Viscoelasticity analytic functions

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. 

Several material properties exhibit a distinct change over the range of Tg. These properties can be classified into three major categories—thermodynamic quantities (i.e., enthalpy, heat capacity, volume, and thermal expansion coefficient), molecular dynamics quantities (i.e., rotational and translational mobility), and physicochemical properties (i.e., viscosity, viscoelastic proprieties, dielectric constant). Figure 34 schematically illustrates changes in selected material properties (free volume, thermal expansion coefficient, enthalpy, heat capacity, viscosity, and dielectric constant) as functions of temperature over the range of Tg. A number of analytical methods can be used to monitor these and other property changes and... 

The Dirac delta function clearly provides one form of spectra which has an analytical transform to the viscoelastic experimental regimes discussed so far. An often overlooked function was developed by Tobolsky6 and Smith.7 They noted that particular forms of the relaxation or retardation spectra have exact analytical transforms. These functions give well defined spectra and provide good fits to experimental data. The relaxation spectrum is defined by the function ... 

A summary of analytic expressions obtained in this manner for all the viscoelastic functions is presented in Table 4 and 5 for the linear and cubic arrays. The well-known phenomenological analogy (8) between dynamic compliance and dielectric permittivity allows the formal use of Eqs. (T 5), (T 6), and (T 11), (T 12) for the dielectric constant, e (co), and loss, e"(co), of the linear and cubic arrays, respectively (see Table 6). The derivations of these equations are elaborated in the next section and certain molecular weight trends are discussed. 

There are several analytical tools that provide methods of extrapolating test data. One of these tools is the Williams, Landel, Ferry (WLF) transformation.14 This method uses the principle that the work expended in deforming a flexible adhesive is a major component of the overall practical work of adhesion. The materials used as flexible adhesives are usually viscoelastic polymers. As such, the force of separation is highly dependent on their viscoelastic nature and is, therefore, rate- and temperature-dependent. Test data, taken as a function of rate and temperature, can be expressed in the form of master curves obtained by WLF transformation. This offers the possibility of studying adhesive behavior over a sufficient range of temperatures and rates for most practical applications. Fligh rates of strain may be simulated by testing at lower rates of strain and lower temperatures. 

The objectives of this test pattern is to analytically resolve these problems into three manageable segments. The first task will be to define the viscoelastic kinetic properties of a material as a function of various reaction temperatures. These properties (viscosity, viscous modulus, elastic modulus, tan delta) define the rate of change in the polymers overall reaction "character" as it will relate to article flow consolidation, phase separation particle distribution, bond line thickness and gas-liquid transport mechanics. These are the properties primarily responsible for consistent production behavior and structural properties. This test is also utilized as a quality assurance technique for incoming materials. The reaction rates are an excellent screening criteria to ensure the polymer system is "behaviorally" identical to its predecessor. The second objective is to allow modeling for effects of process variables. This will allow the material to undergo environmental... 

The linear viscoelastic behavior of the pure polymer and blends has already been described quantitatively by using models of molecular dynamics based on the reptation concept [12]. To describe the rheological behavior of the copolymers in this study, we have selected and extended the analytical approach of Be-nallal et al. [13], who describe the relaxation function G(t) of Hnear homopolymer melts as the sum of four independent relaxation processes [Eq. (1)]. Each term describes the relaxation domains extending from the lowest frequencies (Gc(t)) to the highest frequencies (Ghf( )), and is well defined for homopolymers in Ref [13]. 

Two test cases are used to validate the linear viscoelastic analysis capability implemented in the present finite-element program named NOVA. In the first case, the tensile creep strain in a single eight-noded quadrilateral element was computed for both the plane-stress and plane-strain cases using the program NOVA. The results were then compared to the analytical solution for the plane-strain case presented in Reference 49. A uniform uniaxial tensile load of 13.79 MPa was applied on the test specimen. A three-parameter solid model was used to represent the tensile compliance of the adhesive. The Poisson s ratio was assumed to remain constant with time. The following time-dependent functions were used in Reference 49 to represent the tensile compliance for FM-73M at 72 °C ... 

A further set of tests was conducted in order to evaluate the accuracy of the finite-element code for the case where creep is followed by creep recovery. A qualitative depiction of the loading and the resulting creep strain is given in Figure 11. Rochefort and Brinson O) presented experimental data and analytical predictions on the creep and creep recovery characteristics of FM-73 adhesive at constant temperature. The Schapery parameters necessary to characterize the viscoelastic response of FM-73 at a fixed temperature of 30 °C are obtained by applying a least-squares curve fit to the data presented in Reference 50. The resulting analytical expressions for the creep compliance function D(i ), the shift function and the nonlinear parameters go> 82 presented in Table 4. 

Generally, the approximation methods have an analytical foundation based oh the properties of the integrands of the corresponding exact equations. Such an integrand is usually the product of the viscoelastic function initially known and an... 

Empirical analytical expressions have sometimes been used to represent viscoelastic functions in the transition zone, especially the relaxation modulus 0(/). 

The effects of a number of environmental factors on viscoelastic material properties can be represented by a time shift and thus a shift factor. In Chapter 10, a time shift associated with stress nonlinearities, or a time-stress-superposition-principle (TSSP), is discussed in detail both from an analytical and an experimental point of view. A time scale shift associated with moisture (or a time-moisture-superposition-principle) is also discussed briefly in Chapter 10. Further, a time scale shift associated with several environmental variables simultaneously leading to a time scale shift surface is briefly mentioned. Other examples of possible time scale shifts associated with physical and chemical aging are discussed in a later section in this chapter. These cases where the shift factor relationships are known enables the constitutive law to be written similar to Eq. 7.53 with effective times defined as in Eq. 7.54 but with new shift factor functions. This approach is quite powerful and enables long-term predictions of viscoelastic response in changing environments. 

Our object is to apply (2.8.9) to problems involving loads on viscoelastic halfspaces. For such problems, it is desirable to re-express these equations in an alternative form [Muskhelishvili (1963)] which facilitates reduction to a Hilbert problem. Let the material occupy the upper half-plane > >0 so that (piz, t) is analytic in this region. It is convenient to extend the region of analyticity of (p(z, t) to the lower half-plane also. Then, as we shall see, it is possible to explore the discontinuities in this function across the real axis, which gives a Hilbert problem. Another approach, possibly more direct, is that of Galin, mentioned previously, which leads to problems of the Riemann-Hilbert type. These however are somewhat more difficult to deal with, from a mathematical point of view. 

Rapp, N. V., Romas ko, V.S (1972) Analytical form of the dynamic functions of the theory of linear viscoelasticity. Sov. Phys. Dokl. 16, 784-786... 

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