Alfreys Correspondence Principle

It is possible using transform methods to convert viscoelastic problems into elastic problems in the transformed domain, allowing the wealth of elasticity solutions to be utilized to solve viscoelastic boundary value problems. Although there are restrictions on the applicability of this technique for certain types of boundary conditions (discussed further in Chapter 9), the method is quite powerful and can be introduced here by building on the framework provided by mechanical models. Recall the differential equation for a generalized Maxwell or Kelvin model, 

See the Appendix B for fundamentals on the Laplace transform. Since the transformed stress and transformed strain are no longer part of the summations, the expression may be further rewritten as 

The reader is cautioned that Eq. 5.29 must be used with care in order to include aU initial conditions properly. Significant differences arise depending upon whether the time begins at 0+ or 0-. In most circumstances used herein, f(t) = 0 for t 0 but in creep or relaxation the jump discontinuity at t = 0 must be included. 

The quotient of operators can be thought of as an elastic modulus in transform space and the above equation can be written as. 

This result of the same form as Hooke s law for a linear elastic material 

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Which type of boundary value problem cannot be solved using the standard (or Alfrey) correspondence principle ... 

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.)... 

Explain, describe and/or derive the rationale behind Alfrey s correspondence principle. 

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 ... 

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