A Contact Problem In Poroelasticity

Abstract

The paper examines the axisymmetric problem of the indentation of poroelastic layer which rests in contact with a rough rigid and impermeable base. The indenting punch has a circular planform and the smooth contact is assumed to be either permeable or impermeable. The paper develops the integral equations governing the problem for the generalized case where the the pore-fluid is considered to be compressible. Numerical results are presented to illustrated the influence of layer thickness, drainage conditions, and the compressibility characteristics of the pore fluid on the degree of consolidation settlement of the indenting punch. INTRODUCTION The classical theory of poroelasticity which is applicable to the study of the mechanics of deformable fluid saturated porous medium was first developed by Biot [5]. This theory has been widely applied to the study of various loading and contact problems associated with halfspace and infinite space regions. A majority of these discussions have focussed on the evaluation of the response of the poroelastic medium which is saturated with an incompressible pore fluid. In this paper, attention is focussed on the problem of the indentation of a poroelastic layer by a smooth rigid circular punch. The term thin is intended to signify a layer thickness which is smaller than the diameter of the indenting circular punch. The punch is in smooth contact with the poroelastic layer which exhibits either permeable or impermeable pore pressure boundary conditions over the entire surface. The layer is underlain by an impermeable rigid base 1 Currently at The Infrastructure Laboratory, Institute for Research in Construction, National Research Council of Canada, Ottawa, Ontario, Canada K1A OR6 Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 286 Contact Mechanics and the displacement contact conditions are considered to be adhesive (Figure 1). The paper summarizes the integral equations governing the mixed boundary value problems associated with boundary variables in the Laplace transform domain. The paper illustrates the influences of the relative thickness of the consolidating layer and the Poisson's ratios of the pore fluid on the degree of the consolidation of the poroelastic layer. A permeable or impermeable surface A rigid circular smooth indentor t A poroelastic layer saturated with a compressible pore fluid xA rough rigid and impermeable base Figure 1. An axisymmetric contact problem in poroelasticity GOVERNING EQUATIONS In the ensuing we shall present a brief account of the governing equations referred to a Cartesian tensor notation. The constitutive equations governing the quasi-static response of a poroelastic medium, which consists of an isotropic poroelastic soil skeleton saturated with a compressible pore fluid takes the forms (1) 3(1 2r/,) where cr^ is the total stress tensor, p is the pore fluid pressure, Cij are the soil skeleton strains defined by £.. = !(„..+„,.) (2) where Ui are the corresponding displacement components. In the absence of body forces, the quasi-static equations of equilibrium take the forms °ij,i = 0 (3) The equations governing quasi-static fluid flow are defined by Darcy's law which take the form %, = -/W (4) Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 Contact Mechanics 287 and the continuity equation associated with quasi-static fluid flow is The above governing equations are characterized by the five basic material parameters which are represented by the drained and undrained Poisson's ratios v and z/% respectively, the shear modulus //, Skempton's [7] pore pressure coefficient B, and the K (=fc/7 , where k is the coefficient of permeability and 7%, is the unit weight of pore fluid). From the consideration of the positive definiteness of a strain energy potential, it can be shown that the material parameters should satisfy the following thermodynamic constraints: /j > 0, 0 0 (see. e.g., Rice and Cleary [6]). Avoiding details, it can be shown (Yue [3]) that the following sets of solution representations exist for the field variables in a linear, isotropic, poroelastic medium of layer extent saturated with a compressible pore fluid. In either the temporal domain or the Laplace transform domain and in the cylindrical coordinate systems (r, #, z) and (p, y, z), we have % roc r2ir } u(r, 0,z,t) = — I I -TlawK 27T Jo JQ p I roc /-27T ,(r, 0, z, f ) = — / / n, rA 2?r Vo Vo