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Laplace Transform Solutions

  • Severino P. C. Marques
  • Guillermo J. Creus
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

Abstract

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.

Keywords

Constitutive Relation Radial Displacement Relaxation Function Correspondence Principle Laplace Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    E.J. Barbero, Finite Element Analysis of Composite Materials (CRC Press, Boca Raton, 2008)Google Scholar
  2. 2.
    E.H. Lee, Stress analysis in viscoelastic bodies. Q. Appl. Math. 13, 183–190 (1955)zbMATHGoogle Scholar
  3. 3.
    L.W. Morland, E.H. Lee, Stress analysis for linear viscoelastic materials with temperature variation. Trans. Soc. Rheol. 4, 223–263 (1960)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A.D. Myskis, Advanced Mathematics for Engineers (Mir Publishers, Moscow, 1979)Google Scholar
  5. 5.
    S.W. Park, R.A. Schapery, Methods of interconversion between linear viscoelastic material functions: part I—A numerical method based on Prony series. Int. J. Solids Struct. 36, 1653–1675 (1999)zbMATHCrossRefGoogle Scholar
  6. 6.
    K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University Press, New York, 2006)CrossRefGoogle Scholar
  7. 7.
    M.H. Sadd, Elasticity: Theory, Applications and Numerics (Elsevier Butterworth-Heinemann, Burlington, 2005)Google Scholar
  8. 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
  9. 9.
    R.A. Schapery, S.W. Park, Methods for interconversion between linear viscoelastic material functions: part II—an approximate analytical method. Int. J. Solids Struct. 36, 1677–1699 (1999)zbMATHCrossRefGoogle Scholar
  10. 10.
    M. Schanz, H. Antes, T. Rüberg, Convolution quadrature boundary element method for quasi-static visco- and poroelastic continua. Comput. Struct. 83, 673–684 (2005)CrossRefGoogle Scholar
  11. 11.
    C.R. Wylie, L.C. Barrett, Advanced Engineering Mathematics (McGraw-Hill, New York, 1995)Google Scholar
  12. 12.
    Y. Yeong-Moo, P. Sang-Hoon, Y. Sung-Kie, Asymptotic homogenization of viscoelastic composites with periodic microstructures. Int. J. Solids Struct. 35, 2039–2055 (1998)zbMATHCrossRefGoogle Scholar

Copyright information

© The Authors 2012

Authors and Affiliations

  1. 1.Centro de TecnologiaUniversidade Federal de AlagoasMaceióBrazil
  2. 2.ILEAUniversidade Federal do Rio Grande do SulPorto AlegreBrazil

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