State space solution of non-axisymmetric Biot consolidation problem for multilayered porous media

Abstract In this paper, the non-axisymmetric Biot consolidation problem for multilayered porous media is studied. Taking stresses, pore pressure and displacements at layer interfaces as basic unknown functions, two sets of partial differential equations, which are independent each other, are formulated. Using Fourier expansion, Laplace transforms and Hankel transforms with respect to the circumferential, time and radial coordinates, respectively, the partial differential equations presented are reduced to the ordinary differential equations. Transfer matrices describing the transfer relation between the state vectors for a finite layer are derived explicitly in the transform space. Using the transfer matrices presented, three cases are studied for the lower surface: (1) permeable rough rigid base, (2) impermeable rough rigid base, and (3) poroelastic half space. The explicit solution in the transform space is presented. Considering the continuity condition at layer interfaces, the solutions of the non-axisymmetric Biot consolidation problems for multilayered semi-infinite porous media are presented in the integral form. The time histories of displacements, stresses and pore pressure are obtained by solving a linear equation system for discrete values of Laplace–Hankel transform inversions.

[1]  J. R. Booker,et al.  The consolidation of a finite layer subject to surface loading , 1974 .

[2]  R. F. Rish,et al.  The analysis of cylindrical shell roofs with post tensioned edge beams , 1974 .

[3]  R. E. Gibson,et al.  PLANE STRAIN AND AXIALLY SYMMETRIC PROBLEMS OF THE CONSOLIDATION OF A SEMI-INFINITE CLAY STRATUM , 1960 .

[4]  R. Bellman.  Introduction To Matrix Analysis Second Edition , 1997 .

[5]  R. Rajapakse,et al.  Exact stiffness method for quasi-statics of a multi-layered poroelastic medium , 1995 .

[6]  Hans Bufler,et al.  Theory of elasticity of a multilayered medium , 1971 .

[7]  L. Bahar.  Transfer Matrix Approach to Layered Systems , 1972 .

[8]  Kenny S. Crump,et al.  Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation , 1976, J. ACM.

[9]  Jianguo Wang,et al.  The state vector methods of axisymmetric problems for multilayered anisotropic elastic system , 1999 .

[10]  J. C. Small,et al.  A method of computing the consolidation behaviour of layered soils using direct numerical inversion of Laplace transforms , 1987 .

[11]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[12]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[13]  I. Vardoulakis,et al.  Numerical Laplace‐Fourier transform inversion technique for layered‐soil consolidation problems: I. Fundamental solutions and validation , 1986 .

[14]  Jianguo Wang.  State vector solutions for nonaxisymmetric problem of multilayered half space piezoelectric medium , 1999 .