Shear stress growth coefficient

Out-of-phase component of complex viscosity Shear stress growth coefficient Shear stress decay coefficient Tensile stress growth coefficient Tensile stress decay coefficient... 

In a third test, a step fimction shear rate is suddenly applied att = 0 (start-up flow). In this case, the shear stress a is measrmed as a fiinction of time and the shear stress growth coefficient (stressing viscosity) +( ) can be calculated ... 

Figure 9.9 Normalized shear stress growth coefficient (s, ) and normalized first normal stress growth coefficient (s, ) as a function of the normalized time s at a shear rate of 2s (T = 20°C, 15%-MDMA0-HN03, pH = 3.5). Predictions of the Giesekus model...

The lower limit on the integral is zero rather than minus infinity, since the sample is known to be in a stress-free state at t = 0. The ratio of the stress to the shear rate is called the shear stress growth coefficient and has units of viscosity ... 

In start-up of steady simple shear, the measured stresses are divided by the imposed shear rate or its square to obtain the shear stress growth coefficient and the first normal stress growth coefficient, which are defined as follows ... 

At short times, when the deformation is small, we expect the shear stress growth coefficient to follow the prediction of the Boltzmann superposition principle. And if the shear rate is sufficiently low, the entire transient should be governed by this principle, and rf (t, y) becomes the same as r) t), which was shown as Eq. 4.8 ... 

As in the equation for evolution of fiber orientation, a closure approximation is needed in Eq. 3.102 to express A4 in terms of A. Using Eq. 3.102 it is straightforward to show that the shear stress growth coefficient, +, and the first normal stress growth function, N, are... 

Viscosity T), first normal-stress coefficient /i, second normal-stress coefficient /2 Shear stress growth coefficient T1+, first normal-stress growth coefficient t t[, second normal-stress growth coefficient /J Shear stress decay coefficient Tj", first normal-stress decay coefficient /f, second normal-stress decay coefficient /j Shear creep compliance J... 

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

Figure 13.5 Dependences of the reduced tensile stress growth coefficient ti (i,t)/ar at 170°C on reduced time fay and reduced strain rate iaj for a SAN resin (wAN = 0.264, Mw = 78 kg/mol, Mw/M = 2.8) during the startup of uniaxial extensions flow. Also illustrated (dashed curve) are dynamic shear viscosity data displayed as 3 r7 (c<, 170°C) versus or7 as suggested by Trouton [64]. Reproduced from L. Li, T. Masuda and M. Takahashi, J.Rheol., 34(1), 103(1990), with permission of the American Institute of Physics...

Hence, for Newtonian liquids, the tensile stress growth coefficient is a constant quantity and the extensional viscosity is three times the shear viscosity. This result was verified experimentally in 1906 by TroutonOD and is known as Trouton s rule. In addition, the ratio of the extensional viscosity to the shear viscosity is known as Trouton s ratio. 

Figure 11.10 Predictions of shear stress and first normal stress growth coefficients by the CCR model of Likhtman etal. modified to include chain stretch by Graham eta/. [37] in start-up of steady shearing compared to data [53] for a 7% solution of nearly monodisperse polybutadiene (M = 350,000) at the shear rates shown, where the parameters = 4.156 -10" s,M = 51779, and Gg = (5/4) G 5 = 51770 Pa are obtained from linear viscoelastic measurements, and = 0.1 sets the rate of constraint release.The longest Rouse orientational relaxation time should be given by the theoretical relationship to =Z however, the retraction rate is artificially doubled (equivalent to taking xj, =0.5 Z T )tooffsetthe error introduced by a closure approximation and to give a better fit to the data. From Graham et al. [37].

Figure 11.12 Comparison of predictions of shear and extensional stress growth coefficients rf (f, y) and (f, e) according to the CCR model of Likhtman etal. [28] modified to include chain stretch by Graham et al. [37] with experimental data for 7% solutions in tricresylphosphate at 40 °C of nearly monodisperse polystyrenes of molecular weights (a) 2.89 million and (b) 8.42 million at the rates shown.These samples are the same as described in Figs. 11.8 and 11.9 the shear data are from Pattamaprom and Larson [36] and the extensional data from Ye et al. [52] shifted to 40 °C. The parameters Tg = 1.911 -10 s, = 270,000, and Gg = (5/4) G 5 = 8075 Pa are obtained from linear viscoelastic measurements and c = 0.1 sets the rate of constraint release. The Rouse relaxation time is set to be tj, = 0.5 Tg, as...

Figure 11.20 Uniaxial extensional stress growth coefficient (upper data sets) and shear stress...

Stress growth coefficients in extension and in shear r] t,y) as functions of time... 

The extension and shear rates made dimensionless with Tjfi) are 120 times smaller than those made dimensionless with Tj.The time is made dimensionless using t,(0), and the stress growth coefficients using t IOIGn. From McLeish and Larson (95). 

Figure 11.25 Comparison of theory (lines) and experiment (symbols) for stress growth coefficients in start-up of steady shear (lower curves) and start-up of steady extension (upper curves) fora polyisoprene H polymer with Mg = 20,000 M, = i i,ooO.The extension rates are 0.03 and 1 s" while the shear rates are 0.01,0.03,0.1,0.3,1,and 3 s". From McLeish etal. [97].

Casson models were used to compare their yield stress results to those calculated with the direct methods, the stress growth and impeller methods. Table 2 shows the parameters obtained when the experimental shear stress-shear rate data for the fermentation suspensions were fitted with all models at initial process. The correlation coefficients (/P) between the shear rate and shear stress are from 0.994 to 0.995 for the Herschel-Bulkley model, 0.988 to 0.994 for the Bingham, 0.982 to 0.990 for the Casson model, and 0.948 to 0.972 for the power law model for enzymatic hydrolysis at 10% solids concentration (Table 1). The rheological parameters for Solka Floe suspensions were employed to determine if there was any relationship between the shear rate constant, k, and the power law index flow, n. The relationship between the shear rate constant and the index flow for fermentation broth at concentrations ranging from 10 to 20% is shown on Table 2. The yield stress obtained by the FL 100/6W impeller technique decreased significantly as the fimetion of time and concentration during enzyme reaction and fermentation. 

As discussed in Sect. 4, in the fluid, MCT-ITT flnds a linear or Newtonian regime in the limit y 0, where it recovers the standard MCT approximation for Newtonian viscosity rio of a viscoelastic fluid [2, 38]. Hence a yrio holds for Pe 1, as shown in Fig. 13, where Pe calculated with the structural relaxation time T is included. As discussed, the growth of T (asymptotically) dominates all transport coefficients of the colloidal suspension and causes a proportional increase in the viscosity j]. For Pe > 1, the non-linear viscosity shear thins, and a increases sublin-early with y. The stress vs strain rate plot in Fig. 13 clearly exhibits a broad crossover between the linear Newtonian and a much weaker (asymptotically) y-independent variation of the stress. In the fluid, the flow curve takes a S-shape in double logarithmic representation, while in the glass it is bent upward only. 

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