Elastic-viscoelastic correspondence principle

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/. 

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 ... 

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. 

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). 

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... 

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... 

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. 

Viscoelastic stress analysis of two component systems shows that a broadening of the dispersion zone is to be expected 166,167), even if the disperse phase (filler) is purely elastic 166) and it is not necessary to ascribe different molecular properties to the continuous phase. The simplest way to visualize this mechanical interaction is by the use of phenomenological mechanically equivalent models. The model of Takayanagi (/68) is illustrated in Fig. 16. The elastic solution for this model is easily derived from elementary considerations. By the correspondence principle of viscoelastic stress analysis 169), the viscoelastic solution is obtained simply by substituting complex moduli in place of purely elastic moduli... 

Concentrating initially on time dependence for infinitesimal strains, the correspondence principle relating viscoelastic and elastic behaviour, well established for isotropic systems, may be simply extended to apply to the anisotropic case. There is, however, a difficulty in showing that the compliance matrix Sy will necessarily have the same symmetry properties in the viscoelastic case as in the classically elastic case. This difficulty arises from the thermodynamic nature of part of the argument used in proving symmetry. In the viscoelastic case the proof would depend upon the less well established principles of irreversible thermodynamics. No discussion on this point will be attempted the symmetry properties of Sij as determined in elastic theory will be accepted and its validity examined in the light of the experimental data available. This data shows that there may be systematic deviations from the assumptions in work at finite strains and further work is needed in this area. However, the manner in which these deviations occur does not detract significantly from the utility of the simple formalism in many cases. 

There are many stress-analysis problems involving viscoelastic materials that are of a statically determinate class, i.e., the stresses in the body depend only on the applied forces and moments and not specifically on the elastic properties of the body. Such problems can be solved by invoking the correspondence principle. Then, the time and temperature dependences of the strains and flexures in the body can be obtained through the time temperature-shift properties of the viscoelastic polymer. 

The correspondence principle states that for problems of a statically determinate nature involving bodies of viscoelastic materials subjected to boundary forces and moments, which are applied initially and then held constant, the distribution of stresses in the body can be obtained from corresponding linear elastic solutions for the same body subjected to the same sets of boundary forces and moments. This is because the equations of equilibrium and compatibility that are satisfied by the linear elastic solution subject to the same force and moment boundary conditions of the viscoelastic body will also be satisfied by the linear viscoelastic body. Then the displacement field and the strains derivable from the stresses in the linear elastic body would correspond to the velocity field and strain rates in the linear viscoelastic body derivable from the same stresses. The actual displacements and strains in the linear viscoelastic body at any given time after the application of the forces and moments can then be obtained through the use of the shift properties of the relaxation moduli of the viscoelastic body. Below we furnish a simple example. 

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.)... 

However, as long as the material is linear, the correspondence principle can be used to obtain viscoelastic solutions from the appropriate elastic solution. It is well to note that such shear corrections are more important for polymeric materials than for metals as moduli are smaller and deformations are correspondingly larger. Therefore, shear corrections are typically more important. 

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. 

This chapter will focus on developing the equations, assumptions and procedures one must use to solve two and three dimensional viscoelastic boundary value problems. The problem of an elastic thick walled cylinder will be used as a vehicle to demonstrate how to obtain the solution of a more difficult reinforced viscoelastic thick walled cylinder. In the process, we first demonstrate how the elasticity solution is developed and then apply the correspondence principle to transform the solution to the viscoelastic domain. Several extensions to this problem will be discussed and additional practice is provided in the homework problems at the end of the chapter. 

Note that with Eqs. 9.10, 9.12 and 9.14, we again have the viscoelastic constitutive law represented in the transform domain in a form equivalent to elasticity. These relationships will then allow us to utilize the correspondence principle as in Chapter 8 to solve 2D and 3D viscoelastic boundary value problems based on elasticity solutions. 

Obviously, the above transformed governing equations for a linear viscoelastic material (Eqs. 9.33- 9.36) are of the same form as the governing equations for a linear elastic material (Eqs. 9.25 - 9.28) except they are in the transform domain. This observation leads to the correspondence principle for three dimensional stress analysis For a given a viscoelastic boundary value problem, replace all time dependent variables in all the governing equations by their Laplace transform and replace all material properties by s times their Laplace transform (recall, e.g., G (s) = sG(s)),... 

While it is beyond the scope of this introductory text to fully develop and solve a wide variety of multidimensional stress analysis problems in viscoelasticity, we provide here a classic example to illustrate the use of the correspondence principle to derive a viscoelastic solution from a practical problem in elasticity. We choose here the problem of a Thick Walled Cylinder, often referred to as the Lame Solution. In the following, we first generate the elasticity solution to the classic Lame problem, then extend this elasticity solution to that for a reinforced thick walled cylinder. Subsequently, we use the latter solution to develop the viscoelastic solution via the correspondence principle. 

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