Dashpot element

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

In this model the spring and dashpot elements are connected in parallel as shown in Fig. 2.36. 

The viscoelastic behaviour of a certain plastic is to be represented by spring and dashpot elements having constants of 2 GN/m and 90 GNs/m respectively. If a stress of 12 MN/m is applied for 100 seconds and then completely removed, compare the values of strain predicted by the Maxwell and Kelvin-Voigt models after (a) 50 seconds (b) 150 seconds. 

Figure H3.3.4 Mechanical models are often used to model the response of foods in creep or stress relaxation experiments. The models are combinations of elastic (spring) and viscous (dashpot) elements. The stiffness of each spring is represent by its compliance (J= strain/stress), and the viscosity of each dashpot is represent by a Newtonian viscosity (ri). The form of the arrangement is often named after the person who originally proposed the model. The model shown is called a Burgers model. Each element in the middle—i.e., a spring and dashpot arranged in parallel—is called a Kelvin-Voigt unit.

The mechanical analog of equation (3-2) is the dashpot element (Figure 3-1). This is merely a leaky piston in a cylinder filled with a liquid of viscosity 77 for shear or t)e for extension. (Remember, the details of geometry, etc. are not important we are looking only for qualitative behavior by analogy. Thus we will use the models of Figure 3-1 for both extensional and shear properties.) Integration of equation (3-2b) for constant shear stress cr0, yields... 

The mechanical response of viscoelastic bodies such as polymers is poorly represented by either the spring or the dashpot. J. C. Maxwell suggested that a better approximation would result from a series combination of the spring and dashpot elements. Such a model, called a Maxwell element, is shown on the right in Figure 3-1. In describing tensile response with the Maxwell element, E, the instantaneous tensile modulus, characterizes the response of the spring while rjE, the viscosity of the liquid in the dashpot, defines the viscous... 

In the Maxwell simple combination of mechanical elements, the stress a and the strain e are measured using subscript 1 for the spring element and subscript 2 for the viscous dashpot element. Given the way in which the stress is applied to the whole element, the stress on each element is the same, while the strain in each element is different ... 

The total strain, e, of the model under a given stress, a, is distributed between the spring and the dashpot elements ... 

In the Kelvin or Voigt model the spring and dashpot elements are connected in parallel, as shown in Figure 3.13a. This model roughly approximates the behavior of rubber. When the load is applied at zero time, the elastic deformation cannot occur immediately because the rate of flow is limited by the dashpot. Displacements continue until the strain equals the elastic deformation of the spring and it resists further movement. On removal of the load the spring recovers the displacement by reversing the... 

Hiis equation, rather than the model with siting and dashpot elements, may be regarded as the foundatxm of die theory. 

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. 

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... 

The simplest linear viscous model is Newton s model. This is shown by a piston-dashpot element [10]. The dashpot is an energy dissipation element, and it represents a viscous damping force. It relates the translational and rotational velocity of a fluid (oil) between two points, and an applied load, by using a damping constant. 

Fig. 8.7. Mechanical models of viscoelastic and viscoplastic materials, built as systems containing spring and dashpot elements...

For small stresses, we can use the approximation sinha w a in equation (8.3) so that the strain rate is proportional to the applied stress. In this case, the behaviour is linear and viscous. As stresses are small, the deformation is not plastic, but elastic, for there is a restoring force corresponding to the spring element in figure 8.7(a), whereas equation (8.3) describes the dashpot element of the Kelvin model. The behaviour is thus linear viscoelastic. At larger stresses, deviations from linearity occur, although the behaviour is still viscoelastic. 

At aU technically relevant temperatures, polymers deform by creep. To describe the time-dependence of plastic deformation, we again exploit equation (8.3). In contrast to the viscoelastic deformation, there is no restoring force in viscoplasticity. Equation (8.3) is thus used to describe the dashpot element connected in series in the four-parameter model from figure 8.7(b). 

As explained in section 8.2, the time-dependent behaviour of polymers can be described using spring and dashpot elements. The behaviour of a spring... 

The constant of integration C can be determined by the fact that the strain e is zero at time t = 0 because the dashpot element cannot react immediately to the stress. Thus, we find C = a and... 

The strain increases with time and approaches a value a/E because the dashpot element will have relaxed completely and aU of the stress is transferred by the spring. 

The strain in the dashpot element approaches a value of Eil Ei + E2) t large times. 

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