Viscoelastic effects time-temperature superposition

There are two superposition principles that are important in the theory of Viscoelasticity. The first of these is the Boltzmann superposition principle, which describes the response of a material to different loading histories (22). The second is the time-temperature superposition principle or WLF (Williams, Landel, and Ferry) equation, which describes the effect of temperature on the time scale of the response. 

We expect that the modification creates the free volume (Vf) in wood substance from the similarity of the effect of and n on viscoelasticity. The discussion for wood, however, is impossible on the basis of a concept of the free volume, although the flexibility of molecular motion for synthetic amorphous polymers is discussed. Unfortunately, we can not directly know the created free volume because the time-temperature superposition principle is not valid for wood [19]. The principle is related to WLF equation by which the free volume is calculated. The free volume, however, relates to volumetric swelling as follows. 

Important viscoelastic principles include the time-temperature superposition principle and its resultant WLF equation. These can be applied to understand the relationship between literature values of the glass transition temperature and actual needs. Thus, by using the growing amount of science now available in the field of damping, one can select that polymeric material which will damp most effectively. 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

Non-linear mechanical properties were observed for rubber eomposites and referred to as the Payne effect. The Payne effeet was interpreted as due to filler agglomeration where the filler clusters formed eontained adsorbed rubber. The occluded rubber molecules within filler elusters eould not eontribute to overall elastic properties. The composites behaved similarly to rubber composites with higher filler loading. Uniform and stable filler dispersion is required for rubber composites to exhibit linear viscoelastic behaviour. Payne performed dielectric measurements on SBR vulcanizates containing silica or carbon black. The dielectric data were used to construct time-temperature superposition master curves. The reference temperature increased with crosslinking but not significantly with filler. Comparison of dynamic mechanical and dielectric results for the SBR blended with NR was made and interpreted. ... 

In equation 5, nj is the number of cycles within a block applied at stress level aj and Nj is the total number of cycles to failure at that stress level (obtained from the a-N curve). If validated for a particular material, equation 5 can be used for any arbitrary loading condition. In this way, the fatigue data generated in the laboratory is referred to as an accelerated test and equation 5 is used much in the same way that time-temperature superposition is used to predict the viscoelastic deformation of a material. Note, however, that any predictive methodology like that presented in equation 5 must be validated before it is employed for lifetime estimates. In the case of Minor s law, the sequence of the loading history should have no effect on the total lifetime. That is, if one specimen is loaded for a selected portion of its life at stress level stress level ct2 followed by CTi. 

For a homogeneous viscoelastic material that is confined to a temperature range within which there is no phase change, the effects of time and temperature can be interchangeable. In other words, if we apply a stress or strain to the material, we can choose either to wait for it to relax or creep at the fixed test temperature or we can get the same response faster by raising the test temperature. This principle is called time-temperature superposition and is widely used in linear viscoelasticity. Taking into account any small variation of the plateau modulus, we can express the principle mathematically (say, in steady shear) as... 

The composite natm e of polyurethane elastomers strongly affects their linear viscoelastic properties. It is known that for most polymers, linear viscoelastic moduli (storage modulus, E u,T), and loss modulus, E" u,T)) are characterized by the so-called time-temperature superposition (TTS) (see, e.g. Ferry [74]). Such behavior can be understood if one assumes that E (and E") is always a function of the product ut T), where t(T) is effective relaxation time. 

Also, even when the data are obtainable only over one or two decades of the logarithmic frequency scale at any one time, the viscoelastic functions can be traced out over a much larger effective range by making measurements at different temperatures, and by applying time-temperature superposition (TTS) for flexible homopolymers (see Chapter 6). In many instances, the effect of an increase in temperature is nearly equivalent to an increase in time or a decrease in frequency, as molecular viscoelastic theories suggest (see Chapter 4). When properly applied, TTS yields plots in terms of reduced variables that can be used with considerable confldence to deduce the effect of molecular parameters, and also to predict viscoelastic behavior in regions of the time or frequency scale not experimentally readily accessible (see Chapters 4 and 6). 

Since fibers consist primarily of oriented crystallites, it is unfair to classify them as heterophase. However, the generalizations of time-temperature superposition that work so well with amorphous polymers do not apply to fibers. Fibers do exhibit viscoelasticity qualitatively like the amorphous polymers. It comes as a surprise to some that J. C. Maxwell, who is best known for his work in electricity and magnetism, should have contributed to the mathematics of viscoelasticity. The story goes that while using a silk thread as the restoring element in a charge-measuring device. Maxwell noticed that the material was not perfectly elastic and exhibited time-dependent effects. He noticed that the material was not perfectly elastic and showed time effects. The model that bears his name was propounded to correlate the real behavior of a fiber. 

The measurement of viscoelastic properties on commercial rotational rheometers is hmited to frequencies < 30 Hz as inertial and edge effects are known to become significant at high frequencies [1-3]. Extension of the frequency range to higher frequencies (> 50 Hz) by means of time-temperature superposition (TTS) is not always possible, especially for semi-ciystalline polymers such as polypropylene, as the highest frequency attainable in TTS is limited by the melting point of the polymer [1,4]. 

For concentrated polymer solutions the viscosity is proportional to the 3.4th power of the molecular mass and about the 5th power of the concentration. The effects of temperature and concentration on viscoelastic properties are closely interrelated. The validity of a time-concentration superposition is shown. A method is given for predicting the viscosity of concentrated polymer solutions. 

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