Recent experimental results for concentrated polymer systems and predictions from two time (t′, t″) single integral type constitutive equations are discussed in detail. The experiments covered are stress relaxation (after cessation of steady shear flow), single- and double- step stress relaxation under large shear strains, parallel- and orthogonal- superposed shear flows, oscillatory shear flow with large amplitude and uniaxial extensional flow.
The experimental results obtained by Malkin et al. for recoverable shear strain following steady shear flow, stress development and stress relaxation are also reviewed. Finally, it is shown that the time-temperature superposition principle can also be applied to nonlinear viscoelastic functions.
The following conclusions were drawn: (1) All of the predictions tor stress relaxation calculated from the rate-of-strain dependent model agree very well with the experimental results. (2) None of the three models (rate dependent, strain dependent and stress dependent models) can well explain the recent experimental results for single- and double- step stress relaxation under large shear strains obtained by Osaki et al. (3) The energy needed for the breakage of entanglement network can be calculated from the measurement of recoverable shear strain.

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In 1971 Barrie carried out injection molding experiment in a circular disk mold in which he applied a constant flow rate Q and measured the pressure drop ΔP in the cavity, just before the mold was filled up in the filling stage. The material was a propylene ethylene copolymer. In the present study a scheme was devised for predicting the ΔP from rheological and thermal data. It was assumed that the relation Y_U/Y=γ exp (-t_U/t_M) can be applied over a small segment of the cavity, as assured by Shibayama in 1971 and 1972 for constant-pressure filling, where Y is half the thickness of the cavity, Y_U is half the thickness of the molten layer (or the flowing layer for an amorphous thermoplastic), and t_U is the time measured from the instant the flow front passed the segment.,For a crystalline thermoplastic, γ=(1/2) exp (1/2) and t_M=(ρsωsY²/4κs) (χ_L/χ_S-1)/2 where ρs, am or κs are the density, the specific heat and the thermal conductivity, respectively, of the frozen layer, χ_L is the enthalpy of a unit mass of the molten layer at the effective melt temperature TI* referred to the mass at the mold temperature T_M; and where χ_S=ωs (T_L-T_M)/2 and T_L is the melting point. Because thermal data for the copolymer were not available, a parameter was recalculated from ΔP. The parameter is the thermal characteristic time per unit squared half the thickness (TCTU), which was defined as t_M/Y². If the scheme is satisfactory, the recalculated TCTU should be independent of Y or Q, though it may depend on T_I* and T_M. For easy calculation the inlet radius W_i was neglected on the basis that the non-Newtonian index was fairly different from 1. The recalculated TCTU was often above the upper bound of TCTU, which can be expected from the thermal data of propylene polymers and copolymers, but it falls rapidly as Y or Q gets larger; thus it often fell below the lower bound. The small TCTU suggests an increase in ΔP, that is either an effect due to the tensile viscosity effect as mentioned by Barrie or an apparent effect due to the increase of the pressure drop before the cavity. The larger hydrostatic pressure during injection enhances the viscosity of the melt and then the pressure drop before the cavity. This causes an apparent increase in ΔP because ΔP was measured as the difference between the injection pressure and the open mold injection pressure. The large TCTU suggests possibly a limitation of the present scheme for slow filling which extends over time periods of 2t_M or more.

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The objective of the present study is to confirm the hypothesis presented by one of the authors that the initiating mechanism of melt flow fracture is affected by the behaviors of distribution and vibration of pressure in shear flow field of high density polyethylene melts. It was recognized that the pressure at reservoir wall and in the reservoir, which was measured in rectangular direction with the flow direction, varied with the positions of measurement points i.e., the pressure distribution curves were similar to the profile of shear rate distribution curves under normal flow region or low shear rate region. The mean pressure in a pressure distribution curve increases with the increase of shear rate. The pressure vibrations occurred at particular measurement points, which were presented in the previous paper as fracture region at the critical condition of normal flow to spiralling flow under moderate shear rate. At these positions of measurement points the vibration phenomena generally initiated near the capillary inlet under low shear rate, and moved to the reservoir wall from the reservoir axis with the increase of shear rate and ultimately, disappeared with further increase of shear rate. The larger melt index of high density polyethylene, the larger pressure vibration becomes. Both flow rate excess induced by the melt flow fracture and the effects of L/D of capillaries upon flow rate excess can be explained by the hypothesis utilizing the particular fracture surface in shear flow field.

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A theory of failure conditions, which depends on the deformation history is proposed. The constitutive relations are assumed to have a generalized form of the hypo-elastic type, that is, the stress rate is a function of the stress and a functional for the deformation history. The failure criterion are defined by a singularity condition, and the failure stretching by a null space. An integral inner-product form of the constitutive functional is obtained. Several special failure criteria, a modified von Mises criterion and a modified Huber-Tresca yield criterion, are proposed.

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A propagation constant Γ_l^* of sound in dispersions of spherical shell structures has been calculated in the case of Rayleigh scattering as follows:
Γ_l*=Γ_l{1-(2πN/κ_l²)ΣiⁿΑ_n},
Α₀=-(Q′/Q)(κ_la)³/κ_l,
Α₁=(i/3){(ρ_m-ρ)/ρ-Α³(ρ_m-ρ′)/ρ}(κ_la)³/κ_l,
Α₂=(10μ/9κ)(Γ′/Γ)(κ_la)³/κ_l.
Here Γ_l is the propagation constant of the medium, N is the number of the shell structures in the unit volume, κ_l the wave number for the longitudinal wave in the medium, Q=(3κ′+4μ_m) (3κ_m+4μ)+12Α³(κ′-κ_m) (μ_m-μ), Q′=(κ-κ_m)(3κ′+4μ_m)-Α³(κ′-κ_m) (3κ+4μ_m), Α=a′/a, a and a′ the outer and the inner radii of the shell structure, Γ and Γ′ are the formulae containing terms of the fourth power in μ′, μ_m or μ and the third, fifth, seventh and tenth powers of Α, and ρ, κ, μ are the density, the bulk and the shear moduli of the medium, in which the prime and the subscript m denote the quantities for the inside and the shell regions, respectively.
The ultrasonic velocity V, given by V=ω/ImΓ_l* (ω the angular frequency), and the density ρ were measured by means of a singaround method at a frequency of 3 MHz in suspensions of bovine erythrocyte ghosts. The aqueous medium contained 150 mM NaCl and 5 mM potassium phosphate buffer at pH 7.0 as varying the concentration c of the ghosts and the temperature from 20° to 40°C. A transition was observed in the limiting number of velocity [V] and of density [ρ] between 30° and 34°C, where [V]=lim(V-V_s)/V_sc, [ρ]=lim(ρ-ρ_s)/ρ_sc and the subscript s denotes the medium. The bulk modulus and the density of the membrane are estimated by the use of the above theory.

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An improved version of the Pao theory (J. Polym. Sci., 61, 413 (1962)) is given. The theory is based on a linear tensorial relation between the stress σ and the recoverable strain r^-1 on a corotational system rigidly rotating with the material element. We discard Pao's assumption that the ij-element of stress is independent of kl-element of recoverable strain (k≠i or l≠j) on the fixed orthonormal system. Instead we assume that the recoverable shear strain in steady shear flow is given by half the ratio of the principal normal stress to the shear stress. The steady shear viscosity thus obtained is very close to that of the Cox-Merz empirical law, which is applicable to many polymeric systems

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