Strain velocity

The terms rate of strain, velocity gradient and shear rate are all used synonymously and Newton s dot is normally used to indicate the differential operator with respect to time. For large gaps the rate of strain will vary across the gap and so we should write... 

Usually, investigations deal with uniaxial tension in a cylindrical sample which is easier to produce in experiments than other types of tension (see Fig. 2). In this case the strain velocity tensor is ... 

Rallman RL 16). Viscosity under stationary tension approaches the constant, while under shear at the same strain velocities it decreases rather intensively unfortunately the experiment in extension was carried out within a limited range of strain velocities. 

Tensile Strain Development in Time. Extension at Constant Strain Velocity... 

Fig. 3. Effective viscosity versus time at extension under conditions of constant strain velocities in molten polyisobutylene [23]...

In works where the stationary flow was detected, the viscosity X = cr/x was, normally, either constant 16 20,31> or grew with increase of x 18 19-24 27> above 3r 0 (r 0 is the maximum shear viscosity) in the linear strain region. The intensive growth of viscosity X was detected, apparently, for the first time in Ref. 19). The greatest increase in viscosity X from 3r to 20ri was observed in Refs. I8 21,30>. In the same works the viscosity X started to drop after the growth (this shall be discussed in detail below). Usually the relationship x/rj (rj = a12/y is effective viscosity at stationary shear) is taken at different strain velocities, which could vary by more than an order of magnitude (see, for example, Ref.30>). The data on the existence of stationary flow and behavior of viscosity in this case, as well as descriptions of extended polymers are collected in Ref. 20>. 

Elastic strain in extension under conditions of x = const in most works (see 18-201 24,26i) js a strictiy increasing function of time with smoothly decreasing derivative d(ln a)/dt. In these cases the velocity of irreversible flow ep(t) strictly increases (exclusion detected in Refs.23,24> is discussed below). At stationary flow the elastic strain is constant and the irreversible strain velocity is ep = x. The higher x, the more the share of elastic component is at fixed general strain s. 

Dependencies of Stress and Strain Velocities upon Elastic Strain... 

Let us consider also the regularities common for different types of extension. Dependencies of extension a upon elastic strain a are given in Fig. 5. Continuous lines in Fig. 5 indicate dependencies a(a) in extension at different constant strain velocities x. The higher x, the higher passes the dependency a(cx). The points with maximum a... 

The comparison of dependencies o(oc) obtained under conditions of constant force and constant strain velocity indicates that the strain velocities are approximately similar at the points of their crossing. 

The existence of the functional dependency of Eq. (8) provides a possibility411 to assume that relaxation processes develop in time similarly after different homogeneous extensions (and even after a preliminary partial relaxation of stresses), if the accumulated elastic strain a and irreversible strain velocity ep in the medium are similar at the moment of start of relaxation. According to the above, we can take, for example, instead of parameters (a, e ) parameters (cr, a). If the processes of stress relaxation coincide in time, the processes of retardation occur automatically similarly under the same initial conditions of (at, e ). 

In Refs. l7 18,211 the uniaxial extension of polyethylene at a constant strain velocity was considered. Figure 6 gives the dependencies of effective viscosity a/x (a is tensile stress, x is strain velocity) upon time t obtained in final form in Ref.21K The stationary flow (i.e., a constant, value of a/x) was attained practically for all values of x. The higher x, the higher passes the respective curve. The lower curve 3a12/y (here a12 is... 

The major difference between dependencies of ct/x upon t, in case of polyethylene, and the similar curves in case of polyisobutylene (see Fig. 4) was that even at very low values of tensile strain velocities x in case of polyethylene, the flow failed to remain linear during the entire time interval of the experiment. Dependencies of effective viscosity a(t)/x at low values of x reached a constant value (stationary flow) exceeding 3r during times t, significantly exceeding the similar time under shear. With further increase in strain velocity x, the value t started to decrease in the same way as it was in experiments with polyisobutylene (PIB). 

Thus, on the basis of facts discussed in this section, we may conclude that effective viscosity ct/x, in extension of some polymers, passes the maximum with increasing strain velocity x. The first time t measured reliably in the nonlinear strain region exceeds significantly the respective time 0 in the linear area. Dependency x decreases with further increase in t(x). 

