Spring and dashpot models

The calculation of the contact force between two particles is actually quite involved. A detailed model for accurately computing contact forces involves complicated contact mechanics (Johnson, 1985), the implementation of which is extremely cumbersome. Many simplified models have therefore been proposed, which use an approximate formulation of the interparticle contact force. The simplest one was originally proposed by Cundall and Strack (1979), where a linear-spring and dashpot model is employed to calculate the contact forces (see Fig. 11 and 12). In this model, the normal component of the contact force between two particles a and b can be calculated by... 

Figure 5.62 (a) Four-element spring and dashpot model of viscoelasticity and (b) resulting... 

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

Fig. 15.22. Spring and dashpot models for stress relaxation and creep.

This is the simplest way of applying the spring and dashpot model, but there are others of increasing complexity. For example, the Maxwell model considers the spring and dashpot to be in series, while the so-called standard linear solid has both parallel and series arrangements. While all of these approaches are mathematically useful, they do not have an underlying physical basis in reality there are no springs and no dashpots. However,... 

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,... 

Figure 2.11. Models for two-phase polymer systems. Elements in (a) and (b) form the basic parallel and series models. Combinations shown in (c) and (d) represent two other possible models. Note the similarity to the spring and dashpot models often invoked in explaining homopolymer behavior. (After Takayanagi et ai, 1963.) Part (a) represents an isostrain model, (b) represents an isostress model, and (c) and (d) illustrate combinations of these limiting cases.

The Kelvin model (also called as the Voigt model or the Kelvin-Voigt model) is a parallel connection of the spring and dashpot models (Kelvin, L. (Thompson, W.) 1875 Voigt 1892), representing the anealstic behavior, as given by... 

Adding spring and dashpot models in series and parallel creates viscoelastic models. Several models have been proposed. Figure 11.11 shows the creep behavior for four viscoelastic models. Stress relaxation is a similar phenomenon and is defined as a reduction in stress during a constant deformation. One example of stress relaxation is the use of plastic washers between a nut and bolt. After the screw is secured, the washer deformation is constant, but the stress in the washer diminishes with time (stress relaxation), and the screw is therefore more likely to loosen with time. 

One way to look at this is to consider a spring and dashpot model for a viscoelastic material. This model is illustrated by Fig. 4.7. The spring represents the elastic component G, whereas the dashpot represents the viscous component. The stress due to the spring is proportional to the strain through the proportionality constant G. On the other hand, the stress due to the viscous-fluid behavior of the dashpot is proportional to the strain rate. Thus, elastic-compoueut stress is proportioual to the siue of cot, aud the viscous compo-ueut is proportioual to the cosiue (derivative of siue) of cot. 

Spring-and-dashpot models are extended by the Voigt-Kelvin (V-K) model, which broadens linear viscoelastic concepts. The spring and dashpot are always in parallel. The V-K spring-and-dashpot models are useful for understanding creep behavior [11]. 

Total strain e in a Voigt-Kelvin spring-and-dashpot model [11] is... 

The three-element springs and dashpot model shown is subject to a creep experiment. Show how the length (or strain) increases with time. At time = t, the stress is removed. Show how the sample recovers. 

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