The standard linear solid

The standard linear solid (SLS) is a more complicated model than the two previously considered. It combines series and parallel elements, as shown in fig. 7.9, and can describe both stress-relaxation and creep. For stress-relaxation the spring a remains at the original strain and only E), nd rj are involved in the relaxation. Hence r = r]/E, but the stress relaxes to eE, not to zero. For creep it can be shown that t = (l/ a + 1 Unlike the Voigt model, the SLS exhibits an immediate response, e = a/ E + E, because the two springs in parallel can extend immediately. Thus the SLS is a much better model than either of the simpler models. 

We have seen that the Maxwell model describes the stress relaxation of a viscoelastic solid to a first approximation, and the Kelvin model describes the creep behaviour, but that neither model is adequate for the general behaviour of a viscoelastic solid where it is necessary to describe both stress relaxation and creep. 

Consider again the general linear differential equation, which represents linear viscoelastic behaviour. From the present discussion, it follows that to obtain even an approximate description of both stress relaxation and creep, at least the first two terms on each side of Equation (5.9) must be retained, that is the simplest equation will be of the form 

This will be adequate to a first approximation for creep (when da/dr = 0) and for stress relaxation (when deldt = 0), giving an exponential response in both cases. 

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Another model consisting of elements in series and parallel is that attributed to Zener. It is known as the Standard Linear Solid and is illustrated in Fig. 2.41. The governing equation may be derived as follows. 

Stress, C7 Fig. 2.41 The standard linear solid Stress-Strain Relations As shown earlier the stress-strain relations are (2.44)... 

It may be observed that the governing equation of the standard linear solid has the form... 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

Here the term ik is the retardation time. It is given by the product of the compliance of the spring and the viscosity of the dashpot. If we examine this function we see that as t -> 0 the compliance tends to zero and hence the elastic modulus tends to infinity. Whilst it is philosophically possible to simulate a material with an infinite elastic modulus, for most situations it is not a realistic model. We must conclude that we need an additional term in a single Kelvin model to represent a typical material. We can achieve this by connecting an additional spring in series to our model with a compliance Jg. This is known from the polymer literature as the standard linear solid and Jg is the glassy compliance ... 

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 ... 

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. 

Since neither model adequately describes the behavior of real viscoelastic materials, a combination of the classic elements is often made to gain closer representation. The most common configuration is called the standard linear solid4 configuration, and it is illustrated in Figure 6.6. A more accurate representation of actual behavior can be obtained by a composite of multiple elements of the standard linear solid configuration into a multi-element model (Figure 6.7) with an array of coefficients for each element. 

Figure 6.6 Elements of the standard linear solid (SLS) and response curve.

Another model, attributed to Zener, consists of three elements connected in series and parallel, as illustrated in Figure 3.15, and known as the standard linear solid. Following the procedure already given, we derive the governing equation of this model ... 

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. 

Using springs and dashpots, draw the standard linear solid model. 

Voigt element n. This is a Voight model which is a component, together with other Voight or Maxwell components, of a more complex viscoelastic model system, such as the standard linear solid. 

To calculate the stress relaxation time for the standard linear solid model with a second spring added parallel to the initial spring, use... 

A response closer to that of a real polymer is obtained by adding a second spring of modulus in parallel with a Maxwell unit (Figure 4.12). This model is known as the standard linear solid and is usually attributed to Zener [2]. It provides an approximate representation to the observed behaviour of polymers in their viscoelastic range. In creep, both springs extend, so that... 

The models mentioned above are based on phenomenologieal quahty and are able to describe either relaxation or ereeping. To describe both effeets the models can be combined to the standard linear solid (SLS) model as shown in Fig. 10a. Consequently, the mechanical impedance of the SLS model Zsls results fi om serial and parallel connection of the single elements and is given by... 

The Maxwell and Voigt models are therefore inadequate to describe the dynamic mechanical behaviour of a polymer, as they do not provide an adequate representation of both the creep and stress relaxation behaviour. A good measure of qualitative improvement could be gained, as in the previous discussion of creep and stress relaxation, by using a three-parameter model, e.g. the standard linear solid, and it is an interesting exercise to show that this model gives a more realistic variation in Gi, G2, and tan 5 with frequency. 

This behaviour is shown by the standard linear solid of Figure 5.20(a). Consider in turn the behaviour in the initial unrelaxed state, and in the final relaxed state. 

Figure 5.20 The standard linear solid (a) gives a response in a stress relaxation test shown in...

While Smith s analysis above took the Maxwell model as its starting point, a useful alternative is to take the standard linear solid as the basis. This is the case in the study of polypropylene by Kitagawa, Mori and Matsutani [17] and of polyethylene by Kitagawa and Takagi [38]. The differential equation of the standard linear solid is given by Equation (5.18) ... 

We have shown (Section 5.2.7) tliat the standard linear solid, a three-component spring and dashpot model, provides to a first approximation a description of linear viscoelastic behaviour. Eyring and his colleagues [52] assumed that the deformation of a polymer was a thermally activated rate process involving the motion of segments of chain molecules over potential barriers, and modified the standard linear solid so that the movement of the dashpot was governed by the activated process. The model, which now represents non-linear viscoelastic behaviour, is useful because its parameters include an activation energy and... 

The rate of strain Equation (11.32) defines an activated viscosity, which is then incorporated in the dashpot of the standard linear solid model, and leads to a more complicated relationship between stress and strain than that for the linear model. The activated dashpot... 

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