Linear viscoelasticity 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. 

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 ... 

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. 

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. 

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... 

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. 

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

Figure 11.5 Comparison between experimental data for PTFE (15 vol% glass fiber) and prediotions from linear viscoelasticity theory.

There appears to be no rigorous theoretical scheme for describing anisotropy of creep behaviour in these materials. However, simple extensions of linear viscoelastic theory are presented and shown to be useful though not completely rigorous. Further development is clearly desirable. 

The mechanical spectroscopy is therefore primarily orientated to investigations involving small loads where no irreversible structural changes in the materials occur and the linear viscoelasticity theory is valid. 

The approach is based on a combination of the linear viscoelastic theory with elements of the elasticity theory developed by Erman, Floiy and Monnery in which the free energy of elasticity, AA i, is considered as a sum of the phantom component, AApj, and a component of steric entanglements determined by interlacing chains andjunctions. ... 

Linear viscoelasticity is an extension of linear elasticity and hyperelasticity that enables predictions of time dependence and viscoelastic flow. Linear viscoelasticity has been extensively studied both mathematically (Christensen 2003) and experimentally (Ward and Hadley 1993), and can be very useful when applied under the appropriate conditions. Linear viscoelasticity models are available in all major commercial FE packages and are relatively easy to use. The basic foundation of linear viscoelasticity theory is the Boltzmann s superposition principle, which states, "Every loading step makes an independent contribution to the final state."... 

Comparison between experimental data for GUR 1050 (30 kCy, Y-N2) and predictions from linear viscoelasticity theory with a two-term Prony series. 

Measured some viscoelastic properties of human radicular dentin under constant strain. Linear viscoelastic theory applied. Strengths unique examination of viscoelastic properties, defined orientation of dentinal tubules, storage conditions and testing environment well controlled. Limitations large scatter in Hi(t), mixed data for different teeth. 

The Phenomenology of the Linear Theory of Viscoelasticity. One of the powers of the linear viscoelasticity theory is that it is predictive. The constitutive law that comes from Boltzmann superposition theory requires simply that the material functions discussed above be known for a given material. Then, for an arbitrary stress or deformation history, the material response can be obtained. In addition, the elastic-viscoelastic correspondence principle can be used so that boundary value problems such as beam bending, for which an elastic solution exists, can be solved for linear viscoelastic materials as well. Both of these subjects are treated in this section. 

Boltzmann Superposition and the Constitutive Law for Linear Viscoelasticity. The underlying assumption of the Boltzmann superposition principle is that responses to loads or deformations applied to a material at different times are linearly additive. This set of assumptions leads to the constitutive laws of linear viscoelasticity theory which can be considered as a linear response theory. For discussion purposes, consider a Maxwell material that is subjected to a two-step deformation history. The history is such that a deformation yi = Ayi... 

Experiments can be carried out either in the frequency domain or in the time domain. In the first case, one obtains both storage, G (co), and loss, G"(co), moduli, while the second method gives the relaxation modulus, G(t). In any case, the linear viscoelasticity theory predicts that G (co), G"(o)) or G(t) can be described using a unique function the relaxation time spectrum, H(A) [1]. ff(A) is related to G(t) by... 

If the linear viscoelasticity theory can be applied, the initial forward stress-strain curve is given by equation 56 of Chapter 3 written for extension ... 

Graham, G.A.C., The Correspondence Principle of Linear Viscoelasticity Theory for Mixed Boundary Value Problems Involving Time-... 

In defining the constitutive relations for an elastic solid, we have assumed that the strains are small and that there are linear relationships between stress and strain. We now ask how the principle of linearity can be extended to materials where the deformations are time dependent. The basis of the discussion is the Boltzmann superposition principle. This states that in linear viscoelasticity effects are simply additive, as in classical elasticity, the difference being that in linear viscoelasticity it matters at which instant an effect is created. Although the application of stress may now cause a time-dependent deformation, it can still be assumed that each increment of stress makes an independent contribution. From the present discussion, it can be seen that the linear viscoelastic theory must also contain the additional assumption that the strains are small. In Chapter 11, we will deal with attempts to extend linear viscoelastic theory either to take into account non-linear effects at small strains or to deal with the situation at large strains. 

Smart, J. and Williams, J.G. (1972) A comparison of single-integral non-linear viscoelasticity theories. J. Mech. Phys. Solids, 20, 313. 

Morland, L. W. (1962) A plane problem of rolling contact in linear viscoelasticity theory. J. Appl. Mech. 29, 345-352... 

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