On Features of Exact Solutions of Boundary Value Problems of Elasticity Theory in the Semi-Strip
Authors: Kovalenko M.D., Men’shova I.V. | Published: 26.07.2017 |
Published in issue: #4(73)/2017 | |
DOI: 10.18698/1812-3368-2017-4-52-64 | |
Category: Mathematics and Mechanics | Chapter: Solid Mechanics | |
Keywords: semi-strip, Fadle - Papkovich functions, exact solutions, non-uniqueness of the solution, incompatibility of deformations, residual stresses |
The purpose of this work was to examine the features of the exact solutions of an odd-symmetric boundary-value problem of elasticity theory for the semi-strip with the free longitudinal sides, as an example. First, we constructed solutions in the form of expansions on Fadle - Papkovich functions and determined the coefficients by biorthogonal systems of functions. Then, we found the solution for the problem and showed it in three statements: a) the stresses were given at the end face of the semi-strip; b) the displacements were given at the end face of the semi-strip; c) the end face of the semi-strip was the line of displacement’s discontinuity. The research showed that the solution for the boundary value problem in the semi-strip is not unique, and this results in residual stresses. The deformation compatibility requirement for the solutions obtained failed to be met. In physical terms it means that semi-strip’s sides, which are rectilinear before deformation, curve after deformation. Next, we discussed the connection between non-uniqueness, compatibility conditions and solution for the equivalent inhomogeneous problem. Finally, we identified the residual stresses attributes resulting from the results obtained: 1) reversed stresses, as a consequence of its self-balanced nature; 2) serious, sometimes inexplicably big stresses, e. g., sub-horizontal stresses in the earth’s crust; 3) serious residual stress of free surfaces; 4) fragments of coupon, which arises after its fragmentation and residual stress dropping, moves and turns as a perfectly rigid body; 5) it is impossible to put down the fragments along rupture faces without interlayer gaps.
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