Equation Kelvin-Voigt

The dashpot constant, rj2, for the Kelvin-Voigt element may be determined by selecting a time and corresponding strain from the creep curve in a region where the retarded elasticity dominates (i.e. the knee of the curve in Fig. 2.40) and substituting into equation (2.42). If this is done then r)2 = 3.7 X 10 MN.s/m ... 

If centered differences are also used for the spatial derivatives in Equation (7) and backward differences are used to represent the time derivative describing the viscous Kelvin-Voigt effect (2 ) then the first term on the RHS of Equation (7), in two space dimensions (x,y) become,... 

From data in a stress relaxation experiment (Chapter 3), where the strain is constant and stress is measured as a function oftime, a If), the relaxation time may be estimated from the time necessary for [cr(r)/cr(0)] to become (1/e) = 0.368. Typically, several Maxwell elements are used to fit experimental data, a t). For the Kelvin-Voigt element (Figure 1-8, right) under stress, the equation is ... 

In Equation (3.85), Jm is the mean compliance of all the bonds and Tm is the mean retardation time Tm equals Jmt m where ijm is the mean viscosity associated with elasticity. One can replace the mean quantities with a spectrum of retarded elastic moduli (Gj) and the viscosities (iji), where, J-, = l/G,. Typically, one or two Kelvin-Voigt elements can be used to describe the retarded elastic region. 

Let us calculate sJ(s) for Kelvin-Voigt element. The constitutive equation is... 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

To start with, let us determine the stress and the deformation of a hollow sphere (outer radius J 2, inner radius R ) under a sudden increase in internal pressure if the material is elastic in compression but a standard solid (spring in series with a Kelvin-Voigt element) in shear (Fig. 16.1). As a consequence of the radial symmetry of the problem, spherical coordinates with the origin in the center of the sphere will be used. The displacement, obviously radial, is a function of r alone as a consequence of the fact that the components of the strain and stress tensors are also dependent only on r. As a consequence, the Navier equations, Eq. (4.108), predict that rot u = 0. Hence, grad div u = 0. This implies that... 

The main disadvantage of the Maxwell model is that the static shear modulus p0 vanishes in this model, while the drawback of the Kelvin-Voigt model is that it cannot describe the stress relaxation. The Zener model [131] lacks these disadvantages. This model combines the Maxwell and Kelvin-Voigt models and describes strains closely approximating the actual physical process. The elasticity equation for the Zener model taking account of anomalous relaxation effects can be written as [131]... 

To eliminate the Newtonian simplification, a rheological constitutive equation is replaced in the equations that require it. Or, in the case where viscoelasticity effects are required, the simple Kelvin-Voigt model can be used. In this case, the stress is decomposed into its viscous and elastic components, as shown in the following equation ... 

Any number of extra Kelvin-Voigt elements can be added in series within the Burgers model, and each will add one extra term to the creep equation, so that most practical creep curves can be reasonably described, with the behaviour at the very shortest times captured by Gi and at the longest times in steady flow by q, so... 

The stress relaxation behaviour has been addressed in terms of complex constitutive equations and simpler models based on Maxwell and Kelvin-Voigt elements [131-134]. 

Since the matrix melt has both elastic and viscous properties, it can be described by a constitutive law of viscoelasticity. The non-linear Kelvin-Voigt equation is employed here... 

The initial stress upon application of a constant strain is governed by the Maxwell spring element Ei, and relaxation on a long-term time scale is dominated by the Maxwell dashpot tji, while the Kelvin-Voigt section with and r/2 governs the delayed elastic relaxation. Solving the differential equation by Laplace transformation is, for example, comprehensively described in (Betten 2008). 

Another disadvantage of Kelvin-Voigt model is that it cannot be used to describe the stress relaxation behavior of polymer fibers. Under a constant strain the governing equation of the Kelvin-Voigt model becomes ... 

This is the governing equation for the Kelvin (or Voigt) Model and it is interesting to consider its predictions for the common time dependent deformations. 

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). 

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... 

Describe the differences between the Kelvin and Voigt models for viscoelasticity and identify their corresponding equations. 

The integral form of Eq. (4.18) (Kaelble, 1971) shows that e is an exponential decay function of t/( /G). The dimensions of r /G reduce to seconds (Appendix 4) and the equation reaches a limiting l/e (0.37emax) in t = ti/G seconds. The retardation time (/rd) is the time required for emax of a Voigt-Kelvin fluid (Fig. 2a) to be reduced to 37% of emax after t has been removed (Barnes et al., 1989 Seymour and Carraher, 1981). A long retardation time is characteristic of a more elastic than viscous fluid. 

This equation for the dielectric constant is the analogue of the compliance of a mechanical model, the so-called Jeffreys model, consisting of a Voigt-Kelvin element characterised by Gi and rp and t =t /Glr in series with a spring characterised by Gz- The creep of this model under the action of a constant stress aQ is (Bland, 1960)... 

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... 

An alternative model is obtained if one places the spring and dashpot in parallel and this is known as the Voigt or the Kelvin model (Fig. 5.14(a)). For this model, the strains on the two components are the same and the overall stress is the sum of the stresses on the dashpot and spring. The constitutive equation becomes... 

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