Green's Tensor and 2Dimiensional BIE Method in Static of Massif with Arbitrary Anisotropy

Abstract

Up to now remains open the question of constructing fundamental solutions of the two-dimensional statics of an elastic body with arbitrary anisotropy. Also in the scope of BEM method, the question of calculating stresses in boundary points and points located close to the boundary of the region still remains actual. In this work, fundamental solutions of the static problem for elastic plane with arbitrary anisotropic properties are obtained as the sum of residues with complex variable function. The assessments of fundamental solution and theirs derivatives are presented in closed form. In the distribution space obtained are the regular representations for the Somigliana formulas and the stress calculation formulas. The numerical implementation of the BIE method in direct formulation has been realized in standard way. The test results performed for circular hole in anisotropic plane of rhombic system show a higher compliance with the boundary values of displacements and stresses and with nodes placed close to boundary. The results of analysis of the stress-strain state in the vicinity of rectangular mining chambers located deep from day surface are presented in tables and pictures of isolines.

Tensor Multivariate Trace Inequalities and Their Applications

“…We then apply tensor Golden-Thompson inequality to provide the tail bound for the independent sum of tensors and this bound will play a crucial role in high-dimensional probability and statistical data analysis. We believe our work can be applied to various science and engineering problems involving multivariate variables [23,6].…”