Uniaxial steady elongation

Small-amplitude oscillatory shear Uniaxial steady elongation Biaxial steady elongation Planar steady elongation Elongational stress growth... 

Note 3 The Finger strain tensor for a steady uniaxial homogeneous elongation deformation or flow of an incompressible viscoelastic liquid or solid is... 

Note 2 From the definition of general homogeneous flow (Definition 1.5) G X = GX = constant) in the particular case of steady uniaxial elongation flow... 

Quotient of the difference between the longitudinal stress (cth) and the lateral stress (ctzi) and the elongational strain rate ( ) in steady uniaxial flow... 

Figures 15 to 18 show the predictions of the model for LD at 160°C in steady state and some transient flows in shear and uniaxial elongation.

LDPE, and with polypropylene, PP, was studied In steady state shear, dynamic shear and uniaxial extenslonal fields. Interrelations between diverse rheological functions are discussed In terms of the linear viscoelastic behavior and Its modification by phase separation Into complex morphology. One of the more Important observations Is the difference In elongational flow behavior of LLDPE/PP blends from that of the other blends the strain hardening (Important for e.g. fllm blowing and wire coating) occurs In the latter ones but not In the former. 

In Fig. 19 the steady state uniaxial elongational viscosity, ng/3. Is conqiared with the steady state shear, n> as well as dynamic, n , end complex, n, viscosities. It Is evident that some strain hardening, evident In I.100 (LLDPE-10) Is systematically diluted by the Increasing amount of LPX-30. Thus, Series 1 behaves as a truly miscible system. By contrast, addition of LDPE (11.100) to LPX-30 (1.0) Is generating more a complex variation of properties. The strain hardening, already visible at lOX of LDPE, reaches Its maximum not at lOOX LDPE but rather at 50 50 composition. Note that at e 0.1 (s" ) the maximum strain at break for 11.50 Is ejj > eg 3.2. The blends behave as Immiscible. 

Figure 4.3 shows the steady-state modulus, F , vs. the reduced elongation rate, 63T, computed from (4.17) for uniaxial elongational flow with fixed elongation rate, 63 = const. The modulus deviates from the modulus of the Gaussian system at the elongation rates e r > 0.5. 

Fig. 4.3. Reduced modulus at the steady-state limit, vs. reduced elongation rate, ear, computed from (4.17) for the non-linear and Gaussian systems subjected to uniaxial elongational flow and N = 100...

Fig. 4.4. Reduced chain elongation coefficient, Xs/X s, vs. reduced time, t/r, computed from (4.18), (4.19) for the uniaxial elongational flow with fixed elongation rates and A = 100. Steady-state levels of the elongation coefficients indicated...

Fig. 4.11. Time evolution of the reduced critical cluster volume, g (6, t)/g 6. t = 0), vs. orientation angle, 0, between the initial state and the steady-state calculated for uniaxial elongational flow with fixed elongation rate, esr = 1. Af = 100...

Uniaxial extensional viscosity and shear viscosity 77+ as functions of time after inception of steady straining for lUPAC A low density polyethylene. The open symbols are elongational viscosities the solid and half-open symbols are shear viscosities. Adapted from Meissner (1972). 

For the steady-state uniaxial elongational flow, the relative deformation gradient tensor F,(r( t) can be written... 

Note that Eq. (2.15) defines the rate-of-strain tensor d in steady-state uniaxial elongational flow. If the surfaces transverse to the direction of principal elongation (i.e., the direction of stretching) are unconstrained, we have... 

Let us consider the upper convected Maxwell model given by Eq. (3.4). For steady-state uniaxial elongation flow, for which the rate-of-strain tensor d is defined by Eq. (2.15), we have (see Appendix 3E)... 

Therefore, the steady-state uniaxial elongation viscosity ]g(e) can be calculated from... 

Relaxation spectrum has been calculated from oscillatory measurements (Figure 1) whereas the nonlinear parameters of all models were identified on the steady uniaxial elongational data only (Figure 2). All model parameters for corresponding materials are summarized in Tables 1-2. The model predictions in comparison with experimental data (shear viscosity, first q/i, and second normal sttess coefficients, y/2, uniaxial extensional viscosity, qE,u) are depicted in Figures 2-5 for all materials. 

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