Linear viscoelastic model

The linear viscoelastic model assumes that the stress at the ciurent time depends not only on the current strain, but on the past strains as well. It also assumes a linear superposition. Its general form reads 

The inelastic liquid is recovered with the relaxation modulus function being set to 

The most often used relaxation modulus function is the multi-mode Maxwell memory function  

Simple viscoelastic models can mimic the phenomena mentioned in Table 7.1. Although the models are inadequate at high stress levels, they aid understanding, and are the basis for more complex treatments. They are mechanical analogues of viscoelastic behavioui constructed using the linear mechanical elements shown in Table 7.2. They are linear because the equations relating the force f and the extension x only involve the first power of both the variables. 

The elements can be combined in series or parallel as shown in Figs 7.1 and 5.5. The convention for these models is that elements in parallel undergo the same extension. It is obvious that elements in series experience the same force. Thus, in the Maxwell model, the spring and dashpot in series experience the same force, while in Voigt model the spring and dashpot in parallel experience the same extension x. The total force f across the Voigt model can be written as the differential equation 

This has the same form as Eq. (7.1) the renamed constants are a Young s modulus , and a viscosity 17. It is not possible to directly link these constants to the modulus of the crystalline phase and the viscosity of the amorphous inter-layers in a semi-crystalline polymer. Hence, the Voigt model is an aid to understanding creep, and relating it to other viscoelastic responses, rather than a model of microstructural deformation. 

Creep loading means that the stress is given by 

For the Voigt model of Eq. (7.2), substituting the constant stress o-q for 0, and dividing by 17, gives 

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The application of finite strains and stresses leads to a very wide range of responses. We have seen in Chapters 4 and 5 well-developed linear viscoelastic models, which were particularly important in the area of colloids and polymers, where unifying features are readily achievable in a manner not available to atomic fluids or solids. In Chapter 1 we introduced the Peclet number ... 

Viscoelasticity has already been introduced in Chapter 1, based on linear viscoelasticity. However, in polymer processing large deformations are imposed on the material, requiring the use of non-linear viscoelastic models. There are two types of general non-linear viscoelastic flow models the differential type and the integral type. 

Linear viscosity is that when the function is splitted in both creep response and load. All linear viscoelastic models can be represented by a classical Volterra equation connecting stress and strain [1-9] ... 

The linear viscoelastic models (LVE), which are widely used to describe the dynamic rheological response of polymer melts below the strain limit of the linear viscoelastic response of polymers. The results obtained are characteristic of and depend on the macromolecular structure. These are widely used as rheology-based structure characterization tools. 

D. Aciemo, F. P. La Mantia, G. Marrucci, and G. Titomanlio, A Non-linear Viscoelastic Model with Structure-dependent Relaxation Times I. Basic Formulation, J. Non-Newt. Fluid Meek, 1, 125-146 (1976). 

This assumption of a linear relationship between stress and strain appears to be good for small loads and deformations and allows for the formulation of linear viscoelastic models. There are also non-linear models, but that is an advanced topic that we won t discuss. There are two approaches we can take here. The first is to develop simple mathematical models that are capable of describing the structure of the data (so-called phenomenological models). We will spend some time on these as they provide considerable insight into viscoelastic behavior. Then there are physical theories that attempt to start with simple assumptions concerning the molecules and their interactions and... 

For irrotational and small deformation flows, Equation 3.84 reduces to the general linear viscoelastic model ... 

Filbey equation (7). For cases of small deformation and deformation gradients, the general linear viscoelastic model can be used for unsteady motion of a viscoelastic fluid. Such a model has a memory function and a relaxation modulus. Bird and co-workers (6, 7) gave details of the available models. 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

In order to derive some simple linear viscoelastic models, it is necessary to introduce the mechanical equivalents for a Newtonian and a Hookean body. 

Linear viscoelastic models General linear x + k AxIAt — tjy ... 

This time-temperature superposition of linear viscoelastic data means that all the retardation times t, of the linear viscoelastic model have a common temperature shift factor a(T)... 

When plastics are unloaded, the creep strain is recoverable. This contrasts with metals, where creep strains are permanent. The Voigt linear viscoelastic model predicts that creep strains are 100% recoverable. The fractional recovered strain is defined as 1 — e/cmax, where e is the strain during recovery and Cmax is the strain at the end of the creep period. It exceeds 0.8 when the recovery time is equal to the creep time. Figure 7.9 shows that recovery is quicker for low Cmax and short creep times, i.e. when the creep approaches linear viscoelastic behaviour. 

It is useful to see how the Voigt linear viscoelastic models of Section 7.2 behave with a sinusoidal strain input. When the strain variation equation (Eq. 7.21) is substituted in the model constitutive equation (Eq. 7.2), the stress is given as... 

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). 

Smce the Zener model is a linear viscoelastic model, it obeys the Boltz-mann superposition principle. In this problem we are concerned with a strain history which is a smooth varying function of time, with y undergoing sinusoidal oscillations. Therefore the integral form of the BSP is the most straightforward one to apply ... 

Linear viscoelasticity is an extension of linear elasticity and hyperelasticity that enables predictions of time dependence and viscoelastic flow. Linear viscoelasticity has been extensively studied both mathematically (Christensen 2003) and experimentally (Ward and Hadley 1993), and can be very useful when applied under the appropriate conditions. Linear viscoelasticity models are available in all major commercial FE packages and are relatively easy to use. The basic foundation of linear viscoelasticity theory is the Boltzmann s superposition principle, which states, "Every loading step makes an independent contribution to the final state."... 

Borg, T. and Paakkonen, E.J. (2009) Linear viscoelastic models Part 1. Relaxation modulus and melt calibration. J. Non-Newtonian Fluid Mech., 156 (1-2), 121-128. 

Borg, T. and Paakkonen, E.J. (2009) Linear viscoelastic models Part 111. 

The model of Tobushi et al. [81, 85, 86] takes irreversible deformations into account by adding a friction slider into a (linear) viscoelastic model. The model considers in detail the change of modulus, viscosity, stress relaxation, and other parameters around Ttrans = Tg. With a nonlinear version of the model [86], it was possible to describe well thermomechanical properties, such as Ef, Ei, and the recovery stress for certain polyurethanes. 

Formulas for linear viscoelastic models can be apphed to tensile deformation as well as shear deformation by replacing the shear stress x with tensile stress o, shear strain y with tensile strain e, shear modulus G with Young s tensile modulus E, and newtonian shear viscosity T with Trouton s tensile viscosity iig [11—13]. 

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