Theory of linear viscoelasticity

N, W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior, Springer-Verlag, New York, 1989. 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

Molecular Entanglement Theories of Linear Viscoelastic Behavior... 

These yield a spectrum of relaxation times , according to theories of linear viscoelasticity. 

Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behavior. Springer, Berlin Heidelberg New York... 

Bland DR, "The Theory of Linear Viscoelasticity", Pergamon Press, Oxford, 1960. 

It will often be desirable to convert the results of one type of experiment into the characteristic quantities of another type. Unfortunately, there are no rigorous rules for these conversions. The problem may be approached in two ways. In the first place, exact interrelations can be derived from the theory of linear viscoelasticity. This method has two disadvantages. 

Schwarzl (1970) studied the errors to be expected in the application of this type of equations, starting from the theory of linear viscoelasticity. Flis results are given schematically in Fig. 13.59. For non-linear viscoelastic behaviour, the exactitude of the approximate equations cannot be predicted. 

Researchers have examined the creep and creep recovery of textile fibers extensively (13-21). For example, Hunt and Darlington (16, 17) studied the effects of temperature, humidity, and previous thermal history on the creep properties of Nylon 6,6. They were able to explain the shift in creep curves with changes in temperature and humidity. Lead-erman (19) studied the time dependence of creep at different temperatures and humidities. Shifts in creep curves due to changes in temperature and humidity were explained with simple equations and convenient shift factors. Morton and Hearle (21) also examined the dependence of fiber creep on temperature and humidity. Meredith (20) studied many mechanical properties, including creep of several generic fiber types. Phenomenological theory of linear viscoelasticity of semicrystalline polymers has been tested with creep measurements performed on textile fibers (18). From these works one can readily appreciate that creep behavior is affected by many factors on both practical and theoretical levels. 

We will begin with a brief survey of linear viscoelasticity (section 2.1) we will define the various material functions and the mathematical theory of linear viscoelasticity will give us the mathematical bridges which relate these functions. We will then describe the main features of the linear viscoelastic behaviour of polymer melts and concentrated solutions in a purely rational and phenomenological way (section 2.2) the simple and important conclusions drawn from this analysis will give us the support for the molecular models described below (sections 3 to 6). 

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

At small stresses and strains, glassy PC exhibits linear viscoelastic behavior. The limit of applicability of the theory of linear viscoelasticity has been investigated by Yannas et al. over the temperature range 23 °C-130 °C. The critical strain at which, within the precision of their measurement, deviations from the linear theory occur has been found to diminish from about 1.2% at 23 °C to about 0.7 % at 130 °C. According to Jansson and Yannas the transition from linear to nonlinear viscoelastic behavior is marked by the onset of significant rotation around backbone bonds. 

Accordingly, the phenomenological theory of linear viscoelasticity predicts the same frequency dependence for the loss relaxation modulus of solids and liquids in the terminal region. 

N Tscoegl. The Phenomenological Theory of Linear Viscoelasticity. Berlin Springer, Chaps 3 and 5. 

DR Bland. The Theory of Linear Viscoelasticity. Oxford, UK Pergamon, 1960, Chap 2. 

Rey, A.D. Theory of linear viscoelasticity of chiral liquid crystals. Rheol. Acta 1996, 35 (5), 400-409. 

The accurate applicability of linear viscoelasticity is limited to certain restricted situations amorphous polymers, temperatures near or above the glass temperature, homogeneous, isotropic materials, small strains, and absence of mechanical failure phenomena. Thus, the theory of linear viscoelasticity is of limited direct applicability to the problems encounted in the fabrication and end use of polymeric materials (since most of these problems involve either large strains, crystalline polymers, amorphous polymers in a glass state failure phenomena, or some combination of these disqualifying features). Even so, linear viscoelasticity is a most important subject in polymer materials science—directly applicable in a minority of practical problems, but indirectly useful (as a point of reference) in a much wider range of problems. 

The Doi-Edwards theory of linear viscoelasticity predicts 1.2 for JeG, where is the plateau shear modulus. This value is significantly lower than typical experimental values found in the range 2—4 [3, 71]. This defect of the Doi-Edwards theory, along with its failure of predicting the 3.4 power law for viscosity, has been pointed out by Osaki and Doi [72]. It is associated with the fact that the Doi-Edwards theory yields a relaxation time distribution which is too narrow compared with observed ones [69]. Modifications of the Doi-Edwards theory have been made so as to bring JeG closer to measured values, but no remarkable success has as yet been achieved. 

It has long been recognised that the mechanical properties of polymers are time-dependent. The behaviour at very small strains (less than 0 5%) can be described by the theory of linear viscoelasticity. Conventionally the stress a at time t is related to the strain e at all previous instants by the equation... 

Shankar V, Pasquali M, Morse DC (2002) Theory of linear viscoelasticity of semiflexible rods in dilute solution. J Rheol 46 1111... 

In a manner similar to the application of springs and dashpots to the theory of linear viscoelasticity, we note that for units in parallel the total stress is (T = (Ti -f (T2 + (T3 -f , and that for units in series the total strain is 6 = -f 82 -f 3 -f . Finally, application of Hooke s law, cr = sE, allows the complex modulus E of the Takayanagi models in Figures 2.11a-d to be represented by the following equations, respectively ... 

The theory of linear viscoelasticity is phenomenological there is no attempt to discover the time and frequencty response of the solid in an altogether a priori fashion. The aim is to predict behaviour under certain circumstances, having observed it under others for example, to correlate creep, stress relaxation, and (fynamic properties so that if one of these has been determined then all the others are known. This is closety related to electrical network theory, both in aim and, as will soon be apparent, in method. 

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