Viscoelastic effects principle

The linear viscoelastic materials obey the so-called Boltzmann Superposition Principle. As noted by Tschoegl (13), this was the only foray of the Viennese statistical physicist Ludwig Boltzmann into mechanics. The principle states that in linear viscoelasticity effects are simply additive it matters at which instant an effect is created and it is assumed that each increment of stress makes an independent contribution. 

In defining the constitutive relations for an elastic solid, we have assumed that the strains are small and that there are linear relationships between stress and strain. We now ask how the principle of linearity can be extended to materials where the deformations are time dependent. The basis of the discussion is the Boltzmann superposition principle. This states that in linear viscoelasticity effects are simply additive, as in classical elasticity, the difference being that in linear viscoelasticity it matters at which instant an effect is created. Although the application of stress may now cause a time-dependent deformation, it can still be assumed that each increment of stress makes an independent contribution. From the present discussion, it can be seen that the linear viscoelastic theory must also contain the additional assumption that the strains are small. In Chapter 11, we will deal with attempts to extend linear viscoelastic theory either to take into account non-linear effects at small strains or to deal with the situation at large strains. 

A linear stability analysis is performed in terms of normal modes. For illustrative purpose and so as to be able to proceed analytically we here restrict ourselves to growth under zero gravity and assume that the principle of the exchange of stabilities holds. The viscoelastic effects appear in the momentum balance equation (Eq. 5] and in the Laplace condition [Eq. 12]. The latter contribution has been previously neglected [6,7]. 

Recently, Matadi Boumbimba et al. [12] proposed a temperature- and frequency-dependent version of the rule of mixtures to describe the viscoelastic response, in terms of storage modulus, of PMMA/Cloisite 20A and SOB. In the present work, to predict the effective viscoelastic response of polymer-based nanocomposites, the elastic-viscoelastic correspondence principle [11] is applied to our micromechanical model. The two implicit equations (5) become ... 

There are two superposition principles that are important in the theory of Viscoelasticity. The first of these is the Boltzmann superposition principle, which describes the response of a material to different loading histories (22). The second is the time-temperature superposition principle or WLF (Williams, Landel, and Ferry) equation, which describes the effect of temperature on the time scale of the response. 

There are many types of deformation and forces that can be applied to material. One of the foundations of viscoelastic theory is the Boltzmann Superposition Principle. This principle is based on the assumption that the effects of a series of applied stresses acting on a sample results in a strain which is related to the sum of the stresses. The same argument applies to the application of a strain. For example we could apply an instantaneous stress to a body and maintain that stress constant. For a viscoelastic material the strain will increase with time. The ratio of the strain to the stress defines the compliance of the body ... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

It was initially stated that Cf are Cf were universal constants (Cf 17 Cf 50 K), but Cf can vary between 2 and 50 and Cf between 14 and 250 K (Mark, 1996). Epoxy values have been found in the low part of these intervals Cf 10, Cf 40 15 K (Gerard et al., 1991), whereas unsaturated polyester values can be relatively high Cf/Cf = 15-55 = 73-267 K (Shibayama and Suzuki, 1965). There is, to our knowledge, no synthetic study on the ideality and crosslinking effects on Cfand Cf. The time-temperature equivalence principles will be examined in detail in Chapter 11, which is devoted to elasticity and viscoelasticity. 

With all these models, the simple ones as well as the spectra, it has to be supposed that stress and strain are, at any time, proportional, so that the relaxation function E(t) and the creep function D(t) are independent of the levels of deformation and stress, respectively. When this is the case, we have linear viscoelastic behaviour. Then the so-called superposition principle holds, as formulated by Boltzmann. This describes the effect of changes in external conditions of a viscoelastic system at different points in time. Such a change may be the application of a stress or also an imposed deformation. 

