New Formulations of Geomechanical Problems with Regard to Post-Limit Deformation of Rocks
A. I. Chanyshev1,2, I. M. Abdulin1
1Chinakal Institute of Mining, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630091 Russia
2Novosibirsk State University of Economics and Management, Novosibirsk, 630099 Russia
Keywords: Resistance, deformation, basis of tensor, proper basis, theoretical curves, post-limit deformation, hyperbolic system of equations, functions, Cauchy problem
Abstract
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.
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