Steady-state recoverable strain

The steady-state recoverable compliance, which is a measure of the elastic strain, can be calculated using... 

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. 

Two quantities that play important roles in flow behavior of polymeric liquids are the steady-state viscosity at zero shear rate, rto, and the steady-state recoverable shear compliance, J°. Both are obtained quite directly from creep results, from (To and the shear rate fss in the steady-state region of the creep phase and J° from the total recoil strain (yr) in the recovery phase ... 

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... 

According to results reported in the literature [1-13] if the shear stress is canceled out after steady-state conditions are reached, the time dependence of the recoverable deformation [er(t)—cre(t)/r ] is obtained where e(t) is the shear strain, a is the stress and r is the viscosity (2.6). The higher temperature, the greater the unrecoverable contribution to the shear deformation i.e. the viscous deformation. Figure 2.3 shows the effect of temperature on the strain. 

Two steady states are recognized for the long-time creep compliance of materials. Either the sample is a solid and the compliance becomes time independent or the sample is a liquid and the compliance becomes linear in -time. Once steady state has been achieved in creep, the stress can be removed (a = 0) and the elastic recoil, called creep recovery, can be measured. Recovery strain is defined as 7r(0 s 7(0) — 7(0 for t > 0, where t is defined to be zero at the start of recovery. The recoverable compliance is defined as the ratio of the time-dependent recovery strain 7r(0 and the initially applied stress a, where both 7r and t are now defined to be zero at the start of recovery ... 

The primary creep has a rapidly decreasing strain rate. It is essentially similar in mechanism to retarded elasticity and, as such, is recoverable if the stress is removed. The secondary or steady-state creep is essentially viscous in character and is therefore nonrecoverable. The strain rate during this state is commonly referred to as the creep rate. It determines the useful life of the material. Tertiary... 

Given a sufficiently long time of creep, the velocity of creep will decelerate to zero and y t) attains an equilibrium limit if a viscoelastic solid is being measured. On the other hand, if the material is a viscoelastic liquid, the velocity of creep will decelerate to a finite constant value. Viscoelastic steady state is achieved, and y t) increases indefinitely. The creep experiment has a second part when the stress is set to zero after a period of creeping. A portion or all of the strain accumulated during creeping is then recovered as a function of time for a viscoelastic liquid or solid, respectively.For a viscoelastic liquid, the portion that is permanent deformation and irrecoverable reflects the contribution of viscous flow to the total deformation accumulated during creep. Since a viscoelastic solid does not flow, all of its creep deformation is recoverable. 

Stage I, where under the influencing load there is an initial instantaneous deflection and a delayed deformation but with creep rate decreasing with time. This is apparent when stresses and strains are low, less than 0.2%, and results in a steady state of effectively zero creep rate and on removal of load the total deformation is largely recoverable... 

H. M. Laun and H. Munstedt, Elongational behavior of a LDPE melt. I. Strain rate and stress dependence of viscosity and recoverable strain in the steady state. Comparison with shear data. Influence of interfacial tension, Rheol. Acta 17, 415-425 (1978). 

The recoverable compliance, therefore, increases with dilution. The effect of dilution on J° is best understood physically from its definition in Eq. (3.9) in terms of the recovery phase after steady-state creep. Dilution reduces the concentration of strands in the entanglement network that support the stress. Thus, for a given stress, the coils are more deformed because the stress per strand is higher, so the recoverable strain and hence J° are larger in the diluted system. 

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. 

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... 

Fig. 7.15. PE under steady state shear flow at 150 °C Strain rate dependencies of the viscosity ry, the primary normal stress coefficient and the recoverable shear strain 7e. The dotted line represents Eq. (7.122). Results obtained by Laun [76]...

Laun, H. M., and H. Munstedt, Elongational Behavior of an LDPE Melt. I. Strain Rate and Stress Dependence of Viscosity and Recoverable Strain in Steady State. Comparison with Shear Data. Influence of Interfacial Tension, Rheol Acta, 17, 415 25, 1978. 

To study the elastic response in the near-quiescent state we used a constant stress cone and plate rheometer to measure the recoverable shear strain after steady flow had been achieved. For an experimental liquid crystal polyester the elastic response at low stress is very large suggesting an elastic modulus of the order 400 N m —at least two orders of magnitude lower than a chemically similar polyester melt which does not exhibit liquid crystal phenomena (Fig. 3). However, at a shear stress of about 10 N —at which the conventional polyester is just starting to demonstrate significant elastic response—the high elasticity of the liquid crystal polyester collapses. 

---