Linear viscoelasticity applicability

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

The theories of elastic and viscoelastic materials can be obtained as particular cases of the theory of materials with memory. This theory enables the description of many important mechanical phenomena, such as elastic instability and phenomena accompanying wave propagation. The applicability of the methods of the third approach is, on the other hand, limited to linear problems. It does not seem likely that further generalization to nonlinear problems is possible within the framework of the assumptions of this approach. The results obtained concern problems of linear viscoelasticity. 

Since these double-base proplnts consist essentially of a single phase which bears the total load in any application of force, their mechanical property behavior is significantly different from composite proplnts. In the latter formulations, the hydrocarbon binder comprises only about 14% of the composite structure, the remainder being solid particles. Under stress, the binder of these proplnts bears a proportionately higher load than that in the single phase double-base proplnts. At small strain levels, these proplnts behave in a linear viscoelastic manner where the solids reinforce the binder. As strain increases, the bond between the oxidizer and binder breaks down... 

Firstly, it helps to provide a cross-check on whether the response of the material is linear or can be treated as such. Sometimes a material is so fragile that it is not possible to apply a low enough strain or stress to obtain a linear response. However, it is also possible to find non-linear responses with a stress/strain relationship that will allow satisfactory application of some of the basic features of linear viscoelasticity. Comparison between the transformed data and the experiment will indicate the validity of the application of linear models. 

The constant Tr is called the Trouton ratio10 and has a value of 3 in this experiment with an incompressible fluid in the linear viscoelastic limit. The elongational behaviour of fluids is probably the most significant of the non-shear parameters, because many complex fluids in practical applications are forced to extend and deform. Studying this parameter is an area of great interest for theoreticians and experimentalists. 

The application of finite strains and stresses leads to a very wide range of responses. We have seen in Chapters 4 and 5 well-developed linear viscoelastic models, which were particularly important in the area of colloids and polymers, where unifying features are readily achievable in a manner not available to atomic fluids or solids. In Chapter 1 we introduced the Peclet number ... 

The application of a shear rate to a linear viscoelastic liquid will cause the material to flow. The same will happen to a pseudoplastic material and to a plastic material once the yield stress has been exceeded. The stress that would result from the application of the shear rate would not necessarily be achieved instantaneously. The molecules or particles will undergo spatial rearrangements in an attempt to follow the applied flow field. 

Note 5 If a mass m is attached to the specimen at the point of application of the force, the linear-viscoelastic interpretation of the resulting deformation gives... 

The Eyring analysis does not explicity take chain structures into account, so its molecular picture is not obviously applicable to polymer systems. It also does not appear to predict normal stress differences in shear flow. Consequently, the mechanism of shear-rate dependence and the physical interpretation of the characteristic time t0 are unclear, as are their relationships to molecular structure and to cooperative configurational relaxation as reflected by the linear viscoelastic behavior. At the present time it is uncertain whether the agreement with experiment is simply fortuitous, or whether it signifies some kind of underlying unity in the shear rate dependence of concentrated systems of identical particles, regardless of their structure and the mechanism of interaction. 

Linear Viscoelasticity is usually applicable only for small deformations [2,3,5, 23-25],... 

The analysis viscoelasticity performed by David Roylance [25] is a nice outline about the mechanical response of polymer materials. This author consider that viscoelastic response is often used as a probe in polymer science, since it is sensitive to the material s chemistry and microstructure [25], While not all polymers are viscoelastic to any practical extent, even fewer are linearly viscoelastic [24,25], this theory provide a usable engineering approximation for many applications in polymer and composites engineering. Even in instances requiring more elaborate treatments, the linear viscoelastic theory is a useful starting point. 

Apply the Boltzmann superposition principle for the case of a continuous stress application on a linear viscoelastic material to obtain the resulting strain y(t) in terms of J(t — t ) and ih/dt, the stress history. Consider the applied stress in terms of small applied At,-, as shown on the accompanying figure. 

With all these models, the simple ones as well as the spectra, it has to be supposed that stress and strain are, at any time, proportional, so that the relaxation function E(t) and the creep function D(t) are independent of the levels of deformation and stress, respectively. When this is the case, we have linear viscoelastic behaviour. Then the so-called superposition principle holds, as formulated by Boltzmann. This describes the effect of changes in external conditions of a viscoelastic system at different points in time. Such a change may be the application of a stress or also an imposed deformation. 

Schwarzl (1970) studied the errors to be expected in the application of this type of equations, starting from the theory of linear viscoelasticity. Flis results are given schematically in Fig. 13.59. For non-linear viscoelastic behaviour, the exactitude of the approximate equations cannot be predicted. 

The engineering property that is of interest for most of these applications, the modulus of elasticity, is the ratio of unit stress to corresponding unit strain in tension, compression, or shear. For rigid engineering materials, unique values are characteristic over the useful stress and temperature ranges of the material. This is not true of natural and synthetic rubbers. In particular, for sinusoidal deformations at small strains under essentially isothermal conditions, elastomers approximate a linear viscoelastic... 

Linear viscoelastic behavior is actually observed with polymers only in very restricted circumstances involving homogeneous, isotropic, amorphous specimens subjected to small strains at temperatures near or above Tg and under test conditions that are far removed from those in which the sample may be broken. Linear viscoelasticity theory is of limited use in predicting service behavior of polymeric articles, because such applications often involve large strains, anisotropic objects, fracture phenomena, and other effects which result in nonlinear behavior. The theory is nevertheless valuable as a reference frame for a wide range of applications, just as the thermodynamic equations for ideal solutions help organize the observed behavior of real solutions. 

