Spring and dash pot model

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... 

Fig. 14. Spring and dash-pot model for viscoelastic behaviour. Elastic and viscous element in scries.

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...

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. 

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... 

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

The behavior of linear spring and dash pot elements coupled in series (MaxweU model) can be characterized by the following functional relations, that are depicted graphically in Figure 2.12a. From the analogy to the in-series arrangement of capacities in electrical circuits (Figure 2.11a) it follows that ... 

Figure 2.12 Behavior of two-element models, or shear deformation /at constant stress (a) Maxwell model (spring and dash pot a or shear stress t (top), and tensile stress a...

Accordingly, the behavior of parallel coupUng of spring and dash pot elements (Voigt-Kelvin model, Figure 2.12b) is governed by ... 

The sample-tip-cantilever system can be modeled as a mechanical system with springs and dash-pots 11,12). Solving the motion equations of this model at low frequency (i.e. below the cantilever resonance frequency) and neglecting the damping constants (i.e. neglecting viscoelastic effects in polymers) leads to the following relation for the ratio between the sample modulation amplitude, z, and the tip response amplitude, also called the dynamic elastic response ... 

Based on the above consideration, simphfied models are deemed preferable for the determination of the gravity load distribution, since they are easy to calibrate and their reliability is not necessarily inferior to that of more refined models. Such a simplified physical model is shown in Fig. 11a. The model consists of springs and dash-pots to represent the instantaneous axial stiffness and the viscoelastic (creep) properties of the RC columns and the masonry infill. The gap Uq, shown in Fig. 11a, corresponds to the short-term deformation of the coluirms due to the gravity loads that are applied before the constmction of the infill walls. The spring-and-dashpot assemblages representing the concrete columns and masonry walls can be calibrated with data from... 

The multi-mode Maxwell model is not useful for modeling creep, and in its place the generalized Voigt model is often used. This generalized model consists of a group of Voigt elements (springs and dash pots in parallel) cormected in series as shown in Fig. 4.9. 

An analysis is made with a mechanical spring/dash pot model as shown in Figure 2.17, where a mass M is restrained by a spring of stiffness k and a dashpot with viscosity T). The solution as illustrated in Figure 2.17 is an exponentially decaying cosine wave, that is, the displacement of the mass is given by... 

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. ... 

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. 

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.

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. 

A number of viscoelastic (i.e., rheological) models have been proposed to model steady-state creep in soils. A selection of fom of these models is presented in Figure 8.47. A model incorporating spring constants, and E2 a slider element of resistance x and a dash-pot with viscosity, v, was proposed by Murayama and Shibata (1964), which is shown in Figure 8.47a. The time-dependent deformation is controlled by the slider element Xq. Deformation will only occur for applied stresses in excess of Xq. 

A convenient physical interpretation may be illustrated by simulating mechanical or electronic models. In the mechanical simulation, a spring represents an elastic or Hookean solid (modulus), while a piston moving in an infinite cylinder filled with a viscous liquid (a dash-pot) represents the Newtonian liquid (viscosity). Thus, the deformation of the solid (spring) is completely recoverable, while that of the liquid (dash-pot) is irrecoverable and is converted to heat. See Figures 4-7, 4-8, 4-9. In conclusion, the elastic energy is conserved and recovered while the viscous energy is dissipated. 

FIGURE 4-7 Mechanical models for solids (spring) and liquids (dash-pot)... 

Another viscoelastic model is derived, by combining a spring and a dash-pot in parallel. This is named after Voigt (or Kelvin) and shown in Figure 4-13. 

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