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The text begins with an introduction to the concepts of viscoelasticity. Succeeding chapters explore the foundations of three-dimensional linear viscoelasticity and stress analysis. Sinusoidal oscillation problems, quasi-static problems, and dynamic problems receive particular attention. The final chapter examines model fitting to measured values of complex modulus or compliance. Numerous examples and figures illuminate the text.
Product Details
| ISBN-13: | 9780486816388 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 10/05/2016 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 144 |
| File size: | 31 MB |
| Note: | This product may take a few minutes to download. |
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The Theory of Linear Viscoelasticity
By D. R. Bland
Dover Publications, Inc.
Copyright © 2016 D. R. BlandAll rights reserved.
ISBN: 978-0-486-81638-8
CHAPTER 1
MODELS–AN INTRODUCTION TO THE CONCEPTS OF VISCOELASTICITY
1. Force-extension equations of the simple models
Viscoelasticity, as its name implies, is a generalization of elasticity and viscosity. The ideal linear elastic element is the spring. When a tensile force is applied to it, the increase in distance between its two ends is proportional to the force. The ideal linear viscous element is the dashpot. It has the property that, when a tensile force is applied to it, the sides move apart at a rate that is proportional to the force. A loose fitting piston in a liquid-filled cylinder, arranged so that liquid flows out around the sides of the piston when it moves slowly up the cylinder, is an example of such an element. The two elements are represented diagrammatically in Figs. 1.1 and 1.2.
If F is the force and a is the extension, i.e. the increase in distance between the ends of the spring in the loaded from the unloaded state, then
(1) F = Ea;
E is a constant, known as the modulus of the spring. For the dashpot, the relationship is
(2) F = ηDa
where the operator D denotes differentiation with respect to time; η is a constant, known as the viscosity of the dashpot.
In viscoelasticity the basic elastic and viscous elements are combined. In this chapter we shall be concerned with networks that are essentially one-dimensional. The elements all lie in the same direction and all the forces concerned act in this direction. In this section some of the simpler combination of elements are considered.
The elements can be combined in series. The combination is known as a Maxwell element and is illustrated in Fig. 1.3. Suppose a tensile force F is applied to this element. What is the relationship between the force and the extension? F is the same in the elastic and viscous elements and a is the sum of the extensions in the elastic and viscous elements, i.e.
(3) a = aAB = aAC + aCB
where C is the point of junction of spring and dashpot. aAC is given by eq. (1) and DaCB by eq. (2). aAB and aCB are eliminated from eqs. (1) to (3). Differentiating eq. (3),
(4) [MATHEMATICAL EXPRESSION OMITTED]
on substituting from eqs. (1) and (2). Equation (4) is the required relationship between a and F.
The combination of the elastic and viscous elements in parallel is known as a Voigt element (sometimes as a Kelvin or Kelvin–Voigt element) and is illustrated in Fig. 1.4. The extension a is the same in spring and dashpot. If the tensile force in the spring is FS and that in the dashpot FD, then
(5) F = FS + FD.
Substituting from eqs. (1) and (2),
(6) F = Ea + η Da.
This is the relationship between force and extension for the Voigt element.
If E is infinite in eq. (4), eq. (4) is identical with eq. (2) and, if η is infinite, it is identical with eq. (1). If E is zero in eq. (6), eq. (6) is identical with eq. (2), and, if η is zero, it is identical with eq. (1). In these special cases the Maxwell and Voigt elements degenerate to the simplest elements.
From two springs and one dashpot, two non-degenerate models can be made. They are illustrated in Fig. 1.5.
The left-hand model consists of an elastic and a Voigt element in series. The force F is the same in the two elements and the extension a is the sum of the extensions, a1 and a2, in the two elements, i.e.
a = a1 + a2.
From eq. (1),
F = E1 a1
and, from eq. (6),
F = E2a2 + η2Da2.
Eliminating a1 and a2
(7) [MATHEMATICAL EXPRESSION OMITTED]
Since an equation is unaltered when multiplied throughout by any non-zero quantity, the coefficient of the highest order derivative of F can be taken as unity.
The right-hand model in Fig. 1.5 consists of an elastic and a Maxwell element in parallel. It is left as an exercise for the reader to show that it has a force-extension relationship
(8) [MATHEMATICAL EXPRESSION OMITTED]
Thus for a given force-extension equation of the form
DF + p0F = q1Da + q0a (9)
two distinct model representations exist. The elastic and viscous constants of one model are found by equating the coefficients of eq. (7) to those of eq. (9); those of the other by equating the coefficients of eq. (8) and (9). The two sets of constants are not the same. In fact whenever the non-degenerate model representation of a particular force-extension equation consists of more than two elements, the model representation is not unique.
The reader should now be able to show that two non-degenerate models can be constructed from two dashpots and one spring. Either model is known as the 3-element viscous model and its force-extension equation is of the form
DF + p0F = q2D2a + q1Da.
