticity

ZusammenfassungDie Beschränkungen, welche die Symmetrie des Materials den Nachgiebigkeitsmatrizen anisotroper viskoelastischer Stoffe auferlegt, werden ohne Bezug auf thermodynamisch bedingte Symmetrien abgeleitet. Experimente zur Überprüfung der thermodynamischen Theorien lassen sich auf die Unterschiede gründen, die zwischen den mit oder ohne Annahmen über Matrix-Symmetrie erhaltenen Formen bestehen. Transversale Isotropie und kubische Symmetrie werden im Detail behandelt.

Energy-minimizing deformations of elastic sheets with bending stiffness

Necessary conditions for energy-minimizing deformations are derived for a theory of sheets in which the strain energy function depends on the second derivatives of the deformation as well as its first derivatives. All of these conditions are extensions of well-known necessary conditions in classical calculus of variations. The interpretation of some of these conditions as material stability conditions is explained.

Normal stresses in flow through tubes of non-circular cross-section

ZusammenfassungEine visko-elastische Flüssigkeit fliesst durch ein zylindrisches Rohr von nichtkreisförmigem Querschnitt. Die Theorie zweiter Ordnung der visko-elastischen Flüssigkeiten wird auf die Bestimmung der Variation der auf die Rohrwandung ausgeübten Normalspannungen angewandt. Explizite Lösungen für elliptische, dreieckige und rechteckige Querschnitte werden angegeben.

Smoothing and Unsmoothing

When the output from a physical process is entirely a smoothed or blurred version of the input, we should not expect to be able to recover the input from the output, since unsmoothing presumably is not a stable, continuous process. Irregularities and fine details of the input are lost beyond recovery in the smoothing operation. This means that integral equations of the first kind usually do not have solutions. Those that we considered in the preceding chapter actually involve differential-integral operators that do not have a pronounced smoothing effect.

Wiener—Hopf Equations

Wiener and Hopf1 used a novel technique to solve Milne’s equation, which comes up in the theory of radiative equilibrium of stellar atmospheres:

Fredholm Theory with Integral Norms

The theory discussed in Chapter 1 does not seem to apply to even the trivial equation because the norms of the inhomogeneous term and the kernel are both infinite, if we use the supremum norm. However, most of the results that we have obtained depend only on general properties of norms, not on properties specific to the supremum norm. If we change the definition of the norm, the results remain valid, but with a different meaning.