Stress relaxation, after step shear strain

The breadth of the scope of nonlinear phenomena can be grasped in part by considering the various time-dependent probes of linear viscoelasticity cited in Table 3.3.2 sinusoidal oscillation, creep, constrained recoil, stress relaxation after step strain, stress relaxation after steady shearing, and stress growth after start-up of steady shearing. In the linear regime— that is, at small strains or small strain rates—the experimental results of any one of these probes (in simple shear, for example) can be used to predict results for any of the other probes, not only for simple shearing defor-... 

Calculate the stress relaxation modulus of the Rouse model (Eq. 8.55) by showing that after a small step shear strain 7 at time t 0 the correlation function of normal modes decays as Xpx t)X y(t)) — i kTjkp) exp (- tjxp). 

Stress relaxation after sudden step shear strain [127.128]. 

Next we consider a polymer melt of high molecular weight in which entanglement is very important. To calculate G(t)> it is convenient to consider the stress relaxation after a step strain. Suppose at t = 0 a shear strain y is applied to the system in equilibrium. The strain causes the deformation of the molecular conformation, and creates the stress, which relaxes with time as the conformation of polymers goes back to... 

Analysis of the distribution of lifetimes for the bridges can be used to deduce their affect on the shear stress relaxation after a unit shear strain [40]. A similar approach has been used to study the dynamic response of triblock copolymers, adsorbed via their terminal blocks between two parallel plates, when they are subjected to step and sinusoidal shear [41]. 

In the limit of linear stress-strain relations, the relaxation modulus does not depend on the initial deformation step and the rheological properties are only described by transient functions. Equation 9.11 suggests that the relaxation modulus describes the stress relaxation after the onset of a step function shear strain. In viscoelastic liquids of entangled solutions of rod-shaped micelles, an applied stress is always relaxing to zero after inflnite long periods of time. 

Step-strain stress-relaxation measurements have been frequently used to determine Sr(X) for polymer melts > . Equation (6) shows that if separability of time and strain effects is possible for the melt under consideration, the stress after a step elongational strain can be factored into a time-dependent function, the linear shear relaxation modulus G(t), and a strain-dependent function, the nonlinear strain measure Sr(X). Also other types of experiment may be oerformed to obtain Sr(X), such as constant-strain-rate experiments "", creep under constant stress and constant-stretching-rate experiments but these methods require more involved analytical and/or numerical calculations. 

Fig. 17 Shear stress relaxation modulus for unfilled LDPE and a series of LDPE/LDH nanocomposite melts after a step strain...

Relaxation After a Step Strain for the Lodge Equation Calculate the relaxation of the shear stress and the first normal stress... 

We have carried out standard rheometric tests as done many times in the literature for entangled polymer solutions. These experiments include startup shear, large amplitude oscillatory shear (LAOS) and large step strain. In terms of the rheological features, we observed the same as others. For example, there is a stress overshoot in startup shear in the stress plateau region the apparent G can drop below G" at frequencies of the elastic plateau and amplitudes around and above 100% and relaxation modulus decreases in time after large step strains. 

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 ... 

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