Viscous material Maxwell model

The Standard Linear Solid Model combines the Maxwell Model and a like Hook spring in parallel. A viscous material is modeled as a spring and a dashpot in series with each other, both of which other, both of which are in parallel with a lone spring. For this model, the governing constitutive relation is ... 

Example 2.14 A plastic is subjected to the stress history shown in Fig. 2.45. The behaviour of the material may be assumed to be described by the Maxwell model in which the elastic component = 20 GN/m and the viscous component r) = 1000 GNs/m. Determine the strain in the material (a) after u seconds (b) after 1/2 seconds and (c) after 3 seconds. 

A plastic component was subjected to a series of step changes in stress as follows. An initial constant stress of 10 MN/m was applied for 1000 seconds at which time the stress level was increased to a constant level of 20 MN/m. After a further 1000 seconds the stress level was decreased to 5 MN/m which was maintained for 1000 seconds before the stress was increased to 25 MN/m for 1000 seconds after which the stress was completely removed. If the material may be represented by a Maxwell model in which the elastic constant = 1 GN/m and the viscous constant rj = 4000 GNs/m, calculate the strain 4500 seconds after the first stress was applied. 

Suppose the multiple Maxwell model which describes the material we are interested in is composed of m processes each with an elasticity Gj, a viscous process with a viscosity rjj and a corresponding relaxation time ty. We can form the relaxation function by adding all these models together ... 

The model represents a liquid (able to have irreversible deformations) with some additional reversible (elastic) deformations. If put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components as per the Maxwell Model. First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell model predicts that stress decays exponentially with time, which is accurate for most polymers. It is important to note limitations of such a model, as it is unable to predict creep in materials based on a simple dashpot and spring connected in series. The Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly with time. However, polymers for the most part show the strain rate to be decreasing with time [23-26],... 

As is well known, springs and dashpots represent, respectively, ideal elastic and viscous responses to step stress perturbations. In a similar way, a combination of the two can be used to describe the viscoelastic behavior of materials. The Maxwell model, a spring in series with a dashpot, is the more immediate idealization of this behavior (Fig. 10.1). 

Relaxation Time - Maxwell proposed a model in the 19 century to describe the time-dependent behavior of viscous materials such as pitch or tar. This model has also been applied to plastics and polymers. A parameter has been defined in this model called relaxation time that is a characteristic of the plastic material. Relaxation time is the ratio of viscosity to the Young s modulus of elasticity. 

The static tests considered in this chapter treat the rubber as being essentially an elastic, or rather, high elastic material, whereas it is in fact visco-elastic and hence its response to dynamic stressing is a combination of an elastic response and of a viscous response, so that energy is lost in each cycle. The Maxwell model represents this behavior as spring and dashpot in parallel. 

The Maxwell model. One of the first attempts to explain the mechanical behavior of matmals such as pitch and tar was made by James Clark Maxwell. He argued that when a material can undergo viscous flow and also respond elastically to a stress it should be described by a combination of both the Newton and Hooke laws. This assumes that both contributions to the strain are additive so that e= e ias, + e jsc-Expressing this as the dilfeimtial equation leads to the equation of motion of a Maxwell unit... 

Maxwell model (Maxwell element) n. A concept useful in modeling the deformation behavior of viscoelastic materials. It consists of an elastic spring in series with a viscous dashpot. When the ends are pulled apart with a definite force, the spring deflects instantaneously to its stretched position then motion is steady as the dashpot opens. A simple combination of these two types provides a fair analogic representation of real viscoelastic behavior under stress. 

However, various combinations of eiastic and viscous elements have been used to approximate the material behavior of polymer melts. Some models are combinations of springs and dashpots to represent the elastic and viscous responses, respectively. The most common ones being the Maxwell model for a polymer melt and the Kelvin or Voight model for a solid. One model that represents shear thinning behavior, normal stresses in shear flow and elastic behavior of certain polymer melts is the K-BKZ model [28-29]. 

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 ... 

Now, imagine deforming the Maxwell model by applying a constant strain to it at a time t = 0. The deformation is held constant and the stress is monitored. Figure 2 shows the mechanical response of the Maxwell model to an applied deformation. The first (early time) response is that the material responds only elastically because the viscous damper initially behaves rigidly (at infinite rate of strain). The total deformation of the element remains constant, but it redistributes itself between the spring and the dashpot. This results in stress relaxation that occurs exponentially with time ... 

Creep is related to plastics viscoelastic behavior and can be explained with the aid of a Maxwell model such as that shown in Figure 3-55 [12, 287]. 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. 

The relaxation period defines the behavior of the system, in accordance with the Maxwell model with respect to the timescale of the applied stress. If the time t during which stress is applied is greater than the relaxation period, that is, t > t the system has properties similar to those of a viscous liquid, while at t t the system behaves like an elastic solid. The flow of glaciers and other processes of strain development in mountains and cliffs are representative examples of such behavior. In rheology, the ratio of a material s characteristic relaxation time to the characteristic flow time is referred to as the Deborah number. This parameter plays an important role in describing the response of various materials to different stresses. 

Fig. 13.29. The Maxwell model for a viscoelastic material considers an elastic spring and a viscous dashpot connected in series.

This model was proposed in the 19th century by Maxwell to explain the time-dependent mechanical behaviour of viscous materials, such as tar or pitch. It consists of a spring and dashpot in series as shown in Fig. 5.7(a). Under the action of an overall stress cr there will be an overall strain e in the system which is given by... 

Through the dashpot a viscous contribution was present in both the Maxwell and Voigt models and is essential to the entire picture of viscoelasticity. These have been the viscosities of mechanical units which produce equivalent behavior to that shown by polymers. While they help us understand and describe observed behavior, they do not give us the actual viscosity of the material itself. 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

Another approach that has physical merit is to model the behavior of viscoelastic materials as a series of springs (elastic elements) and dashpots (viscous elements) either in series or parallel (see Figure 8.1). If the spring and dashpot are in series, which is described as a Maxwell mechanical element, the stress in the element is constant and independent of the time and the strain increases with time. 

The Maxwell element (elastic deformation plus flow), represented by a spring and a dashpot in series. It symbolises a material that can respond elastically to stress, but can also undergo viscous flow. The two contributions to the strain are additive in this model, whereas the stresses are equal ... 

Maxwell fltiid A constitutive model in which the response of the material to applied stress includes both an elastic and viscous response in series. In response to a constant applied force, the material will respond elastically at first, then flow. At fixed deformation, the stresses in the material will relax to zero. 

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