Stress growth function

Figure 4.15 The stress growth function for a Maxwell model with a relaxation time tr...

For strain rates lower than 8x10 s, it was found that the rheological behaviour is nearly linear viscoelastic Fig. 5 shows the tensile stress-growth function CT (0,t) = EXT]E (E,t) at 123°C for three different strain rates in the linear range after about 1000s, the stress reaches a... 

Figure 5. Tensile stress-growth function for sample SI at 123T.

Figure 11. Tensile stress-growth function of sample SI at 123°C. Symbols data of Fig. 5. Solid lines Rouse model.

The proposed method of data treatment has two advantages (1) It allows assessment of the status of blend miscibility In the melt, and (11) It permits computation of any linear viscoelastic function from a single frequency scan. Once the numerical values of Equation 20 or Equation 21 parameters are established Che relaxation spectrum as well as all linear viscoelastic functions of the material are known. Since there Is a direct relation between the relaxation and Che retardation time spectra, one can compute from Hq(o)) the stress growth function, creep compliance, complex dynamic compliances, etc. 

The behavior of LLDPE blends at constant rate of stretching, e, was examined at 150°C. The results are shown In Fig. 13 for Series I and II as well as in Fig. 14 for Series III. The solid lines In Fig. 13 represent 3n calc values computed from the frequency relaxation spectrtmi by means of Equation (36), while triangles Indicate the measured in steady state 3n values at y = 10 2 (s ), I.e. the solid lines and the points represent the predicted and measured linear viscoelastic behavior respectively. The agreement Is satisfactory. The broken lines In Fig. 13 represent the experimental values of the stress growth function In uniaxial extension, nE 3he distance between the solid and broken lines Is a measure of nonlinearity of the system caused by strain hardening, SH. 

Figure 13. Stress growth function for blends of linear low density polyethylene, LLDPE, with (left) another type of LLDPE (miscible, Series-I) and (right) with low density polyethylene (immiscible. Series II). Broken lines experimental data in elongation triangles experimental data in steady state shearing at y - O.Ols". Solid lines were computed from the frequency relaxation spectrum. (Adapted from ref. 14.)...

Figs. 13 and 14 also provide information on sample poly dispersity. The initial slope of the stress growth function can be expressed as ... 

Figure 14. Stress growth function In uniaxial extension vs. time for LPX-30/LPX-24 blends (Serles-III) at T-150 C and e = 0.001,0.01, 0.1 and 1 (s b.

Figure 15. Initial slopes of the stress growth functions of LLDPE s (containing butene, hexene or octene comonomer) as function of resin polydlsperslty.

Figure 16. Compositional dependence of the Initial slope of the elongational stress growth function for LPX-30/LLDPE 10 (top) and LPX-30/LDFE (bottom). The broken lines were computed assuming miscibility.

Figure 17. Initial slope of the stress growth function for LPX-30/LPX-24 blends vs. composition.

The extenslonal flow. The stress growth functions In shear (n, labelled RMS) and extension, ng, are shown In Fig. 24. The dependencies for all five materials are similar. There Is an excellent agreement between ng and 3ti . Note the absence of strain hardening, SH, clearly visible (see Figs. 13 and 14) In blends of... 

The isochronal stress-strain curve can be found by analysis of the stress-growth function cr(t y) as given by the Boltzmann superposition relationship of equation (2-46). For a generalized Maxwell model, the result is the familiar... 

Two types of rheological phenomena can be used for the detection of blend s miscibility (1) influence of polydispersity on the rheological functions, and (2) the inherent nature of the two-phase flow. The first type draws conclusions about miscibility from, e.g., coordinates of the relaxation spectmm maximum cross-point coordinates (G, CO ) [Zeichner and Patel, 1981] free volume gradient of viscosity a = d(lnT]) / df the initial slope of the stress growth function S = d(lnr +g)/dlnt the power-law exponent n = d(lnOj2)/dlny = S, etc. The second type involves evaluation of the extrudate swell parameter, B = D/D, strain (or form) recovery, apparent yield stress, etc. 

Two contributions to the tensile stress growth function, r j+, should be distinguished one due to the linear viscoelastic response, Tl+gL the other originating in the structural change of the specimen during deformation, The first... 

Extensional Flows The yield stress also occurs in extensional flows [Kamal et al., 1984 Utracki, 1988]. Yield stress is apparent in two related dependencies (i) as a vertical displacement in the stress growth function at decreasing strain rates, contrasting with the normal linear viscoelastic behavior of single-phase polymeric melts, and (ii) as a deviation from the theory that Lim tie(e) = 3... 

Thus, once the four parameters of Eq 7.42 are known, the relaxation spectrum, and then any linear viscoelastic function can be calculated. For example, the experimental data of the dynamic storage and loss shear moduli, respectively G and G , or the linear viscoelastic stress growth function in shear or uniaxial elongation can be computed from the dependencies [Utracki and Schlund, 1987] ... 

Blends of LLDPE/PP = 50 50, with or without compatibilizing ethylene-propylene copolymer, EPR, were studied by Dumouhn et al. [1984]. In spite of the expected immiscibihty, the blends showed additivity of properties with good superposition of the stress growth functions in shear and elongation, as well as with the zero deformation rate Trouton ratio, Rj, = 1. In earlier work, blends of medium density PE (MDPE) with small quantities of ultra-high molecular weight polyethylene (UHMWPE) were studied in shear and extension. Again, SH and Rj, = 1 were observed. 

Fig. 7.19. Shear stress growth function at several shear rates for solution of polybutadiene M = 3.5 x 10 ). In this sample is about 1 sec. Reproduced from...

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