A particularly useful model is that shown in Fig. 1.6 with three more representations equivalent to it.
Using a self-evident notation
a = a1 + a2 + a3.
From eq. (1)
F = E1a1,
from eq. (2)
F = η2Da2
and from eq. (6)
F = E3a3 + η3Da3.
Eliminating a1, a2 and a3,
(10) [MATHEMATICAL EXPRESSION OMITTED]
This is the force-extension relationship for the four element model of the first type.
2. Creep and relaxation behaviour
A creep test consists of applying a force to a model previously unloaded and then maintaining this force constant. The extension is measured as a function of the time. A relaxation test consists of imposing a constant extension and measuring the force required to do so as a function of the time. The response of the simple models to creep and relaxation will be considered.
The problem of creep can be formulated in mathematical terms thus: given F = CH(t) and a relationship between F and a, find a. C is constant and H(t) is the Heaviside unit function.
For the elastic element, the F–a relation is eq. (1) and so a(t) is given by
a(t) = 1/E F(t) = C/E H(t).
For the viscous element, eq. (2) gives
Da(t) = C/η H(t).
On integration
a(t) = C/η tH(t), since a = 0 at t = 0.
For the Maxwell element, eq. (4) gives
Da = C/E δ(t) + C/η H(t)
where δ(t) is the Dirac δ-function. On integration
(11) a(t) = C(1/E + 1/ηt) H(t) since a = 0 at t = 0.
For the Voigt element, eq. (6) gives
Ea + ηDa = CH(t).
Multiplying through by the integrating factor exp {(E/η)t},
[MATHEMATICAL EXPRESSION OMITTED]
On integration
[MATHEMATICAL EXPRESSION OMITTED]
The various extension responses are plotted in Fig. 1.7.
The extension response of the Maxwell element is the sum of the extension responses of the viscous and elastic elements. This is because it consists of these two elements in series — the stress is the same in each element and the overall extension is the sum of the extensions in the two elements. This property, the extension response of elements combined in series, can be used to determine the extension response of more complicated models. For example, the four element model of Fig. 1.6 consists of a Maxwell and a Voigt element combined in series. Its extension response is therefore, using eqs. (11) and (12),
[MATHEMATICAL EXPRESSION OMITTED]
This response is plotted in Fig. 1.7.
Consider a model consisting of a large number of Voigt elements and a spring and a dashpot all connected in series. This is known as a generalized Voigt model and is illustrated in Fig. 1.8.
Its extension response will be
[MATHEMATICAL EXPRESSION OMITTED]
The extension response to unit force is sufficiently important to be given a special name. When the components corresponding to instantaneous elasticity and long term viscous flow are excluded, the response is known as "a creep function", denoted by ψ(t). Hence for the generalized Voigt model
(13) [MATHEMATICAL EXPRESSION OMITTED]
From eq. (12) it is seen that the time, τ, taken for the creep function of the Voigt element to reach a proportion 1 - (1/e) of its final value is
τ = η/E.
τ is known as the retardation time of the Voigt element. The reciprocal of the modulus of the spring is known as the compliance and is denoted by J:
J = 1/E.
Equation (12) can now be written
(14) ψ(t) = J{1 - exp (-t/τ)}H(t).
Similarly if τr = ηr/Er and Jτ = 1/Eτ, eq. (13) can be written
(15) [MATHEMATICAL EXPRESSION OMITTED]
Thus the generalized Voigt model has a discrete spectrum of retardation times τr. The compliances Jr determine the magnitude in the creep function of the terms corresponding to each value of τr.
Some writers include the terms J1H(t) and 1/η2tH(t) in the creep function ψ(t). They correspond to τ1 = 0 and τ2 = ∞ respectively. The latter case arises because E2 = 0. As E2 -> 0, τITL2 -> ∞ and
[MATHEMATICAL EXPRESSION OMITTED]
A brief examination of the form of the creep function for a few particular cases in a given paper or book will show which of the alternative definitions of ψ(t) the author has adopted.
If the number of Voigt elements in the generalized model increases indefinitely, in such a way that Jr -> 0 for all r but [MATHEMATICAL EXPRESSION OMITTED] constant > 0, then the creep function becomes
(16) [MATHEMATICAL EXPRESSION OMITTED]
j(τ) is known as the distribution function of retardation times or more briefly the "retardation spectrum".
It is possible to visualize a model which has both a discrete and a continuous spectrum of relaxation times. For example, a new model could be made by placing one with a discrete spectrum and one with a continuous spectrum in series. In this case the creep function can be represented by a Stieltjes integral
(17) [MATHEMATICAL EXPRESSION OMITTED]
where α(τ) is an increasing function of τ with derivatives at all but a finite number of points. At all points, except this finite number, the integrand is of the form of that ineq. (16); at the other points it contributes terms of the form of those in eq. (15).
Now consider relaxation tests. The problem of relaxation can be formulated in mathematical terms thus: given a = KH(t) and a relationship between F and a, find F. K is constant. The force responses for the simple models will now be found.
