WIAS Preprint No. 1387, (2008)

Stochastic analysis of an elastic 3D half-space respond to random boundary displacements: Exact results and Karhunen--Loéve expansion


  • Shalimova, Irina
  • Sabelfeld, Karl

2010 Mathematics Subject Classification

  • 65C05 65C20 76S05

2008 Physics and Astronomy Classification Scheme

  • 02.60.Cb 02.70.Lq


  • Boundary white noise, Karhunen-Loève expansion, Poisson integral formula, boundary random excitations, 3D Lamé equation


A stochastic response of an elastic 3D half-space to random displacement excitations on the boundary plane is studied. We derive exact results for the case of white noise excitations which are then used to give convolution representations for the case of general finite correlation length fluctuations of displacements prescribed on the boundary. Solutions to this elasticity problem are random fields which appear to be horizontally homogeneous but inhomogeneous in the vertical direction. This enables us to construct explicitly the Karhunen-Loève (K-L) series expansion by solving the eigen-value problem for the correlation operator. Simulation results are presented and compared with the exact representations derived for the displacement correlation tensor. This paper is a complete 3D generalization of the 2D case study we presented in J. Stat. Physics, v.132 (2008), N6, 1071-1095.

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