Maxwell model for viscoelastic

In the Maxwell model for viscoelastic deformation, it is assumed that the total strain is equal to the elastic strain plus the viscous strain. This is expressed in the two following differential equations from Equations 14.2 and 14.3. 

It is rather complex to describe viscoelastic behaviors mathematically. We will only focus here the simple, but representative Maxwell model for viscoelastic liquids. A mechanical analogue of a Maxwell liquid model is obtained by a serial combination of a spring and a dashpot see Fig. lc(l). If the individual strain rates of the spring and the dashpot, respectively, are 4oiid liquid > then the total strain rate e is given by the sum of these two components ... 

Buildings and Bridges Equipped with Passive Dampers Under Seismic Actions Modeling and Analysis, Fig. 5 Generalized Maxwell model for viscoelastic dampers... 

Figure 5 Schematic illustration of generalized Maxwell model for viscoelastic fluids.

The grade of polypropylene whose creep curves are given in Fig. 2.5 is to have its viscoelastic behaviour fltted to a Maxwell model for stresses up to 6 MN/m and times up to ICKX) seconds. Determine the two constants for the model and use these to determine the stress in the material after 900 seconds if the material is subjected to a constant strain of 0.4% throughout the 900 seconds. 

David and Augsburger (63) studied the decay of compressional forces for a variety of excipients, compressed with flat-faced punches on a Stokes rotary press. They found that initial compressive force could be subject to a fairly rapid decay and that this rate was dependent on the deformation behavior of the excipient for the materials studied, they found that maximum loss in compression force was for compressible starch and MCC, which was followed by compressible sugar and DCP. This was attributed to differences in the extent of plastic flow. The decay curves were analyzed using the Maxwell model of viscoelastic behavior. Maxwell model implies first order decay of compression force. 

In fact, Equation 5.281 describes an interface as a two-dimensional Newtonian fluid. On the other hand, a number of non-Newtonian interfacial rheological models have been described in the literature. Tambe and Sharma modeled the hydrodynamics of thin liquid films bounded by viscoelastic interfaces, which obey a generalized Maxwell model for the interfacial stress tensor. These authors also presented a constitutive equation to describe the rheological properties of fluid interfaces containing colloidal particles. A new constitutive equation for the total stress was proposed by Horozov et al. ° and Danov et al. who applied a local approach to the interfacial dilatation of adsorption layers. 

Since the response of a liquid to application of an external stress is dependent upon the rate of application of that stress, viscoelasticity can occur over a wide range of viscosities. For common rates of stress application, these viscosities lie in the region of the glass transformation range, particularly in the range from 10 to 10 Pas. The most common basic model for viscoelasticity, known as the Maxwell model, is shown in Figure 6.2. The sample is considered to consist of an elastic... 

Figure 6.2 The Maxwell model for relaxation of a viscoelastic material...

Figure 1.17. The Maxwell model for a viscoelastic body. The spring constant and dashpot viscosity are both variable.

The Maxwell model integrates the elastic and viscous behavior of a thermoplastic by combining the spring and dashpot, providing a simple model for viscoelastic polymers. The spring and dashpot are in series [10],... 

Figure 2.13 Three-element model for viscoelastic behavior of liquids (general relaxation model) as a combination of Maxwell and Voigt-Kelvin models.

For polyacrylamide there are two rheological effects which can be explained in terms of its random coil structure. Firstly, it was discussed above that polyacrylamide is much more sensitive than xanthan to solution salinity and hardness. This is explained by the fact that the salinity causes the molecular chain to collapse, which results in a much smaller molecule and hence in a lower viscosity solution. The second effect which can be explained in terms of the polyacrylamide random coil structure is the viscoelastic behaviour of this polymer. This is shown both in the dynamic oscillatory measurements and in the flow through the stepped capillaries (Chauveteau, 1981). When simple models of random chains are constructed, such as the Rouse model (Rouse, 1953 Bird et al, 1987), the internal structure of these bead and spring models gives rise to a spectrum of relaxation times, Analysis of this situation shows that these relaxation times define response times for the molecule, as indicated in the simple Maxwell model for a viscoelastic fluid discussed above. Thus, because of the internal structure of a flexible coil molecule, one would expect to observe some viscoelastic behaviour. This phenomenon is discussed in much more detail by Bird et al (1987b), in which a range of possible molecular models are discussed and the significance of these to the constitutive relationship between stress and deformation rate and deformation history is elaborated. 

While the exponential stress relaxation predicted by the viscoelastic analog of the Mawell element, ie., a single exponential, is qualitatively similar to the relaxation of polymeric liquids, it does not describe the detailed response of real materials. If, however, it is generalized by assembling a number of Maxwell elements in parallel, it is possible to fit the behavior of real materials to a level of accuracy limited only by the precision and time-range of the experimental data. This leads to the generalized, or multi-mode. Maxwell model for linear viscoelastic behavior, which is represented mathematically by a sum of exponentials as shown by Eq. 4.16. 

The Maxwell class of viscoelastic constitutive equations are described by a simpler form of Equation (1.22) in which A = 0. For example, the upper-convected Maxwell model (UCM) is expressed as... 

Creep modeling A stress-strain diagram is a significant source of data for a material. In metals, for example, most of the needed data for mechanical property considerations are obtained from a stress-strain diagram. In plastic, however, the viscoelasticity causes an initial deformation at a specific load and temperature and is followed by a continuous increase in strain under identical test conditions until the product is either dimensionally out of tolerance or fails in rupture as a result of excessive deformation. This type of an occurrence can be explained with the aid of the Maxwell model shown in Fig. 2-24. 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

One feature of the Maxwell model is that it allows the complete relaxation of any applied strain, i.e. we do not observe any energy stored in the sample, and all the energy stored in the springs is dissipated in flow. Such a material is termed a viscoelastic fluid or viscoelastic liquid. However, it is feasible for a material to show an apparent yield stress at low shear rates or stresses (Section 6.2). We can think of this as an elastic response at low stresses or strains regardless of the application time (over all practical timescales). We can only obtain such a response by removing one of the dashpots from the viscoelastic model in Figure 4.8. When a... 

We have developed the idea that we can describe linear viscoelastic materials by a sum of Maxwell models. These models are the most appropriate for describing the response of a body to an applied strain. The same ideas apply to a sum of Kelvin models, which are more appropriately applied to stress controlled experiments. A combination of these models enables us to predict the results of different experiments. If we were able to predict the form of the model from the chemical constituents of the system we could predict all the viscoelastic responses in shear. We know that when a strain is applied to a viscoelastic material the molecules and particles that form the system gradual diffuse to relax the applied strain. For example, consider a solution of polymer... 

For a viscoelastic solid the situation is more complex because the solid component will never flow. As the strain is applied with time the stress will increase continually with time. The sample will show no plateau viscosity, although there may be a low shear viscous contribution. This applies to both a single Maxwell model and one with a spectrum of processes ... 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

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