Models strain dependence

FIGURE 30.9 Modeling the strain dependence of complex modulus. 

Lines in Figure 30.12 were drawn with parameters obtained when fitting data with Equation 30.3. It is fairly obvious that, outside the experimental window, data would not necessarily conform to such a simple model, which in addition cannot meet the inflection at 100% strain. All results were nevertheless fitted with the model essentially because correlation coefficient were excellent, thus meaning that the essential features of G versus strain dependence are conveniently captured through fit parameters. Furthermore any data can be recalculated with confidence within the experimental strain range with an implicit correction for experimental scatter. Results are given in Table 30.1 note that 1/A values are given instead of A. 

A Maxwell model is a good candidate and we can incorporate the strain dependence by multiplying by a function f(y) ... 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

In a later study [56], the effect of gas volume fraction (foam rheology was investigated. Two models were considered one in which the liquid was confined to the Plateau borders, with thin films of negligible thickness and the second, which involves a finite (strain-dependent) film thickness. For small deformations, no differences were observed in the stress/strain results for the two cases. This was attributed to the film thickness being very much smaller than the cell size. Thus, it was possible to neglect the effect of finite film thickness on stress/strain behaviour, for small strains. 

In view of an illustration of the viscoelastic characteristics of the developed model, simulations of uniaxial stress-strain cycles in the small strain regime have been performed for various pre-strains, as depicted in Fig. 47b. Thereby, the material parameters obtained from the adaptation in Fig. 47a (Table 4, sample type C60) have been used. The dashed lines represent the polymer contributions, which include the pre-strain dependent hydrodynamic amplification of the polymer matrix. It becomes clear that in the small and medium strain regime a pronounced filler-induced hysteresis is predicted, due to the cyclic breakdown and re-aggregation of filler clusters. It can considered to be the main mechanism of energy dissipation of filler reinforced rubbers that appears even in the quasi-static limit. In addition, stress softening is present, also at small strains. It leads to the characteristic decline of the polymer contributions with rising pre-strain (dashed lines in... 

On the other hand the estimation of the 3e>/3R quantities entails somewhat more effort. There are effectively two methods of approach of which the simpler uses the unsophisticated electrostatic model. On this basis Dq, and hence e, shows a 1/R5 distance dependence so that direct differentiation then indicates that at any given metal-ligand distance, R, de-JdR — -5 (ex/R). In justification of this rather naive model it may be noted that Bums and Axe13) found the strain dependence of Dq to be rather well described by the 1/R5 variation. 

The discussion in the Introduction led to the convincing assumption that the strain-dependent behavior of filled rubbers is due to the break-down of filler networks within the rubber matrix. This conviction will be enhanced in the following sections. However, in contrast to this mechanism, sometimes alternative models have been proposed. Gui et al. theorized that the strain amplitude effect was due to deformation, flow and alignment of the rubber molecules attached to the filler particle [41 ]. Another concept has been developed by Smith [42]. He has indicated that a shell of hard rubber (bound rubber) of definite thickness surrounds the filler and the non-linearity in dynamic mechanical behavior is related to the desorption and reabsorption of the hard absorbed shell around the carbon black. In a similar way, recently Maier and Goritz suggested a Langmuir-type polymer chain adsorption on the filler surface to explain the Payne-effect [43]. 

The strain-dependency of the number N(y0) of surviving clusters can be modeled, in a certain approximation, by the rate at Eq. (15), introduced by Kraus [36]. As already discussed in Sect. 2, the Kraus equation considers a rate equilib-... 

Derived firom e Lodge s rubbeilike Hquid theory, the Wagner model is based on a concept of separabiHty, since it is assumed that the memory function is the product of a time-dependent hnear function by a strain-dependent nonlinear fimction. 

This latter approximation shows that the strain dependence of the Doi-Edwards equation is softer than that of the temporary network model roughly by the factor 1 -p (7i — 3)/5, There is also a differential approximation to the Doi-Edwards equation (Marmcci 1984 Larson 1984b) ... 

The deformation dependence of the stress in the Edwards tube model is the same as in the classical models [Eqs (7.32) and (7.33)] because each entanglement effectively acts as another crosslink junction in the network. Therefore, the Edwards tube model is unable to explain the stress softening at intermediate deformations, demonstrated in Fig. 7.8. The reason for the classical functional form of the stress strain dependence is that the confining potential is assumed to be independent of deformation. 

The deformation dependence of the confining potential [Eq. (7.62)] results in a non-classical stress strain dependence of the non-affine tube model. The prediction of this model for the stress-elongation relation in tension is qualitatively similar to the Mooney-Rivlin equation [Eg. (7.59)]... 

Initial moduli at room temperature were obtained with an Instron Model 4206 at a strain rate of 2/min ASTM D638 type V specimens were used. The Instron was also used in the creep experiments, in which deformation under a 1 NPa tensile load was continuously monitored for 10 sec, followed by measurement of the recovered length 48 h after load removal. Strain dependence of the elastic modulus was determined by deforming specimens to successively larger tensile strains and, at each strain level, measuring the stress after relaxation after it had become invariant for 30 min. 

Bell SA, Kuntze I, Caputo A, Chatelain R (2002) Strain-dependent in vitro and in vivo effects of oleic acid anilides on splenocytes and T cells in a murine model of the toxic oil syndrome. Food Chem Toxicol, 40 19-24. 

Berking C, Hobbs MV, Chatelain R, Meurer M, Bell SA (1998) Strain-dependent cytokine profile and susceptibility to oleic acid anilide in a murine model of the toxic oil syndrome. Toxicol Appl Pharmacol, 148 222-228. 

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