Viscoelastic: materials parameters

Generalization of Hooke s law shows that in the range of linear viscoelasticity, material parameters become a function of time ... 

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Model (material) parameters used in viscoelastic constitutive equations... 

A viscoelastic material also possesses a complex dynamic viscosity, rj = rj - - iv( and it can be shown that r = G jiuj-, rj = G juj and rj = G ju), where CO is the angular frequency. The parameter Tj is useful for many viscoelastic fluids in that a plot of its absolute value Tj vs angular frequency in radians/s is often numerically similar to a plot of shear viscosity Tj vs shear rate. This correspondence is known as the Cox-Merz empirical relationship. The parameter Tj is called the dynamic viscosity and is related to G the loss modulus the parameter Tj does not deal with viscosity, but is a measure of elasticity. 

Many of the new plastics, blends, and material systems require special, enhanced processing features or techniques to be successfully injection molded. The associated materials evolution has resulted in new plastics or grades, many of which are more viscoelastic. That is, they exhibit greater melt elasticity. The advanced molding technology has started to address the coupling of viscoelastic material responses with the process parameters. This requires an understanding of plastics as viscoelastic fluids, rather than as purely viscous liquids, as is commonly held... 

We expect that the classical framework of linear viscoelasticity also applies at the gel point. The relaxation spectrum for the critical gel is known and can be inserted into Eq. 3-3. The resulting constitutive equation will be explored in a separate section (Sect. 4). Here we are mostly concerned about the material parameters which govern the wide variety of critical gels. 

To test the foregoing dimensionless relationship, two powders (Avicel Pl 1101, a ductile, viscoelastic material, and Emcompress, a brittle material, blended with 0.5% magnesium steaiate) were compressed on the PressterTw, a single-station mechanical replicator of rotary tablet presses. In the first set of experiments, a 16-station Manesty Betapress (a research-scale press) was simulated at two speeds, 60 and 100 rpm. In the second set, a 36-station Fette P2090 (a medium-scale production press) was simulated at two speeds, 55.8 and 70 rpm. It should be noted that 100 rpm of the Beta-press corresponds to 55.8 rpm of the Fette 2090 in terms of the linear speed of the turret. Basic parameters for the two tablet presses arc presented in Table 3. 

Hooke s Law, which states that a proportional relationship exists between stress and strain, usually holds for a viscoelastic material at a small strain. This phenomenon is called linear viscoelasticity (LVE). Within the LVE region, the viscoelastic parameters G and G" remain constant when the amplitude of the applied deformation is changed. Consequently, parameters measured within the LVE region are considered material characteristics at the observation time (frequency). 

The quantities r and r] in equation (8.34) depend on the invariants of the tensor rik in accordance with equation (8.32). We ought to note that the behaviour of a non-linear viscoelastic liquid in a non-steady state would be different, if a dependence of the material parameters r and r] on the tensor velocity gradients or on the stress tensor is assumed. This is a point which is sometimes ignored. In any case, if r and r) are constant, equation (8.34) belongs to the class of equations introduced and investigated by Oldroyd (1950). 

Beside the consideration of the up-cycles in the stretching direction, the model can also describe the down-cycles in the backwards direction. This is depicted in Fig. 47a,b for the case of the S-SBR sample filled with 60 phr N 220. Figure 47a shows an adaptation of the stress-strain curves in the stretching direction with the log-normal cluster size distribution Eq. (55). The depicted down-cycles are simulations obtained by Eq. (49) with the fit parameters from the up-cycles. The difference between up- and down-cycles quantifies the dissipated energy per cycle due to the cyclic breakdown and re-aggregation of filler clusters. The obtained microscopic material parameters for the viscoelastic response of the samples in the quasi-static limit are summarized in Table 4. 

In view of an illustration of the viscoelastic characteristics of the developed model, simulations of uniaxial stress-strain cycles in the small strain regime have been performed for various pre-strains, as depicted in Fig. 47b. Thereby, the material parameters obtained from the adaptation in Fig. 47a (Table 4, sample type C60) have been used. The dashed lines represent the polymer contributions, which include the pre-strain dependent hydrodynamic amplification of the polymer matrix. It becomes clear that in the small and medium strain regime a pronounced filler-induced hysteresis is predicted, due to the cyclic breakdown and re-aggregation of filler clusters. It can considered to be the main mechanism of energy dissipation of filler reinforced rubbers that appears even in the quasi-static limit. In addition, stress softening is present, also at small strains. It leads to the characteristic decline of the polymer contributions with rising pre-strain (dashed lines in... 

Consider two experiments carried out with identical samples of a viscoelastic material. In experiment (a) the sample is subjected to a stress CT for a time t. The resulting strain at f is ei, and the creep compliance measured at that time is D t) = e la. ln experiment (b) a sample is stressed to a level CT2 such that strain i is achieved immediately. The stress is then gradually decreased so that the strain remains at f for time t (i.e., the sample does not creep further). The stress on the material at time t will be a-i, and the corresponding relaxation modulus will be y 2(t) = CT3/C1. In measurements of this type, it can be expected that az> 0 > ct and Y t) (D(r)) , as indicated in Eq. (11-14). G(t) and Y t) are obtained directly only from stress relaxation measurements, while D(t) and J(t) require creep experiments for their direct observation. Tliese various parameters can be related in the linear viscoelastic region described in Section 11.5.2. 

Craze growth at the crack tip has been qualitatively interpreted as a cooperative effect between the inhomogeneous stress field at the crack tip and the viscoelastic material behavior of PMMA, the latter leading to a decrease of creep modulus and yield stress with loading time. If a constant stress on the whole craze is assumed then time dependent material parameters can be derived by the aid of the Dugdale model. An averaged curve of the creep modulus E(t) is shown in Fig. 13 as a function of time, whilst the craze stress is shown in Fig. 24. 

The same parameters can also be determined by applying a constant shear stress to the interface and measuring the resulting shear strain as a function of time (see fig. 3.40), so-called interfacial creep tests. At t = 0, a shear stress is suddenly applied, and kept constant thereafter. For ideally viscous monolayers a steady increase of the shear strain with t will be observed, while for an elastic material the observed strain will be instantaneous and constcmt in time. For a viscoelastic material, as in fig. 3.40, there is first am Instantaneous increase AB in the strain, the elastic response followed by a delayed elastic response BC and a viscous... 

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