Laplace Transform Solutions
Laplace Transform is a useful tool in solving important problems in different areas of science and engineering. Usually, it is employed to convert differential or integral equations into algebraic equations, simplifying the problem solutions. Particularly, in linear nonageing viscoelasticity, interesting applications have been found for Laplace transform techniques. Many computational solutions are also based on the use of Laplace transforms. As already mentioned, an important task in viscoelasticity consists of determining relations between the different constitutive viscoelasticity functions of a material. In this chapter, we show procedures based on Laplace transforms that allow us to obtain relaxation function given the corresponding creep function, or vice versa. Also, we show equivalence conditions between the integral and differential representations of the constitutive viscoelastic relations. In many practical situations, we know the creep function, which is evaluated in uniaxial tension or compression tests, and we need to determine the viscoelastic constitutive functions for multiaxial states of stress or strain. This problem is also focused in the present chapter. Finally, using the similarity between the mathematical formulations of the linear elastic and linear viscoelastic mechanical problems in the Laplace domain, the Correspondence Principle is stated and applied.
KeywordsConstitutive Relation Radial Displacement Relaxation Function Correspondence Principle Laplace Domain
- 1.E.J. Barbero, Finite Element Analysis of Composite Materials (CRC Press, Boca Raton, 2008)Google Scholar
- 4.A.D. Myskis, Advanced Mathematics for Engineers (Mir Publishers, Moscow, 1979)Google Scholar
- 7.M.H. Sadd, Elasticity: Theory, Applications and Numerics (Elsevier Butterworth-Heinemann, Burlington, 2005)Google Scholar
- 8.R.A. Schapery, Approximate methods of transform inversion for viscoelastic stress analysis. Proceedings of 4th U.S. National Congress of Applied Mechanics, ASME, 1962, p. 1075Google Scholar
- 11.C.R. Wylie, L.C. Barrett, Advanced Engineering Mathematics (McGraw-Hill, New York, 1995)Google Scholar