Revisiting displacement functions in three-dimensional elasticity of inhomogeneous media

M. Kashtalyan, J. J. Rushchitsky

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

The paper presents a three-dimensional solution to the equilibrium equations for linear elastic transversely isotropic inhomogeneous media. We assume that the material has constant Poisson’s ratios, and its Young’s and shear moduli have the same functional form of dependence on the co-ordinate normal to the plane of isotropy. We show, apparently for the first time, that stresses and displacements in such an inhomogeneous transversely isotropic elastic solid can be represented in terms of two displacement functions which satisfy the second- and fourth-order partial differential equations. We examine and discuss key aspects of the new representation; they include the relationship between the new displacement functions and Plevako’s solution for isotropic inhomogeneous material with constant Poisson’s ratio as well as the application of the new representation to some important classes of three-dimensional elasticity problems. As an example, the displacement function is derived that can be used to determine stresses and displacements in an inhomogeneous transversely isotropic half-space which is subjected to a concentrated force normal to a free surface and applied at the origin (Boussinesq’s problem).
Original languageEnglish
Pages (from-to)3463-3470
Number of pages8
JournalInternational Journal of Solids and Structures
Volume46
Issue number18-19
DOIs
Publication statusPublished - Sep 2009

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Inhomogeneous Media
Elasticity
elastic properties
Poisson ratio
Transversely Isotropic
Three-dimensional
Elastic moduli
Poisson's Ratio
Partial differential equations
equilibrium equations
Elasticity Problem
isotropic media
Isotropy
isotropy
half spaces
Free Surface
partial differential equations
Half-space
Fourth Order
Modulus

Keywords

  • elasticity
  • linear elasticity
  • three-dimensional elasticity solution
  • analytical solutions
  • functionally graded materials
  • transversely isotropic material

Cite this

Revisiting displacement functions in three-dimensional elasticity of inhomogeneous media. / Kashtalyan, M.; Rushchitsky, J. J.

In: International Journal of Solids and Structures, Vol. 46, No. 18-19, 09.2009, p. 3463-3470.

Research output: Contribution to journalArticle

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AB - The paper presents a three-dimensional solution to the equilibrium equations for linear elastic transversely isotropic inhomogeneous media. We assume that the material has constant Poisson’s ratios, and its Young’s and shear moduli have the same functional form of dependence on the co-ordinate normal to the plane of isotropy. We show, apparently for the first time, that stresses and displacements in such an inhomogeneous transversely isotropic elastic solid can be represented in terms of two displacement functions which satisfy the second- and fourth-order partial differential equations. We examine and discuss key aspects of the new representation; they include the relationship between the new displacement functions and Plevako’s solution for isotropic inhomogeneous material with constant Poisson’s ratio as well as the application of the new representation to some important classes of three-dimensional elasticity problems. As an example, the displacement function is derived that can be used to determine stresses and displacements in an inhomogeneous transversely isotropic half-space which is subjected to a concentrated force normal to a free surface and applied at the origin (Boussinesq’s problem).

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