DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media.

*(English)*Zbl 1398.74099Summary: Based on the Porous Media Theory presented by de Boer, the governing differential equations for a layered space-axisymmetrical fluid-saturated porous elastic body are firstly established, in which the suitable interface conditions between layers are presented. Then, a differential quadrature element method (DQEM) is developed, and the DQEM and the second-order backward difference scheme are applied to discretize the governing differential equations of the problem in the spatial and temporal domain, respectively. In order to show the validity of the present analysis, the dynamic response of a fluid-saturated porous medium is analyzed, and the obtained numerical results are directly compared with the existing analytical results. The effects of the numbers of the elements and grid points on the convergence of the numerical results are considered. Finally, the dynamic characteristics of a layered fluid-saturated elastic soil cylinder subjected to a water pressure or a dynamic loading are studied, and the effects of material parameters are considered in detail. From the above numerical results, it can be found that the DQEM has advantages, such as little amount in computation, good stability and convergence as well as high accuracy, so it is a very efficient method for solving the problems in soil mechanics, especially such problems with discontinuities.

##### MSC:

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

76S05 | Flows in porous media; filtration; seepage |

Full Text:
DOI

##### References:

[1] | Biot M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941) · JFM 67.0837.01 |

[2] | Bowen R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20, 697–735 (1982) · Zbl 0484.76102 |

[3] | de Boer R.: Theory of Porous Media: Highlights in the Historical Development and Current State. Springer, Berlin (2000) · Zbl 0945.74001 |

[4] | de Boer R.: Theoretical poroelasticity–a new approach. Chaos Soliton Fract. 25, 861–878 (2005) · Zbl 1071.74019 |

[5] | Schrefler B.A.: Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solution. Appl. Mech. Rev. 55, 351–388 (2002) |

[6] | Placidi L., Dell’lsola F., Ianiro N., Sciarra G.: Variational formulation of pre-stressed solid-fluid mixture theory, with an application to wave phenomena. Eur. J. Mech. A-Solid. 27, 582–606 (2008) · Zbl 1146.74012 |

[7] | Schanz M.: Poroelastodynamics: linear models, analytical solutions, and numerical methods. Appl. Mech. Rev. 62, 030803-1–030803-15 (2009) |

[8] | Ghaboussi J., Wilson E.L.: Variational formulation of dynamics of fluid-saturated porous elastic solids. J. Eng. Mech. Div. 98, 947–963 (1972) |

[9] | Zeinkiewicz O.C., Shiomi T.: Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution. Int. J. Numer. Anal. Met. Geomech. 8, 71–96 (1984) · Zbl 0526.73099 |

[10] | Zienkiewicz O.C., Chan A.H.C., Pastor M., Paul D.K., Shiomi T.: Static and dynamic behaviour of soils: a rational approach to quantitative solutions. I. Fully saturated problems. Proc. R. Soc. Lond. A 429, 285–309 (1990) · Zbl 0725.73074 |

[11] | Chen J.: Time domain fundamental solution to Biot’s complete equations of dynamic poroelasticity. Part I: two dimensional solution. Int. J. Solid Struct. 31, 1447–1490 (1994) · Zbl 0945.74669 |

[12] | Chen J.: Time domain fundamental solution to Biot’s complete equations of dynamic poroelasticity. Part II: three dimensional solution. Int. J. Solid Struct. 31, 169–202 (1994) · Zbl 0816.73001 |

[13] | Soares D. Jr, Telles J.C.F., Mansur W.J.: A time-domain boundary element formulation for the dynamic analysis of non-linear porous media. Eng. Anal. Bound. Elem. 30, 363–370 (2006) · Zbl 1187.74248 |

[14] | Khalili N., Yazdchi M., Valliappan S.: Wave propagation analysis of two-phase saturated porous media using coupled finite–infinite element method. Soil Dyn. Earthq. Eng. 18, 533–553 (1999) |

[15] | Karim M.R., Nogami T., Wang J.G.: Analysis of transient response of saturated porous elastic soil under cyclic loading using element-free Galerkin method. Int. J. Solids Struct. 39, 6011–6033 (2002) · Zbl 1032.74585 |

[16] | Bowen R.M., Reinicke K.M.: Plane progressive waves in a binary mixture of linear elastic materials. J. Appl. Mech. 45, 493–499 (1978) · Zbl 0399.73037 |

[17] | Bowen R.M.: Plane progressive waves in a heat conducting fluid saturated porous material with relaxing porosity. Acta Mech. 46, 189–206 (1983) · Zbl 0517.76093 |

[18] | de Boer R., Ehlers W., Liu Z.: One-dimensional transient wave propagation in fluid-saturated incompressible porous media. Arch. Appl. Mech. 63, 59–72 (1993) · Zbl 0767.73013 |

[19] | Breuer S.: Quasi-static and dynamic behaviors of saturated porous media with incompressible constituents. Transp. Porous Med. 34, 285–303 (1999) |

[20] | Breuer S., Jagering S.: Numerical calculation of the elastic and plastic behavior of saturated porous media. Int. J. Solid Struct. 36, 4821–4840 (1999) · Zbl 0935.74024 |

[21] | Yang X., Cheng C.J.: Gurtin variational principle and finite element simulation for dynamical problems of fluid-saturated porous media. Acta Mech. Solida Sin. 16, 24–32 (2003) |

[22] | Hu Y.J., Zhu Y.Y., Cheng C.J.: DQM for dynamic response of fluid-saturated visco-elastic porous media. Int. J. Solids Struct. 46, 1667–1675 (2009) · Zbl 1217.74144 |

[23] | Bellman R., Casti J.: Differential quadrature and long-term integration. J. Math. Anal. Appl. 34, 235–238 (1971) · Zbl 0236.65020 |

[24] | Bellmam R., Kashef B.G., Casti J.: Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys. 10, 40–52 (1972) · Zbl 0247.65061 |

[25] | Bert C.W., Malik M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1–28 (1996) |

[26] | Striz A.G., Chen W.L., Bert C.W.: Free vibration of plates by the high accuracy quadrature element method. J. Sound Vib. 202, 689–702 (1997) · Zbl 1235.74124 |

[27] | Wang X.W., Gu H.Z.: Static analysis of frame structures by the differential quadrature element method. Int. J. Numer. Meth. Eng. 40, 759–772 (1997) · Zbl 0888.73078 |

[28] | Liu F.L., Liew K.M.: Static analysis of Reissner–Mindlin plates by differential quadrature element method. J. Appl. Mech. 65, 705–710 (1998) |

[29] | Hu Y.J., Zhu Y.Y., Cheng C.J.: DQEM for large deformation analysis of structures with discontinuity conditions and initial displacements. Eng. Struct. 30, 1473–1487 (2008) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.