Steady state compliance

A constant strain rate test may be used to determine the relaxation modulus and a constant stress-rate test may be used to find the creep compliance. Steady state oscillation tests may also be used to determine the viscoelastic properties of polymers. These details and the interrelation between various test approaches are given in Chapter 5. 

Model equations can be augmented with expressions accounting for covariates such as subject age, sex, weight, disease state, therapy history, and lifestyle (smoker or nonsmoker, IV drug user or not, therapy compliance, and others). If sufficient data exist, the parameters of these augmented models (or a distribution of the parameters consistent with the data) may be determined. Multiple simulations for prospective experiments or trials, with different parameter values generated from the distributions, can then be used to predict a range of outcomes and the related likelihood of each outcome. Such dose-exposure, exposure-response, or dose-response models can be classified as steady state, stochastic, of low to moderate complexity, predictive, and quantitative. A case study is described in Section 22.6. 

Compliance with U.S. EPA s design performance standards can be demonstrated through one-dimensional, steady-state flow calculations, instead of field tests. For detection sensitivity, the calculation of flow rates should assume uniform top liner leakage. For detection time, factors such as drain spacing, drainage media, bottom slope, and top and bottom liners should all be considered, and the worst-case leakage scenario calculated. 

Nere- Xf is the weight-average molecular weight of the blend, and Afwl is the weight-average molecular weight of component /. The steady-state creep compliance 7, of the blend is... 

Chain branching affects the viscosity, the longest relaxation time, and the steady-state compliance and therefore influences creep and stress relaxation (19,163- 167). The effect is difficult to quantify because the length... 

Star-shaped polymer molecules with long branches not only increase the viscosity in the molten state and the steady-state compliance, but the star polymers also decrease the rate of stress relaxation (and creep) compared to a linear polymer (169). The decrease in creep and relaxation rate of star-shaped molecules can be due to extra entanglements because of the many long branches, or the effect can be due to the suppression of reptation of the branches. Linear polymers can reptate, but the bulky center of the star and the different directions of the branch chains from the center make reptation difficult. 

Now as t -> 0 the spectral function L reduces to the area under the distribution. This is the steady state compliance Je ... 

Note 5 Creep is sometimes described in terms of non-linear viscoelastic behaviour, leading, for example, to evaluation of recoverable shear and steady-state recoverable shear compliance. The definitions of such terms are outside the scope of this document. 

The greater melt viscosities observed for some branched polymers, as compared with linear ones of the same MW, are not accounted for by current theories, as indicated in Section 5. The greater values of the steady state compliance mentioned above is also unexpected theory (128) would suggest a difference in the opposite sense. 

Portions of the literature on viscoelasticity in concentrated polymer systems of narrow distribution have been reviewed recently (15, 16, 152, 153). The following discussion concerns three principal characteristics, the viscosity-molecular weight relation, the plateau modulus, and the steady-state compliance. 

The steady-state compliance data have been analyzed in reduced form... 

Fig. 5.14. Reduced compliance vs molecular weight for undiluted polystyrenes of narrow molecular weight distributions. Symbols are O from creep recovery (163), Cr from G (w) (192), O- from flow birefringence (180), (X from (189), 9 from G (a>) (M>105 only) (124), jO extrapolated from steady state creep (191), -O from stress relaxation (165), and...

Fig. 5.16. Reduced compliance vs cMw for solutions and undiluted samples of cis-polyisoprene. Symbols are undiluted samples from steady state creep (166, 196), undiluted samples from Nt (197), and O solutions extrapolated from steady state creep (196)...

Studies have been made of the stresses produced in several non-steady flow histories. These include the buildup to steady state of a and pu — p22 at the onset of steady shearing flow (355-35 ) relaxation of stresses from their steady state values when the flow is suddenly stopped (356-360) stress relaxation after suddenly imposed large deformations (361) recoil behavior when the shear stress is suddenly removed after a steady state in the non-linear region has been reached (362) and parallel or transverse oscillations superimposed on steady shearing flow (363-367). Experimental problems caused by the inertia and compliance of the experimental apparatus are much more severe than in steady state measurements (368,369). Quantitative interpretations must therefore still be somewhat tentative. Nevertheless, the pattern of behavior emerging is suggestive with respect to possible molecular flow mechanisms. 

Steady state compliance for a monodisperse polymer of molecular weight . 

Holmes, L. A., Ferry, J.D. Dependence of the steady-state compliance on concentration and molecular weight in polymer solutions. J. Polymer ScL Pt C 23,291-299 (1968). 

Kusamizu,S., Holmes,L.A., Moore,A.A., Ferry,J.D. The steady-state compliance of polymer solutions. Trans, Soc. RheoL 12,559-571 (1968). 

Values of the Reduced Steady-State Compliance. Validity of... 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

Eq. (2.2) follows from eq. (2.3) when, according to Hopkins and Hamming (49), for the creep compliance a limiting expression is used, which holds only for the steady state, viz ... 

This means that for a linear elastico-viscous liquid, the steady-state shear compliance Je must be equal to the constrained shear recovery sx which follows on a steady shear flow at unity shear stress. From this one deduces the following relation between and Je ... 

---