Viscoelasticity Maxwell element

We can get a first approximation of the physical nature of a material from its response time. For a Maxwell element, the relaxation time is the time required for the stress in a stress-strain experiment to decay to 1/e or 0.37 of its initial value. A material with a low relaxation time flows easily so it shows relatively rapid stress decay. Thus, whether a viscoelastic material behaves as a solid or fluid is indicated by its response time and the experimental timescale or observation time. This observation was first made by Marcus Reiner who defined the ratio of the material response time to the experimental timescale as the Deborah Number, Dn-Presumably the name was derived by Reiner from the Biblical quote in Judges 5, Song of Deborah, where it says The mountains flowed before the Lord. ... 

For viscoelastic materials combinations of these two models can be used, e.g. a spring and a dashpot in series or parallel. The first combination is called the Maxwell element, its response under constant stress is the sum of that of its two components ... 

Both models, the Maxwell element and the Kelvin-Voigt element, are limited in their representation of the actual viscoelastic behaviour the former is able to describe stress relaxation, but only irreversible flow the latter can represent creep, but without instantaneous deformation, and it cannot account for stress relaxation. A combination of both elements, the Burgers model, offers more possibilities. It is well suited for a qualitative description of creep. We can think it as composed of a spring Ei, in series with a Kelvin-Voigt element with 2 and 772. and with a dashpot, 771... 

The models described so far provide a qualitative illustration of the viscoelastic behaviour of polymers. In that respect the Maxwell element is the most suited to represent fluid polymers the permanent flow predominates on the longer term, while the short-term response is elastic. The Kelvin-Voigt element, with an added spring and, if necessary, a dashpot, is better suited to describe the nature of a solid polymer. With later analysis of the creep of polymers, we shall, therefore, meet the Kelvin-Voigt model again in more detailed descriptions of the fluid state the Maxwell model is being used. 

The viscoelastic quantities rj, ye, G, xFi and t were defined at the beginning of this chapter. For a system with only one relaxation time, e.g. for a Maxwell element, the following interrelations exist between these rheological quantities ... 

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

In the first case [P t) is known], let us consider a standard solid that is viscoelastic in shear (spring in parallel with a Maxwell element) but elastic in compression. By assuming a step input, we obtain in the usual way... 

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 ... 

The relaxation time r is a fundamental dynamic property of all viscoelastic liquids. Polymer liquids have multiple relaxation modes, each with its own relaxation time. Any stress relaxation modulus can be described by a series combination of Maxwell elements. 

The mechanical response of viscoelastic bodies such as polymers is poorly represented by either the spring or the dashpot. J. C. Maxwell suggested that a better approximation would result from a series combination of the spring and dashpot elements. Such a model, called a Maxwell element, is shown on the right in Figure 3-1. In describing tensile response with the Maxwell element, E, the instantaneous tensile modulus, characterizes the response of the spring while rjE, the viscosity of the liquid in the dashpot, defines the viscous... 

These equations are often used in terms of complex variables such as the complex dynamic modulus, E = E + E", where E is called the storage modulus and is related to the amount of energy stored by the viscoelastic sample. E" is termed the loss modulus, which is a measure of the energy dissipated because of the internal friction of the polymer chains, commonly as heat due to the sinusoidal stress or strain applied to the material. The ratio between E lE" is called tan 5 and is a measure of the damping of the material. The Maxwell mechanical model provides a useful representation of the expected behavior of a polymer however, because of the large distribution of molecular weights in the polymer chains, it is necessary to combine several Maxwell elements in parallel to obtain a representation that better approximates the true polymer viscoelastic behavior. Thus, the combination of Maxwell elements in parallel at a fixed strain will produce a time-dependent stress that is the sum of all the elements ... 

As Equation 14.7 shows, the Maxwell element is merely a linear combination of the behavior of an ideally elastic material and pure viscous flow. Now let us examine the response of the Maxwell element to two typical experiments used to monitor the viscoelastic behavior of polymer. 

A physical insight into the viscoelastic character of a material can be obtained by examining the material response time. This can be illustrated by defining a characteristic time for the material — for example, the relaxation time for a Maxwell element, which is the time required for the stress in a stress relaxation experiment to decay to e (0.368) of its initial value. Materials that have low relaxation times flow easily and as such show relatively rapid stress decay. This, of course, is indicative of liquidlike behavior. On the other hand, those materials with long relaxation times can sustain relatively higher stress values. This indicates solidlike behavior. Thus, whether a viscoelastic material behaves as an elastic solid or a viscous liquid depends on the material response time and its relation to the time scale of the experiment or observation. This was first proposed by Marcus Reiner, who defined the ratio of the material response time to the experimental time scale as the Deborah number, D . That is. 

