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AN APPROACH TO STUDY ELASTIC VIBRATIONS OF FRACTAL CYLINDERS

LEV STEINBERG

Department of Mathematical Sciences, University of Puerto Rico, Mayagüez Campus, Puerto Rico 00681-9018, USA

E-mail Address: lev.steinberg@upr.edu

Corresponding author.

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and

MARIO ZEPEDA

Department of Mathematical Sciences, University of Puerto Rico, Mayagüez Campus, Puerto Rico 00681-9018, USA

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https://doi.org/10.1142/S0218348X1650050XCited by:1 (Source: Crossref)

Abstract

This paper presents our study of dynamics of fractal solids. Concepts of fractal continuum and time had been used in definitions of a fractal body deformation and motion, formulation of conservation of mass, balance of momentum, and constitutive relationships. A linearized model, which was written in terms of fractal time and spatial derivatives, has been employed to study the elastic vibrations of fractal circular cylinders. Fractal differential equations of torsional, longitudinal and transverse fractal wave equations have been obtained and solution properties such as size and time dependence have been revealed.

Keywords:

References

1. A. Carpinteri and N. Pugno, Are scaling laws on strength of solids related to mechanics or to geometry?, Nat. Mater. J. 14(01) (2005) 88–94. Google Scholar
2. B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman & Co, San Francisco, 1982). Google Scholar
3. B. Mandelbrot, