Four element model

The Maxwell model does not represent the creep behavior, while the Kelvin-Voigt does not describe the stress relaxation behavior. This problem can be addressed by 

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Example 2.12 An acrylic moulding material is to have its creep behaviour simulated by a four element model of the type shown in Fig. 2.38. If the creep curve for the acrylic at 14 MN/m is as shown in Fig. 2.40, determine the values of the four constants in the model. 

Figure 4 illustrates the creep and recovery of a four-element model with the following constants ... 

Derive equation (6) for the elongation t of a four-element model. 

The total deformation in the four-element model consists of an instantaneous elastic deformation, delayed or retarded elastic deformation, and viscous flow. The first two deformations are recoverable upon removal of the load, and the third results in a permanent deformation in the material. Instantaneous elastic deformation is little affected by temperature as compared to retarded elastic deformation and viscous deformation, which are highly temperature-dependent. In Figure 5.62b, the total viscoelastic deformation is given by the curve OABDC. Upon unloading (dashed curve DFFG),... 

Notice that the compliance is inversely proportional to the modulus. Equations for the time-dependent strain can be developed for the four-element model shown above, or for any combination of elements that provide a useful model. The corresponding time-dependent compliance can then be determined using Eq. (5.75), or the time-dependent modulus using an analogous equation. 

The mechanical analogs corresponding to Equations (3.94) and (3.95) are shown in Figure 3-42. It is important to note that an association was established between the components of the ice cream and the components of the rheological model. This in turn permitted the justification of a six element model for the mix and a four element model for the melted ice cream. 

FIGURE 3.14 (a) Four-element model, (b), (c) Responses of the model under time-dependent modes of... 

FIGURE 13.13 Schematic representation of a creep curve a, initial elastic response b, region of creep c, irrecoverable viscous flow. This curve can be represented by the four-element model shown in Figure 13.14. 

Fig. 6.10 Illustration of a series connection of the Maxwell model and the Kelvin model fw the four-element model to describe the viscoelastic creep behaviors of polymers...

A series crmnection of the Maxwell and Kelvin models makes the four-element model, known as the Burger s model (Burgers 1935), which can describe the viscoelastic creep behaviors of polymers, as given by... 

Polymers undergo deformation under an applied stress over their lifetime some deformations which are irrecoverable once the source of stress is removed are referred to as creep. An understanding of the mechanical response of a polymer can be obtained by investigating the viscoelastic properties using creep experiments, where the behaviour is monitored under small deformational stress. Creep behaviour is an important consideration if the properties and dimensions are to be maintained. Experimental creep behaviour can be quantified using the four-element model with some limitations evident in the viscoelastic transitional region. ... 

A few viscoelastic behaviors can be modeled adequately by a two-element model, but usually it is necessary to combine Maxwell and Kelvin elements. A series arrangement of the two elements, known as the four-element model, is the simplest model that exhibits all the features of viscoelasticity (Fig. 15). It is beyond the scope of this introductory chapter to derive the mathematical equations that describe the various models. Several excellent texts exist and can be consulted (66-68). 

Figure 15 The four-element model and the creep behavior predicted hy this model.

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. 

Figure 10.5 (a) The four-element model, (b) Creep behavior as predicted by this model. At f, the stress is relaxed, and the model makes a partial recovery. 

The equation for the four-element model (Figure 10.5a) may now be written for the condition of constant stress ... 

It is of interest to compare the retardation time with the relaxation time. The retardation time is the time required for E2 and % in the Kelvin element to deform to 1 - lie, or 63.21% of the total expected creep. The relaxation time is the time required for 1 and 1/3 to stress relax to lie or 0.368 of ob, at constant strain. Both Ti and T2, to a first approximation, yield a measure of the time frame to complete about half of the indicated phenomenon, chemical or physical. A classroom demonstration experiment showing the determination of the constants in the four-element model is shown in Appendix 10.2. 

Draw stress relaxation curves for the four-element model. 

The objective of the experiment is to measure the viscoelastic characteristics of cheese, and interpret the data in terms of the four-element model. 

The time required is one hour. The level is junior or senior standing. The principles to be investigated are the apphcabihty of the four-element model to polymers in general, and cheese in particular. 

Figure A10.2.2 Resolution of the experimental data according to the four-element model. Curve 1 represents flow according to rja. Curve 2 illustrates the behavior of the middle element, which is really a Kelvin element. The strain at zero time up to the interception of curve 1 represents the instantaneous defomiation according to the spring Ei.

The data were analyzed in Rgure AIO.2.2 according to the four-element model. First, at zero time, the strain yields Fj. The straight-line portion at long times, separated as curve 1, yields Then, by subtraction, curve 2 was obtained. By simple curve fittings E2, r 2, and T2 can be determined. The retardation time for Veiveeta cheese was found to be about 7 to 9 minutes (see Table AIO.2.1). 

Fig. 3.22 Spring and damper arrangements for three and four element models.

Figure 16.23. Schematics of (A) the four-element model, and (B) the corresponding creep-recovery curve.

The four-element model shown below is subject to a creep experiment. Show how the length increases with time. At time = t, the stress is removed. Show how the length recovers. 

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