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1986 | Buch | 2. Auflage

Introduction Kapitel lesen Erstes Kapitel lesen

Lectures on Viscoelasticity Theory

verfasst von: A. C. Pipkin

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

This book contains notes for a one-semester course on viscoelasticity given in the Division of Applied Mathematics at Brown University. The course serves as an introduction to viscoelasticity and as a workout in the use of various standard mathematical methods. The reader will soon find that he needs to do some work on the side to fill in details that are omitted from the text. These are notes, not a completely detailed explanation. Furthermore, much of the content of the course is in the problems assigned for solution by the student. The reader who does not at least try to solve a good many of the problems is likely to miss most of the point. Much that is known about viscoelasticity is not discussed in these notes, and references to original sources are usually not give, so it will be difficult or impossible to use this book as a reference for looking things up. Readers wanting something more like a treatise should see Ferry's Viscoelastic Properties of Polymers, Lodge's Elastic Liquids, the volumes edited by Eirich on Rheology, or any issue of the Transactions of the Society of Rheology. These works emphasize physical aspects of the subject. On the mathematical side, Gurtin and Sternberg's long paper On the Linear Theory of Viscoelasticity (ARMA II, 291 (I962» remains the best reference for proofs of theorems.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
This book contains notes for a one-semester course on viscoelasticity given in the Division of Applied Mathematics at Brown University. The course serves as an introduction to viscoelasticity and as a workout in the use of various standard rnathematical methods
A. C. Pipkin
Chapter I. Viscoelastic Response in Shear
Abstract
In the classical linearized theory of elasticity, the stress in a sheared body is taken to be proportional to the amount of shear. The Navier-Stokes theory of viscosity takes the shearing stress to be proportional to the rate of shear. In most materials, under appropriate circumstances effects of both elasticity and viscosity are noticeable. If these effects are not further complicated by behavior that is unlike either elasticity or viscosity, we call the material viscoelastic.
A. C. Pipkin
Chapter II. Fourier and Laplace Transforms
Abstract
The simplest methods of determining J, given G, or vice versa, are based on the use of Laplace transforms. Fourier and Laplace transforms also find a variety of other applications in connection with viscoelasticity theory. Let us briefly review these transform methods.
A. C. Pipkin
Chapter III. Relations Between Modulus and Compliance
Abstract
Connections between some of the gross features of a modulus and the corresponding compliance are easy to obtain by using the reciprocal relation between their s-multiplied transforms. In the present chapter we first consider some exact relations that can be obtained by considering the behavior of the transforms for small values of the transform parameter. We then consider some approximate but more detailed relations that can be obtained by using linear approximations to the response functions on a log-log plot. The main object of these considerations is to develop enough qualitative understanding that, given one response function in graphical or numerical form, graphs of all others can be sketched immediately with fair quantitative accuracy. More refined numerical techniques that can give arbitrarily accurate results are not discussed because they do not lead to any qualitative understanding of the situation.
A. C. Pipkin
Chapter IV. Some One-Dimensional Dynamical Problems
Abstract
We now consider some dynamical problems involving especially simple geometries: torsional oscillations of a rod, and propagation of plane waves in a semi-infinite medium. We have several purposes in mind. First, these problems illustrate some important similarities between the behavior of viscoelastic materials and the behaviors of materials that are purely viscous or purely elastic, and they also illustrate some important distinctions. Second, we wish to illustrate that viscoelasticity problems can be analyzed to significant depth even if the basic response functions are given only graphically or numerically, in the form of data. Third, the mathematical techniques used in these problems are of interest in themselves because they find applications in many areas.
A. C. Pipkin
Chapter V. Stress Analysis
Abstract
We now consider some problems involving small, gradually changing deformations. The stresses and their changes over long periods are the matters of interest. Deformations can also be of interest in themselves if the structure creeps, i.e., the deformation continues to change for a long time.
A. C. Pipkin
Chapter VI. Thermal Effects
Abstract
Thermal effects are important in viscoelasticity in the same way that they are important in the special cases of classical elasticity and fluid dynamics. But in addition, in viscoelasticity theory there are really drastic temperature effects of a kind never considered in the classical theories. The mean relaxation time of a material depends very strongly on the temperature. Generally, the higher the temperature is, the quicker the material relaxes or complies. This kind of temperature-dependence is, of course, absent from the theories that involve no relaxation time.
A. C. Pipkin
Chapter VII. Large Deformations with Small Strains
Abstract
The solution of a problem of finite deformation may require knowledge of material properties not embodied in the linear stress-relaxation moduli. Then again, it may not. In some problems of finite deformation, no material element is distorted very much even though displacements and rotations are large. In such cases we need to know only those material properties that entered into the description of response in infinitesimal deformations. However, if there are large geometry changes, it is necessary to take these into account in the kinematics.
A. C. Pipkin
Chapter VIII. Slow Viscoelastic Flow
Abstract
Constitutive equations based on the linear response functions are not adequate for most problems of viscoelastic flow. In the present chapter we first discuss the general problem of choosing a constitutive equation that can describe the stress response adequately in whatever specific problem is at hand. We then turn to a specific class of flows for which the Navier-Stokes equation is a good first approximation. These flows are slow in the sense that nothing changes much in one relaxation time, but may be arbitrarily fast if by “fast” one means something to do with the Reynolds number. We discuss the nature of approximations based on this notion of slowness and show how to solve flow problems within this approximation scheme.
A. C. Pipkin
Chapter IX. Viscometric Flow
Abstract
On the edge of the flow diagnosis diagram where ωT = 0, the shear amplitude A is YT, the amount of shear in one relaxation time. We can inch away from the Newtonian corner by using slow viscoelastic flow approximations, but as we have seen in the case of tube flow, this can become very laborious. We need a constitutive equation that deals directly with steady shearing motions at arbitrary shear rates.
A. C. Pipkin
Backmatter
Metadaten
Titel
Lectures on Viscoelasticity Theory
verfasst von
A. C. Pipkin
Copyright-Jahr
1986
Verlag
Springer New York
Electronic ISBN
978-1-4612-1078-8
Print ISBN
978-0-387-96345-7
DOI
https://doi.org/10.1007/978-1-4612-1078-8