Model Burgers

The Burgers model, also called a linear liquid of four elements, is a combination of the Maxwell model with a Kelvin-Voigt element (see Fig. 10.9). For a stress input, the strains are additive, 

After eliminating 8i, 82, and 83 between these equations, one obtains 

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Figure H3.3.4 Mechanical models are often used to model the response of foods in creep or stress relaxation experiments. The models are combinations of elastic (spring) and viscous (dashpot) elements. The stiffness of each spring is represent by its compliance (J= strain/stress), and the viscosity of each dashpot is represent by a Newtonian viscosity (ri). The form of the arrangement is often named after the person who originally proposed the model. The model shown is called a Burgers model. Each element in the middle—i.e., a spring and dashpot arranged in parallel—is called a Kelvin-Voigt unit.

Figure H3.3.5 The creep response of a food (circles) was fitted to a Burger model with one Kelvin-Voight unit. The goodness of fit is shown as the continuous curve and the standard error. The values of compliance and viscosity of the respective springs and dashpots were outcomes of the fitting process.

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... 

Summarizing The basic idea, mentioned in chapter 6, that creep of solid polymers could be represented by a simple four-parameter model (the Burgers model), composed of a Maxwell and a Kelvin-Voigt model in series, appears to be inadequate for three reasons ... 

These three complications are schematically shown in Figure 7.11, using creep isochrones as a reference (a) a Kelvin-Voigt element has been chosen with a spring in series (a Burgers model without irreversible flow). 

Brittle temperature, 461,467-9 Bulk/Bulkiness, 877 compliance, 385 modulus, 385, 395,396,405 Burgers model, 413... 

Figure 8-13 (A) Voigt-Kelvin, (B) Maxwell, and (C) Burgers Models...

The viscoelasticity properties are also important, because they can supply information directly related to the form of the macromolecules. The models of the linear viscoelasticity are developed from two elements a spring and a dashpot. Two of those elements in line constitute the Maxwell model and in parallel the Kelvin model (or Vogt).20 Normally, those models don t represent the behavior of complex materials satisfactorily. Other models such as the Burgers model, where the Maxwell and Kelvin models are connected in line, are used to determine the modulus of elasticity (Yj and Y2) and the coefficients of viscosity ( and t]2).21... 

The viscoelastic properties of films were determined by stress relaxation tests with a texture analyzer TA.XT2i (SMS). The films were cut into strips of 15 mm width and 100 mm lengths and affixed to the instrument. The initial grip separation and crosshead speed were 80 mm and 0.9mm/sec, respectively. The instrument was set for a deformation of 1 %, which was held constant for 70 sec. The force required to maintain this deformation was monitored by a microcomputer in real time. The viscoelastic properties were calculated according to the Burgers model (Equation 1) ... 

The coefficients of viscosity (fy) and the modulus of elasticity (Y,) for the Maxwell (i = 1) and Kelvin units (i = 2) according to the Burgers Model, had the same behavior (Figures 1-4), increasing with thickness of all the films, according to a power law model, with important variation in the thin films, excepting the Y, that present a low value of regression coefficient (R2 = 0.33). 

The time-dependent deformation of the foundation rock, caused by loadings, is described by a Burgers model, which is composed of a Kelvin model and Maxwell model in series, as showed in Figure 2, The partial strain expression of Burgers model is... 

Figure 9. Four-parameter or Burger model for the deformation behaviour of polymers.

The Burger model provides a correct graphic description of the elongation-time behavior of most plastics in a first approximation. The spring 1 results in spontaneous elastic load application and relaxation elongation, 1 + 2 in parallel cause creep during load application and creep recovery (delayed viscoelastic reverse deformation) after relaxation, damper 2 results in residual elongatimi. 

Fig. 19 Extended, more realistic model (four-parameter or Burger model) and its elongation-time behavior...

Real materials exhibit a much more complex behavior compared to these simplified linear viscoelastic models. One way of simulating increased complexity is by combining several models. If, for instance, one combines in series a Maxwell and a Voigt model, a new body is created, called the Burger model (Figure 4-15). 

The Burger model represents the behavior above Tg in the elastomeric state, while the single dash-pot simulates melt flow at higher temperatures. Around the transition between the glassy and elastomeric states, the modulus of elasticity drops 10 to 10 fold, and the material becomes flexible and... 

There are other models based on springs and dashpots such as the simple Kelvin-Voigt model for viscoelastic solid and the Burgers model. Reader is referred to Refs. [1-5] for details. Other elementary models are the dumbbell, bead-spring representations, network, and kinetic theories. However, the most notable limitation of all these models is their restriction to small strain and strain rates [2, 3]. 

The second model (Figure 4.15d) describes the complicated viscoelastic behaviour of bitumen. Upon application of stress, the model immediately presents elastic deformation and continues to deform at a non-linear rate. Thus, for a given temperature, if a constant stress (oi) is applied, the strain (e) after time (t) could be calculated using the Burgers model by the following equation ... 

Apart from the Kelvin-Voigt and Burgers models, many other more complicated models have been developed, aiming at a better simulation of the recovery curve. All models developed constitute combinations of the Kelvin-Voigt and Burgers models. More information can be found in Meyers and Chawla (1999) and Ward (1983). 

An interesting three-parameter model (the Burger model has four parameters) was proposed by Hsueh [6] and is shown in Fig. 3b. He demonstrated that for a Hookean elastic element (Ei) in series with a Kelvin solid (E2,ry), the stress-strain rate relations for constant strain rate and constant stress creep tests are,... 

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