Relaxation modulus strain-dependent

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

Using this factorizability of response into a time-dependent and a strain-dependent function. Landel et ai. (61,62) have proposed a theory that would express tensile stress relaxation in the nonlinear regime as the product of a time-dependent modulus and a function of the strain ... 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

The temperature dependence of the compliance and the stress relaxation modulus of crystalline polymers well above Tf is greater than that of cross-linked polymers, but in the glass-to-rubber transition region the temperature dependence is less than for an amorphous polymer. A factor in this large temperature dependence at T >> TK is the decrease in the degree of Crystallinity with temperature. Other factors arc the reciystallization of strained crystallites ipto unstrained ones and the rotation of crystallites to relieve the applied stress (38). All of these effects occur more rapidly as the temperature is raised. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Stress relaxation is the time-dependent change in stress after an instantaneous and constant deformation and constant temperature. As the shape of the specimen does not change during stress relaxation, this is a pure relaxation phenomenon in the sense defined at the beginning of this section. It is common use to call the time dependent ratio of tensile stress to strain the relaxation modulus, E, and to present the results of the experiments in the form of E as a function of time. This quantity should be distinguished, however, from the tensile modulus E as determined in elastic deformations, because stress relaxation does not occur upon deformation of an ideal rubber. 

The data are not usually reported as a stress/time plot, but as a modulus/time plot. This time-dependent modulus, called the relaxation modulus, is simply the time-dependent stress divided by the (constant) strain (Equation 13-71) ... 

It is worth mentioning that the strain function is not temperature dependent and that the influence of temperature is only applied on the memory function or relaxation modulus through the shortening of the relaxation times with increasing temperatures. 

One convenient manner of studying viscoelasticity is by stress relaxation where the time-dependent shear stress is studied for step increase in strain. In Figure 1-7, the stress relaxation of a Hookean solid, and a viscoelastic solid and liquid are shown when subjected to a strain instantaneously and held constant. The relaxation modulus can be calculated as ... 

Consider imposing a step strain of magnitude 7 at time t = 0 (see Fig. 7.20). If the material between the plates is a perfectly elastic solid, the stress will jump up to its equilibrium value Gj given by Hooke s law [Eq. (7.98)] and stay there as long as the strain is applied. On the other hand, if the material is a Newtonian liquid, the transient stress response from the jump in strain will be a spike that instantaneously decays to zero. For viscoelastic materials, the stress after such a step strain can have some general time dependence a(t). The stress relaxation modulus G(t) is defined as the ratio of the stress remaining at time t (after a step strain was applied at time t = 0) and the magnitude of this step strain 7 ... 

For viscoelastic liquids, the Maxwell model can be used to qualitatively understand the stress relaxation modulus. In the step strain experiment, the total strain 7 is constant and Eqs (7.101)-(7.103) can be combined to give a first order differential equation for the time-dependent strain in the viscous element ... 

Initial moduli at room temperature were obtained with an Instron Model 4206 at a strain rate of 2/min ASTM D638 type V specimens were used. The Instron was also used in the creep experiments, in which deformation under a 1 NPa tensile load was continuously monitored for 10 sec, followed by measurement of the recovered length 48 h after load removal. Strain dependence of the elastic modulus was determined by deforming specimens to successively larger tensile strains and, at each strain level, measuring the stress after relaxation after it had become invariant for 30 min. 

The conversion of strain mismatch into stress is a function of the stress relaxation modulus exhibited by the polymer. A predictive stress model must incorporate the complex dependencies of the modulus and stress relaxation behavior on temperature, glass transition temperature, degree of cure, crosslink density, solvent-plasticization, and reaction kinetics. 

Figure 2. Time dependence of the linear stress relaxation modulus for the Si02 - PBA50K hybrid at a strain of 0.02 is shown. The stress relaxation data for strain values below 0.04 were identical and exhibited solid-like behavior for 80,000 s after application of the step strain.

For continuous straining the integral in Eq. (6.30) can be approximated as a series of step strains, each described by Eq. (6.32). From these equations the stress for arbitrary strain history can be predicted, although the calculation requires that both the stress relaxation function and the strain-dependence of the modulus be determined in separate experiments. This general approach to predicting the rheology of polymer melts has met with good success (Tanner, 1988). 

Under infinitesimal strain, the thermal motion of the chain activating the relaxation coincides with the motion at equilibrium. For this case, the stress o(f) under the step strain y ( 1) is proportional to y, and G(f) is independent of y and depends only on t. In this linear viscoelastic regime, the stress o(f) due to a strain y(f ) of arbitrary history at f < f is expressed as a convolution of the relaxation modulus G(f) and the strain rate dy/df (Ferry, 1980) ... 

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