Dash-pot

We shall follow the same approach as the last section, starting with an examination of the predicted behavior of a Voigt model in a creep experiment. We should not be surprised to discover that the model oversimplifies the behavior of actual polymeric materials. We shall continue to use a shear experiment as the basis for discussion, although a creep experiment could be carried out in either a tension or shear mode. Again we begin by assuming that the Hookean spring in the model is characterized by a modulus G, and the Newtonian dash-pot by a viscosity 77. ... 

The Maxwell and Voigt models of the last two sections have been investigated in all sorts of combinations. For our purposes, it is sufficient that they provide us with a way of thinking about relaxation and creep experiments. Probably one of the reasons that the various combinations of springs and dash-pots have been so popular as a way of representing viscoelastic phenomena is the fact that simple and direct comparison is possible between mechanical and electrical networks, as shown in Table 3.3. In this parallel, the compliance of a spring is equivalent to the capacitance of a condenser and the viscosity of a dashpot is equivalent to the resistance of a resistor. The analogy is complete... 

Table 3.3 Comparison of Mechanical and Electrical Models Consisting of Different Arrangements of Springs and Dash-pots or Their Equivalents, Capacitance and Resistance, Respectively...

Real polymers require more elaborate systems of springs and dash-pots to describe them. This approach of polymer rheology can be developed to provide criteria for design with structural polymers. At present, this is rarely done instead, graphical data (showing the creep extension after time t at stress a and temperature T) are used to provide an estimate of the likely deformation during the life of the structure. 

A foreed-damped system is shown in Figure 5-17. This system has a mass M, whieh is suspended on a spring K with a spring eonstant and a dash pot to produee damping. The viseous damping eoeffieient is c... 

Stoss, m. impulse, thrust, push, blow, stroke impact, collision, percussion, shock, jolt, blimp recoil pile, heap, (of papers) file joint blast (of a horn), stossartig, a. jolting, jerky, intermittent. Stossbutter, /. farm butter, stossdlimpfend, a. shock-absorbing, cushioning. Stoss ddmpfer, m. shock absorber dash pot. -dauer, /. duration of collision, of impact, etc. (see Stoss). 

Tsuji et al. (1990) have modeled the flow of plastic pellets in the plug mode with discrete dynamics following the behavior of each particle. The use of a dash pot/spring arrangement to account for the friction was employed. Their results show remarkable agreement with the actual behavior of real systems. Figure 28 shows these flow patterns. Using models to account for turbulent gas-solid mixtures, Sinclair (1994) has developed a technique that could have promise for the dense phase transport. 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Dash pots are correctly charged time adjustments and levels are identified. 

The polyurethane can be considered to consist of two components, somewhat like the physics spring and dash pot model for viscous materials. The elastic component (spring) stores and returns the energy. The second or viscous area (pot) converts the retained energy into heat. This is an important property in the design and selection of polyurethanes. The design of the... 

The two modeling elements, spring and dash pot, are combined in various ways to demonstrate the deformation of a polymer subjected to the application of stress as shown in Fig. 14.19. 

Figure 14.19 Stress-strain plots for (a) a Hookean spring where E is the slope (6) a Newtonian dash pot where s is constant, (c) stresstime plot stress for relaxation in the Maxwell model, and (d) stresstime plot stress for a Voigt-Kelvin model.

We can generalize the analogy by considering the viscoelastic materials as a continuum where the theory of transmission lines can be applied. In this way, a continuous distribution of passive elements such as springs and dash-pots can be used to model the viscoelastic behavior of materials. Thus the relevant equations for a mechanical transmission line can be written following the same patterns as those in electrical transmission lines. By representing the impedance and admittance per unit of length by g and j respectively, one has... 

Dash pot. A type of buffer used in some recoil systems to cushion the movement of the recoiling parts as they near the end of recoil and again during return to battery Ref Glossary of Ord (1959), 88-R... 

After the stress has been removed (point D in Fig. 13A), the recovery phase follows a pattern mirroring the creep compliance curve to some degree First, there is some instantaneous elastic recovery (D-E return of spring 1 into its original shape Fig. 13A, B). Second, there is a retarded elastic recovery phase (E-F slow movement of the Kelvin unit into its original state Fig. 13A, B). However, during the Newtonian phase, links between the individual structural elements had been destroyed, and viscous deformation is non-recoverable. Hence, some deformation of the sample will remain this is in the mechanical model reflected in dash-pot 2, which remains extended (Fig. 13B). 

A single weightless Hookean, or ideal, elastic spring with a modulus of G and a simple Newtonian (fluid) dash pot or shock absorber having a liquid with a viscosity are convenient to use as models illustrating the deformation of an elastic solid and an ideal liquid. Because polymers are often viscoelastic solids, combinations of these models are used to demonstrate deformations resulting from the application of stress to an Isotropic solid polymer. 

Fig. 3.8. Maxwell (a) and Kelvin (b) model as linear or parallel combination of spring and dash-pot...

Newkirk (12) found that if the balance mechanism of the Chevenard thermobalance was not properly thermally shielded, the oil in the dash pots became warm, causing an apparent mass-gain due to the decreased buoyancy of the oil. In the latest model of this balance, the oil dash pots have been replaced by a magnetic damping device. 

Note that for very large values of N, die hyperbolic sine may be approximated by a single term, which reduces the model to a dash-pot . 

---