The Viscoelastic Correspondence Principle

As an example, consider here the deflection of a beam under arbitrary loading. For small deformations, the deflection 8 at some point is a product of three functions  

By use of the viscoelastic correspondence principle, the linear viscoelastic solution for a polymer can be immediately written as 

The student should realize the importance of viscoelastic correspondence principle. All of the mechanics of materials solutions for linear elastic materials have a corresponding solution for linear viscoelastic materials New solutions do not need to be derived. Only the transformation (l/ ) /(f) needs to be made. 

What if the load on the cantilever beam changes with time Let P = Pit) and use the hereditary integral formulation. Thus, 

An aluminum cantilever beam is replaced by a polymer beam of the same dimensions. The maximum defection of the aluminum beam was 0.01 in. What will be deflection of the polymer beam after 2 s if J t) = 0.5(10 ) psi when r = 2s  

---

Using the viscoelastic correspondence principle (Section 3.8), valid for small deflections on which Equation 4.6 is based, the buckling load for a viscoelastic column can be calculated by replacing (l/ ) by 7(t), thus... 

A theory of thermoviscoelasticity that includes the temperature dependence of the relaxation or retardation functions is necessarily nonlinear, and consequently the elastic-viscoelastic correspondence principle is not applicable. Nevertheless, a linear theory of thermoviscoelasticity can be developed in the framework of rational thermodynamics with further constitutive assumptions (Ref. 5, Chap. 3 see also Ref. 10). 

Equations (17.20) are Laplace transforms of the equations of viscoelastic beams and can be considered a direct consequence of the elastic-viscoelastic correspondence principle. The second, third, and fourth derivatives of the deflection, respectively, determine the forces moment, the shear stresses, and the external forces per unit length. The sign on the right-hand side of Eqs. (17.20) depends on the sense in which the direction of the strain is taken. 

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. 

And integration of the hereditary integrals for strain and deflection gives the solution to any applied history of the moment M t). A note of caution, however, arises for mixed conditions in which the interface between the stress and the displacement boundaries is not constant. In such cases the elastic-viscoelastic correspondence principle is not applicable and the solutions become more difficult (21). 

Having evaluated the frequency response function F(v) of the deflection of the undamped plate, light damping is built in by an alternative quadrature type of the elastic-viscoelastic correspondence principle, /4/, /5/. 

Equations (1.8.19), (2.1.1) are formally identical to the equations governing the elastic problem, in FT form, where the regions B and the specified functions C/(r, t), rf/(r, t) are the same but where the elastic moduli are replaced by the complex moduli /i y /(cu). If solutions to the elastic problem are known, one can obtain a solution of the corresponding viscoelastic problem by replacing the moduli by the complex moduli in the expressions for the displacements and stresses and calculating the inverse transforms. This observation is the content of the Classical Correspondence Principle, which allows us to use the vast catalogue of known elastic solutions to generate viscoelastic solutions. 

The Classical Correspondence Principle was enunciated in reasonably general form by Read (1950) and Lee (1955) among others and discussed rigorously by Sternberg (1964) for the more general non-isothermal case. Sternberg (1964) reviews the older literature in some detail. Tao (1966) discusses correspondences between elastic and viscoelastic inertial problems in terms of Laplace transforms, essentially generalizing the work of Lee (1955). 

The conclusion is also valid for viscoelastic bodies - if the non-inertial approximation applies. This follows immediately by invoking the Classical Correspondence Principle. Our object in this section is to generalize the result to the case of two viscoelastic bodies in contact. 

I. Viscoelastic Solutions in Terms of Elastic Solutions. The fundamental result is the Classical Correspondence Principle. It is based on the observation that the time Fourier transform (FT) of the governing equations of Linear Viscoelasticity may be obtained by replacing elastic constants by corresponding complex moduli in the FT of the elastic field equations. It follows that, whenever those regions over which different types of boundary conditions are specified do not vary with time, viscoelastic solutions may be generated in terms of elastic solutions that satisfy the same boundary conditions. In practical terms this method is largely restricted to the non-inertial case, since then a wide variety of elastic solutions are available and transform inversion is possible. 

We will write down the displacement-traction relationship on the boundary that will form the basis of the considerations of this chapter. This is essentially the solution of the stress boundary value problem, discussed in Sect. 3.2 in the plane case. We shall neglect surface shear, however, so that the required relationship is a generalization to Viscoelasticity of the classical Boussinesq relationship. Its form follows directly from the elastic result by invoking the Classical Correspondence Principle. A more explicit derivation may be found in Hunter (1961) and also Golden (1978), who includes a shear traction term. Letting... 

In chapter 1, the properties of the viscoelastic functions are explored in some detail. Also the boundary value problems of interest are stated. In chapter 2, the Classical Correspondence Principle and its generalizations are discussed. Then, general techniques, based on these, are developed for solving non-inertial isothermal problems. A method for handling non-isothermal problems is also discussed and in chapter 6 an illustrative example of its application is given. Chapter 3 and 4 are devoted to plane isothermal contact and crack problems, respectively. They utilize the general techniques of chapter 2. The viscoelastic Hertz problem and its application to impact problems are discussed in chapter 5. Finally in chapter 7, inertial problems are considered. 

Recently, Matadi Boumbimba et al. [12] proposed a temperature- and frequency-dependent version of the rule of mixtures to describe the viscoelastic response, in terms of storage modulus, of PMMA/Cloisite 20A and SOB. In the present work, to predict the effective viscoelastic response of polymer-based nanocomposites, the elastic-viscoelastic correspondence principle [11] is applied to our micromechanical model. The two implicit equations (5) become ... 

In all three equations Ex and E > are now the complex moduli the storage and loss moduli for the blend are obtained by direct substitution into these equations and separation of the real and imaginary parts to obtain separate mixture rules for each. Analytical expressions have been obtained for these, but they are lengthy and cumbersome. All the calculations described, therefore, were carried out by computer. The substitution of complex moduli into the solution of the equivalent purely elastic problem is justified by the correspondence principle of viscoelastic stress analysis (6). 

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

The generalized stress-strain relationships in linear viscoelasticity can be obtained directly from the generalized Hooke s law, described by Eqs. (4.85) and (4.118), by using the so-called correspondence principle. This principle establishes that if an elastic solution to a stress analysis is known, the corresponding viscoelastic (complex plane) solution can be obtained by substituting for the elastic quantities the -multiplied Laplace transforms (8 p. 509). The appUcation of this principle to Eq. (4.85) gives... 

Horizontal shifts of the isotherms obtained in aging processes combined with suitable vertical shifts give master curves that permit prediction of the viscoelastic behavior of aged systems over a wide interval of time. The timeaging time correspondence principle for poly(vinyl chloride) (26) is shown in Figure 12.23. The retardation times in these creep experiments are related to the aging time, ta, by means of the expression... 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

Until now, we have considered only elastic beams. To generalize the elastic results to the viscoelastic case is relatively easy. Actually, the correspondence principle (5) indicates that if E tends to E " then G approaches G, where the asterisk indicates a complex magnitude. Then, according to Eqs. (17.75) and (17.78), we can write... 

According to the correspondence principle, the equation describing a viscoelastic beam under transversal and longitudinal effects is given by... 

Let us take as the origin of coordinates the middle point of the bar, which will be assumed fixed. It is more convenient to start by solving the elastic case and then consider the viscoelastic one by making use of the correspondence principle. 

---