KELViN-solid

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Analyses of the results obtained depend on the shape of the specimen, whether or not the distribution of mass in the specimen is accounted for and the assumed model used to represent the linear viscoelastic properties of the material. The following terms relate to analyses which generally assume small deformations, specimens of uniform cross-section, non-distributed mass and a Voigt-Kelvin solid. These are the conventional assumptions. 

Note 4 Damping curves are conventionally analysed in terms of the Voigt-Kelvin solid giving a decaying amplitude and a single frequency. 

Note 5 Given the properties of a Voigt-Kelvin solid, a damping curve is described by the equation... 

Note 2 The logarithmic decrement can be used to evaluate the decay constant, p. From the equation for the damping curve of a Voigt-Kelvin solid. 

Note 2 For a Voigt-Kelvin solid, with P(D)=1 and Q y)=a+pD, where a is the spring constant and P the dashpot constant, the equation describing the deformation becomes... 

Note 7 Notes 2 and 5 show that application of a sinusoidal uniaxial force to a Voigt-Kelvin solid of negligible mass, with or without added mass, results in an out-of-phase sinusoidal uniaxial extensional oscillation of the same frequency. 

Note 1 For a Voigt-Kelvin solid of negligible mass, the absolute modulus can be evaluated from the ratio of the flexural force (/o) and the amplitude of the flexural deflection (y) with... 

Note 3 A material specimen which behaves as a Voigt-Kelvin solid under forced oscillation , with a mass added at the point of application of the applied oscillatory force... 

Numerous attempts have been made to fit simplified mechanical models to the two behavior patterns described by Eq. (6). One can picture the elastic element as a spring-anayed network parallel with the viscous element to give essentially a (Kelvin) solid with retarded elastic behavior, wherein ... 

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

Pharmaceutical materials are rarely described by simple mechanical equivalents such as the Kelvin (solid-like behavior) or the Maxwell (liquid-like behavior) model. 

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,... 

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,... 

Burns, M.L. et al. (1984) Analysis of compressive creep behavior of the vertebral unit subjected to uniform axial loading using exact parametric solution equations of Kelvin solid models Part I. /. Biomech., 17, 113-130. 

Differential Stress-Strain Relations and Solutions for a Kelvin Solid... 

The Kelvin model is also frequently used to describe the phenomena of creep. Recall the Kelvin solid from Fig. 3.21. 

By eliminating various elements in the four-parameter model the response of a Maxwell fluid, Kelvin solid, three-parameter solid (a Kelvin and a spring in series) can be obtained and the model can be used to represent thermoplastic and/or thermoset response as illustrated in Fig. 3.13. For example the creep response of a three-parameter solid is obtained by eliminating the free damper in Eq. 3.44 and gives the creep and creep recovery response shown in Fig. 3.13 for a crosslinked polymer. 

Referring to the solution under creep for a Kelvin material given in Chapter 3, quite obviously the solution of the three-parameter model for the case of creep is simply the superposition of the solution for creep of a spring and creep of a Kelvin solid. 

Again, the solution is left as an exereise for the reader (see problem 5.4). However, it should be noted that the solution of the differential equation for a four-parameter fluid in the ease of ereep is the superposition of creep of a Maxwell fluid and creep of a Kelvin solid (refer to Chapter 3). 

A Generalized Kelvin Solid is composed of a number of Kelvin elements in series as shown in Fig. 5.10a. 

However, this model still has no instantaneous elasticity and a free spring is normally included in series with the generalized Kelvin solid with the result (sometimes referred to as the Voigt-Kelvin model),... 

Fig. 5.11 Generalized Voigt-Kelvin solid with a free damper.

Give an equation that would represent the creep response for a generalized Kelvin Solid. 

Applying this example to a material which is well represented by a Kelvin solid, where,... 

Using Boltzman s superposition integral, find the strain output for a Kelvin solid for the given stress input. 

For a viscoelastic bar, Eq. 8.13 can be used in the Laplace domain, simplified and inverted, or Eq. 8.14 can be used directly in the time domain as illustrated here. In either case, the time dependent compliance of the material must be chosen to determine the solution. Here, choose a simple Kelvin solid such that... 

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