Non-Linear Effects of Viscoelasticity

Abstract The discussion of relaxation and diffusion of macromolecules in very concentrated solutions and melts of polymers showed that the basic equations of macromolecular dynamics reflect the linear behaviour of a macromolecule among the other macromolecules, so that one can proceed further. Considering the non-linear effects of viscoelasticity, one have to take into account the local anisotropy of mobility of every particle of the chains, introduced in the basic dynamic equations of a macromolecule in Chapter 3, and induced anisotropy of the surrounding, which will be introduced in this chapter. In the spirit of mesoscopic theory we assume that the anisotropy is connected with the averaged orientation of segments of macromolecules, so that the equation of dynamics of the macromolecule retains its form. Eventually, the non-linear relaxation equations for two sets of internal variables are formulated. The first set of variables describes the form of the macromolecular coil - the conformational variables, the second one describes the internal stresses connected mainly with the orientation of segments. 

Although the microscopic theory remains to be the real foundation of the theory of relaxation phenomena in polymer systems, the mesoscopic approach has and will not lose its value. It will help to understand the laws of diffusion and relaxation of polymers of various architecture. The information about the microstructure and microdynamics of the material can be incorporated in the form of constitutive relation, thus, allowing to relate different linear and non-linear effects of viscoelasticity to the composition and chemical structure of polymer liquid. 

The investigation of viscoelasticity of dilute blends confirms that the reptation dynamics does not determine correctly the terminal quantities characterising viscoelasticity of linear polymers. The reason for this, as has already been noted, that the reptation effect is an effect due to terms of order higher than the first in the equation of motion of the macromolecule, and it is actually the first-order terms that dominate the relaxation phenomena. Attempts to describe viscoelasticity without the leading linear terms lead to a distorted picture, so that one begins to understand the lack of success of the reptation model in the description of the viscoelasticity of polymers. Reptation is important and have to be included when one considers the non-linear effects in viscoelasticity. 

The first two factors manifest themselves in the existaice of superpc ition temperature-frequency-filler concentratirm . The latter factor depicting the dynamical mechanical properties leads to non-linear effects of dianges in viscoelastic properties. 

Thus, one can see that the single-mode approximation allows us to describe linear viscoelastic behaviour, while the characteristic quantities are the same quantities that were derived in Chapter 6. To consider non-linear effects, one must refer to equations (9.52) and (9.53) and retain the dependence of the relaxation equations on the anisotropy tensor. 

Pokrovskii VN, Pyshnograi GV (1990) Non-linear effects in the dynamics of concentrated polymer solutions and melts. Fluid Dyn 25 568-576 Pokrovskii VN, Pyshnograi GV (1991) The simple forms of constitutive equation of polymer concentrated solution and melts as consequence of molecular theory of viscoelasticity. Fluid Dyn 26 58-64... 

Pyshnograi GV (1996) An initial approximation in the theory of viscoelasticity of linear polymers and non-linear effects. J Appl Mech Techn Phys 37(1) 123—128 Pyshnograi GV (1997) The structure approach in the theory of flow of solutions and melts of linear polymers. J Appl Mech Techn Phys 38(3) 122—130 Pyshnograi GV, Pokrovskii VN (1988) Stress dependence of stationary shear viscosity of linear polymers in the molecular field theory. Polym Sci USSR 30 2624—2629 Pyshnograi GV, Pokrovskii VN, Yanovsky YuG, Karnet YuN, Obraztsov IF (1994) Constitutive equation on non-linear viscoelastic (polymer) media in zeroth approximation by parameter of molecular theory and conclusions for shear and extension. Phys — Doklady 39(12) 889-892... 

In defining the constitutive relations for an elastic solid, we have assumed that the strains are small and that there are linear relationships between stress and strain. We now ask how the principle of linearity can be extended to materials where the deformations are time dependent. The basis of the discussion is the Boltzmann superposition principle. This states that in linear viscoelasticity effects are simply additive, as in classical elasticity, the difference being that in linear viscoelasticity it matters at which instant an effect is created. Although the application of stress may now cause a time-dependent deformation, it can still be assumed that each increment of stress makes an independent contribution. From the present discussion, it can be seen that the linear viscoelastic theory must also contain the additional assumption that the strains are small. In Chapter 11, we will deal with attempts to extend linear viscoelastic theory either to take into account non-linear effects at small strains or to deal with the situation at large strains. 

Ward has discussed attempts to extend linear viscoelastic theory to take into account non-linear effects at small strains and to deal with the situation of large strains. 

Other ideas proposed to explain the 3/4 power-law dependence include effects due to viscoelasticity, non-linear elasticity, partial plasticity or yielding, and additional interactions beyond simply surface forces. However, none of these ideas have been sufficiently developed to enable predictions to be made at this time. An understanding of this anomalous power-law dependence is not yet present. 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

The mechanical response of polypropylene foam was studied over a wide range of strain rates and the linear and non-linear viscoelastic behaviour was analysed. The material was tested in creep and dynamic mechanical experiments and a correlation between strain rate effects and viscoelastic properties of the foam was obtained using viscoelasticity theory and separating strain and time effects. A scheme for the prediction of the stress-strain curve at any strain rate was developed in which a strain rate-dependent scaling factor was introduced. An energy absorption diagram was constructed. 14 refs. 

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