Steady-state recoverable shear compliance

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. 

With all of the viscoelastic functions it is important to note the limiting values or forms which are qualitatively independent of the molecular structure. For a viscoelastic liquid, lini, /(f) = Jg, lim, /(f) = tlr], and lim,, Jr t) = J t) -thr] = Jg + Jd = /j.The last Umiting value Js is called the steady-state recoverable shear compliance. It is the maximum recoverable strain per unit stress, which reflects the maximum configurational orientation achievable at the present stress. 

In the behavior of polymeric liquids two quantities are important. These are steady-state recoverable shear compliance, (as shown above) and steady-state viscosity at zero shear rate, r o- These quantities are related ... 

Figure 6.10 Creep compliance (/) as a function time (t). Calculation of zero-shear-rate viscosity and steady-state recoverable shear compliance.

Flow properties are very strongly dependent on molecular architecture, i.e. molar mass and chain branching. Figures 6.13 and 6.14 illustrate the effect of molar mass on the zero-shear-rate viscosity (//q) and on the steady-state recoverable shear compliance. 

Figure 6.14 The logarithm of the steady-state recoverable shear compliance plotted versus the logarithm of molar mass. Schematic curve.

Alternatively, from steady shearing experiments, which yield rio directly in the limit of low shear rate, the steady-state recoverable compliance can be obtained from the first normal stress coefficient, which is the ratio of the first normal stress difference to the square of the shear rate, measured at low shear rate... 

For a Newtonian low molar mass liquid, knowledge of the viscosity is fully sufficient for the calculation of flow patterns. Is this also true for polymeric liquids The answer is no under all possible circumstances. Simple situations are encountered for example in dynamical tests within the limit of low frequencies or for slow steady state shears and even in these cases, one has to include one more material parameter in the description. This is the recoverable shear compliance , usually denoted and it specifies the amount of recoil observed in a creep recovery experiment subsequent to the unloading. Jg relates to the elastic and anelastic parts in the deformation and has to be accounted for in all calculations. Experiments show that, at first, for M < Me, Jg increases linearly with the molecular weight and then reaches a constant value which essentially agrees with the plateau value of the shear compliance. 

The two material parameters which characterize polymeric fluids at low strain rates, the viscosity, j, and the recoverable shear compliance, Je, can be directly determined, rj follows from the measurement of the torque under steady state conditions, Jq shows up in the reverse angular displacement subsequent to an unloading, caused by the retraction of the melt. From the discussion of the properties of rubbers we know already that simple shear is associated with the building-up of normal stresses. More specifically, one finds a non-vanishing... 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

Similarly, the steady-state recoverable compliance /e = yt/rioy (with rigf being the steady-state shear stress), which characterizes a strain recovered (recoiled) on removal of the stress from a steady fiow state, is determined from the G and G" data at a f/rj as... 

The corresponding relationships are noted for the storage and loss moduli, zero-shear viscosity, steady-state recoverable compliance, and average relaxation times (cf. eqns [32]-[35]) ... 

The steady-state compliance shows a strong dependence on the molecular heterodispersity. Thus the value of for a mixture of two fractions of the same polymer, one of low and the other of high molecular weight, may be up to 10 times as high as that of each component. This behavior can be explained by taking into account that 4 is the total recoverable deformation per unit of shear stress. The chains of high molecular weight have a very... 

In the course of tensile creep, the form of the time dependence of strain (as expressed by the stretch ratio X, for example) depends on the magnitude of tensile stress at high stresses." " Recovery is considerably more rapid than would be predicted from the Boltzmann superposition principle, as illustrated in Fig. 13-23 for polyisobutylene of high molecular weight. " The course of recovery is predicted successfully by the theory of Bernstein, Kearsley, and Zapas. 2 - 22 -pije stress-dependent recoverable steady-state compliance D = which is equal to Z) at low stresses, decreases with increasing Ot- This effect, moderate when the tensile strain e is defined as X — 1, is more pronounced when it is replaced by the Hencky strain, defined as In X. The stress dependence of steady-state compliance in shear will be discussed in Chapter 17. The reader is referred to the review by Petrie" for more details. 

Figure 11.4.4 shows that experimentally t](y) and ti ico) and /7 (cu) areallvery similarfunctions. Other predictions for some of the linear viscoelastic functions are recoverable steady state shear compliance... 

Thus, creep recovery leads to the same material function as creep (as long as we are in the linear viscoelastic regime). The ultimate recoil or recoverable shear is the recovered shear in the limit of long time when recoil has ended, and this quantity is directly related to the steady-state compliance. 

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