# MODELING OF FINITE INHOMOGENEITIES BY DISCRET SINGULARITIES

### Abstract

This work focuses on development of a mathematical apparatus that allows to perform an approximate description of inhomogeneities of finite sizes in a continuous bodies by arranging the sources given on sets of smaller dimensions. The structure and properties of source densities determine the adequacy of the model. The theory of differential forms and generalized functions underlies this study. The boundary value problems with nonsmooth coefficients are formulated. The solutions of such problems is sought in the form of weakly convergent series and as an alternative - an equivalent recurrent set of boundary value problems with jumps. A feature of this approach is the ability to consistently improve the adequacy of the description of inhomogeneity. This is important because it allows to qualitatively assess the impact of real characteristic properties on the accuracy of the model description. Reducing the dimensions of inhomogeneities allows the use of efficient methods such as the Green's function and boundary integral equations to obtain a semi-analytic solution for direct and inverse problems. The work is based on a number of partial problems that demonstrate the proposed approach in modeling of inhomogeneities. The problems of modeling of the set of finite defects in an oscillating elastic beam, the set of inhomogeneities of an arbitrary shape in an oscillating plate, fragile cracks in a two-dimensional elastic body under static loading are considered.

### References

Clouet E., Varvenne C., Jourdan T. Elastic modeling of point-defects and their interaction. Computational Materials Science. 2018, 147, pp.49–63.

Khludnev A.M. On thin inclusions in elastic bodies with defects. Z. Angew. Math. Phys. 70, 45. 2019. https://doi.org/10.1007/s00033-019-1091-5

Itou H., Khludnev A.M. On delaminated thin Timoshenko inclusion inside elastic bodies. Math. Methods Appl. Sci. 39. 2016, 17 4980–4993 pp.

Rubio L., Fernández-Sáez J., Morassi A. The full nonlinear crack detection problem in uniform vibrating rods. Journal of Sound and Vibration. 2015, 339, 99–111.

Eshelby J.D. The elastic field outside an ellipsoidal inclusion. Proc. Roy. Soc. Lond. A 252, 1959, 561–569. https://doi.org/10.1098/rspa.1959.0173

Zrazhevsky G.M., Zrazhevska V.F. The Extension Method for Solving Boundary Value problem of the Theory of Oscillations of Bodies with Heterogeneity. 2020/3/22, World Journale of Engineering Research and Technology. Vol. 6, Iss. 2, pp. 503–514.

Zrazhevsky G.M., Golodnikov A.N., Uryasev S.P., Zrazhevsky A.G. Application of Buffered Probability of Exceedance in Reliability Optimization Problems. Cybernetics and Systems Analysis. 2020. Vol. 56, pp. 476–484.

Zrazhevsky G.M., Zrazhevska V.F. Usage of generalized functions formalism in modeling of defects by point singularity. Bulletin of Taras Shevchenko National University of Kyiv. Series: physical and mathematical sciences. 2019. Iss. 1, pp. 58–61.

Aki K., Richards P. Quantitative Seismology, 2nd Ed. University Science Books, 2002.

Gelfand I.M. Generalized Functions, Vol. 1, AMS Chelsea Publishing, 1964.

Crouch S.L., Starfield A.M. Boundary Element Methods in Solid Mechanics. George Allen & UNWIN, 1983. London.