We expect that the modification creates the free volume (Vf) in wood substance from the similarity of the effect of and n on viscoelasticity. The discussion for wood, however, is impossible on the basis of a concept of the free volume, although the flexibility of molecular motion for synthetic amorphous polymers is discussed. Unfortunately, we can not directly know the created free volume because the time-temperature superposition principle is not valid for wood [19]. The principle is related to WLF equation by which the free volume is calculated. The free volume, however, relates to volumetric swelling as follows. 

Important viscoelastic principles include the time-temperature superposition principle and its resultant WLF equation. These can be applied to understand the relationship between literature values of the glass transition temperature and actual needs. Thus, by using the growing amount of science now available in the field of damping, one can select that polymeric material which will damp most effectively. 

With the long term objective of treating the effects of moisture and other plasticizers on the mechanical properties of materials, a new scheme that yields a complete constitutive model of viscoelastic materials has been developed. The time-temperature principle is an integral part of this modeling with a quantitative description of the glass transition behavior of pol3nmers. 

The Rivlin-Ericksen constitutive equation gives a good account of some characteristics of both the time dependence of the viscoelastic behavior and the normal stress effects. This relationship is based on the assumption that the stress depends not only on the velocity (x ) and the shear rate gradient (dxi/dx ) but also on derivatives of higher order (%, dXp/dXq. .. 8xf /8xi). As a consequence of the principle of material... 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

In linear elasticity or viscoelasticity, the superposition principle states that the resulting effects of the different causes (stress or displacements), acting separately, can be superposed to give the total values due to these combined causes. This principle is a consequence of the linearity of the equations governing the stress, strain, and displacements. 

According to the correspondence principle, the equation describing a viscoelastic beam under transversal and longitudinal effects is given by... 

Overall, the semi-quantitative agreement between the linear viscoelastic spectra and first-principles MCT calculations is very promising. Yet, crystallization effects in the data prevent a closer look, which will be given in Sect. 6.2, where data from a more polydisperse sample are discussed. 

The time-temperature equivalence principle makes it possible to predict the viscoelastic properties of an amorphous polymer at one temperature from measurements made at other temperatures. The major effect of a temperature increase is to increase the rates of the various modes of retarded conformational elastic response, that is, to reduce the retarding viscosity values in the spring-dashpot model. This appears as a shift of the creep function along the log t scale to shorter times. A secondary effect of increasing temperature is to increase the elastic moduli slightly because an equilibrium conformational modulus tends to be proportional to the absolute temperature (13). 

As discussed earlier for a Hookean solid, stress is a linear function of strain, while for a Newtonian fluid, stress is a linear function of strain rate. The constants of proportionality in these cases are modulus and viscosity, respectively. However, for a viscoelastic material the modulus is not constant it varies with time and strain history at a given temperature. But for a linear viscoelastic material, modulus is a function of time only. This concept is embodied in the Boltzmann principle, which states that the effects of mechanical history of a sample are additive. In other words, the response of a linear viscoelastic material to a given load is independent of the response of the material to ary load previously on the material. Thus the Boltzmann principle has essentially two implications — stress is a linear function of strain, and the effects of different stresses are additive. 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

Another very fast and effective method for measuring rubber processing properties is to perform a stress relaxation test. A stress relaxation decay curve can quickly quantify the viscoelastic properties of both raw rubbers and mixed stocks. The Maxwell model, shown in Fig. 35. illustrates this principle with a spring and dashpot in series [125]. A sudden... 

First, we need a rule to predict the effect of time-varying loads on a viscoelastic model. When a combination of loads is applied to an elastic material, the stress (and strain) components caused by each load in turn can be added. This addition concept is extended to linear viscoelastic materials. The Boltzmann superposition principle states that if a creep stress ai is... 

Generalizations of the Newton s flow law [7.2.3] for polymeric liquids are aimed to describe in more or less details the features of their rheological behavior. The most important among these features is the ability to accumulate elastic deformation during flow and thus to exhibit the memory effects. At first we restrict ourselves to the case of small deformation rates to discuss the basic principles of the general linear theory of viscoelasticity... 

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