For the elastic element, eq. (1),
F(t) = Ea(t) = KEH(t);
For the viscous element, eq. (2),
F(t) = ηDa(t) = Kηδ(t).
For the Maxwell element, eq. (4),
(18) F(t) = KE exp (-E/η t) H(t)
and for the Voigt element, eq. (6),
(19) F(t) = KEH(t) + Kηδ(t).
Two of the force responses just obtained contain a δ-function; the stress goes to infinity at the instant of application of the strain. The cause is that a dashpot, unlike a spring, cannot give a finite instantaneous extension response to a finite instantaneous change in force. Consequently, if a finite instantaneous extension is to be imposed on a dashpot, it requires an infinite force. For a model containing more than one element a finite instantaneous extension can be produced by a finite force only if there are one or more springs free to expand or contract without at the same time expanding or contracting a dashpot. For example the spring in the Maxwell element is free, that in the Voigt element is not. Consequently eq. (18) does not contain a δ-function, eq. (19)does. A model that can respond with an instantaneous finite change in extension to an instantaneous change in force is said to exhibit "instantaneous elasticity".
If reference is made back to the extension responses in creep, it will be seen that, as t -> ∞, a(t) either tends to a limit or a(t) ÷ Ct/η2 tends to unity. In the latter case a dashpot is free to flow unrestricted by any spring and the model is said to exhibit "long term viscous flow". The viscous element and the Maxwell element have this property, the elastic and the Voigt do not. In relaxation a dashpot, if one exists free from restriction by a spring, will eventually take up the whole extension and the force will drop to zero. The reader can check from the above equations for F(t) that the force eventually falls to zero for the viscous and Maxwell elements but not for the elastic and Voigt.
Just as the constancy of the force enabled the extension responses in creep to be found by addition for models built up from simpler models in series so the constancy of the extension enables the force response in relaxation to be found by addition for models built up from simpler models in parallel. For example, the force response of the Voigt element is the sum of those for the elastic and for the viscous elements.
Consider a model consisting of a large number of Maxwell elements, a spring and a dashpot all in parallel, Fig. 1.9.
The force response in relaxation of such a model will be
[MATHEMATICAL EXPRESSION OMITTED]
The force response to unit extension, a(t) = H(t), but excluding the constant and δ-function components, is known as "the relaxation function", denoted by χ(t). Hence for the generalized Maxwell model
(20) [MATHEMATICAL EXPRESSION OMITTED]
If relaxation times τ'r are defined by
τ'r = η'r/E'r,
then
(21) [MATHEMATICAL EXPRESSION OMITTED]
Thus the generalized Maxwell model has a discrete spectrum of relaxation times τr'. The elasticities Er' determine the magnitude in the relaxation function of the terms corresponding to each value of τr'.
Some authors include the terms E1'H(t) + ηITL2' δ(t) in the relaxation function. The term E1'H(t) corresponds to τ1' = ∞, i.e. η1' = ∞; and the term η2' δ(t) corresponds to τ2' = 0, i.e. E2' = ∞.
If the number of Maxwell elements in the generalized model increase indefinitely, in such a way that Er' -> 0 but [MATHEMATICAL EXPRESSION OMITTED] constant > 0, then the relaxation function becomes
(22) [MATHEMATICAL EXPRESSION OMITTED]
y(τ) is known as the distribution function of relaxation times or more briefly the "relaxation spectrum". A model which has both a discrete and a continuous spectrum of relaxation times has a relaxation function represented by a Stieltjes integral
(23) [MATHEMATICAL EXPRESSION OMITTED]
where β(τ) is an increasing function of τ with derivatives at all but a finite number of points.
3. Complex modulus and compliance
Let the model be subject to a sinusoidally oscillating force of radian frequency ω. After sufficient time has elapsed for the effect of the initial conditions to be negligible, the extension also will be of radian frequency ω. If R[z] denotes the real part of z, then for sinusoidal oscillations
(24) F = R[F0(ω) exp (iωt)]
and
(25) a = R[a0(ω) exp (iωt)],
where F0(ω) and a0(ω) are independent of t and in general complex.
(Continues...)
Excerpted from The Theory of Linear Viscoelasticity by D. R. Bland. Copyright © 2016 D. R. Bland. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
This concise introduction to the concepts of viscoelasticity focuses on stress analysis. Three detailed individual sections present examples of stress-related problems. In addition, it explains procedures for model fitting to measured values of complex modulus or compliance.The text begins with an introduction to the concepts of viscoelasticity. Succeeding chapters explore the foundations of three-dimensional linear viscoelasticity and stress analysis. Sinusoidal oscillation problems, quasi-static problems, and dynamic problems receive particular attention. The final chapter examines model fitting to measured values of complex modulus or compliance. Numerous examples and figures illuminate the text.
Dover republication of the Pergamon Press, Oxford, 1960 edition.