Kelvin element

Voigt-Kelvin model Voigt-Kelvin element... 

Note 2 The retardation time of a Voigt-Kelvin element is r = 1/g o = pia = (dashpot constant)/(spring constant). 

Note 5 The retardation spectrum (spectrum of retardation times) describing creep in polymers may be considered as arising from a group of Voigt-Kelvin elements in series. 

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... 

Because we are modeling the stress-strain behavior of tissues in which all tissues are under internal tension conditions where the stress is a function of time, then this element is not particularly useful for even superficial modeling of the stress-strain behavior of ECMs. The other element is a spring in parallel with a dashpot, termed the Voigt or Kelvin element. In... 

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)... 

These two basic elements may be combined in series or parallel, giving the Maxwell-element and the Voigt-Kelvin element. 

The Voigt-Kelvin element (retarded elastic response), represented by a spring and a dashpot in parallel. The elastic response is not instantaneous but retarded by a viscous resistance. The two contributions to the stress are additive in this model whereas the strains are equal ... 

In a creep experiment the Voigt-Kelvin element is instantaneously subjected to a stress [Pg.414]

The time dependent compliance of a Voigt-Kelvin element is... 

The limiting value of the creep is equal to 00 = a0/E = Voigt-Kelvin element is only able to describe qualitatively the creep behaviour of rubberlike materials with a limited creep and not the creep of an elastic liquid. In general the creep compliance may be expressed as... 

In a limited way, the Maxwell element describes a liquid. Similarly, the Kelvin (Voight) element describes a solid. As the relaxation time, Tj, is defined for the Maxwell element, the retardation time, T2, is defined for the Kelvin element. For the Kelvin element under stress. 

Fig. 11-16. Simple mechanical models of viscoelastic behavior, (a) Voigt or Kelvin element and (b) Maxwell element.

Neither simple mechanical model approximates the behavior of real polymeric materials very well. The Kelvin element does not display stress relaxation under constant strain conditions and the Maxwell model does not exhibit full recovery of strain when the stress is removed. A combination of the two mechanical models can be used, however, to represent both the creep and stress relaxation behaviors... 

In a Voigt (or Kelvin) element tlie spring and dashpot are parallel. If a stress is suddenly applied the spring cannot respond immediately because of the resistance caused by the viscous flow (delayed elasticity). Monolayers with a two-dimensional network and viscous material between the cross-links will display such behaviour. So, the increase of the strain is retarded. Eventually the maximum strain / K° is attained, see fig. 3.52a. After cessation of the strain the energy stored in the spring relaxes, again with a rate determined by the parallel viscosity, till AA— 0. Behaviour like this is semi-solid. In the limit of r] - 0 the block diagram of fig. 3.49b is retrieved. 

The differential equation of a Kelvin element is obtained by directly inserting the elastic Equation (32) and the viscous Equation (33) into (37) ... 

Since, as we saw above, the Maxwell element is not perfect, it seems logical to consider a parallel arrangement of the spring and the dashpot. This is the so-called Voigt or Voigt-Kelvin element... 

From this it follows through integration that (index K characterizes the Kelvin element)... 

The experimental creep function is often analyzed as a nnm-ber of Kelvin elements in series (Figure 40.32), each having the property of a spring and dashpot in parallel (Genevaux, 1989 Martensson, 1992 Mohager and Toratti, 1993 Hanhijarvi, 1999 Passard and Perre, 2005). In the case of uniaxial load, this leads to... 

FIGURE 40.32 Modeling the viscoelastic behavior of wood with the number of Kelvin elements in series. 

This relation describes a parallel arrangement of an elastic spring g and a kind of dashpot which is responsible for the stress relaxation according to do/d(ln t) = -X. This arrangement resembles the classical Kelvin element, as is demonstrated by comparing the creep response of this element... 

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). 

Figure 13 The Maxwell and Kelvin elements, representing simple series and parallel arrays of springs and dashpots.

Figure 14 Creep behavior of the Maxwell and Kelvin elements. The Maxwell element exhibits viscous flow throughout the deformation, whereas the Kelvin element reaches an asymptotic limit to deformation.

In the Maxwell element both the spring and the dashpot are subjected to the same stress but are permitted independent strains. The inverse is true for the Kelvin element, which is equivalent to saying that the horizontal connecting portions on the right-hand side of Figure 10.3 are constrained to remain parallel. 

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