Linearized extended superposition

The superposition approximation (SA) suggested in Refs. Ill and 259 is essentially a nonlinear theory that cannot be represented in the form of Eqs. (3.707). The same is true for the extended version of SA [260]. For this reason, we focus on two derivatives of these theories linearized near the equilibrium the linearized superposition approximation (LSA) and the linearized extended superposition approximation (LESA). It was found that LSA developed in a number of works [139,175,255,260] isinfact identical to IET (see Table VIII). They both have the same concentration-independent kernel S(j ). As for LESA, it was, strictly speaking, created for the reactions in the ground state [241,242], but can be easily extended to the case of equal lifetimes, uA = uc-... 

S A was also applied to the reversible reactions considered in Section XII.C.4. Since the results were not satisfactory, an extended superposition approach (ESA) was developed, then linearized, and later known as LESA [241]. Independently, a similar linearization over deviations from equilibrium was also made in Ref. 242. Although the asymptotic description of the quenching kinetics is improved, it was recognized [242] that LESA is not valid with a large equilibrium constant K because the superposition approach worsens when K increases [241]. This is especially true at earlier times when the deviations from equilibrium are not small. However, the authors who constructed LESA claimed that it is applicable at all times [241]. Therefore, it was taken for comparison with other approximations. In the irreversible limit (K —> oo), the kernels obtained in both works [241,242] coincide with that listed as LESA in Table V. 

As many nonlinear approaches are beyond the intended level and scope of this text, the focus will be on single integral mathematical models which are an outgrowth of linear viscoelastic hereditary integrals and lead to an extended superposition principle that can be used to evaluate nonlinear polymers. The emphasis will be on one-dimensional methods but these can be readily extended to three dimensions using deviatoric and dilatational stresses and strains as was the case for linear viscoelastic stress analysis as discussed in Chapters 2 and 9. 

In this Section we consider again the kinetics of bimolecular A + B -A 0 recombination but instead of the linearized approximation discussed above, the complete Kirkwood superposition approximation, equation (2.3.62) is used which results in emergence of two new joint correlation functions for similar particles, Xu(r,t), v = A,B. The extended set of the correlation functions, nA(f),nB(f),Xfi,(r,t),Xa(r,t) and Y(r,t) is believed to be able now to describe the intermediate order in the particle spatial distribution. 

The linear superposition given as the scalar product ( T ) 1B) IT ) ICL)MC(T ) C(T )] represents an entangled state. This form is used to remind that whatever we do with this model, only the amplitudes would change. In an extended base set ( T ) -k laj) T )11 ) T ) 0B) T ) +k, ldetailed description becomes possible. In what follows, we discuss situations with the "inner" part of this extended base set, but implicitly, unentanglement/entanglement processes require mechanisms changing the amplitudes. 

Any of equations (2-45), (2-46), (2-49), or (2-50) is sufficient as a statement of the Boltzmann superposition principle for linear viscoelastic response of a material. Often in particular applications, however, it is more convenient to use one form than another. All can be extended to three dimensions by using the same forms with the strains given by equation (2-18). Thus, for example, equation (2-46) becomes ... 

Another interesting aspect of linear viscoelasticity is that it can be extended to enable predictions at different temperatures. The basis for this approach is based on a time-temperature superposition principle.This approach has been shown to work well in a restricted temperature range, but does not change the requirement of small strains. 

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... 

If linear, a sine excitation input results in a sine response. However, the immittance concept can be extended to nonlinear networks, where a sine wave excitation leads to a nonsinusoidal response. Including a separate immittance value for each harmonic component of the response performs the necessary extension. In the linear region, the principle of superposition is valid. This means, for example, that the presence of strong harmonics in the applied current or voltage would not affect immittance determination at the fundamental frequency or a harmonic (Schwan, 1963). Some lock-in amplifiers can measure harmonic components, making it possible to analyze nonlinear phenomena and extend measurement to nonsinusoidal responses. 

If the material is subjected to a time-dependent strain, the situation becomes more complicated. However, in the case of a linear viscoelastic material (like many food products) the superposition principle can be applied the response of the stress to a strain increment is independent of the already existing strain. The effect of the strain as a function of time can therefore be integrated, and the generalized Hooke s law can be extended to describe the stress-strain behavior of linear viscoelastic materials relatively easily. 

Although the probability of finding the particle at a given position varies with the position, the eigenfunctions given by Eq. (2.24) extend over the full length of the box. Wavefunctions for a particle that is more localized in space can be constructed from linear combinations of these eigenfunctions. A superposition state formed in... 

Figures 9.4 and 9.5 show G and G" for two linear polymers, a poly(vinyl methyl ether) (PVME) with a molecular weight of 138,000 and a polystyrene (PS) with a molecular weight of 123,000, respectively. The data for each polymer have been moved horizontally along the frequency axis until they form a single curve. There is a substantial region of overlap, extending over three decades of frequency, so the superposition is clearly established. The shift factors needed to obtain overlap of the curves are shown as inserts. The reference temperature for each case was taken to be 84 °C this temperature has no significance other than being a convenient value for the particular application for which the data were obtained, which was a study of phase separation in blends of the two polymers. One of the significant uses of time-temperature superposition is made evident by focusing on the open and closed symbols in the PVME curves. The dynamic moduli are available over five orders...

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. 

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