General Viscoelastic Correspondence Principle

Graham, G.A.C., Golden, J.M. (1988) The generalized partial correspondence principle in linear viscoelasticity. Q. Appl. Math, (to appear)... 

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... 

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... 

R. A. Schapery, Correspondence principles and a generalized J integral for large deformation and fracture analysis of viscoelastic media, Int. J. Tract. 25, 195-223 (1984). 

What is now known as the correspondence principle for converting viscoelastic problems in the time domain into elastic problems in the transform domain was first discussed by Turner Alfrey in 1944. As a result, the principle is sometimes referred to as Alfrey s correspondence principle. Later in 1950 and in 1955 the principle was generalized and discussed by W.T. Read and E. H. Lee respectively. (See bibliography for references.)... 

The fact that Eq. 8.3 can be considered as the equivalent of Hooke s law in the transform domain leads to a general method to solve many practical viscoelastic boundary value problems in a simple manner. This procedure is often attributed to Turner Alfrey and is sometimes referred to as Al-frey s correspondence principle. Simply stated the procedure is as follows ... 

Returning to the general expression Eq. 8.65, using the correspondence principle for a polymer beam that is viscoelastic, the solution in the transform domain will be,... 

The various approaches to the solution of viscoelastic boundary value problems discussed in the last chapter for bars and beams carry over to the solution of problems in two and three dimensions. In particular, if the solution to a similar problem for an elastic material already exists, the correspondence principle may be invoked and with the use of Laplace or Fourier transforms a solution can be found. Such solutions can be used with confidence but one must be cognizant of the general equations of elasticity and the methods of solutions for elasticity problems in two and three dimensions as well as any assumptions that might often be applied. To provide all of the necessary information and background for multidimensional elasticity theory is beyond the scope of this text but the procedures needed will be outlined in the following sections. 

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. 

However, formally speaking, non-commutative algebra presents no insuperable barriers and various standard results may readily be generalized to aging material. The oldest results provide a correspondence between non-inertial viscoelastic and appropriate elastic solutions for materials that have only one independent relaxation function. These results have been described by Arutyunyan (1952), Predeleanu (1965) and Bazant (1975) and are extensions of ideas due to McHenry (1943). They are generalizations to the aging case of results that would follow from the correspondence principle for materials with one relaxation function. 

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. 

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