This section will deal with suppression of flow in extension of molten polymers in the region of significant elastic strains21 24 . The study of polyisobutylene 11-20 23,35) failed to reveal such phenomena (the velocity of irreversible strain ep = d(ln[3)/dt increased strictly with time). Retardation of polymer fluid flow is considered on the example of homogeneous extension at constant strain velocity and force. Most experiments were carried out with commercial low-density polyethylene (LDPE) with molecular weight MW 105. Figure 7 gives experimental dependencies of tensile force F/S0 and irreversible strain In 3 ( 3 = e/ot) upon time t at different... 

Fig 7a-d. Tensile force F, total (lne) and irreversible (In P) strains versus time t at different strain velocities [23, 24] (see text for explanation)... 

In case of extension of polyisobutylene (see Fig. 1) the hardening effects were not observed in the investigated region of strain velocities at room temperature. Dependencies of F(t) had one maximum (as in case of polyethylene at low values of x) and then decreased strictly while the velocity of irreversible strain was strictly increasing (ep = d(ln P)/dt). 

Let us consider now the effect of flow retardation in extension under constant force 25). The above-mentioned experiments wer,e carried out with low-density polyethylene at 125 °C and with polyisobutylene 11-20 at 44 °C. It should be noted right away that at the above-specified temperatures these melts have approximately similar characteristics under shear strain maximum viscosity is t) 3x 105 Pa s and relaxation time is 0 102 s. Flow curves in the investigated region of shear strain velocities also did not differ significantly from one another. 

Despite the proximity of flow curves, as well as values r) and 0 for these polymers (values r 0 and 0 are usually subject to dimensionless representation, normalization ofo0 and t), the time during which the maximum possible strain is attained In max = 2.8 for polyethylene and exceeds that of polyisobutylene at the same o0 by factor of 6. In this case the dependency In e(t) in the region of measurement for polyisobutylene is characterized by increasing strain velocity x = d(ln e)/dt in contrast to which x(t) decreases strictly in low-density polyethylene within a significant section of s and becomes approximately constant at high values of t. 

Division of the total tensile strain under conditions of F = const into several components 25,6R,69) produced interesting results (see Fig. 8). It has been found that the behavior of molten low-density polyethylene (Fig. 8a) is qualitatively different from polyisobutylene (Fig. 8 b) the extension of which was performed under temperature conditions where the high-elasticity modulus, relaxation time, and initial Newtonian viscosity practically coincided (in the linear range) in the compared polymers. Flow curves in the investigated range of strain velocities were also very close to one another (Fig. 21). It can be seen from the comparison of dependencies given in Fig. 8a,... 

Fig. 22. Variation of total strain (In e) and its reversible (In a) and irreversible (In P) components with time (t) in extension under conditions of constant strain velocity (x = const)...

The approach to analysis of biaxial extension of melts in the simulation of the sleeve inflation process was developed by Pirson and Petrie in 1966-1970 with the use of ideas of the thin shell theory which allows to substitute sleeve film by flat film in analysis. The problem was formulated more accurately and completely and solved in works by Han et al. The author made several conclusion the velocity of material extension changes in the main direction of sleeve motion while effective longitudinal viscosity may increase, decrease, or remain constant depending on the nature of material and the range of strain velocities under consideration longitudinal viscosity of the material at fixed process parameters decreases with temperature rise (the behavior of longitudinal velocity is described more strictly above, in Sect, 2.2.6). 

In all above- and below-cited publications in this field (e.g. 84 ) the problem was solved in order to calculate the tensors of strain velocity and stress, to prognosticate alteration of longitudinal viscosity, profile of alteration of the thickness of material over the height of the film sleeve (by coordinate on the central line of the sleeve counted from the outlet face of the extrusion head) and configuration of the sleeve ( bubble ) and also to solve thermal problems in order to determine the dependency of melt temperature upon height (or time) and to forecast the position of the crystallization... 

In 77) the authors give dependencies of the maximum Newtonian viscosity upon amplitude of periodic strain velocity q0 = f(e) for polyethylene and polystyrene. It has been also revealed that the dependency of normalized viscosity upon the velocity of stationary shear T /r 0 = f(y q0) obtained at r. = 0 coincides with a similar dependency when acoustic treatment is employed, i.e., at e 0. In other words, the effect of shear vibrations and velocity of stationary shear upon valuer] can be divided, representing the role of the first factor in form of dependency q0(s0) and that of the second in form of dependency (n/q0) upon (y r 0) invariant in relation to e. 

Fig. 16. Maximum viscosity versus periodic strain velocity e...

For low strain velocities a linear approach suggests itself... 

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