Relaxation shear stress

The authors [6] suggest the term shearability , Sm, for the maximum shear strain that a homogeneous crystal can withstand. It is defined by Sm = argmax o-(s), where a(s), is the resolved shear stress and s is the engineering shear strain in a specified slip system. The relaxed shear stress, (7 in Table 4.2 is normalized by Gr. In this table, experimental and calculated values of the relaxed shear vales of Gr are given. For details on these calculations, refer to the work of Ogata et al. [6]. 

Fig. 4.21 (Color) Relaxed shear stress-strain curves of 22 materials, rescaled such that all have unit slope initially and reach maximum at 1. The renormalized Frenkel model Eq. (4.8) is shown (in heavy black line) for comparison [6]. With kind permission of Dr. Yip...

Suppose we divide the flow segments into classes according to relaxation times and index the various states by the subscript i. Thus the relaxation time and the component of shear stress borne by the segments in class i are and Fj, respectively. The applied shear force is related to the Fj s through... 

This is the fundamental differential equation for a shear stress relaxation experiment. The solution to this differential equation is an equation which gives a as a function of time in accord with experiment. 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

The reason for this unphysical behavior lies in the assumption that v remains constant. A large value of N should result in very rapid shear stress relaxation which, in turn, reduces v (Figure 7.5). 

Cooling rates can affect product properties in a number of ways. If the polymer melt is sheared into shape the molecules will be oriented. On release of shearing stresses the molecules will tend to re-coil or relax, a process which becomes slower as the temperature is reduced towards the Tg. If the mass solidifies before relaxation is complete (and this is commonly the case) frozen-in orientation will occur and the polymeric mass will be anisotropic with respect to mechanical properties. Sometimes such built-in orientation is deliberately introduced, such as... 

Not only are the creep compliance and the stress relaxation shear modulus related but in turn the shear modulus is related to the tensile modulus which itself is related to the stress relaxation time 0. It is therefore in theory possible to predict creep-temperature relationships from WLF data although in practice these are still best determined by experiment. 

For dynamical studies of diffusion, conformational and transport behavior under shear stress, or kinetics of relaxation, one resorts to dynamic models [54,58,65] in which the topological connectivity of the chains is maintained during the simulation. 

Figure 10 shows that upon cessation of shear flow of the melt, shear stress relaxation of LLDPE is much faster than HP LDPE because of the faster reentangle-... 

Figure 10 Relaxation of shear stress with time upon cessation of steady flow. (O) Resin A, ( ) Resin B, (O) Resin C, (A) Resin D, (A) Resin E. (Refer to Table 2 for symbol code.) Source Ref. 56.

The effect of driving shear stresses on the dislocations are studied by superimposing a corresponding homogeneous shear strain on the whole model before relaxation. By repeating these calculations with increasing shear strains, the Peierls barrier is determined from the superimposed strain at which the dislocation starts moving. 

The solidihed layer yields and returns to the liquid phase if the shear stress excesses the critical value, which initiates the sliding. When the stress is relaxed as a result of slip, the solid phase resumes again. The periodic transition between the solid and liquid states has been interpreted in the literature as a major cause of the stick-slip motion in lubricated sliding. Understanding the stick-slip and static friction in terms of solid-liquid transitions in thin films makes a re-... 

Cheremisinoff and Davis (1979) relaxed these two assumptions by using a correlation developed by Cohen and Hanratty (1968) for the interfacial shear stress, using von Karman s and Deissler s eddy viscosity expressions for solving the liquid-phase momentum equations while still using the hydraulic diameter concept for the gas phase. They assumed, however, that the velocity profile is a function only of the radius, r, or the normal distance from the wall, y, and that the shear stress is constant, t = tw. ... 

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. 

The transition from ideal elastic to plastic behaviour is described by the change in relaxation time as shown by the stress relaxation in Fig. 66. The immediate or plastic decrease of the stress after an initial stress cr0 is described by a relaxation time equal to zero, whereas a pure elastic response corresponds with an infinite relaxation time. The relaxation time becomes suddenly very short as the shear stress increases to a value equal to ry. Thus, in an experiment at a constant stress rate, all transitions occur almost immediately at the shear yield stress. This critical behaviour closely resembles the ideal plastic behaviour. This can be expected for a polymer well below the glass transition temperature where the mobility of the chains is low. At a high temperature the transition is a... 

If the applied shear stress varies during the experiment, e.g. in a tensile test at a constant strain rate, the relaxation time of the activated transitions changes during the test. This is analogous to the concept of a reduced time, which has been introduced to model the acceleration of the relaxation processes due to the deformation. It is proposed that the reduced time is related to the transition rate of an Eyring process [58]. The differential Eq. 123 for the transition rate is rewritten as... 

Consequently, A is called the relaxation time it is the time taken for the shear stress to fall to lie times the initial value. 

Now if we divide the shear stress in Equation (4.13) by the applied strain we obtain an expression in the form of a shear modulus. This term G(t) is described as the relaxation function ... 

In order to proceed with the evaluation of the time-dependent Poisson ratio v(0, both sets of relaxation behaviour are required. Now from Chapter 2 we know the Poisson ratio is the ratio of the contractile to the tensile strain and that for an incompressible fluid the Poisson ratio v = 0.5. Suppose we were able to apply a step deformation as we did for a shear stress relaxation experiment. The derivation then follows the same course as that to Equation (4.69) ... 

Figure 6.3 Plot of a simple non-linear viscoelastic response for (a) the stress relaxation as a function of the applied strain, (b) stress as a function of time at a shear strain y = 1 and (c) viscosity as a function of shear stress. (r (0) = 33Pas, rj(co) = 3 Pas, a = 1, P = 0.1, m = 0.35 and t = Is). Continued overleaf...

It is fairly clear that as re approaches rd the role of Rouse relaxation is significant enough to remove the dip altogether in the shear stress-shear rate curve. As the relaxation process broadens, this process is likely to disappear, particularly for polymers with polydisperse molecular weight distributions. The success of the DE model is that it correctly represents trends such as stress overshoot. The result of such a calculation is shown in Figure 6.23. 

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...

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