Flows involving different types of chain branches have been modelled as functions of the uniaxial elongation using the recently generated constitutive model and molecular dynamics for linear viscoelasticity of polymers. Previously control theory was applied to model the relationship between the relaxation modulus, dynamic and shear viscosity, transient flow effects, power law and Cox-Merz rule related to the molecular weight distribution (MWD) by melt calibration. Temperature dependences and dimensions of statistical chain tubes were also modelled. The present study investigated the elongational viscosity. We introduced earlier the rheologically effective distribution (RED), which relates very accurately and linearly to the viscoelastic properties. The newly introduced effective strain-hardening distribution (RED H ) is related to long-chain branching. This RED H is converted to real long-chain branching distribution by melt calibration and a simple relation formula. The presented procedure is very effective at characterizing long-chain branches, and also provides information on their structure and distribution. Accurate simulations of the elongational viscosities of low-density polyethylene, linear low-density polyethylene and polypropylene, and new types of MWDs are presented. Models are presented for strain-hardening that includes the monotonic T. Borg (B) increase and overshoot effects. Since the correct behaviour at large Hencky strains is still unclear, these theoretical models may aid further research and measurements.
Start-up and transient shear-stress flows are modelled here using the recently generated constitutive model for linear viscoelasticity of polymers. The relation derived from control theory and the melt calibration procedure has been developed between the relaxation modulus, dynamic viscosity and molecular weight distribution (MWD). This study extends the start-up, decay, and transient effects of linear viscoelasticity on stress and viscosity from the novel viewpoint of the classical Boltzmann superposition principle. We show that shear viscosity is not only a "viscometric function" but also linearly viscoelastic. Moreover, for a constant MWD, application of the method to melt calibration allows interconversions between rheological functions that depend on frequency, shear rate and time. A Cox-Merz rule and power law are also verified. The shear viscosity measurements of well-characterized classical low-density polyethylene (LDPE IUPAC A) with a known MWD were used to obtain time-dependent stress and viscosity transitions. The developed formulas model the start-up situation with an overshoot effect, shear-stress growth, and decay coefficients. Simulations were performed for relaxation modulus measurements at different shear histories defined by the effective viscosity. Dynamic moduli components 1 To whom correspondence should be addressed. 2 were modelled, with the results compared with measurement data. As an example of practical applications, capillary flow and injection moulding for producing a mobile-phone cover were modelled to obtain the pressure loss and orientation level for every element in finite element models, which predicted the shrinking and warping of the end products.
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