This paper attempts to study the mechanical behavior of an elastoplastic rock mass around an underground circular excavation by using gradient theory. The problem (in plane conditions) is reduced to that of a circular hole within an elastoplastic geomaterial which is subjected to a far-field uniform compression. At the perimeter of the hole an all-around uniform pressure is applied. The constitutive law adopted for the elastic region is a special form of gradient elasticity while for the plastic region around the tunnel; the Tresca failure criterion is used. Appropriate boundary conditions (classic and extra) are taken in order to solve the corresponding boundary value problem, and the analytical expressions of stresses, strains and displacements in the elastic and plastic regions are presented. The solution for the elastoplastic boundary is achieved numerically and the observed size effect, i.e. the dependence of the elastoplastic boundary on the tunnel radius, is described.

1 Introduction

Classical continuum solid mechanics theories (linear/nonlinear elasticity, plasticity, damage), have been used in a wide range of fundamental problems and applications in civil and geotechnical engineering. Even though the scales that these theories were initially designed for were ranging roughly from millimeter to meter, they were also used in the last century to describe phenomena evolving at atomistic scales (elastic theory of dislocations), earth scales (faults and earthquakes) and astronomic scales (relativistic elastic solids). In particular, standard elasticity formulae have been used to characterize deformation behavior at the nanoscale. It was concluded that classical models should be generalized for describing material behavior in this wide range of scales [Aifantis (1992), Aifantis (1999)]. In the middle of the 1980s a simple model of gradient plasticity for strain softening materials, was proposed by Aifantis in order to determine the width of shear bands. The simplicity of this formulation relies on the fact that only one additional constitutive constant is required. Next, in the beginning of the 1990s, the same author proposed another simple model with only one additional constant to extend classical elasticity. In general, both strain and stress gradients may affect the constitutive response of materials. A systematic way to account for this is to employ the framework of implicit constitutive equations. In the case of gradient dependent elasticity models, this idea is based on considering an implicit constitutive equation which involves both stress and strain and their Laplacians [Aifantis (2003), Aifantis (2011)]. Several issues related to the form and sign of the gradient terms and associated gradient coefficients, the corresponding extra boundary conditions and their physical meaning, the elimination of elastic singularities and the prediction of size effects, as well as numerical aspects and experimental validation, are still open and need further consideration [Askes and Aifantis (2011)]. In contrast to classical elasticity, it is possible to simulate size dependent mechanical behavior with gradient elasticity. This has been demonstrated for a number of geometries and it has been found that the gradient effects are most pronounced if the material length scale parameter is of the same order of magnitude as the dimension of the geometric feature (hole, indentor or otherwise) that triggers gradient activity. There is a variety of gradient-dependent constitutive equations that can be used to consider the important issue of size effect, i.e. the dependence of strength and other mechanical properties on the size of the specimen. The particular constitutive form to be adopted depends on the material properties, as well as on the particular configuration at hand. This may be physically understood on the basis that higher-order gradients in the constitutive variables is a measure of the heterogeneous character of deformation field, the overall effect of which may depend on the specimen size. In fact, solution of boundary value problems based on higher-order governing equations for the strain field bring in the size of the specimen in a nontrivial manner and, thus, related size effects may be captured accordingly [Aifantis (1999), Aifantis (2003), Aifantis (2011)].

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