Stress relaxation function

Figure 5.18 This figure shows how the properties of a glass polyalkenoate cement change as it ages. S is the compressive strength, E the modulus, a a stress-relaxation function, and c a strain-conversion function from elastic to plastic strain (Paddon Wilson, 1976).

The transition strongly affects the molecular mobility, which leads to large changes in rheology. For a direct observation of the relaxation pattern, one may, for instance, impose a small step shear strain y0 on samples near LST while measuring the shear stress response T12(t) as a function of time. The result is the shear stress relaxation function G(t) = T12(t)/ < >, also called relaxation modulus. Since the concept of a relaxation modulus applies to liquids as well as to solids, it is well suited for describing the LST. 

Finally it is worth noting an alternate form for the stress dependence of a series of strains. Some microstructural models utilise the memory function m t). This is the rate of change of the stress relaxation function ... 

This is the stress relaxation function, so the slope plotted as a function of time provides us with G(t). Now in the limit of short times we find the exponential tends to unity ... 

For a viscoelastic liquid (7(0) = 0. These expressions transform the stress relaxation function to the storage and loss moduli. Being Fourier trans-... 

The data has been superimposed by dividing the relaxation function G(t) by G(t = 0), the limiting short time value, and the time has been divided by the characteristic relaxation time Tr. The first feature to notice is that the stress relaxation function overshoots and shows a peak. This is an example of non-linear behaviour. It is related to both the material and the instrumental response (Section 4.5.1). The general shape of the curves (excluding the stress overshoot) can be described using two approaches. 

The stress relaxation function GR(t) for a single chain is Rouse-like and given by... 

The function fj(y) represents the non-linear strain dependence of the deformed tube. The non-linear stress relaxation function in the reptation zone is thus... 

Figure 1 The shear stress relaxation function, C(t), obtained from a molecular dynamics simulation of500 SRP spheres at a reduced temperature of 1.0 and effective volume fraction of 0.45. Note that n = 144 and 1152 (from Equation (1)) cases are superimposable with the analytic function of Equation (4) ( Algebraic on the figure) for short times, t (or nt here)...

The rheology of many of the systems displayed gel-like viscoelastic features, especially for the long-range attractive interaction potentials, which manifested a non-zero plateau in the shear stress relaxation function, C/t), the so-called equilibrium modulus, which has been considered to be a useful indicator of the presence of a gel. The infinite frequency shear rigidity modulus, was extremely sensitive to the form of the potential. Despite being the most short-... 

Figure 5 Time evolution of the shear stress relaxation function, C t), for the 36 18 potential at 4> = 0.2 and T = 0.3. The waiting times are = 3 and 162 for the two curves...

In common with glasses, the dynamics of particle gels can be strongly dependent on its history of formation. One finds that the relaxation dynamics become increasingly slower with age or waiting time from the quench, The stress relaxation function now depends on two times, t -I- tj. The larger the... 

Fig. 14. The strain dependent part of the shear stress relaxation function for different values of the tube constraint parameter z...

Thus, the stress relaxation function can be approximated by a stretched exponential form... 

Note Added In Proof This question has now been answered. Stress relaxation functions for LJ argon obtained in recent simulations (S. M. Rekhson, D. M. Heyes, C. J. Montrose and T. A. Litovitz, J. Non-Cryst. Solids 38-39, 403 (1980) D. M. Heyes, J. J. Kim, C. J. Montrose and T. A. Litovitz, J. Chem. Ptyrs. 73(8), 3987 (1980)) show that for argon also the approach to equilibrium is non-exponential. Remarkably enough, the decay function proves to be almost indistinguishable in form from that measured in the laboratory for Si02 glass. 

While considering tendons and ligaments as simple nonlinear elastic elements (Table 48.6) are often sufficient, additional accuracy can be obtained by incorporating viscous damping. The quasi-hnear viscoelastic approach [Fung, 1981] introduces a stress relaxation function, G(t), that depends only on time, is convoluted with the elastic response, T (A,), that depends only on the stretch ratio, to yield the complete stress response, K X, t). To obtain the stress at any point in time requires that the contribution of all preceding deformations be assessed ... 

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). 

Equation (7.198) agrees with the empirical equation proposed by Bernstein, Kearseley, and Zapas (BKZ), who found that the stress response for various flow histories can be predicted by eqn (7.198) using the stress relaxation function ap(t, E) determined experimentally. Subsequent experiments done by many authors " revealed that the BKZ equation is one of the most successful empirical constitutive equations. 

As was shown in Section 7.5, the empirical stress relaxation function 4>ap t, E) is in good agreement with the reptation theory for linear polymers of narrow molecular weight distribution. This, together with the success of the BKZ equation, indicates that the constitutive equation... 

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