Discontinuous solutions of mixed problems and block elements
UDC
539.3Abstract
A number of mixed boundary-value problems in the theory of elasticity are not traditional in the sense that unstable states of the system arise, leading to destruction. These include mixed problems with discontinuous boundary conditions, in which the behavior of contact stresses and displacements, which indicate the destruction of the medium, appear. In some cases, such boundary problems have unlimited energy. Examples of such mixed boundary problems are the contact problems for two rigid dies, which have approached the contact state but have not merged into one stamp, as well as two adjacent cracks, which disappear with a small distance between them. It is shown that such problems, arising in seismology, theory of strength, construction, have singular components, in some cases with unlimited energy, and can be solved by topological methods with pointwise convergence, in particular, by the block element method. Numerical methods based on the application of the energy integral to such problems are not applicable in connection with its divergence. In the case of cracks, taking into account the investigation of the properties of solutions of integral equations obtained earlier, it is proved that a set of cracks lying in the same plane whose vertices are removed by some distance will uncontrollably merge with logarithmic growth when the distance between the vertices reaches a certain minimum.
Keywords:
block element, topology, integral and differential factorization methods, exterior forms, block structures, boundary problems, starting earthquakesAcknowledgement
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Copyright (c) 2017 Babeshko V.A., Evdokimova O.V., Babeshko O.M., Gladskoy I.B., Gorshkova E.M., Zaretskaya M.V., Mukhin A.S.
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