Kelvin models

Here we derive expressions for D and D of a Voigt-Kelvin model consisting of z elements assuming a sinusoidal strain application. Applying equation (3-22) to the Voigt-Kelvin model experiencing a strain in the yth element given by 

Solving equations (3-36) and (3-37) for the strain in theyth element gives 

It is clear that an inversion of this equation will not yield a simple result. 

---

It may be seen that the simple Kelvin model gives an acceptable first approximation to creep and recovery behaviour but does not account for relaxation. The Maxwell model can account for relaxation but was poor in relation to creep... 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

During a test on a polymer which is to have its viscoelastic behaviour described by the Kelvin model the following creep data was obtained when a stress of 2 MN/m was applied to it. 

In Chapter 2 when the Maxwell and Kelvin models were analysed, it was found that the time constant for the deformations was given by the ratio of viscosity to modulus. This ratio is sometimes referred to as the Relaxation or Natural time and is used to give an indication of whether the elastic or the viscous response dominates the flow of the melt. 

FIGURE 28.7 Viscoelastic Kelvin Model. (Redrawn from Bhattacharya, S.N., Rheology Fundamentals and Measurements, RMIT University, Melbourne, Australia, 2004.)... 

The constitutive equations for these models become more complex. When we apply a stress to the parallel elements of a Kelvin model both elements will respond. Thus a linear addition of the stresses describes the constitutive equation ... 

We have developed the idea that we can describe linear viscoelastic materials by a sum of Maxwell models. These models are the most appropriate for describing the response of a body to an applied strain. The same ideas apply to a sum of Kelvin models, which are more appropriately applied to stress controlled experiments. A combination of these models enables us to predict the results of different experiments. If we were able to predict the form of the model from the chemical constituents of the system we could predict all the viscoelastic responses in shear. We know that when a strain is applied to a viscoelastic material the molecules and particles that form the system gradual diffuse to relax the applied strain. For example, consider a solution of polymer... 

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

Here the term ik is the retardation time. It is given by the product of the compliance of the spring and the viscosity of the dashpot. If we examine this function we see that as t -> 0 the compliance tends to zero and hence the elastic modulus tends to infinity. Whilst it is philosophically possible to simulate a material with an infinite elastic modulus, for most situations it is not a realistic model. We must conclude that we need an additional term in a single Kelvin model to represent a typical material. We can achieve this by connecting an additional spring in series to our model with a compliance Jg. This is known from the polymer literature as the standard linear solid and Jg is the glassy compliance ... 

As with the elastic solid we can see that as the stress is applied the strain increases up to a time t = t. Once the stress is removed we see partial recovery of the strain. Some of the strain has been dissipated in viscous flow. Laboratory measurements often show a high frequency oscillation at short times after a stress is applied or removed just as is observed with the stress relaxation experiment. We can replace a Kelvin model by a distribution of retardation times ... 

Note 7 There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in Definition 5.2. (See, for example, [11].) Such definitions have certain advantages regarding their mathematical generality. However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models). 

Voigt-Kelvin model Voigt-Kelvin element... 

Note 1 The Voigt-Kelvin model is also known as the Voigt model or Voigt element. 

Note 3 The relationship defining the Voigt-Kelvin model may be written... 

Note 4 Comparison with the general definition of linear viscoelastic behaviour shows that the polynomial /"(D) is of order zero, 0(D) is of order one, ago = a and a = p. Hence, a material described by the Voigt-Kelvin model is a solid (go > 0) without instantaneous elasticity (/"(D) is a polynomial of order one less than 0(D)). 

Note 5 The Voigt-Kelvin model may be represented by a spring and a dashpot filled with a Newtonian liquid in parallel, in which case a is the spring constant (force = a extension) and P is the dashpot constant (force = pxdAQ of extension). 

Voigt-Kelvin element Voigt-Kelvin model Voigt element Voigt model volume compression vorticity tensor width of the resonance curve Young s modulus zero-shear viscosity... 

FIGURE 14.2 Stress-strain plot for stress relaxation for the Maxwell model (a) and Voigt-Kelvin model (b). 

In the Voigt-Kelvin model for viscoelastic deformation, it is assumed that the total stress is equal to the sum of the viscous and elastic stress, 5 = + So, so that... 

Voigt-Kelvin model or element Model consisting of an ideal spring and dashpot in parallel in which the elastic response is retarded by viscous resistance of the fluid in the dashpot. 

The static tests considered in Chapter 8 treat the rubber as being essentially an elastic, or rather high elastic, material whereas it is in fact viscoelastic and, hence, its response to dynamic stressing is a combination of an elastic response and a viscous response and energy is lost in each cycle. This behaviour can be conveniently envisaged by a simple empirical model of a spring and dashpot in parallel (Voigt-Kelvin model). 

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... 

A second approximation of the tensile creep function is the Voigt-Kelvin model... 

Figure 14.19 Stress-strain plots for (a) a Hookean spring where E is the slope (6) a Newtonian dash pot where s is constant, (c) stresstime plot stress for relaxation in the Maxwell model, and (d) stresstime plot stress for a Voigt-Kelvin model.

If it is now assumed that all constants except Po (=G) and Pi(=ri) are zero, the so-called Kelvin model results ... 

---