Newtonian dashpot element

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. 

The Maxwell model integrates the elastic and viscous behavior of a thermoplastic by combining the spring and dashpot, providing a simple model for viscoelastic polymers. The spring and dashpot are in series [10], 

T] = viscosity of newtonian fluid (oil) in dashpot, Pa s Total strain e = strain in the spring + strain in the dashpot 

The Voigt model spring and dashpot are parallel. The model is a conventional concept for understanding stress and strain relationships when load is applied to a viscoelastic material [10]. 

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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.

A simple way to illustrate the viscoelastic properties of materials subjected to small deformations is to evaluate the stress that results from combining a linear spring that obeys Hooke s law and a simple fluid that obeys Newton s law of viscosity. An example of such combination is the mathematical representation of the Maxwell element. Even though this model is inadequate for quantitative correlation of polymer properties, it illustrates the quahtative nature of real behavior. Furthermore, it can be generahzed by the concept of a distribution of relaxation times so that it becomes adequate for quantitative evaluation. Maxwell s element is a simple one combining one viscous parameter and one elastic parameter. Mechanically, it can be visualized as a Hookean spring and a Newtonian dashpot in series ... 

By means of these models the strong dependency of the stress-strain behavior of polymers on strain, strain rate and temperature can be represented. Most of the published works deal with linear spring-dashpot systems. However, by introducing a conventional Maxwell element, which is represented by a spring and a Newtonian dashpot in series, the range of non-linear viscoelasticity cannot be covered. Therefore, Schmachtenberg proposed in the early 1980s to use a flow law for the dashpots ... 

Suppose we consider a spring and dashpot connected in series as shown in Fig. 3. 7a such an arrangement is called a Maxwell element. The spring displays a Hookean elastic response and is characterized by a modulus G. The dashpot displays Newtonian behavior with a viscosity 77. These parameters (superscript ) characterize the model whether they have any relationship to the... 

Maxwell element or model Model in which an ideal spring and dashpot are connected in series used to study the stress relaxation of polymers, modulus Stress per unit strain measure of the stiffness of a polymer, newtonian fluid Fluid whose viscosity is proportional to the applied viscosity gradient. 

The ideal viscous element can be represented by a dashpot filled with a Newtonian fluid, whose deformation is linear with time while the stress is applied, and is completely irrecoverable (Newton element). In a dynamic mechanical experiment the stress is exactly 90° out of phase with the strain [Pg.412]

In the second element, the strain is analogous to the viscosity of the material within the dashpot and is time dependent. The stress/strain relationship that exists is o = r /t. This is a non-Newtonian behaviour. 

The Maxwell Model. In the above development, discussion moves from elastic behavior to viscoelastic descriptions of material behavior. In a simple sense, viscoelasticity is the behavior exhibited by a material that has both viscous and elastic elements in its response to a deformation or load. In early days, this was often represented by elastic or viscous mechanical elements combined in different ways (9-12). The simplest models are two element models that contain a viscous element (dashpot) and an elastic element (spring). The dashpot is assumed to follow a Newtonian fluid constitutive law in which the stress is related directly to the strain rate by the following expression ... 

Similarly, linear viscous response can be modelled using a dashpot. A plunger moving through a very viscous Newtonian liquid physically represents this. The response of this element is described mathematically by the equation... 

When this model is subjected to a constant stress, the response includes an instantaneous elastic strain caused by spring 1, retarded elastic strain by the Kelvin component, viscous flow by dashpot 1, instantaneous elastic strain on unloading from spring 1, retarded strain recovery from the Kelvin element and permanent deformation in dashpot 1. The multiparameter model response is shown in Figure 4.13. This model can be described as the combined response of a Hookean elastic element, a Kelvin retarded-elastic solid and a Newtonian viscous fluid. 

Materials, and, in particular, polymers, that show viscoelastic behaviour, can be modelled by a combination of perfectly elastic Hookean springs and Newtonian viscous dashpots. For many polymers the behaviour at temperatures above when strains are small 1 per cent) is approximately represented by the so-called standard linear substance. This consists of a dashpot and spring in series (called a Maxwell element) and this combination in parallel with a second spring of different elastic modulus. In deformation, the strain of the Maxwell element and of the spring will be the same, say e. At a time t let Oi be the stress in the Maxwell element and 02 that in the spring. Then, if Ei is the Young s modulus of the Maxwell element and E2 that of the spring ... 

The spring element is considered to behave according to Hooke s law in a linear elastic manner while the mechanical behavior of the dashpot equates linear viscous flow of a Newtonian fluid. Following the models of, for example. Lord Kelvin and Woldemar Voigt... 

It may be shown that any combination of linear elements must be linear, so any models based on these linear elements, no matter how complex, can represent only linear response. Just how realistic is linear response Its most conspicuous shortcoming is that it permits only Newtonian behavior (constant viscosity) in equlibrium viscous flow. For most polymers at strains greater than a few percent or so (or rates of strain greater than 0.1 s" -), linear response is not a good quantitative description. Moreover, even within the limit of linear viscoelasticity, a fairly large number of linear elements (springs and dashpots) are usually... 

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