M. Schanz Institute of Applied Mechanics, Technical University Braunschweig, D-38023 Braunschweig, Germany A. H.-D. Cheng Department of Civil and Environmental Engineering, University of Delaware, Newark, DE 19716 Dynamic Analysis of a One-Dimensional Poroviscoelastic Column The response due to a dynamic loading of a poroviscoelastic one-dimensional column is treated analytically. Biot’s theory of poroelasticity is generalized to poroviscoelasticity using the elastic-viscoelastic correspondence principle in the Laplace domain. Damping effects of the solid skeletal structure and the solid material itself are taken into account. The fluid is modeled as in the original Biot’s theory without any viscoelastic effects. The solution of the governing set of two coupled differential equations known from the purely poroelastic case is converted to the poroviscoelastic solution using the developed elastic- viscoelastic correspondence in Laplace domain. The time-dependent response of the col- umn is achieved by the ‘‘Convolution Quadrature Method’’ proposed by Lubich. Some interesting effects of viscoelasticity on the response of the column caused by a stress, pressure, and displacement loading are studied. DOI: 10.1115/1.1349416 1 Introduction For a wide range of fluid infiltrated materials, such as water saturated soils, oil impregnated rocks, or air filled foams, the elas- tic theory is a crude approximation. Due to presence of a second, interacting continuum, a different theory is necessary. The theory of porous materials containing a viscous fluid, known as the theory of poroelasticity, was introduced by Biot 1. In subsequent years, this theory was extended to the anisotropic case 2, and also to dynamics 3. Following this development, the dynamic as well as the quasi-static analysis of a fully saturated porous continuum is possible. A comprehensive review of the quasi-static theory in rock mechanics can be found in the work of Detournay and Cheng 4. In addition to the effect of the viscous fluid diffusion in the pores, the solid constituent, its skeleton, and its interaction with partially entrapped fluid can introduce time-dependent behavior as viscoelastic material. Further on, the rheology of pore fluid can exhibit viscoelastic behavior as well. This effect, however, will not be taken into account in the study here. The implementation of the solid viscoelastic effects in the theory of poroelasticity was first introduced by Biot 5. Further work on this topic was done in the quasi-static case in 6 and in dynamics in 7, to cite a few. The last cited paper generalized Biot’s theory to partially satu- rated continua. Recently, a representation of the poroviscoelastic theory based on rheological modeling at micromechanical level was published by Abousleiman et al. 8. It was argued that to have a physically consistent model, the rheology for the solid constituent and the skeletal structure should be clearly separated, and then combined to form a bulk continuum model. Based on this model, originally in quasi-static range, the current work examines its dynamic re- sponses. The set of the governing differential equations for the dynamic case are deduced for a one-dimensional column. The corresponding analytical solution for one-dimensional column for the poroelastodynamic case has been presented by Schanz and Cheng 9. The extension to poroviscoelasticity of this solution will be done in Laplace domain with the help of the elastic- viscoelastic correspondence principle. With this solution, the frequency-dependent response of this column due to an impulsive load can be studied with respect to the influence of the viscoelasticity by taking the real part of the com- plex Laplace variable to zero. Then, the response of an arbitrary dynamical loaded system in time domain is given by the convo- lution integral of the impulse response function and the time- dependent loading. This convolution integral is numerically evalu- ated by the so-called ‘‘Convolution Quadrature Method’’ proposed by Lubich 10. The weights of this quadrature formula are determined from the Laplace transformed impulse response function and a linear multistep method. In this method, no solu- tion in time domain of the original problem is necessary. Through a series of stringent tests that includes a comparison with the highly acclaimed Dubner-Abate-Durbin-Crump method e.g., 11 or 12, our experience indicates that the Lubich method is one of the most robust in performing the inversion of wave-like functions that involves a significant number of cycles resulting from impact loading. This method has been, among other applications, suc- cessfully applied to a time domain formulation of the boundary element method 13. 2 Governing Equations Following Biot’s approach to model the behavior of porous media, the constitutive equations can be expressed as 1  ij =2 G ij +  K - 2 3 G   kk  ij -  ij p (1a)  = kk +  2 R p , (1b) in which  ij denotes the total stress, p the pore pressure,  ij the strain of the solid frame,  the variation of fluid volume per unit reference volume, and  ij the Kronecker delta. In the above, the sign conventions for stress and strain follow that of elasticity, namely, tensile stresses and strains are denoted positive. The Latin indices takes the values 1, 2, 3 or 1, 2 in three-dimensional or two-dimensional cases, respectively, where summation conven- tion is implied over repeated indices. The bulk material is defined by the material constants shear modulus G and the drained bulk compression modulus K. Biot’s effective stress coefficient , the porosity , and R complete the set of material parameters. Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, De- cember 12, 1999; final revision, July, 2000. Associate Editor: D. A. Siginer. Discus- sion on the paper should be addressed to the Editor, Professor Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS. 192 Õ Vol. 68, MARCH 2001 Copyright © 2001 by ASME Transactions of the ASME