Rheology Kelvin model

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

As seen in Figure 3.21, a complete rheological curve contains four characteristic regions. Region I corresponds to low stresses under which the system may demonstrate a solid-like behavior with high viscosity (Kelvin model). This case is characteristic of the already mentioned bentonite clays. The studies of relaxation structures in moderately concentrated suspensions of bentonite clays indicated the appearance of elastic aftereffect at low shear stresses. This effect has an entropic nature, as it is associated with the... 

The key point in the rheological classification of substances is the question as to whether the substance has a preferred shape or a natural state or not [19]. If the answer is yes, then this substance is said to be solid-shaped otherwise it is referred to as fluid-shaped [508]. The simplest model of a viscoelastic solid-shaped substance is the Kelvin body [396] or the Voigt body [508], which consists of a Hooke and a Newton body connected in parallel. This model describes deformations with time-lag and elastic aftereffects. A classical model of viscoplastic fluid-shaped substance is the Maxwell body [396], which consists of a Hooke and a Newton body connected in series and describes stress relaxation. 

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 ... 

The data is taken from Zhang and Soong (1992). Two rheological models describing the d5mamic behaviour of dampers were applied in the calculations the Kelvin fractional model and the Maxwell fractional model. 

To account for the fact that neither ideal solids nor ideal liquids exist in the real word, the rheological behavior of real materials can be approximated by a combination of the individual model elements, either in Unear two-element models of MaxweU and Voigt-Kelvin types, or in linear three-element models. In many real cases, nonlinear models have to be invoked (see Section 2.4.1.3). 

Other quantities are also needed by finite element software, such as the tangent matrix for the use of Newton-Raphson methods, and are also function of the number N of Kelvin-Voigt elements. In the case of limiting the rheological model to N = 2 the tangent matrix is ... 

The processes involved in the elastic aftereffect can be described by a simple rheological model consisting of two Kelvin s elements connected in series, as shown in Figure 3.32a. It is worth emphasizing here that this model is applicable only in the region of low shear stresses, below the onset of Schwedow s creep. From this model, one gets the following values for the slow and fast elastic strain moduli ... 

FIGURE 3.32 The corresponding rheological models describing two stages of the elastic aftereffect. The two Kelvin elements connected in series (a) G, and strains are replaced with a single elasticity modulus, G i (b). 

The origin of the theory of viscoelasticity may be traced to various isolated researchers in the last decades of the nineteenth Century. This early stage of development is essentially due to the work of Maxwell, Kelvin and Voigt who independently studied the one dimensional response of such materials. The linear constitutive relationships introduced therein are the base of rheological models which are still used in many applications [121]. Their works led to Boltzmann s [122] first formulation of three dimensional theory for the isotropic medium, which... 

Earlier in the theory of viscoelasticity many rheological models with combinations of the Maxwell and the Voigt - Kelvin bodies were considered (see Freudental and Geiringer [1]). These models have constitutive laws for stresses o j and strains eij which include time derivatives of arbitrary order. 

Time response of different rheological systems to applied forces. The Maxwell model gives steady creep with some post stress recovery, representative of a polymer with no cross-linking. The Kelvin-Voigt model gives a retarded viscoelastic behavior expected from a cross-linked polymer. 

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