Visco-elastic models

Visco-elastic models have been developed for the nonlinear mechanical properties of fluids and solids. For a viscous fluid in simple shear flow, the shear stress, r y y), is a function of the effective viscosity, rj(-y) and the shear rate, y, as follows ... 

Two forms of specimen are commonly nsed to determine the material parameters in the models outlined below (1) bulk tensile tests and (2) thick adheiend shear tests (TAST). There are two common forms of modelling creep at the macroscopic level. The first is through visco-elastic models, which can be visnalized as a combination of spring and dashpot elements. The simplest of these is the Voigt model shown in Fig. 2. The constitutive equation for this model and its solution for the conditions of creep (constant stress) are given" respectively as... 

More complex models can be formed by connecting a number of such elements in series. However, even these are only linear visco-elastic models in which the rate of straining is directly proportional to the stress. For polymers at typical structural levels of load, the stress-strain rate is often highly non-linear. A molecular interpretation of this can be found in a thermally activated rate process model involving motion of chain segments... 

Although it is possible to introduce non-linear visco-elastic models, the second common approach is to utilize existing metals creep models. There are a number of models and all express the creep strain rate as a function of stress and other parameters. Popular forms are given below for primary and secondary creep, respectively. 

Buckley et al. (2007) demonstrated the shape memory behaviors of polyurethanes based on a linear, visco-elastic model, in which the SME was ascribed to the variation... 

Visco-elastic substances can be described with the spring/shock-absorber model of Kevin and Voigt, and have phase displacements of 0° to 90°. In analogy to other time dependent processes in physics, the oscillation tests are evaluated with complex arithmetics. Obtained are the complex quantities ... 

There is strong interest to analytically describe the fzme-dependence of polymer creep in order to extrapolate the deformation behaviour into otherwise inaccessible time-ranges. Several empirical and thermo-dynamical models have been proposed, such as the Andrade or Findley Potential equation [47,48] or the classical linear and non-linear visco-elastic theories ([36,37,49-51]). In the linear viscoelastic range Findley [48] and Schapery [49] successfully represent the (primary) creep compliance D(t) by a potential equation ... 

The dynamic elastic modulus (E ), the loss modulus (E"), and the tan 8 were measured simultaneously by the Rheovibron dynamic visco-elastometer model DDV-II (Toyo Measuring Instruments Co., Ltd.). Measurements were made at a frequency of 110 Hz, starting from —140°C and heating at 1°-2°C per minute until the samples became too soft to be tested. The readings were taken every 4°-8°C except in the transition zone when the readings were taken every 2°C. The sample chamber was kept dry by a stream of moisture-free nitrogen. 

Magnitudes of binary and ternary force interactions required in Eq. (62) are calculated by simple elastic or visco-elastic constitutive equations and then projected into force vectors. Elastic model is the direct implementation of the Hook s law with the stress between micro-elements A and B dependent linearly on the strain eAB ... 

The visco-elastic interactions of connected micro-elements of semicrystalline polyolefins are approximated by the Maxwell model, which formally consists of the spring with modulus E and the viscous dashpot characterized by the viscosity rf connected in series,... 

Fig. 21. Morphogenesis model of porous polyolefin particle based on the elastic and visco-elastic interactions among micro-elements (from Grof et al., 2005a).

A similar approach has also been developed by Susteric [108], who compares the behavior during low-amplitude deformation of rubbers, loaded with aggregated carbon black, with the visco-elastic behavior of macromolecules undergoing high-frequency deformation. The specific features of the breaking of carbon black aggregates defined by the deformation amplitude of loaded rubbers are described by the above author by a mathematical model developed for the description of the dynamic, visco-elastic behavior of polymer molecules. This approach revealed... 

At this voliime fraction, the viscosity diverges because the shear stress is now given by the particle-particle contact in the tightly packed structure. As a result, we obtain a fluid with visco-elastic properties similar to polymeric solids. In ceramic processing, we extrude and press these pastes into green shapes. As a result, the rheology of ceramic pastes is of importance. The rheology of very concentrated suspensions is not particularly well developed, with the exception of model systems of monodisperse spheres. This section first discusses visco-elastic fluids and second the visco-elastic properties of ceramic pastes of monodisperse spheres. The material on visco-elastic fluids draws heavily from the book Colloidal Dispersions by Russel, Saville, and Schowalter [31]. 

For viscoelastic fluids, the formalism of a viscous fluid and an elastic solid are mixed [31]. The equations for the effective viscosity, dynamic viscosity, and the creep compliance are given in Table 12.4 for a viscous fluid, an elastic solid, and a visco-elastic solid and fluid. For the viscoelastic fluid model the dynamic viscosity, >j (tu), and the elastic contribution, G (ti)), are plotted as a function of (w) in Figure 12.31. With one relaxation time, X, the breaks in the two curves occur at co. 

FIGURE 12J31 Shear modulus and d3mamic viscosity as a fiuiction of frequency. Mechanic behavior of a visco.elastic fluid and visco-elastic solid from models in Table 12.4. 

FIGURE 12.32 Shear moduli and dynamic viscosities measured for silica spheres at = 0.46, a = 28 2nm, O + a = 76 2nm(Mellemaetal. [68]). The broken lines correspond to the infinite shear viscosities (de Kruif et al. [43]) and the solid curves to the frequency dependence predicted by the visco-elastic fluid model of Table 12.4 with the measured values of 170,171 , and Gi. Redrawn from Russel et al. [31]. Reprinted with the permission of Cambridge University Press. 

The theory of relaxation spectra in polarized luminescence for various dynamic models of a flexible polymer chain has been developed by several groups of workers. Wahl has proposed a theory for the model of Gaussian subchains. The authors and coworkers used dynamic chain models consisting of rigid or deformable elements with continuous visco-elastic mechanism of mobility and rotational-isomeric lattice chain... 

Therefore, when Jenike developed his methods to mathematically model the flow of bulk solids, he concluded that a bulk solid must be modeled as a plastic, and not a visco-elastic, continuum of solid particles (1). This approach included the postulation of a flow-no-flow criterion that states the bulk solid would flow from a bin when the stresses applied to the bulk solid exceed the strength of the bulk solid. The terms stress and strength are further discussed in this section on cohesive strength tests below. The flow properties test methods discussed are used to obtain the equipment parameters required to provide consistent, reliable flow. 

The swelling and bending behaviour of hydrogels results from the equilibrium of different forces osmotic pressure forces, electrostatic forces, visco-elastic restoring forces, etc. To describe the different phenomena occurring in the gels and between the gel and solution phase adequately, the modelling can be performed on different scales (Fig. 3) ... 

A study of numerical modeling of rutting in flexible pavement foundations under cyclic loading was reported by Huang Yu et al., 2001. On the basis of a coupled dynamic soil-pore water system theory a visco-elastic dynamic model has been developed to predict the non-linear response of pavement structure. 

The consideration of thermal effects leads to the development of thermoplastic models. Furthermore also transient effects will play a role both in the barrier material and in the host rock under elevated temperature conditions. This requires an extension of the material models regarding visco-elasticity or visco-plasticity. 

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