## 1 INTRODUCTION

Cavity expansion theory has been extensively developed and widely used for the study of many engineering problems. Since its early application to geotechnical problems in the 1960s [1], many analytical solutions have been proposed using increasingly sophisticated constitutive soil models. Vesic [2] presented an approximate solution for spherical cavity expansion in an infinite soil mass using a compressible Mohr–Coulomb material. The analysis was applied to evaluate the bearing capacity factors of deep foundations. Carter *et al.* [3] derived closed-form solutions for cavity expansion from zero initial radius in an ideal cohesive-frictional material with a small-strain restriction. The deformations in the elastic region were assumed to be infinitesimal, and the convected term of the stress rate was neglected in the governing equation, which provided an approximate limit pressure solution. Yu and Houlsby [4] provided a unified analytical solution of cavity expansion in dilatant elastic-plastic soils, using the Mohr–Coulomb yield criterion with a non-associated flow rule. The complete large-strain analysis, with the aid of a series expansion, was introduced to derive a rigorous closed-form solution without any additional restrictions or assumptions. The limitation of their analysis was that the material properties were assumed to be constant and independent of stress–strain history. Salgado *et al.* [5] reported a cylindrical cavity expansion solution and produced a stress rotation analysis for the interpretation of the cone penetration test (CPT). A numerical formulation was used in the plastic region to achieve a variable stiffness, friction angle, and dilation angle.

As reviewed by Yu [6], the cavity expansion theory has mainly been applied in the geotechnical engineering areas of in-situ soil testing [5, 7-11], deep foundations [12-15], tunnels and underground excavations [16-18], and recently, for an interaction analysis between tunnels and piles [19, 20]. Despite the wide application of the theory to geotechnical problems, very little work has been carried out to consider the effect of distinct soil regions within the framework of cavity expansion analyses. The work of Xu and Lehane [21] is notable for its use of a numerical analysis of spherical cavity expansion for investigating pile or probe resistance in two-layered soil profiles.

Analytical cavity expansion solutions for two concentrically layered media were developed by Bernard [22, 23] for the study of projectile penetration. The analysis considered an incompressible material as well as the assumption of a finite locking strain and was used to solve for dynamic solutions of penetration depth and impact velocity. Sayed and Hamed [24] were the first to apply analytical cavity expansion analyses of concentrically layered media to the field of geomechanics. However, in their analysis, the medium was assumed to be a frictionless linear-elastic solid and did not account for the plastic behaviour of soils. In this paper, the analytical solution described by Yu and Houlsby [4] is extended in order to consider a cavity embedded within a profile of two different concentric regions of soil. The soil is treated as an isotropic dilatant elastic-perfectly plastic material with a Mohr–Coulomb yield criterion and a non-associated flow rule. Large-strain quasi-static expansion of both spherical and cylindrical cavities is considered. The complete large-strain expansion for non-associated Mohr–Coulomb materials in two concentric media has not previously been presented in the literature. The development of an analytical cavity expansion method for application to geotechnical problems involving aspects of soil layering is the main motivation for the work described in this paper. The focus here is on the development of the analytical method; its application to practical geotechnical problems will be explored in future publications.

The paper begins with a general definition of the problem and the necessary geometric parameters. The following section considers the most general expansion problem for a cavity embedded in two different concentric regions of soil and derives expressions for stresses, strains, and displacements within elastic and plastic zones. The cavity expansion solution is then validated against results obtained using the Finite Element (FE) method. Further results and parametric analyses are then presented with focus placed on the resulting pressure–expansion curves and the development of plastic regions within the two soil zones. A discussion of the application of the proposed method and its limitations is provided, followed by concluding remarks.