Deformation of Spherical Cavities and Inclusions in Fluid-infiltrated Elastic Materials

The problem of a spherical cavity which is embedded in a linear, fluid-infiltrated, elastic porous medium and which is subjected to the sudden quasi-static application of a stress at the cavity boundary is solved. It is demonstrated that the deformation of the cavity is homogeneous regardless of the boundary condition imposed on the pore fluid at the cavity wall. For the case in which the pore pressure vanishes at the cavity wall, the time dependence of the cavity strain is evaluated explicitly and is shown to vary between the limits of the ordinary linear elastic response based on the short-time (undrained) and on the long-time (drained) properties of the fluid-saturated solid. The results are then used to obtain a relation between the uniform stress or strain applied at infinity and the stress and strain in a highly permeable, possibly non-linear spherical inclusion. The application of this relationship to a study of earthquake premonitory processes based on the deformation of a rock mass with a spherical weakened zone is outlined. It is argued that the fluid coupling effects serve to stabilize the weakened rock against rapid fracture. and give rise instead to a precursory period of accelerating but initially quasi-static straining which ultimately leads to dynamic instability. INTRODUCTION fund-i~~ltratio~ of an otherwise elastic porous solid introduces a time dependence into the response to applied loads. For deformation which is much slower than the characteristic time for the diffusion of pore fluid, the local pore &id pressure in each material element remains constant and the response is said to be drained. Conversely, when load alterations are rapid by comparison to the diffusion time, the local fluid mass content in each material element remains constant, and the response is undrained and elastically stiffer than the drained response. This time dependence has been proposed as a possible factor in accounting for several features of earth-faulting processes: migration of aftershocks[l,2], stabilization of incipient faulting[3], fault creep[4], and premonitory events for earthquakes [5-71. It is also relevant to a wide range of geotechnical problems including hydraulic fracture [e.g. 3, 81 and soil consolidation [9, lo]. An exceptional instance in which the response is time-independent even in the presence of ~uid-infiltration is the quasi-static shear of a homogeneous body. If, however, the body contains a cavity or other inhomogeneity, the response is time-dependent. In this paper, we will examine this feature in detail by deriving the solution for the time-dependent strain of a spherical cavity in a fluid-infiltrated elastic porous solid subjected to a suddenly applied shear stress at the cavity boundary. In the course of obtaining the full solution, we will demonstrate that the cavity deforms homogeneously, a result which is analogous to that of Eshelbyll I] for the ellipsoidal inclusion embedded in an elastic matrix. The particular relevance of this solution is that it enables us to write an expression which, again analogously with the results of Eshelby[ll], relates in a simple way the stress and strain in a spherical inhomogeneity to the applied far-field stress and strain. By using the results of Eshelby, Rudnicki[l2,133 has investigated models for the inception of earth faulting in which the inhomogeneity is considered to be a zone of material weakened by fissuring and past faulting. The present results make it possible to include in these considerations the time-dependent response of the elastic material surrounding the weakened zone. Our analysis suggests that this time-dependent response may be an important factor in leading to an earthquake precursory period of accelerating but initially stable and quasi-static straining prior to instability. We begin by reviewing the governing constitutive relations and field equations. Then, after deducing the form of the solution from considerations of symmetry and linearity, we establish directly its spatial dependence. Although the solution for the full time dependence involves tDivision of Engineering. Brown University. Providence, RI 02912, U.S.A. tSeismologica1 Laboratory. California htslitute of Technology, Pasadena, CA 91125. U.S.A. _‘%I J. R. R~ct rr al. much numerical computation, the time dependence of the strain of the cavity boundary, which is the feature of the solution of greatest interest for applications, will be evaluated explicitly. GOVERNING EQUATIONS The constitutive relations for a linear fluid-infiltrated solid were established by Biot[lO]. In order to effect a formulation in terms of easily interpretable parameters, Rice and Cleary[3] exploited the observation that the response of the fluid-saturated solid has the form of the usual linear elastic response in the limits of drained and undrained behavior. For an isotropic linear elastic material the expression for the stress u,j depends on the displacement gradient u,,,( = au;/Jxj) and the alteration of pore fluid pressure p as follows: where A and p are the Lam6 moduii appropriate for drained deformation and 6, is the Kronecher delta; C = IK/K;, where K( = A + 2~/3) is the drained bulk modulus and K: may, in certain circumstances, be identified with the bulk modulus of solid constituents, but, more generally, must be regarded as an empirical constant[3]. A second constitutive relations is needed for the alteration of fluid mass (per unit volume) m from its reference value mo, and this may be put in the form m m0 = ii~~[uk.k + 5ph A )I (2) where pa is the reference value of the density of the homogeneous pore fluid and A, is the value of the Lam6 constant for undrained deformation. The latter satisfies m > A,, > A, where the upper limit is attained for separately incompressible constituents. For undrained deformation 111 = III,) and inserting