Viscoelasticity 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),... 

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). 

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. 

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 Four-Element ModeF. The behavior of viscoelastic materials is complex and can be better represented by a model consisting of four elements, as shown in Figure 5.62. We will not go through the mathematical development as we did for the Maxwell and Kelvin-Voigt models, but it is worthwhile studying this model from a qualitative standpoint. 

Figure 5.62 (a) Four-element spring and dashpot model of viscoelasticity and (b) resulting... 

Stoner et al. (1974) have proposed a mechanical model for postmortem striated muscle it is shown in Figure 8-28. The model is a combination of the Voigt model with a four-element viscoelastic model. The former includes a contractile element (CE), which is the force generator. The element SE is a spring that is passively elongated by the shortening of the CE and thus develops an... 

In practice, viscoelastic properties can be determined by static and dynamic tests. The typical static test procedure is the creep test. Here, a constant shear stress is applied to the sample over a defined length of time and then removed. The shear strain is monitored as a function of time. The level of stress employed should be high enough to cause sample deformation, but should not result in the destruction of any internal structure present. A typical creep curve is illustrated in Fig. 13A together with the four-element mechanical model that can be used to explain the observations. The creep compliance represents the ratio between shear strain rate and constant stress at any time t. 

At aU technically relevant temperatures, polymers deform by creep. To describe the time-dependence of plastic deformation, we again exploit equation (8.3). In contrast to the viscoelastic deformation, there is no restoring force in viscoplasticity. Equation (8.3) is thus used to describe the dashpot element connected in series in the four-parameter model from figure 8.7(b). 

Spring and damper elements can be combined in a variety of arrangements to produce a simulated viscoelastic response. Early models due to Maxwell and Kelvin combine a linear spring in series or in parallel with a Newtonian damper as shown in Fig. 3.21. Other basic arrangements include the three-parameter solid and the four-parameter fluid as shown in Fig. 3.22. 

In another research, the thermo-mechanical behavior of SMPs was described by both linear and nonlinear viscoelastic theories [4]. In this woik four element mechanical units consisting of spring, dashpot and frictional device were used to derive a constitutive differential equation. In order to determine the material properties by a constitutive differential equation the modulus, viscosity and other parameters were assumed to decay exponentially with temperature. Liu et al. [5] developed a constitutive equation for SMPs based on thermodynamic concepts of entropy and internal energy. They adopted the concept of frozen strain and demonstrated the utility of the model by simulating the stress and strain... 

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 ... 

Based on the theoretical analysis about the morphology and the etfeet of crystallinity on the development of residual stresses for injection molded crystalline polymers, the residual stresses have been simulated by means of a new four elements viscoelastic mechanical model. Considering the crystalline orientation of injection molded parts, we view the parts as orthotropic solids to simulate the development of residual stresses in both longitudinal and transverse directions of the parts. 

With the new four elements viscoelastic model, the constitutive equations of simulating residual stresses for injection molded crystalline polymers can be sinplified to be orthotropic elastic equations under the conditions of plane stress state. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The acoustic response of resonant viscoelastic fluid structures to a pressure wave may be simulated by a four-dimensional calculation, three dimensions in space and one in time. The Lagrangian, primitive finite element and Eulerian finite difference schemes form the basis for two models presented in this paper which are able to simulate a wide range of fluid structures containing inclusions of arbitrary spacing, shape and composition. 

An example of creep deflection in a tensile bar for an epoxy at different temperatures is shown in Fig 5.12. It will be noticed that the creep response for a temperature of 155° C still has a positive slope after seven hours. Without knowing the type of material, one might expect the response to be that of a viscoelastic fluid. The creep response for 165° C and 170° C clearly have reached a limit and has the character of a thermoset. Because of the nature of the response, the epoxy could be best characterized by a viscoelastic fluid model such as the four-parameter fluid for both the 155° C and 160° C data. On the other hand, the epoxy could best be characterized by a viscoelastic solid model such as the three-parameter solid for temperatures above 160° C. To characterize the material over all time and temperature ranges would require a generalized model with a large number of elements. Methods to accomplish this will be discussed in subsequent sections. 

FIGURE 15.1 Linear viscoelastic models (a) linear elastic (b) linear viscous (c) Maxwell element (d) Voigt-Kelvin element (e) three-parameter (f) four-parameter. 

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