Journal of Mathematical Sciences

, Volume 97, Issue 1, pp 3777–3795 | Cite as

On local singularities in mathematical models of physical fields

  • V. T. Grinchenko
  • A. F. Ulitko


In constructing mathematical models of physical fields there are various widely used hypotheses that include assumptions on the smoothness of the field, the nature of the boundary surfaces of the region of existence of the field and the properties of the interaction between the object being studied and surrounding objects. In many cases the use of convenient mathematical model representations leads to a situation in which some characteristics of the field do not have bounded values at individual points. The problem of the appearance of such local singularities in physical fields of different natures is of fundamental importance in the study of these fields by the methods of mathematical modeling, from the point of view of general understanding of the possibilities and content of mathematical modeling. The appearance of singularities in the characteristics of the field is a consequence of the contradictions arising when the properties of the medium and the nature of the boundary and the boundary conditions are modeled independently. In the present work we discuss a unified methodological approach to determining the nature of the singularity in various types of physical fields. The approach is based on introducing the concept of a general solution of the boundary-value problem and the use of such a solution in the vicinity of singular points. It is shown that the solutions with singularities give a reliable qualitative and quantitative description of the fields outside a small zone in a neighborhood of the singularity. Analysis of the solutions of boundary-value problems with local singularities shows the nature of the difficulties in interpreting the physical content of the solution near singularities. Typical situations are those in which interpenetration of different parts of the body is observed, the values of the displacements or velocities of certain points of the boundary depend on the direction of the limiting approach, and others. In a number of cases a priori knowledge of the nature of the singularity makes it possible to develop new approaches to the construction of effective solutions of boundary-value problems.


Boundary Condition Mathematical Model General Solution Singular Point Model Representation 
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Literature Cited

  1. 1.
    V. M. Abramov, “The problem of contact of an elastic half-plane with an absolutely rigid base taking account of friction,”Dokl. Akad. Nauk SSSR,17, No. 4, 73–178 (1937).Google Scholar
  2. 2.
    S. S. Grigoryan, “On the nature of superdeep' penetration of rigid microparticles into solid materials,”Dokl. Akad. Nauk SSSR,292, No. 6, 1319–1323 (1987).MathSciNetGoogle Scholar
  3. 3.
    V. T. Grinchenko, “Development of a method of solving problems of radiation and scattering of sound in noncanonical regions,”Gidromekhanika, No. 70, 27–40 (1996).Google Scholar
  4. 4.
    A. L. Gol'denveizer,Theory of Elastic Thin Shells [in Russian], Nauka, Moscow (1976).Google Scholar
  5. 5.
    N. E. Kachalovs'ka and A. T. Ulitko, “Singularities of the stressed state of an elastic medium containing a rigid ‘needle-shaped’ inclusion,”Dop. Akad. Nauk Ukr. RSR, No. 5, 44–48 (1987).Google Scholar
  6. 6.
    S. G. Mazur, “The refined theory of bending of thin plates by concentrated forces,”Prikl. Mekh.,15, No. 4, 86–89 (1979).MATHGoogle Scholar
  7. 7.
    G. D. Malyuzhinets, “Mathematical statement of problems of forced harmonic vibrations in an arbitrary region,”Dokl. Akad. Nauk SSSR,78, No. 3, 439–442 (1951).MATHGoogle Scholar
  8. 8.
    F. Morse and G. Feshbach,Methods of Theoretical Physics [Russian translation], Vol. 2, Foreign Literature, Moscow (1960).Google Scholar
  9. 9.
    N. I. Muskhelishvili,Some Basic Problems of the Mathematical Theory of Elasticity [in Russian], Nauka, Moscow (1966).Google Scholar
  10. 10.
    G. Neiber,Stress Concentration, [Russian translation], Gostekhizdat, Moscow (1947).Google Scholar
  11. 11.
    V. V. Panasyuk,Mechanics of Quasi-Brittle Fracture of Materials [in Russian], Naukova Dumka, Kiev (1991).Google Scholar
  12. 12.
    D. N. Parfinenko and A. F. Ulitko, “Penetration into an elastic half-space by a stamp having the shape of a circular arc in a plane,”Izv. Ross. Akad. Nauk. Mekh. Tver. Tela No. 6, 32–41 (1994).Google Scholar
  13. 13.
    Ya. S. Pidstryhach,Vibrations of Plates [in Ukrainian], Naukova Dumka, Kiev (1995).Google Scholar
  14. 14.
    Yu. N. Podil'chuk, “Strain of an axisymmetrically loaded elastic spheroid,”Prikl. Mekh.,1, No. 6, 85–91 (1965).Google Scholar
  15. 15.
    S. P. Timoshenko and J. Goodier,Theory of Elasticity [Russian translation], Nauka, Moscow (1975).Google Scholar
  16. 16.
    A. F. Ulitko,The Vector-Valued Eigenfunction Method in Three-Dimensional Problems of the Theory of Elasticity [in Russian], Naukova Dumka, Kiev (1979).Google Scholar
  17. 17.
    A. F. Ulitko, “A semi-infinite cut along the boundary of rigidly joined half-planes of different materials,” in:Modern Problems of Solid-State Mechanics [in Russian], Rostov-na-Donu (1995), pp. 185–193.Google Scholar
  18. 18.
    Ya. S. Uflyand,Integral Transforms in Problems of the Theory of Elasticity [in Russian], Nauka, Leningrad (1967).Google Scholar
  19. 19.
    H. Henle, A. Maue, and K. Westphal,Theory of Diffraction [Russian translation], Mir, Moscow (1964).Google Scholar
  20. 20.
    M. Comnionou and D. Schmueser, “The interface crack in a combined tension-compression and shear field,”J. Appl. Mech.,46, 345–348 (1979).Google Scholar
  21. 21.
    A. A. Griffith, “The phenomena of rupture and flow in solids,”Phil. Trans. Roy. Soc. London A,221, 163–198 (1921).Google Scholar
  22. 22.
    G. R. Irwin, “Linear fracture mechanics, fracture transition, and fracture control,”Eng. Fract. Mech.,1, No. 2, 241–257 (1968).MathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • V. T. Grinchenko
  • A. F. Ulitko

There are no affiliations available

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