Mechanical models Maxwell model

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. 

The simplest model that can show the most important aspects of viscoelastic behaviour is the Maxwell fluid. A mechanical model of the Maxwell fluid is a viscous element (a piston sliding in a cylinder of oil) in series with an elastic element (a spring). The total extension of this mechanical model is the sum of the extensions of the two elements and the rate of extension is the sum of the two rates of extension. It is assumed that the same form of combination can be applied to the shearing of the Maxwell fluid. 

The meaning of "model" was to become far less concrete in the next couple of decades. In 1929, Irving Langmuir criticized mechanical models, like those of Lord Kelvin and Maxwell, on the grounds that the relationships of their parts are restricted to what is already known in mechanics, electricity, or magnetism, limiting the possibility of new insights into new phenomena. "Mathematical relationships are far more flexible," he claimed, and "the mathematical theory is a far better model of the atom than any of the mechanical... 

Most reported zeolite/polymer mixed-matrix membranes, however, have issues of aggregation of the zeolite particles in the polymer matrix and poor adhesion at the interface of zeolite particles and the polymer matrix. These issues resulted in mixed-matrix membranes with poor mechanical and processing properties and poor separation performance. Poor compatibility and poor adhesion between the polymer matrix and the zeolite particles in the mixed-matrix membranes resulted in voids and defects around the zeolite particles that are larger than the micropores of the zeolites. Mixed-matrix membranes with these voids and defects exhibited selectivity similar to or even lower than that of the continuous polymer matrix and could not match that predicted by Maxwell model [59, 60]. 

Over a century ago Maxwell used this combination of dashpot and spring to explain the mechanical behavior of pitch. The Maxwell model may be expressed mathematically as follows ... 

Figure 8.1. Diagram showing Maxwell mechanical model of viscoelastic behavior of connective tissues. In this model an elastic element (spring) with a stiffness Em is in series with a viscous element (dashpot) with viscosity T m. This model is used to represent time dependent relaxation of stress in a specimen bold of fixed length.

When the examination was over and the report submitted, a new tumult was raised. Kelvin opposed the theory in general. He could -understand nothing, he said, which could not be translated into a mechanical model. For this reason he had likewise rejected Maxwell s electromagnetic theory of light. Only the Dane submitted an enthusiastic judgment of the... 

Viscoelasticity Models For characterization with viscoelasticity models, simulation models have been developed on the basis of Kelvin, Maxwell, and Voigt elements. These elements come from continuum mechanics and can be used to describe compression. 

Figure 12.10 shows the mechanical response as a funetion of time of two structures formed by combining a spring and a dashpot in series (Maxwell model) and in parallel (Voigt model). Creep is slow deformation of a viseoelastic material under constant stress (o), while relaxation is the time response of the stress after imposing a constant deformation (e). Thus, a simple mathematieal expression accounts for the relation between structure (combination of elements) and a property (creep or relaxation). Evidently, more complex responses ean be obtained by eombining several elements in series and in parallel. 

Fig. 11-16. Simple mechanical models of viscoelastic behavior, (a) Voigt or Kelvin element and (b) Maxwell element.

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

Neither simple mechanical model approximates the behavior of real polymeric materials very well. The Kelvin element does not display stress relaxation under constant strain conditions and the Maxwell model does not exhibit full recovery of strain when the stress is removed. A combination of the two mechanical models can be used, however, to represent both the creep and stress relaxation behaviors... 

Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on pol rmers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 

For the very simplified situation that the sphere behaves electrically as a pure capacitor, and the solution as a pure resistance, the relaxation can be described by a Maxwell-Wagner mechanism, with T = e e/K, see (1.6.6.321. Although some success has been claimed by Watillon s group J to apply this mechanism for a model, consisting of shells with different values of e and K, generally a more detailed double layer picture is needed. In fact, this Implies stealing from the transport equations of secs. 4.6a and b. generedizing these to the case of a.c. fields. 

Figure 1-8 Maxwell Model (Left) and Kelvin-Voigt Model (Right) Illustrate Mechanical Analogs of Viscoelastic Behavior.

It is well known that the behavior of electrical transmission lines can be represented in terms of distributed passive elements. As we mentioned at the beginning of this chapter, there exists an analogy between the electrical and mechanical behavior of the systems. Returning to the Maxwell model, one has... 

The simplest mechanical model which can describe a viscoelastic solution is called Maxwell element. It consists of a spring and a viscous element (dashpot) connected in series. The spring corresponds to a shear modulus Gq and the dashpot to a viscosity r). The behavior of the Maxwell element under harmonic oscillations can be obtained from the following equations ... 

Because of the interaction of the two complicated and not well-understood fields, turbulent flow and non-Newtonian fluids, understanding of DR mechanism(s) is still quite limited. Cates and coworkers (for example, Refs. " ) and a number of other investigators have done theoretical studies of the dynamics of self-assemblies of worm-like micelles. Because these so-called living polymers are subject to reversible scission and recombination, their relaxation behavior differs from reptating polymer chains. An additional form of stress relaxation is provided by continuous breaking and repair of the micellar chains. Thus, stress relaxation in micellar networks occurs through a combination of reptation and breaking. For rapid scission kinetics, linear viscoelastic (Maxwell) behavior is predicted and is observed for some surfactant systems at low frequencies. In many cationic surfactant systems, however, the observed behavior in Cole-Cole plots does not fit the Maxwell model. 

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