The authors construct an exact solution to the problem on the stress–strain behavior of rock mass at the boundary of an underground excavation of an arbitrary geometry if the vectors of the Cauchy ...stresses and displacements are assigned simultaneously at this boundary. All explicit components of stress and strain tensors, as well as the components of rotation vector are determined as functions of the elastic characteristics of rocks, values of the preset functions and differential properties of the boundary.
The author solves the problem connected with determination of shape of pillars which remain stable under any compression due to barrel distortion. The analysis of cylindrical structures uses the ...known Leibenson–Ishlinsky stability criterion. The boundary conditions of the problem and its solution are obtained: elasticity in the form of the critical load dependence on the height/radius ratio of pillars. The found asymptote to the curves is associated with the optimized shape of pillars.
In this study, the gauge invariance and phase transition are considered to develop an elasto-plastic model of geomaterials with dissipation. Displacements and plastic distortions are selected as the ...independent variables. The initial Lagrangian is constructed by the requirement of invariance of the Lagrangian with respect to translational transformation. To take into account the continuous structural phase transition effect of plastic deformation of geomaterials, fourth- and sixth-power terms of the distortion tensor are added to the initial Lagrangian. The differential motion equations of media with dissipation and the corresponding boundary conditions are obtained based on Hamilton’s principle. The generalized Hooke’s laws are obtained on the basis of the kinematic variational principle. One special case, dilatation-compaction deformation case, is discussed, and the obtained equation of motion is applied for modeling the deformation waves and zonal disintegration.
Rock testing data are used to determine proper bases of tensors where strains along the unit vectors are only governed by stresses along them. The obtained curves along the unit vectors—one curve is ...proportional and the other curve is nonlinear, and both are independent of loading history and mechanism—are used to solve geomechanical problems. In planar post-limit deformation, these curves lead to a hyperbolic system of differential equations with four real functions and four relations to find four unknown functions: average stress, maximum shear stress, rotation angle and angle of directions of principal stress tensor axes. For finding their boundary values, the Cauchy stress vector and the displacement vector are assigned simultaneously at one and the same boundary. The authors propose an algorithm of finding these four functions within the post-limit deformation domain.
Abstract
The problem of penetration of an absolutely rigid round cone shaped indenter into a weighted rock (an axisymmetric plasticity problem) is being solved. Limit load, depending on the angle of ...internal friction of the rock, adhesion, angular opening at the top of a cone, rock weight, is determined. The dependence of loading on these parameters is given.
Abstract
For the media with periodic changes in Young’s modulus and yield strength, the problems of stress, strain and displacement distribution around single excavations of spherical and cylindrical ...shape are solved. Block structure of the medium is determined by the difference between the elastic modules and the yield strength in the blocks and interblock space. In each case analytical solutions are obtained. The influence of blocks quantity, difference in block properties and interlayers on the nature of changes in stresses, strains and displacements was studied. It is noted that block structure is one of the factors for forming zonal disintegration around excavations.
Displacements associated with deformations are considered. The deformation vector at an arbitrary area with the normal n→ is defined as the ratio of the displacement vector of the site points to the ...distance from the site to the displacement reference point. The deformation vectors related to an arbitrary element of the medium have properties: their principal vector and principal moment are zero. For the strain vector, Mohr's circles are constructed similarly to the stress vector. It is shown that, for an arbitrary area with a normal n→, the work of forces corresponding to the stress vector on the relative displacements determined by the deformation vector depends, in the general case, on the loading path even in the case of applying the relations of Hooke's law. The only areas where this effect does not exist are the stresses and deformations of the site that are equally inclined to the main axes. On this basis, a block model of materials (such as metals) is constructed, where the elements have the form of octahedrons.
The authors determine stress and deformation in a heterogeneous rock mass at the preset displacement and Cauchy stress vector at the boundary of an underground excavation. The influence of ...coordinates on Young's modulus, shear modulus and ultimate strength is shown. It is found that regions of tension and compression alternate at the excavation boundary-i.e. zonal rock disintegration phenomenon is observed.
The problem of determining the stress-strain state in the vicinity of an opening with an arbitrary shape using the measurements of the Cauchy stress vector and displacement vector is solved. The ...states of elasticity, plasticity, and post-limiting straining are considered. The obtained results allow rapid determining of the resource capabilities of rock mass resistance to failure on the boundary both in a buried opening and in opencast mining.
Abstract
The mathematical model of plastic deformation in an initially anisotropic medium assumes that principal directions of strain in elasticity remain principal directions of strain in plasticity ...and fracture as well. The modeling considers penetration of a bit represented by a cylindrical tool with a wedge-shaped tip. The maximal penetration depth of the bit is calculated at the preset initial rate of penetration. The influence of the anisotropy of the medium and the bit parameters on the bit penetration depth is studied.
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