Advanced Viscoelastic Models

The Maxwell fluid and Kelvin solid model with one spring and one dashpot actually fits very little experimental data well. Experimental results of creep and stress relaxation often flts with a Prony series that is a series of Maxwell fluid elements in parallel or a series of Kelvin solids in series. It is typical to utilize one element per decade of time. While this is not always necessary, it has historically been standard convention. The Prony series for the Maxwell fluid and Kelvin solid are 

For example, examination of the data in homework problem 3.1 shows that a single element does not result in an adequate flt. The underlying explanation for this is that the material responds with multiple timescales. The Prony series result in a pair of parameters and P for each decade of time flt. 

While there can be some rationalization of the Prony series to the effect that the response to mechanical stimuli has different time scales it is somewhat of a brute force methodology both mathematically and philosophically. There are more advanced viscoelastic predictions based on both more in depth mathematics and physics rational. If one takes experimental results and attempts to flt only a single Maxwell or Kelvin viscoelastic element it is seen that the fit is neither good nor adequate. They will often also occur with the Prony series. The first model to step beyond the Prony series is that which uses a stretching exponent for the creep or stress relaxation modulus. 

One step in sophistication and certainly complexity is to utilize a fractional calculus-based model. In the fractional calculus model the dashpot is replaced by an element that has the properties of both a spring and dashpot. This element is called a spring-pot with the following constimtive relation 

FIGURE 3.11 Four-parameter solid model with spring pot. 

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This assumption of a linear relationship between stress and strain appears to be good for small loads and deformations and allows for the formulation of linear viscoelastic models. There are also non-linear models, but that is an advanced topic that we won t discuss. There are two approaches we can take here. The first is to develop simple mathematical models that are capable of describing the structure of the data (so-called phenomenological models). We will spend some time on these as they provide considerable insight into viscoelastic behavior. Then there are physical theories that attempt to start with simple assumptions concerning the molecules and their interactions and... 

In recent years, the SPH methods in particular have gone through major improvements, and their application was expanded into a wider range of engineering problems. These include both more advanced physical models and more advanced engineering processes. For example, SPH was successfully used to simulate non-Newtonian fluid flows and viscoelastic materials. It has been also used for the analysis of fluid-stmcture interaction problems, fluid flow in porous media and fractures, heat transfer, and reacting flow problems. 

In a recent study (Nguyen et al. 2014), starting with the diffusive impedance of a conducting polymer actuator, electrical, mechanical, and viscoelastic properties of a tri-layer conjugated polymer actuator are combined into an advanced mafliematical model, which describes the relationship between the eurvature of the actuator and an applied voltage, expressed as... 

Figure 5. Representation of stress at advancing crack tip, and below, the state of viscoelastic strain indicated by Voigt models (35).

In addition, advanced models as those calculating viscoelasticity and the 3D model have been developed. They will be described in the following. 

In the last twenty years, major advances in the characterisation of polymer melt viscoelasticity has taken place and in addition applied mathematicians have produced numerical codes that enable viscoelastic fluids to be modelled for processing conditions. Within the last few years it has become possible to reasonably accurately predict the way in which a viscoelastic polymer will flow into, within and out of an extrusion die. The accurate prediction of die swell is nearly possible and advances are being made to predict the onset of extrusion instabilities. 

Several excellent treatments of molecular viscoelasticity are available. (See the references of Chapter 1.) The book by Professor Ferry, in particular, is an exhaustive and complete exposition. The question may then be asked, why the necessity for still another text and one restricted to bulk amorphous polymers, at that Such a question must send each of the authors scurrying in quest of an "apologia pro vita sua." The answer to the question lies in the use of the word "introduction" in the title. What we have attempted to do is to provide a detailed grounding in the fundamental concepts. This means, for example, that all derivations have been presented in great detail, that concepts and models have been presented with particular attention to assumptions, simplifications, and limitations, and that problems have been provided at the end of each chapter to illustrate points in the text. The level of mathematical difficulty is such that the average baccalaureate chemist should be able to readily grasp it. Where more advanced mathematical techniques are required, such as transform techniques, the necessary methods are developed in the text. 

The preceding equations provided a reasonable foundation for predicting DE behavior. Indeed the assumption that DEs behave electronically as variable parallel plate capacitors still holds however, the assumptions of small strains and linear elasticity limit the accuracy of this simple model. More advanced non-linear models have since been developed employing hyperelasticity models such as the Ogden model [144—147], Yeoh model [147, 148], Mooney-Rivlin model [145-146, 149, 150] and others (Fig. 1.11) [147, 151, 152]. Models taking into account the time-dependent viscoelastic nature of the elastomer films [148, 150, 151], the leakage current through the film [151], as well as mechanical hysteresis [153] have also been developed. 

BaschnagelJ, Binder K, Doruker P, et al Advances in polymer science viscoelasticity, atomistic models, Stat Chem 152 41—156, 2000. 

The flow domain of interest has to be subdivided into a number of volumes, or cells, to form the computational grid. The conditions at all the boimdaries of the grid must then be specified—f/ze boundary conditions, e.g. in-flow, outflow, no-slip, velocity or pressure. The next step is to specify the properties of the liquid flowing through the domain—this is called the constitutive equation, which might be as simple as a power-law description of viscosity/shear-rate or might be as complicated any of the most advanced models that account for viscoelasticity and time effects such as thixotropy, see below. The completion of these three steps makes it possible to solve the flow of the liquid throughout the domain. 

The previous sections give a brief review of some elementary concepts of solid mechanics which are often used to determine basic properties of most engineering materials. However, these approaches are sometimes not adequate and more advanced concepts from the theory of elasticity or the theory of plasticity are needed. Herein, a brief discussion is given of some of the more exact modeling approaches for linear elastic materials. Even these methods need to be modified for viscoelastic materials but this section will only give some of the basic elasticity concepts. 

Principle. The quantity, E (s), in transform space is analogous to the usual Young s modulus for a Unear elastic materials. Here, the Unear differential relation between stress and strain for a viscoelastic polymer has been transformed into a linear elastic relation between stress and strain in the transform space. It will be shown in the next chapter that the same result can be obtained from integral expressions of viscoelasticity without recourse to mechanical models, so that the result is general and not limited to use of a particular mechanical model. Therefore, the simple transform operation allows the solution of many viscoelastic boundary value problems using results from elementary solid mechanics and from more advanced elasticity approaches to solids such as two and three dimensional problems as well as plates, shells, etc. See Chapters 8 and 9 for more details on solving problems in the transform domain. 

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