The Maxwell Model

To find the relationship between total stress and total strain, Equation (5.14a) can be written as 

The Maxwell model is of particular value in considering a stress relaxation experiment. In this case. 

This equation shows that the stress decays exponentially with a characteristie time constant 

Secondly, the stress relaxation behaviour cannot usually be represented by a single exponential decay term, nor does it necessarily decay to zero at infinite time. 

FIGURE13-88 Schematic diagram depicting the Maxwell model. 

Maxwell then assumed viscous forces that were Newtonian, hence (Equation 13-84)  

If the rate of strain is then the linear sum of these two components, which is what the mechanical model represents (Equation 13-85)  

The Maxwell model does a far more interesting job of modeling stress relaxation, however. If we again start with the basic equation (Equation 13-85) and impose the constant strain condition dzldt = 0 we get Equation 13-86  

FIGURE 13-89 The Maxwell model gives the type of strain/dme plot shown in (A) whereas real data looks like the plot shown in (B). 

For stress-relaxation at constant strain e is independent of t, so that 

A Maxwell element consists of an elastic spring of modulus = 10 Pa and a dashpot of viscosity lO Pa s. Calculate the stress at time t = 100 s in the following loading programme (i) at time t = 0 an instantaneous strain of 1 % is applied and (ii) at time t = 30 s the strain is increased instantaneously from 1% to 2%. 

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Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

In the Maxwell model, the two units are connected in series, so that each bears the full stress individually and the deformations of the elastic and viscous components are additive ... 

It is interesting to note that the Voigt model is useless to describe a relaxation experiment. In the latter a constant strain was introduced instantaneously. Only an infinite force could deform the viscous component of the Voigt model instantaneously. By constrast, the Maxwell model can be used to describe a creep experiment. Equation (3.56) is the fundamental differential equation of the Maxwell model. Applied to a creep experiment, da/dt = 0 and the equation becomes... 

The Maxwell model thus predicts a compliance which increases indefinitely with time. On rectangular coordinates this would be a straight line of slope I/77, and on log-log coordinates a straight line of unit slope, since the exponent of t is 1 in Eq. (3.69). 

As we did in the case of relaxation, we now compare the behavior predicted by the Voigt model—and, for that matter, the Maxwell model—with the behavior of actual polymer samples in a creep experiment. Figure 3.12 shows plots of such experiments for two polymers. The graph is on log-log coordinates and should therefore be compared with Fig. 3.11b. The polymers are polystyrene of molecular weight 6.0 X 10 at a reduced temperature of 100°C and cis-poly-isoprene of molecular weight 6.2 X 10 at a reduced temperature of -30°C. 

The Maxwell Model consists of a spring and dashpot in series at shown in Fig. 2.34. This model may be analysed as follows. 

This is the governing equation of the Maxwell Model. It is interesting to consider the response that this model predicts under three common time-dependent modes of deformation. 

From Fig. 2.35 it may be seen that for the Maxwell model, the strain at any time, t, after the application of a constant stress, Cg, is given by... 

This indicates an exponential increase in strain from zero up to the value, (To/, that the spring would have reached if the dashpot had not been present. This is shown in Fig. 2.37. As for the Maxwell Model, the creep modulus may be determined as... 

It may be seen that the simple Kelvin model gives an acceptable first approximation to creep and recovery behaviour but does not account for relaxation. The Maxwell model can account for relaxation but was poor in relation to creep... 

Solution The spring element constant, i, for the Maxwell model may be obtained from the instantaneous strain, . Thus... 

Solution From Section 2.11 for the Maxwell model, the strain up to 100s is given by... 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

Example 2.14 A plastic is subjected to the stress history shown in Fig. 2.45. The behaviour of the material may be assumed to be described by the Maxwell model in which the elastic component = 20 GN/m and the viscous component r) = 1000 GNs/m. Determine the strain in the material (a) after u seconds (b) after 1/2 seconds and (c) after 3 seconds. 

This will in fact be constant for all values of M3 because the Maxwell Model cannot predict changes in strain if there is no stress. The overall variation in strain is shown in Fig. 2.46. 

Maxwell and Kelvin-Voigt models are to be set up to simulate the creep behaviour of a plastic. The elastic and viscous constants for the Kelvin-Voigt models are 2 GN/m and 100 GNs/m respectively and the viscous constant for the Maxwell model is 200 GNs/m. Estimate a suitable value for the elastic constant for the Maxwell model if both models are to predict the same creep strain after 50 seconds. 

This will be a non-linear response for all non-zero values of n. (2.27) For the 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. 

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... 

The range of frequencies used to calculate the moduli are typically available on many instruments. The important feature that these calculations illustrate is that as the breadth of the distributions is increased the original sigmoidal and bell shaped curves of the Maxwell model are progressively lost. A distribution of Maxwell models can produce a wide range of experimental behaviour depending upon the relaxation times and the elastic responses present in the material. The relaxation spectrum can be composed of more than one peak or could contain a simple Maxwell process represented by a spike in the distribution. This results in complex forms for all the elastic moduli. 

In principle this integral could be applied directly to the Maxwell model to predict the decay of stress at any point in time. We can simplify this further with an additional assumption that is experimentally verified, i.e. that the function in the integral is continuous. The first value for the mean theorem for integrals states that if a function f(x) is continuous between the limits a and b there exists a value f(q) such that... 

It has been reported in the literature that the properties of mixed-matrix membranes can be predicted by using a Maxwell model [40]. The Maxwell model equa-hon is as follows ... 

In Eq. (11.1), P is permeability, < z is the volume fraction of the dispersed zeolite, the MMM subscript refers to the mixed-matrix membrane, the P subscript refers to the continuous polymer matrix and the Z subscript refers to the dispersed zeolite. The permeabiUty of the mixed-matrix membrane (Pmmm) can be estimated by this Maxwell model when the permeabilities of the pure polymer (Pp) and the pure zeoUte (Pz), as well as the volume fraction of the zeoUte (< ) are known. The selectivity of the mixed-matrix membrane for two molecules to be separated can be calculated from the Maxwell model predicted permeabiUties of the mixed-matrix membrane for both molecules. 

Figure 11.2 Selection of proper zeolite material for a mixed-matrix membrane (MMM) using the Maxwell model.

The Maxwell model can also guide the selection of a proper polymer material for a selected zeolite at a given volume fraction for a target separation. For most cases, however, the Maxwell model cannot be applied to guide the selection of polymer or zeolite materials for making new mixed-matrix membranes due to the lack of permeabihty and selectivity information for most of the pure zeolite materials. In addition, although this Maxwell model is well-understood and accepted as a simple and effective tool for estimating mixed-matrix membrane properties, sometimes it needs to be modified to estimate the properties of some non-ideal mixed-matrix membranes. 

Note 2 The relationship defining the Maxwell model may be written... 

Note 4 The Maxwell model may be represented by a spring and a dashpot filled with a Newtonian liquid in series, in which case Ha is the spring constant (force = 1/a) extension) and 1// is the dashpot constant (force = (l/y9) rate of extension). 

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. 

Thus, according to Equation 14.8 for the Maxwell model or element, under conditions of constant strain, the stress will decrease exponentially with time and at the relaxation time t = T, s will equal 1/e, or 0.37 of its original value, So-... 

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