A new gradient elasticity model for the elastic boreholes

A new gradient elasticity model is employed to discuss strain gradient effects and its ability in predicting size effects on an elastic rock mass with microstructure, around an axisymmetric borehole under internal pressure and remote isotropic compressive stress. The constitutive equation of the model involves the Laplacian of the strain tensor multiplied by the gradient coefficient. The formulated boundary value problem is solved analytically to derive stress, strain, and displacement distributions and discuss respective gradient effects on the mechanical behavior of the rock mass and the corresponding borehole stability. The paper concludes with the employment of the Rankine failure criterion to investigate size effects on the stress concentration factor at the perimeter of the borehole, and the comparison with another gradient elasticity model which involves the Laplacian of the hydrostatic part of the strain tensor. © Published under licence by IOP Publishing Ltd

An elastoplastic axisymmetric borehole problem using a deformation theory of gradient plasticity

A deformation theory of gradient plasticity is employed to study elastoplastic axisymmetric boreholes subjected to far-field biaxial tension. The gradient dependence is introduced in the constitutive framework by assuming that the flow stress (i.e., an effective stress measure) depends not only on an effective strain measure but also on the Laplacian thereof. The classical theory of linear elasticity is adopted for the elastic deformations while strain softening is assumed for the plastic part, where an equivalent work hypothesis is used to associate the stress state to the final total strain. Representative stress distributions are illustrated and compared in the context of size effects, which originate from the presence of the gradient term in the governing equations. The influence of the borehole radius on the initiation of plastic deformation, the nominal stress-strain response and the evolution of the elastoplastic interface are examined. Moreover, a maximum principal strain failure criterion is employed to discuss size effects on the onset of macroscopic fracture. The obtained results show that narrower boreholes have higher plastic and failure strength. © 2020 Informa UK Limited, trading as Taylor & Francis Group

Strain gradient and wavelet interpretation of size effects in yield and strength

The problem of yield stress and ultimate strength dependence on the specimen size is considered. This dependence, in otherwise geometrically similar specimens, is herein interpreted on the basis of gradient plasticity and wavelet analysis which introduce an internal length scale in the stress vs. strain relation. In particular, gradient plasticity within a simple strength of materials approach and wavelets within a simple scale-dependent constitutive equations approach, are used to derive expressions for the yield stress and the ultimate strength of solid bars subjected to torsion and bending. It is shown that these expressions depend on the size of the specimen. When the dimensions of the specimen become comparable to the internal length scale, size effects become important, while for larger specimens such dependence becomes negligible. Comparisons of the theoretical predictions to available experimental data are given. © 2002 Elsevier Science Ltd. All rights reserved

Underground circular openings in elastoplastic rocks: Strain gradient and size effects

A special form of gradient elasticity and a deformation version of gradient plasticity are employed to derive analytical solutions for the problem of an unsupported underground circular opening in an elastoplastic rock subjected to an isotropic far-field compressive stress. In the gradient elasticity model, the classical Hooke’s law is extended to include the Laplacian of the hydrostatic part of the strain tensor, while the gradient plasticity model is based on the modification of the flow stress (an effective stress measure) to include the Laplacian of a corresponding effective strain. The boundary value problem is solved analytically and exact expressions for the displacement, strain and stress distributions are obtained. The presence of size effects which contribute to the circular opening stability is illustrated and discussed. This includes size-dependent yield stresses and plastic region diameters, while a maximum strain failure criterion is adopted to discuss size effects on the macroscopic fracture of the rock medium. © International Society for Rock Mechanics and Rock Engineering Norwegian Group for Rock Mechanics

Deformation vs. flow and wavelet-based models of gradient plasticity: Examples of axial symmetry

The modest goal of this article - the first in a series of three dedicated to Kirk Valanis - is to discuss and compare (in terms of their constitutive and variational formulation, boundary conditions and size-dependent solutions to axially symmetric problems) a class of deformation and flow models of gradient plasticity derived from the same basic assumption of a gradient-dependent flow stress: That is, the dependence of the flow or effective stress σ̄ on the Laplacian ∇2ε̄ of the effective strain ε̄. This assumption is also used to derive, through wavelet analysis, a scale-dependent constitutive model involving a scale or gage length parameter instead of a gradient term. All models exhibit size effects which may differ from one model to another according to the nature of the boundary conditions and the values of the gradient coefficients or the scale parameter used. A weak formulation of stress equilibrium is employed to derive corresponding extra boundary conditions for the deformation and flow models which are then examined in view of their solutions for problems of axial symmetry, and also in connection with the corresponding solutions of the scale-dependent model for which extra boundary conditions are not required. In particular, results for the stress/strain distributions and the description of the associated size effects for internally pressurized thick-walled cylinders are provided. A comparison with some available experimental results for the yielding behaviour of internally pressurized hollow cylinders is also made. © 2006 Elsevier Ltd. All rights reserved

Circular tunnel in a gradient elastoplastic rock mass

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. © 2020 ISRM