Nonlinear viscoelastic responses

Sui, C., McKenna, G.B., and Puskas, J.E. Nonlinear viscoelastic response of dendritic (arborescent) polyisobutylenes in single- and reversing double-step shearing flows, J. Rheol, 51, 1143, 2007. 

There is a wealth of microstructural models used for describing nonlinear viscoelastic responses. Many of these relate the rheological properties to the interparticle forces and the bulk of these consider the action of continuous shear rate or stress. We will begin with a consideration of the simplest form of potential, a hard rigid sphere. 

There are a wealth of experiments, papers and texts covering the nonlinear viscoelastic response of polymers. In this book we cannot hope to cover the area in as much detail as the many admirable texts in this field. 

Figure 3. Stress-strain curves in nonlinear elastic and nonlinear viscoelastic responses...

Characteristics of the Hysteresis Loop and Stress Wave in the Nonlinear Viscoelastic Response to the Sinusoidal Straining. Figure 3 is a schematic of a hysteresis loop obtained when a nylon 6 monofilament was subjected to a sinusoidal straining with yo = 1% and Ay = 1% at 90°C under a frequency of 10 cycles per sec. 

Fourier transform rheology (FTR) [Wapner and Forsman, 1971] was used for the analysis of CPNCs. The experiments were performed in the ARES using the software developed by Wilhelm [2002]. It was expected that the method may be suitable for characterization of the nonlinear viscoelastic response of CPNC. The FTR analysis is based on the expression... 

General Regimes of Response. The nonlinear viscoelastic response of polymers, of course, follows some of the same classifications as does the linear response. Hence, the behavior above the glass temperature and into the terminal zone is fluid behavior, and often follows time-temperature superposition. The phenomenology of polymer melts and solutions is commonly described by... 

Below the glass temperatin-e, the nonlinear viscoelastic response of polymeric materials has been much less widely studied than has the behavior of melts and solutions. One reason for this is the lack of an adequate theory of behavior. Therefore the discussion about amorphous materials below the glass tem-peratiu e focuses on recent measin-ements of the nonlinear response as well as... 

Nonlinear Viscoelastic Response of Solid-like Polymers. The study of the nonlinear viscoelastic response of solid or solid-like polymers is one that has... 

One important issue in dealing with the nonlinear viscoelastic response of materials is the amount of data needed to determine the material parameters... 

The Schapery Model. One of the earliest models of the nonlinear viscoelastic response of pol5nners to use the concept of a reduced time is due to Schapery (147-149). The model is based on thermodynamic considerations and has a form similar to the Boltzmann superposition principal described previously. The model time dependences, except for the shift factors, are the same as those obtained in the linear response regime. Hence, the model is relatively easy to implement and to determine the relevant material parameters. It results in a generalization of the generalized superposition principal developed by Leaderman (150). 

Plasticity and Viscoplasticity and Other Models. As discussed above, the alternative representation of the nonlinear viscoelastic response of polymers is that of plasticity and viscoplasticity. In some respects, these models could be recast as viscoelastic models and they would be equivalent to some of the models discussed above. However, the perspective that glassy polymers are really fluids and do follow time-temperature superposition is lost with these models. Hence, the physical interpretation of material parameters, in this author s opinion, becomes very qnestionable. Therefore, only references to the major papers on polymer plasticity and viscoplasticity are given (174-177). 

G. C. Papanicolaou, S. P. Zaoutsos, and A. H. Cardon, Prediction of the Nonlinear Viscoelastic Response of Unidirectional Fiber Composites Compos. Sci. Technol. 59, 1311-1319 (1999). 

C. Hiel, A. H. Cardon, and H. F. Brinson, The Nonlinear Viscoelastic Response of Resin Matrix Composite Laminates, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA (March 1983). 

This quantity has many uses. For example, to is approximately the time required for final equilibration of flow-induced stress in the liquid. As shown below, to also locates the onset of nonlinear viscoelastic response in steady-shear flows. 

Rheology is a branch of physics concerned with the time-dependent deformation of solids and the viscous flow of liquids. Rheological models can be used to illustrate the nonlinear viscoelastic response of rPET polymer concrete. These models are mechanical comparisons that demonstrate the interrelationship between the elastic and viscous response of polymers. Simple and complex models can be proposed to... 

Block copol3nners form a new class of molecular composite materials by the phase separation of incompatible hard and soft segments which form their macro-molecular structure. Thermoplastic elastomers where the soft segments form the continuous phase have been extensively investigated by means of an adsorption-interdiffusion (A-I) model for the interfacial phase which bonds the hard and soft phases. The molecular structure and rheological activity of the interfacial phase in thermoplastic elastomer block copolymers is shown to play a dominant role in nonlinear viscoelastic response, mechanical hysteresis and energy absorption. Creation of elastomeric microphases in epoxy structural adhesives has been recently identified with in situ block copol3nnerization between carboxy terminated nitrile (CTBN) rubber and the diepoxide. 

Smith, T.L. (1962) Nonlinear viscoelastic response of amorphous elastomers to constant strain rates. Trans. Soc. Rheol., 6,61. 

Crook, R.A. (1993) Damage and the nonlinear viscoelastic response of glassy polycarbonate and LaRC-TPI. Polym. Eng. Sci., 33, 56. 

General Regimes of Response. The nonlinear viscoelastic response of polymers, of course, follows some of the same classifications as does the linear response. Hence, the behavior above the glass temperature and into the terminal zone is fluid behavior, and often follows time-temperature superposition. The phenomenology of polymer melts and solutions is commonly described by constitutive laws that relate the stress and strain histories to each other (59,69). A brief description of the K-BKZ model (70-72) is provided as it seems to capture most of the behaviors of polymer melts and solutions subjected to large deformations or high deformation rates. At the same time the nonlinear form of the reptation... 

Nonlinear Viscoelastic Response of Solid-like Polymers. The study of the nonlinear viscoelastic response of solid or solid-like polymers is one that has been relatively neglected. One reason is that there is no real molecular framework for the description of these materials, particularly when they are amorphous. The other reason is that many workers in the field have adopted the framework of metal plasticity and then made modifications to try to adapt it to, for example, the fact that amorphous polymers do not readily admit to treatment with the physics of dislocations. In the case of semicrystalline polymers, the... 

One important issue in dealing with the nonlinear viscoelastic response of materials is the amount of data needed to determine the material parameters in the models. As noted above, even the general finite elasticity theory requires significant work to obtain the material parameters over the full three-dimensional deformation space. This is one reason that the VL framework is so attractive, when it works. Therefore, it is of interest to investigate whether or not the model can be extended to include compressibility. Pesce and McKenna (146) performed torsional tests on polycarbonate as described above. They then asked whether the VL function could be used to predict the tension and compression responses of the material. An important assumption in their approach was that the VL function determined from the torsional measurements using equations 45, 46, 47, 48, 49, 50, 51 (described immediately above) could be used to predict uniaxial data. When the incompressible equations 50 were applied to try to estimate the uniaxial stress-deformation data (isochronal), these equations did not work. However,... 

The Knauss-Emri Model. There have been several works in the literature in which volume- or ee-ooZume-dependent clocks were used to describe the nonlinear viscoelastic response of polymeric glasses. The chief success among these is the ICnauss-Emri model (163) in which the reduced time was defined in terms of a shift factor that depended on temperature, stress, and concentration of small molecules in such a way that the responses depended on the free volume induced by each of these parameters. For an isothermal single phase and homogeneous material, the equations are... 

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