The early observations of Bagge et al. [1977] led them to suggest that the neutrophil behaves as a simple viscoelastic solid with a Maxwell element (an elastic and viscous element in series) in parallel with an elastic element. This elastic element in the model was thought to pull the unstressed cell into its spherical shape. Subsequently, Evans and Kukan [1984] and Evans and Yeung [1989] showed that the cells flow continuously into a pipette, with no apparent approach to a static limit, when a constant suction pressure was applied. Thus, the cytoplasm of the neutrophil should be treated as a liquid rather than a solid, and its surface has a persistent cortical tension that causes the cell to assume a spherical shape. 

The equation for solvent transport consists of a diffusional term and a term due to osmotic pressure. The osmotic pressure term arises by using linear irreversible diermodynamics arguments (20). The osmotic pressure is relat to the viscoelastic properties of the polymer through a constitutive equation. In our analysis, the Maxwell element has been used as the constitutive model. Thus, the governing equations for solvent transport in the concentrated regime are... 

Maxwell model (Maxwell element) n. A concept useful in modeling the deformation behavior of viscoelastic materials. It consists of an elastic spring in series with a viscous dashpot. When the ends are pulled apart with a definite force, the spring deflects instantaneously to its stretched position then motion is steady as the dashpot opens. A simple combination of these two types provides a fair analogic representation of real viscoelastic behavior under stress. 

Relaxation time n. Of a viscoelastic material under constant strain (specifically, one behaving as a Maxwell element), the time required for the stress to diminish to He (=0.368) of its initial value. Compare retardation time. 

Nonlinear models of rheological behavior can be approximated by step functions, whereby the existence of a finite yield stress G plays a dominant role. Three typical nonlinear models include the Saint-Venant model of ideal plastic behavior, the Prandtl-Reuss model of an elastoplastic material, and the Bingham model of viscoelastic behavior. The first model can be mechanically approximated by a sliding block, the second by a Maxwell element and a sliding block in series, and the third by a dash pot damping element and a sliding block in parallel (Figure 2.14). 

The Four-Element Model While a few problems in viscoelasticity can be solved with the Maxwell or Kelvin elements alone, more often they are used together or in other combinations. Figure 10.5 illustrates the combination of the Maxwell element and the Kelvin element in series, known as the four-element model. It is the simplest model that exhibits all the essential features of viscoelasticity. 

The various models were invented explicitly to provide a method of mathematical analysis of polymeric viscoelastic behavior. The Maxwell element expresses a combination of Hooke s and Newton s laws. For the spring,... 

The two basic viscoelastic models include the Kelvin-Voigt (K-V) and the Maxwell elements. The K-V element behaves as a solid when sheared, since the deformed material regains its initial state after the applied stress is relaxed. The components of equivalent shear modulus and equivalent viscosity, respectively, are... 

A simple way of understanding the behavior of viscoelastic liquids is by analogy to mechanical models. The stress response t of a spring of modulus G connected in series to a dashpot having a damping constant t] (the combination is often called a Maxwell element) is given in one dimension by the equation... 

Here the d)mainic response of a viscoelastic Uquid over a range of frequencies can be modelled by choosing a range of Maxwell elements with appropriate values of G and x chosen to cover the range used in the experiment for which G and G" values are available, as say x = lO s, lO s,... 1(P s for a frequency range of 1(P down to lO rad/s, together with the appropriate values of G . It is obvious that at very high... 

At very short times this simplifies to elastic behaviour. Then at t = r, the stress is 1/e of its value at steady state, where it is cr= 77/, i.e. purely steady-state viscous behaviour. The start-up of real viscoelastic liquids may need to be modelled using a number of Maxwell elements. However, for most realistic experiments using this kind of test, the response quickly enters the non-linear region since the strain is continually increasing. 

The viscoelastic functions exhibited by the Maxwell element can be easily derived and are summarized as follows ... 

A group of Maxwell elements in parallel represents a discrete spectrum of relaxation times, each time r,- being associated with a spectral strength G/. Since in a parallel arrangement the forces (or stresses) are additive, it can readily be shown that for the Maxwell model. Fig. 1-9, the viscoelastic functions G(/), G (aj), G"(w), and r] ((x)) are obtained simply by summing the expressions in equations 2 to 5 over all the parallel elements thus, if there are n elements,... 

Fig. 9.3 Variation of the radial stress and hoop stress with position in the viscoelastic reinforced cylinder loaded with a step input of internal pressure. Parameters used are K /Go=3, t=1000, where the viscoelastic cylinder has an elastic bulk modulus and is a single Maxwell element in shear modulus. Response parameterized with time from the initial application of load at t=0 to asymptotic response at long times.

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