Elastic-viscoelastic correspondence principle linear viscoelasticity

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. 

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). 

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. 

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. 

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. 

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)),... 

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. 

The correspondence principle following the Boltzmann superposition principle allows the conversion of the common mechanical relationships of linear elasticity theory into linear viscoelasticity simply by replacing cr by time-dependent a t) and e by time-dependent e(t). Young s modulus E or the relaxation modulus Ej (f)= cr(f)/e is accordingly transformed to the creep modulus c(f) = cile t) orthe creep compliance/(f) = s(f)/(7,respectively. These time-dependent parameters can be determined from tensile creep and relaxation experiments. In compression or shear mode, the corresponding parameters of moduli are calculated in a similar manner. 

Equation (5.12) would be identical to the linear viscoelastic constitutive equation if the kernels Lq and L] were independent of f(5). Because f(C) is present in the equation, linear transforms cannot be utilized on the constitutive equation to demonstrate that the assumed linear elastic solution for the stresses was valid. If it can be shown that equation (5.12) will satisfy the equations of equilibrium and compatibility whenever the linear elastic solution is valid, a type of correspondence principle will have been developed similar to what has already been done in the theory of linear viscoelasticity. The validity of the elastic solution can be demonstrated by substituting the constitutive equation directly into the equations of equilibrium and compatibility. It should be immediately obvious that no complications can arise in such a procedure since the only spatially dependent quantities in the constitutive equation are the stresses and the strains ij. For the two dimensional problem only one equation of compatibility is present,... 

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. 

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