Viscoelastic theory

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 basic viscoelastic theory assumes a timewise linear relationship between stress and strain. Based on this assumption and using mechanical models thought to represent the behavior of a plastic material, it can be shown that the stress, at any time t, in a plastic held at a constant strain (relaxation test), is given by ... 

Note that the term y in Eqs. 2-15 and 2-16 has a different significance than that in Eq. 2-14. In the first equation it is based on a concept of relaxation and in the others on the basis of creep. In the literature, these terms are respectively referred to as a relaxation time and a retardation time, leading for infinite elements in the deformation models to complex quantities known as relaxation and retardation functions. One of the principal accomplishments of viscoelastic theory is the correlation of these quantities analytically so that creep deformation can be predicted from relaxation data and relaxation data from creep deformation data. 

Using viscoelastic theory, it is possible to demonstrate that ... 

Pipkin AC (1986) Lectures on viscoelasticity theory, 2nd edn. Springer, Berlin Heidelberg, New York... 

There are many types of deformation and forces that can be applied to material. One of the foundations of viscoelastic theory is the Boltzmann Superposition Principle. This principle is based on the assumption that the effects of a series of applied stresses acting on a sample results in a strain which is related to the sum of the stresses. The same argument applies to the application of a strain. For example we could apply an instantaneous stress to a body and maintain that stress constant. For a viscoelastic material the strain will increase with time. The ratio of the strain to the stress defines the compliance of the body ... 

This is an extended exponential. It operates within the remit of linear viscoelastic theory. So for example for a simple exponential we can show that the integral under the relaxation function gives the low shear viscosity ... 

This section is primarily concerned with the behaviour of simple homo-polymers. The development of viscoelastic theory was intimately linked with the study of polymeric species. This area of activity has led the way in the development of rheological models and experimental design and so is a very important area for the proto-rheologist to understand. So far in this chapter we have taken the approach of developing phase diagrams from a rheological perspective in order to understand linear viscoelastic... 

The mechanical response of polypropylene foam was studied over a wide range of strain rates and the linear and non-linear viscoelastic behaviour was analysed. The material was tested in creep and dynamic mechanical experiments and a correlation between strain rate effects and viscoelastic properties of the foam was obtained using viscoelasticity theory and separating strain and time effects. A scheme for the prediction of the stress-strain curve at any strain rate was developed in which a strain rate-dependent scaling factor was introduced. An energy absorption diagram was constructed. 14 refs. 

We need to consider this question of scale more broadly when we wish to apply elasticity (or viscoelasticity) theory to real materials. Consider the following solids ... 

Finally, it is instructive to compare the temperature effect on the tensile strength of the SBS and SIS block polymers. As noted previously (Figure 6) the tensile strength of an elastomer vulcanizate can be related to the difference between the test temperature and the Tg of the elastomer, in accordance with the viscoelastic theory of tensile strength. Since the Tg values for polyisoprene ( — 65°C) and polybutadiene ( —95°C) differ... 

A general method of applying viscoelasticity theory to unstable (changing) materials in varying temperature fields was proposed in a number of publications (see, for example Ref.135). In this approach, the state of a material is represented by the factor jr, which is a function of a set of "structural" parameters... 

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. 

D. C. Bogue, An Explicit Constitutive Equation Based on an Integrated Strain History, Ind. Eng. Chem. Fundam., 5, 253-259 (1966) also I. Chen and D. C. Bogue, Time-Dependent Stress in Polymer Melts and Review of Viscoelastic Theory, Trans. Soc. Rheol., 16, 59-78 (1972). 

Ham, J. S. Viscoelastic theory of branched and cross-linked polymers. J. Chem. Phys. 26, 625-633 (1957). 

The ability to correctly reproduce the viscosity dependence of the dephasing is a major accomplishment for the viscoelastic theory. Its significance can be judged by comparison to the viscosity predictions of other theories. As already pointed out (Section II.C 22), existing theories invoking repulsive interactions severely misrepresent the viscosity dependence at high viscosity. In Schweizer-Chandler theory, there is an implicit viscosity dependence that is not unreasonable on first impression. The frequency correlation time is determined by the diffusion constant D, which can be estimated from the viscosity and molecular diameter a by the Stokes-Einstein relation ... 

In pure liquids, short-range repulsive forces are responsible for most of the dephasing. The viscoelastic theory describes the interaction of these forces with the diffusive dynamics of the liquid (Section IV.D). The resulting frequency modulation is in the fast limit in low-viscosity liquids but can reach the slow-modulation limit at higher viscosities. This type of dephasing was seen in supercooled toluene (Section IV.C). 

Linear Viscoelasticity Theory. FTMA is based on linear viscoelasticity theory. A one dimensional form of constitutive equation for linear viscoelastic materials which are isotropic, homogeneous, and hereditary (non-aging) is given by (21) ... 

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. 

M. Berg, "Comparison of a Viscoelastic Theory of Solvation Dynamics to Tune-Resolved Experiments in a Nonpolar Solution, Chem. Phys. Lett, in press. 

There is much more to tell about the rheology of filled melts, but space limitations preclude further discussion here. The interested reader is directed to the articles by Khan and Prud homme (1987), Metzner (1985), amd White (1982), the book by Han (1981), and references therein. Viscoelastic theories for filled melts, especially for rubbers containing carbon black, can be found in Montes and White (1993), Witten et al. (1993), and references therein. 

AC Pipkin. Lectures on Viscoelasticity Theory. New York Springer-Verlag, 1972. 

Linear viscoelasticity theory predicts that one component of a complex viscoelastic function can be obtained from the other one by means of the Kronig-Kramers relations (10-12). For example, the substitution of G t) — Ge given by Eq. (6.8b) into Eq. (6.3) leads to the relationship... 

Camera-Roda G and Sarti GC. Non-Fickian mass transport through polymers A viscoelastic theory. Transp. Theory Statist. Phys. 1986 15 1023-1031. 

Before proceeding, we remark that according to phenomenological viscoelasticity theory, the viscosity rj can be written as the sum of products of shear rigidities and relaxation times Tj (7), one term for each relaxation process contributing to viscous flow ... 

Pbticolas, W. L. Introduction to the molecular viscoelastic theory of polymers and its application. Rubber Chem. Technol. 36, 1422 (1963). 

Viscoelastic constitutive equations are used to model material properties. Viscoelastic theory combines the elements of elasticity and Newtonian fluids. The theory of viscoelasticity was developed to describe the behavior of materials which show intermediate behavior between solids and fluids. 

This expression when substituted into Eq. (7.25) will give for small-amplitude oscillatory motion exactly the expression for >7 in Eq. (7.17). Linear viscoelasticity theory cannot, however, predict the normal stress behavior. 

Pipkin, A. C., Lectures on Viscoelasticity Theory, Applied Mathematical Sciences, Vol. 7, Springer-Verlag, New York, 1972. 

The basic foundation of linear viscoelasticity theory is the Boltzmann s superposition principle which states ... 

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