Schelokova M.A., Slobodian S.B., Dyrda V.I. Fractal approach to solid fracture mechanics

Details

Parent Category: Geo-Technical Mechanics, 2018

Category: Collection of scientific papers «Geotekhnicheskaya Mekhanika» Issue 138

Geoteh. meh. 2018, 138, 227-259

DOI: https://doi.org/10.15407/geotm2018.01.227

Fractal approach to solid fracture mechanics

Schelokova M.A., Slobodian S.B., Dyrda V.I.

Authors:

Shelokova MA, Ph.D. tech. Sci., Associate Professor, Associate Professor of the Department of Applied Mathematics, Zaporizhzhya National Technical University, Zaporizhia

Slobodyan SB, Ph.D. tech. Sciences, Associate Professor, Podolsk State Agrarian Technical University, Kamenets-Podolsk, Ukraine

Dyrda V.I., Dr. Tech. sciences, professor (IGTM NAS of Ukraine)

UDC 539.3

Language:Russian

Abstract.

The authors present an overview of how the concept of solid destruction mechanisms has been forming since the end of the nineteenth century. Early works of A.F. Ioffe, A.A. Griffiths, Irvine, G.V. Kolosov and others are considered with focusing on popular Griffiths destruction criterion, wihich is based on the energy balance and used for calculating the cracks movement. The overview also includes E.M. Morozov approaches, Cherepanov-Rice J-integral, works of A.A. Lebedev, V.Z. Parton, A.N. Guz, I.A. Miklashevich, G.I. Baranblatt, destruction criteria of Dagdale and Leonov-Panasyuk, and others. It is shown that it is expedient to use fractal models to describe micro-features of real cracks. Therefore, such important issues as: general scheme of fractal approach, generalized fractal model of real crack, effect of the crack fractal dimension on value of the stress intensity factor, mathematical description of synergetic model of the fractal crack are considered.

At the microlevel, profile of rough crack is approximated by fractal object; though at the macro level, the crack features a smooth contour. Therefore, classical formulations of fracture problems remain valid, and additional parameter - fractal dimension – is a carrier of the fractal micro-features of the crack. For estimating real “length” of the crack, an entropic a-dimensional measure was constructed. With the help of this approach, fractal generalization of energy criterion for destruction of crack-contained solids with fractal specificity at the micro level becomes possible. Rate of elastic energy is determined at the macrolevel; surface energy needed for creating two fractal surfaces is recorded with taking into account the fact that at the microlevel, the crack features fractal roughness. Such approach makes it possible to establish a link between micro- and macrolevels, and allows the concept of structure as such to be translated to the higher level of validation.

In the conclusion of the article, fractal generalization of the energy concept of solid destruction is considered. In particular, a problem of damage accumulation and fractal analysis of rubbers during long cyclic destruction is considered. Fractal dimension of the fracture surface was found for the concrete rubber in its initial state and after operation under extreme cyclic loading for more than 30,000 hours. A change of Poisson’s ratio for original rubber and rubber “loosened” due to prolonged fatigue is shown.

Keywords:

deformation, strength, fractal, destruction, crack, fracture energy.

References:

1. Bulat, A.F. and Dyrda, V.I. (2005), Fraktaly v geomekhanike [Fractals in geomechanics], Naukova dumka, Kiev, Ukraine.

2. Bulat, A.F. and Dyrda, V.I. (2003), “Fractal nature of destruction of elastomers under prolonged cyclic loading”, Geo-Technical Mechanics, no. 45, pp. 137-144.

3. Shcholokova, M.O. (2007), “Fractal generalization of the energy criterion for quasi-violent destruction of solids”, Abstract of Ph.D. dissertation, National Academy of Sciences of Ukraine, Zaporizhia, Ukraine.

4. Shchelokova, M.A. (2004a), “Fractal generalization of the Griffith equation”, Innovative Materials and Technologies in Metallurgy and Mechanical Engineering, no. 2, p. 86-89.

5. Shchelokova, M.A. (2004b), “The application of fractal geometry to the description of the mechanism of destruction”, Problems of computational mechanics and strength of structures, no. 8, p. 137-144.

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https://doi.org/10.1007/BF01524344

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About the authors

Schelokova Marina Alexandrovna, Candidate of Technical Sciences, Associate Professor of Department of Applied Mathematics in Zaporozhye National Technical University, Zaporizhia, Ukraine, This email address is being protected from spambots. You need JavaScript enabled to view it.

Slobodyan Sergey Borisovich, Candidate of Technical Sciences, Associate Professor of Department of General Technical Disciplines and Physics, State Agrarian and Engineering University in Podilya, Kamianets-Podilskyi, Ukraine, This email address is being protected from spambots. You need JavaScript enabled to view it.

Dyrda Vitaly Illarionovich, Doctor of Technical Sciences (D. Sc.), Professor, Head of Department of Elastomeric Component Mechanics in Mining Machines, Institute of Geotechnical Mechanics named by N. Polyakov of National Academy of Science of Ukraine (IGTM NASU), Dnipro, Ukraine, This email address is being protected from spambots. You need JavaScript enabled to view it.

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