The experimental ranges of strain rates (or strains) are summarized in Table 2 for the various types of experiments. Time-temperatiire superposition was successfully applied on the various steady shear flow and transient shear flow data. The shift factors were foimd to be exactly the same as those obtained for the dynamic data in the linear viscoelastic domain. Moreover, these were found to be also applicable in the case of entrance pressure losses leading to an implicit appUcation to elongational values. 

At small stresses and strains, glassy PC exhibits linear viscoelastic behavior. The limit of applicability of the theory of linear viscoelasticity has been investigated by Yannas et al. over the temperature range 23 °C-130 °C. The critical strain at which, within the precision of their measurement, deviations from the linear theory occur has been found to diminish from about 1.2% at 23 °C to about 0.7 % at 130 °C. According to Jansson and Yannas the transition from linear to nonlinear viscoelastic behavior is marked by the onset of significant rotation around backbone bonds. 

In addition, other measurement techniques in the linear viscoelastic range, such as stress relaxation, as well as static tests that determine the modulus are also useful to characterize gels. For food applications, tests that deal with failure, such as the dynamic stress/strain sweep to detect the critical properties at structure failure, the torsional gelometer, and the vane yield stress test that encompasses both small and large strains are very useful. 

It is now easy to understand why the viscoelastic behavior of filled vulcanizates at large strains is so complex and why it is so difficult to investigate in definitive manner. None of the processes responsible for stress-softening occur instantaneously (16,182), nor can they generally be expected to obey the time-temperature relationships of linear viscoelasticity. Since time-temperature superposition is no longer applicable,... 

Any of equations (2-45), (2-46), (2-49), or (2-50) is sufficient as a statement of the Boltzmann superposition principle for linear viscoelastic response of a material. Often in particular applications, however, it is more convenient to use one form than another. All can be extended to three dimensions by using the same forms with the strains given by equation (2-18). Thus, for example, equation (2-46) becomes ... 

The accurate applicability of linear viscoelasticity is limited to certain restricted situations amorphous polymers, temperatures near or above the glass temperature, homogeneous, isotropic materials, small strains, and absence of mechanical failure phenomena. Thus, the theory of linear viscoelasticity is of limited direct applicability to the problems encounted in the fabrication and end use of polymeric materials (since most of these problems involve either large strains, crystalline polymers, amorphous polymers in a glass state failure phenomena, or some combination of these disqualifying features). Even so, linear viscoelasticity is a most important subject in polymer materials science—directly applicable in a minority of practical problems, but indirectly useful (as a point of reference) in a much wider range of problems. 

The theory behind linear viscoelasticity is simple and appealing. It is important to realize, however, that the applicability of the model for fluoropolymers is restricted to strains below the yield strain. One example comparing predictions based on linear viscoelasticity and experimental data for PTFE with 15 vol% glass fiber in the very small strain regime is shown in Fig. 11.4. 

Tn many practical applications of viscoelastic materials under cyclic straining conditions the strain amplitude which the material experiences in the applications is too large to allow the assumption of linear viscoelasticity. For example, the tire cord experiences a strain amplitude of about 1% or more (1) while the tire for a passenger car runs on the road under a normal condition. Under the strain amplitude of this magnitude viscoelastic behavior of the material deviates from linearity significantly, and therefore analysis of the viscoelasticity must consider the nonlinearity. 

Our discussion of the viscoelastic properties of polymers is restricted to the linear viscoelastic behavior of solid polymers. The term linear refers to the mechanical response in whieh the ratio of the overall stress to strain is a function of time only and is independent of the magnitudes of the stress or strain (i.e., independent of stress or strain history). At the onset we concede that linear viscoelastie behavior is observed with polymers only under limited conditions involving homogeneous, isotropie, amorphous samples under small strains and at temperatures close to or above the Tg. In addition, test conditions must preclude those that ean result in specimen rupture. Nevertheless, the theory of linear viseoelastieity, in spite of its limited use in predicting service performance of polymeric articles, provides a useful reference point for many applications. 

Just as real materials may have behaviour close to ideal yielding behaviour but with some features similar to those of viscoelastic materials, materials that exhibit behaviour close to the ideal linear viscoelastic may exhibit features similar to those of yield, particularly for high stresses and long times of application. If the term e i) in equation (7.4) is not zero, a linear viscoelastic material will not reach a limiting strain on application of a fixed stress at long times the strain will simply increase linearly with time. The material may also depart from linearity at high or even moderate stresses, so that higher stresses produce disproportionately more strain. These are features characteristic of yield. 

These observations suggest that, for real materials, there need not be a very clear-cut distinction between yield and creep. In fact polymers present a complete spectrum of behaviours between the two ideal types. Fortunately, however, many polymers under conditions of temperature and strain-rate that are within the ranges important for applications do have behaviours close to one of these ideals. It is therefore useful both from a practical and from a theoretical point of view to try to understand these approximately ideal behaviours before attempting to study the more complicated behaviours exhibited by other materials. In chapter 7 this approach is used in discussing creep and linear viscoelasticity and in the present chapter it is used in discussing yield. 

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