Spring-dashpot models

Here t, is still a ratio of a viscosity to a modulus, as in the spring-dashpot model of Figure 1, but each sprint has the same (shear) modulus, pRTfM and the steady-flow viscosity T] of equation (16) is the sum of the viscosities of the individual submolecules. Molecular theories are discussed more fully in Section X. 

A 2D soft-sphere approach was first applied to gas-fluidized beds by Tsuji et al. (1993), where the linear spring-dashpot model—similar to the one presented by Cundall and Strack (1979) was employed. Xu and Yu (1997) independently developed a 2D model of a gas-fluidized bed. However in their simulations, a collision detection algorithm that is normally found in hard-sphere simulations was used to determine the first instant of contact precisely. Based on the model developed by Tsuji et al. (1993), Iwadate and Horio (1998) incorporated van der Waals forces to simulate fluidization of cohesive particles. Kafui et al. (2002) developed a DPM based on the theory of contact mechanics, thereby enabling the collision of the particles to be directly specified in terms of material properties such as friction, elasticity, elasto-plasticity, and auto-adhesion. 

The last three are used as models for viscoelastic materials, and are often presented in the form of spring-dashpot models which have the same constitutive equations. 

The master creep curve for the above data is generated by sliding the individual relaxation curves horizontally until they match with their neighbors, using a fixed scale for a hypothetical curve at 204°C. Since the curve does not exist for the desired temperature, we can interpolate between 208.6°C and 199.4°C. The resulting master curve is presented in Fig. 1.34. The amount each curve must be shifted from the master curve to its initial position is the shift factor, log (aT). The graph also shows the spring-dashpot models and the shift factor for a couple of temperatures. 

Typically, an accelerometer (or g-cell) consists of a proof mass that is suspended with a spring, or compliant beam, in the presence of some damping, and anchored to a fixed reference. Coupling this mass-spring-dashpot model (Fig. 7.1.2)... 

The time-temperature equivalence principle makes it possible to predict the viscoelastic properties of an amorphous polymer at one temperature from measurements made at other temperatures. The major effect of a temperature increase is to increase the rates of the various modes of retarded conformational elastic response, that is, to reduce the retarding viscosity values in the spring-dashpot model. This appears as a shift of the creep function along the log t scale to shorter times. A secondary effect of increasing temperature is to increase the elastic moduli slightly because an equilibrium conformational modulus tends to be proportional to the absolute temperature (13). 

The operator form is useful for setting up the differential equations for more complex spring-dashpot models. If a constant strain is applied to the Maxwell model, Eq. (5.44) becomes... 

Mechanical models consisting of combinations of springs and dashpots are very popular in numerous disciplines. Spring-dashpot models have been used to model the mechanical behavior of viscoelastic... 

In Eigures 4.15 and 4.16, it is possible to predict the behaviour of a combination of temperature and stress ratios by using spring-dashpot models that can accurately simulate separate but simpler combination of these constraints. 

When all is said and done, probably the best definition of a linear material is simply one that follows Boltzmann s principle. Thus, spring-dashpot models, which are linear, automatically follow Boltzmann s principle. However, it is important not to infer a dependence of the Boltzmann principle on spring-dashpot models. The Boltzmann principle applies to a hnear response regardless of whether it can be described with a spring-dashpot model. All that is needed are experimental GXt) or Jdt) data. Models are used here simply as a matter of convenience to illustrate application of the principle. 

In order to investigate the correlation between tensile drawing data and Reversed Charpy results, true stress vs. true strain properties of PE are measured at several strain rates and temperatures [6]. Each stress-strain curve is represented using a Haward-Thackray [7] spring-dashpot model, whose applicability to polyethylene has been established empirically by a number of previous studies [5, 7] although the physical basis for the model remains... 

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