Maxwells Model

The stress and strain can be related by differentiating the strain equation and writing the spring and dashpot strain rates in terms of the stress  

This is the so-called governing equation for the Maxwell model. 

In a stress relaxation test, the polymer fiber is stretched to a constant strain e, i.e., 

Another major drawback of the Maxwell model is that it cannot be used to describe the creep behavior. In a creep test, the constant stress is applied to the polymer fiber and the governing equation becomes  

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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... 

A spring and dashpot in series is called a Maxwell model. 

Figure 3.7 Maxwell models consisting of a spring and dashpot in series (a) single unit and (b) set of units arranged in parallel.

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. 

Evaluate G(t) for integral powers of 10 between 10 ° and 10 sec. Use the same table entries to evalute G(t) for a two-element Maxwell model consisting of elements 1 and 4 above. On the same graph plot both sets of results as log G(t) versus log t. Comment on the similarities and differences between the two curves. 

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... 

Example 2.13 A plastic which can have its creep behaviour described by a Maxwell model is to be subjected to the stress history shown in Fig. 2.43(a). If the spring and dashpot constants for this model are 20 GN/m and 1000 GNs/m respectively then predict the strains in the material after 150 seconds, 250 seconds, 350 seconds and 450 seconds. 

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. 

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. 

A plastic component was subjected to a series of step changes in stress as follows. An initial constant stress of 10 MN/m was applied for 1000 seconds at which time the stress level was increased to a constant level of 20 MN/m. After a further 1000 seconds the stress level was decreased to 5 MN/m which was maintained for 1000 seconds before the stress was increased to 25 MN/m for 1000 seconds after which the stress was completely removed. If the material may be represented by a Maxwell model in which the elastic constant = 1 GN/m and the viscous constant rj = 4000 GNs/m, calculate the strain 4500 seconds after the first stress was applied. 

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. 

Fig. 2-24 Maxwell model used to explain viscoelastic behavior.

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. 

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