The network theory of Yamamoto (J. Phys. Soc. Japan, 11, 413 (1956)) is briefly reviewed and then applied to concentrated polymer systems in shearing flow, with the assumption that the probability of chain-breakage is proportional to the square of the end-to-end distance in the chain. It is shown that the theory can explain at least qualitatively some of the nonlinear viscoelastic behavior. The shear-rate dependence of the steady viscosity is similar to the frequency dependence of the absolute value of the complex viscosity. The so-called stress overshoot at the beginning of shearing flow, the stress relaxation after the stoppage of flow, the ordinary stress relaxation under large deformation, and the superposition of a small oscillation upon steady shearing flow are treated; the results are in good qualitative agreement with the experiment. The rate-dependent and the deformation-dependent relaxation spectra are derived from the time dependence of the stresses in the two kinds of stress relaxation.
A critical review of the recent phenomenological and experimental studies on the nonlinear viscoelastic phenomena in concentrated polymer systems is presented. Differential type constitutive equations are classified into three groups: the extension of the 3-dimensional linear Maxwell model, of the Jeffreys model and of the constitutive equation for general viscous fluid. Applicability and limitations of these equations are discussed. On the other hand, integral type constitutive equations are classified into two groups: multi-integral expansion type and approximate closed form type equations. The last type equations are classified further into two groups: the one time (t′)-single integral type (WJFLMB, Bird-Carreau, Yamamoto, BKZ, special forms of BKZ) and the two times (t′, t″)-single integral type (Carreau, Yamamoto, Takahashi-Masuda-Onogi). The memory function in the one time (t′)-single integral type constitutive equation depends on the invariants of either the rate of strain tensor at t′ or the relative strain tensor between t′ and t. It cannot express the dependences of relaxation times on the rate of strain or the relative strain. The memory function in the two times (t′, t″)-single integral type constitutive equation includes the time (t″) integration of the function of the invariants of either the rate of strain tensor at t″ (t′≤t″≤t), the stress tensor at t″, or the relative strain tensor between t″ and t′. It can express the dependences of relaxation times on the rate of strain, relative strain or stress. Comparisons between recent experimental results and the predictions of the two times (t′, t″)-single integral type constitutive equations for specific flows (steady shear flow, stress overshoot, and interrupted flow) are discussed in detail. Among others, it is emphasized that the integral type constitutive equations with either the rate of strain or the relative strain dependent memory function cannot explain the recent experimental data obtained by van Es and Christensen.
Applicability of a strain-dependent (S-type) constitutive equation was examined for flow properties of a polystyrene solution in diethyl phthalate. Viscoelastic quantities studied were the relaxation modulus G (t, s), transient stresses at the start and cessation of steady shear flow, and the steady shear viscosity η(κ). Here t is the time, s is the magnitude of shear, and κis the rate of shear. All these quantities were measured with rheometers of cone-and-plate type. The memory function for the S-type equation was determined from experimental results on G(t, s) and therefrom were calculated the stresses associated with steady shear flow. Calculated results were in good agreement with the observed over whole range of rate of shear investigated. The capability of the S-type equation was compared with that of a rate-dependent (R-type) constitutive equation on the basis of present results and some published results. It was concluded that the S-type equation is very useful in some types of flow including steady shear flow and the type of flow for measurement of relaxation modulus.
Stress relaxation after application of double-step shear strains was measured on two concentrated solutions of polystyrene in diethyl phthalate with a cone-plate type relaxometer. The first strain s1 was applied to the sample at time t=-t1, the second strain s2 was added at t=0 either in the same direction as the first or in the opposite direction, and then the shear stress was measured as a function of time t. A simple case of this type of deformation in which s1=-s and s2=s>0 was found to be useful to examine the applicability of various models of single-integral type constitutive equations such as proposed by Carreau, Yamamoto, and Takahashi et al. No constitutive equation of this type was able to explain the experimental results quantitatively, except in the case of very small strains. The discrepancy between theoretical and experimental values of stress became more marked as the value of time interval t1 was smaller. A new type of strain-dependent constitutive equation presented here, however, was able to represent quantitatively the stresses obtained for the type of deformation history investigated here, unless the value of t1 was very small. This equation is of the same form as Yamamoto's, but contains the invariant of strain-tensor defined on the reference time different from that of Yamamoto's equation.
A non-linear constitutive equation is derived through the modification of the memory function in the Lodge equation. It is characterized by the following two assumptions. One is the assumption that the disentanglement rate depends on the time elapsed of entanglement points as well as on the deformation rate of segmental chains. The other is the assumption that the deformation rate is expressed in terms of the time derivative of the first invariant of the relative Finger tensor. Practically, the disentanglement process is assumed to be expresosed by the first order reaction model with the rate coefficient _??_. Here, λ is the relaxation time in the linear region and λ0 is the characteristic time peculiar to nonlinear region. Ic-1(t; t′) is the first invariant of the relative Finger tensor.“·” denotes the differential with t at constant t′. The main results obtained from the constitutive equation are as follows. (1) Relaxation shear modulus at finite strain shows that the strength of the relaxation spectrum depends apparently on shear strain. (2) Stress relaxation after the cessation of simple shearing flow shows that the strength of relaxation spectrum depends apparently on shear rate. (3) The experimental results for “double-step” stress relaxation are described more appropriately by the proposed equation than the constitutive equation with “strain dependent relaxation spectrum”. (4)“Overshoot” of shear stress and normalstress difference are predicted at the beginning of simple shearing flow. Finally it may be concluded that the proposed constitutive equation describes more appropriately the observed non-linear viscoelastic phenomena than any other theories, and the two assumptions are appropriate.
The validities of the linear and nonlinear constitutive equations for polymeric materials are discussed from non-steady experiments with a Weissenberg rheogoniometer model R-17 of Sangamo Controls. Lodge's linear constitutive equation is well supported by the measurements of normal and shear stresses under oscillatory shear flow. On the other hand, the so-called “rate type” nonlinear constitutive equation could not explain the experimental results of the stress growth and relaxation after the onset and the cessation of steady simple shear flow. A discussion on the failure of the theories for stress-overshoot phenomena is given based on the idea that a transition of the internal structure due to external force does not occur at the moment when the external force is given to the material but occurs after a certain amount of strain is accumulated in the material. Moreover, a method to estimate the energy for entanglement destruction in polymeric materials is proposed.