Spring-dashpot

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. 

Fig. 11. Graphical representation of the linear spring-dashpot soft-sphere model. From Hoo-mans, Ph.D. thesis, University of Twente (2000).

We can follow a similar procedure for the tangential spring-dashpot system. So, the tangential damping coefficient is determined by... 

To perform simulations of relatively large systems for relatively long times, it is essential to optimize the computational strategy of discrete particle simulations. Obviously, the larger the time step 5t, the more efficient the simulation method. For the soft-sphere model, the maximum value for 5t is dictated by the duration of a contact. Since there are two different spring-dashpot systems in our current model, it is essential to assume that tcontact>n — tcontacUU so that... 

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. 

Stress relaxation master curve. For the poly-a-methylstyrene stress relaxation data in Fig. 1.33 [8], create a master creep curve at Tg (204°C). Identify the glassy, rubbery, viscous and viscoelastic regions of the master curve. Identify each region with a spring-dashpot diagram. Develop a plot of the shift factor, log (ax) versus T, used to create your master curve log (ot) is the horizontal distance that the curve at temperature T was slid to coincide with the master curve. What is the relaxation time of the polymer at the glass transition temperature ... 

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. 

Constitutive equations of the Maxwell-Wiechert tjq)e have received a lot of attention as far as their ability to describe the linear viscoelastic behaviour of pol3maer melts is concerned. From a phenomenological point of view [1-4], these equations can be easily understood and derived using the multiple spring-dashpot mechanical analogy leading to the linear equation ... 

Clearly other kinds of dumbbells can be imagined spring-dashpot combinations, inverse Langevin-spring connectors, and bending-spring connectors have in fact been used by various investigators. 

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)... 

Figure 3-1. Spring, dashpot, and Maxwell model. In subsequent depictions of the dashpot, the "fluid" will be removed for simplicity.

Figures 3-3 illustrate the behavior of the Maxwell model in a stress-relaxation experiment. Note that these functions have been plotted on log scales as well as the usual rectilinear scales. At times considerably shorter than the relaxation time of the spring/dashpot combination, the element behaves as if it were a spring alone. At times very long compared to the relaxation time of the dashpot, the model behaves as if it were a dashpot alone, that is, the stress...

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 result is re-expressed in terms of the resonant frequency coq = k/m of the undamped system, and the characteristic damping time r = dk of the spring-dashpot combination, giving... 

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... 

By inspection of Figure 4.14(a) we see the first term of eqn 4.50 to be the time-independent deflection of the spring the second term is the time-dependent deflection of the parallel-spring dashpot element. 

This viscosity might, of course, in its turn again be represented by a series of little spring dashpot systems of the type shown in Volume I, Chapter I, all having a very short relaxation time. 

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... 

Rheological Models One-dimensional constitutive models for viscoelasticity based on spring, dashpot, and spring-pot elements. The elements maybe cormected in series or in parallel. In models where the elements are cormected in series the strain is additive while fire stress is equal in each element. In parallel cormections, the stress is additive while the strain is equal in each element. 

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