Maxwell body

The Maxwell body is appropriate for the description of stress relaxation, while the Voigt element is more suitable for creep deformation. In a stress relaxation experiment, a strain yo is imposed atr = Oand held constant thereafter (dy/r// = 0) while r is monitored as a function of t. Under these conditions, Eq. (11-29) for a Maxwell body behavior becomes... 

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. 

The subscript s and d refer to the elastic spring and viscous dashpot, respectively. The differential equation for a Maxwell body is obtained by first differentiating the sum of the strains Equation (35). This allows the viscous Equation (33) to be directly inserted in the differentiated form of Equation (35). Then by differentiating elastic Equation (32) it can also be inserted into the differentiated form of Equation (35) giving the final result ... 

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. 

Figure 3-2. Creep response of a Maxwell body displayed using linear (left) and...

Figure 3-3. Maxwell body behavior using stress relaxation conditions (a) linear plot, (b) linear-log plot, and (c) log-log plot.

The application of sinusoidal stress and strain is similar to that for a Maxwell body. The results are summarized in Table 3-1 along with the previously derived results for a Maxwell element. Figure 3-6 displays the frequency dependence of D and D" for the Voigt element in tension. The response in shear would be identical with J replacing D. 

Develop expressions for the complex modulus and compliance for a Maxwell body... 

Figure 3-3c is a plot of the relaxation behavior of a Maxwell body on a log-log scale. Thus we want to calculate... 

The time ti / G is denoted as retardation time, applicable to bodies other than a Maxwell body. The relaxation curve according to Eq. (3.2) or its spectrum (the process may be complex and can have more than one relaxation time), was applied for fitting the pressure decay obtained after a sudden stop of the compressing of a monolayer at a certain surface pressure. A typical experimental result is shown in Fig. 3.2 (Tabak et al. 1977). 

The response to an applied strain of 200% (e = 2) was then studied. As shown, the logarithmic term of Eq. (6) may be neglected for large N, and the response of the model then becomes equivalent to that of a generalized Maxwell body (Weichert body) with 48 discrete relaxation times, in parallel with a spring. 

The equations can be generalized for both shear and tension, and G can be replaced by E. The mechanical analogue for the Maxwell unit can be represented by a combination of a spring and a dashpot arranged in series so that the stress is the same on both elements. This means that the total strain is the sum of the strains on each element as expressed by Equation 13.19. A typical stress-strain curve predicted by the Maxwell model is shown in Figure 13.12(a). Under conditions of constant stress, a Maxwell body shows instantaneous elastic deformation first, followed by a viscous flow. 

Maxwell bodies are obtained if Hookean and Newtonian bodies are connected in series (Figure 11-11). The Kelvin or Voigt model, on the other hand, contains Hookean and Newtonian bodies in a parallel arrangement (Figure 11-11). The Maxwell body is a model for relaxation phenomena and the Kelvin body is a model for retardation processes. 

A Maxwell body whose modulus is twice as much. 

Write the relaxation equation of a Maxwell body. Derive the following ... 

Equation (2) reveals that in the process of formation of a fault net, i.e. in mega- and macro-fracture of rock mass in natural conditions, some common fracture laws appear without dependence on degree of tectonic activation and tectonic history of evolution. Comparison between the data obtained in geological investigations and the experimental data for Maxwell bodies, reveals that the character of the behavior of Maxwell bodies in fracture is identical to that of the distribution of the faults with different lengths in regions with different tectonic evolution history and different degree of activation. Hence, it can be concluded that in the process of formation of a fault net, the Earth crust behaves as a Maxwell body, i.e. as elasto-viscous body. 

Though equation (25) is derived from phenomenological point of view, observations in situ and experimental research completely prove its validity. If we identify the law of deformation and fracture of geological medium to that of the Maxwell body, then in the analysis of occurrence of earthquakes we can consider the reoccurrence period of earthquakes to be proportional to the relaxation time of rock mass. According to the statistics of Sadovsky and et al. (1987), the reoccurrence period T of earthquakes and their energy have the next correlation relation (Fig. 3) ... 

Solving this equation for a Maxwell body leads to the following equations of moduli, which include the parameter x ... 

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