Compliance shear

Before discussing tire complex mechanical behaviour of polymers, consider a simple system whose mechanical response is characterized by a single relaxation time x, due to tire transition between two states. For such a system, tire dynamical shear compliance is [42]... 

Figure C2.1.14. (a) Real part and (b) imaginary part of tire dynamic shear compliance of a system whose mechanical response results from tire transition between two different states characterized by a single relaxation time X.

Polymers owe much of their attractiveness to their ease of processing. In many important teclmiques, such as injection moulding, fibre spinning and film fonnation, polymers are processed in the melt, so that their flow behaviour is of paramount importance. Because of the viscoelastic properties of polymers, their flow behaviour is much more complex than that of Newtonian liquids for which the viscosity is the only essential parameter. In polymer melts, the recoverable shear compliance, which relates to the elastic forces, is used in addition to the viscosity in the description of flow [48]. 

The example shows that the shear compliance of the Voigt material is given... 

Figure 3.11 Time-dependent shear compliance [as J(t)/J(°°)] versus time (as t/r) (a) linear coordinates and (b) log-log coordinates.

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. 

Other types of linear viscoelastic experiments may be used. Dynamic shear compliance measurements provide the storage and loss compliances J (co) and J"(co). An equation analogous to Eq.(3.12) is available for determining the initial modulus from J"(co) ... 

Einaga,Y., Osaki,K., Kurata,M, Tamura,M. Creep behavior of polymer solutions. II. Steady-shear compliance of concentrated polystyrene solutions. Macromolecules 4, 87-92 (1971). 

Sakai, M.,Fujimoto,T.,Nagasawa,M. Steady flow properties of monodisperse polymer solutions. Molecular weight and polymer concentration dependences of steady shear compliances at zero and finite shear rates. Macromolecules 5,786-792 (1972). 

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 ... 

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 ... 

The corresponding value of the reduced steady-state shear compliance reads ... 

Unfortunately, Fixman has not yet given a value for the reduced steady-state shear compliance. However, from a comparison of eqs. (3.60a), (3.64) and (3.66) the impression is obtained that the theory of Ptitsyn and Eizner overestimates the influence of the excluded volume on 0 and JeR. As will be shown in the experimental section of this chapter, this impression is supported by flow birefringence measurements on solutions in 0- solvents and in good solvents. 

This is the well-known front factor of the reduced steady-state shear compliance as quoted by Ferry (113). In this form polydispersity factor p can be compared with the diverse molecular weight averages, as obtained from equilibrium ultracentrifugation, light-scattering and osmometry. 

Fig. 4.3. Reduced steady-state shear compliance J,B vs. volume fraction p or concentration c, according to a combination of eq. (4.10) with eqs. (4.6) and (4.9), respectively (smoothed curves). Figures below full lines... MjMp-values, figures above dashed lines. . . M/if-values (theoretical curves)...

Fig. 4.4. Concentration dependence of reduced steady-state shear compliance J,B for a series of anionic polystyrenes, as mostly provided by Pressure Chem. Corp., Pittsburgh, Pa. Except for the solutions of the three lowest concentrations of S 111 (Dow Chem. Corp.), which were prepared with methyl (4-bromo-phenyl) carbinol and used at various temperatures, all solutions were prepared with mono-bromo-benzene and used at 25° C (32). Measurement temperatures for the melts varied from 196 to 240° C (59). For the molecular weights of the polymers see Table 4.1...

The deformation of a material when subjected to a constant stress is, as discussed, usually time-dependent. At times of c. 10 6 s and less all materials, including liquids, have shear compliances (i.e shear/shear stress) of c. 10 n to 10"9 m2 N-1. This is because there is only sufficient time available for an alteration of interatomic distances and bending of bond angles to take place, and the response of all materials is of the same order of magnitude in this respect. Hie time required for the various structural units of a material to move into new positions relative to one another depends on the size and shape of the units and the strength of the bonds between them. 

The molecules of a liquid start to move relative to one another and the shear compliance increases rapidly after the shearing force has been applied for only c. 10 6 s. On the other hand, hard solids, such as diamond, sodium chloride crystals and materials at a low enough temperature to be in the glassy state, show only the above rapid elastic deformation, even after the shearing stress has been applied for a considerable time. 

According to a thorough calculation by Janeschitz-Kriegl (General references, 1983, Appendix A3) the elastic shear compliance is equal to... 

According to Eq. (15.80) recoverable shear strain is proportional to the elastic shear compliance, J°. The elastic shear compliance depends on the molecular-weight distribution in the following way... 

These conclusions are in agreement with results reported by Janeschitz-Kriegl (1969), who measured the shear compliance JeR of highly concentrated to very dilute solutions of a series of polystyrenes with narrow molecular weight distributions. For the melt down to moderately concentrated solutions JeR appeared to be equal to 0.4, which is the value to be expected for free-draining solutions. In very dilute solutions JeR tended to decrease to the non-draining case, where /eR=0.205. 

The first normal stress coefficient at low shear rates, vPio, and the elastic shear compliance, /g, are related by Eq. (15.78) which reads... 

The ratio Sq, which is defined to be equal to 1 /i /rj1, is equal to the elastic shear compliance /° for shear rates approaching zero. Both Th and 77 are shear rate dependent, but it is previously not clear whether it increases or decreases with increasing shear rate. In Fig. 16.20 both 77/r]0 and Sq //( are plotted vs. qi0 on double logarithmic scales for solutions and melts of poly (a-methyl styrene) as well (Sakai et al., 1972). First, it appears that the... 

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