Viscoelasticity Burgers model

Another possibility to work with the parameters of the viscoelastic Burgers models in evaluating and predicting relaxation phenomena is by using a recursive definition of the relaxation function (r t). Therefore, the total period of time to be considered is first divided into ttf incremental steps and the calculation of strain for each consecutive step is based on the value of the preceding one. 

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 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 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. 

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). 

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 ... 

In more detail, the flow of glass is more complex due to the combined elastic and viscous response to any type of applied stress, known as viscoelasticity. Several models have been proposed to describe viscoelasticity. Among them. Burger s model has been shown to characterize reasonably well the behavior of inorganic glasses [5]. In this version, illustrated in Fig. 3a, viscous (771) and elastic (El) elements are combined in series with a Kelvin solid, where two other elements (772, 2) are arranged in parallel and reflect the slow elastic properties. The rate of deformation under constant tensile stress a and zero initial deformation is made up from the rate of Newton s viscous deformation,... 

Fig. 4. Creep and recovery response for a Burgers model (insert), showing the four regions of polymer-like viscoelastic response as discussed in text. After Rosen (13), with permission.

The Maxwell and Kelvin-Voigt models may be expanded to give either multiple Maxwell models, which are usually combined in parallel, or else multiple Kelvin-Voigt models, which are usually combined in series. However, if we combine a Maxwell model and a Kelvin-Voigt model in series, we obtain a Burgers model, named in honour of the Dutch physicist (1895 - 1981) of that name. This model is the most complicated that we need to consider, since it describes aU the basic features of interest to us. Below we will see how aU these various combinations of springs and dashpots begin to describe real viscoelastic materials. 

The behaviour normally seen for typical viscoelastic liquids—represented for convenience and simplicity by a Burgers model—in a creep test is shovm in figure 3. 

Creep response for viscoelastic materials according to the Burgers model... 

The creep response according to the Burgers model in Fig. 34.4 covers all elementary aspects of time-dependant viscoelastic behavior including instantaneous elastic strain, secondary steady state creep in the long-term area, and a delayed elastic strain transition behavior that can be, for example, fitted to experimental data according to the choice of the t/i, Ei, ijz, and Ez parameters. 

The extension to the four-parameter Burgers model provides again a more detailed description of relaxation in viscoelastic polymer materials. The initial differential equation and the assignments of parameters are as follows ... 

The line connected by data points in Fig. 34.15 represents the experimental data while the continuous curve relates to the numerically simulated behavior based on the four parameter Burgers model for viscoelasticity. The load history consists of repeatedly loading for 24 h followed by 24 h of recovery. 

The Burgers Model. As a preview to the viscoelastic behavior of polymers, we next consider the four-element Burgers model that captures a minimum set of behaviors that is seen in polymeric materials and as discussed here (13). The insert in Figure 4 shows the Burgers model as a Maxwell model in series with a Kelvin-Voigt model. As shown in Figure 4, upon application of a constant stress Ti2 for a time ti followed by its removal, the model captures the folloAving aspects of polymer viscoelasticity ... 

Pol3rmers, as seen later, show more complex viscoelastic behaviors than does the Burgers model, but the essentials are in the Binders model. The instantaneous elasticity can be thought of as the glass-like response the molecular slip is the terminal response, as the long pol3rmer molecules slide past each... 

The simplest model that can be used for describing a single creep experiment is the Burgers element, consisting of a Maxwell model and a Voigt-Kelvin model in series. This element is able to describe qualitatively the creep behaviour of viscoelastic materials... 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

Figure 5.17 Burgers (four-parameter) viscoelastic model.

For the Burger s viscoelastic model, derive an expression for the strain under the application of a uniaxial tensile stress. 

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... 

Fig. 19.19. Idealized viscoelastic material response (Burger s model) to constant, protracted tensile stress starting at followed by sudden release of the stress at t2.

---