Loading dashpot

For equilibrium of forces it can be seen that the applied load is supported jointly by the spring and the dashpot, so... 

When a load is applied to the system, shown diagrammatically, the spring will deform to a certain degree. The dashpot will first remain stationary under the applied load, but if the same load continues to be applied, the viscous fluid in the dashpot will slowly leak past the piston, causing the dashpot to move. Its movement corresponds to the strain or deformation of the plastic material. 

In a recovery test after all the load is removed at time / the creep is all recoverable except for the flow that occurred in the dashpot with viscosity %-... 

In Chapter 4, the response of these models to dynamic (i.e., sinusoidal) loads or strains is illustrated. In Chapter 5, the stress-strain response in constant rate experiments is described. Models with nonlinear springs and nonlinear dashpots (i.e., stress not proportional to strain or to strain rate)... 

Next to this case of creep under constant loading, we also consider the stress relaxation which occurs when the deformation is kept constant. At t = 0 the model is jumpwise deformed to a strain e. The instantaneous response is a stress <7o = Ee the spring is strained, the dashpot does not yet respond. The dashpot is, at t = 0, subjected to the same stress, so it starts flowing, while it takes over an increasing part of the imposed strain so that the strain in the spring, and also the stress, decrease. 

Stress relaxation is the decrease in load of a material held at a fixed displacement. Figure 15.22 shows the spring and dashpot in series that can be used to model the stress relaxation behavior. Using this model one... 

Fig. 14 Simplified Mason circuit (a) close to Fig. 13d. Since tan(fcqfiq) is large close to the resonance and, further, since this element is in parallel to the small load AZl, it may be neglected, b Close to resonance we have cot(fcqfiq 0) and the element - 2L4Zq cot(fcqfiq) can be approximated by a spring, a mass, and a dashpot. c Using the electromechanical analogy, the spring, the mass, and the dashpot may also be represented as a motional capacitance, Ci, a motional inductance, L, and a motional resistance, R ...

Loading with a Mass in Series with a Dashpot. 158... 

Fig. 2 Different circuits to be inserted for the load in Fig. 1. The conversion from the physical situation (right) to the equivalent circuits (left) entails a complication because networks are depicted such that the electrical Kirchhoff rules apply. Elements which are placed in series, physically, are represented as parallel circuit elements and vice versa (cf. Fig. 5 in Chap. 2 in this volume). For instance, the forces exerted by the spring and the dashpot in e are additive. In order to let the corresponding voltages in the electrical circuit also be additive, the circuit elements have to be placed in series. In the literature on polymer rheology, networks of springs and dashpots are drawn according to the physical situation (right-hand-side in this figure), which comes down to a different set of Kirchhoff rules...

The extension of the previous models to a sphere coupled to the plate via a spring and a dashpot is straightforward. The coupling can be achieved either via a Voigt-type circuit (viscoelastic solid, Fig. 2e) or via a Maxwell-type circuit (viscoelastic liquid, Fig. 2f). Below, we assume that the object is so heavy that it does not take part in the motion. When the mass is infinite, the inertial term drops out of the load impedance. An infinite mass is graphically depicted as a wall. For Voigt-type couphng we find ... 

Immediately the load is applied, the specimen elongates corresponding to an instantaneous elastic modulus. This is followed by a relatively fast rate of creep, which gradually decreases to a smaller constant creep rate. Typically this region of constant creep in thermoplastics essentially corresponds to viscous flow. In terms of the spring and dashpot model, the retardation is dominated by the viscous liquid in the dashpot. As before,... 

The Maxwell model consists of a spring and dashpot coimected in series (Figure 3.10a). When a load is applied, the elastic displacement of the spring occurs immediately and is followed by the viscous flow of liquid in the dashpot which requires time. After the load is removed, the elastic displacement is recovered immediately, but the viscous displacement is not recovered. 

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... 

If the standard linear solid (SLS) is unloaded from a constant stress, the spring (modulus ,) closes immediately and the elastic strain is removed. The anelastic strain then decays to zero as the second spring closes the dashpot, i.e., there is complete recovery. Under the action of a constant strain, the SLS model will also show stress relaxation but, in this case, the time constant, Tf =rf /(E +E2). In applying a constant stress to the SLS model, the strain can be considered to lag behind the stress, both on loading and unloading. This lag concept is also very important in considering the effect of a dynamic stress or strain. 

The experimental creep function is often analyzed as a nnm-ber of Kelvin elements in series (Figure 40.32), each having the property of a spring and dashpot in parallel (Genevaux, 1989 Martensson, 1992 Mohager and Toratti, 1993 Hanhijarvi, 1999 Passard and Perre, 2005). In the case of uniaxial load, this leads to... 

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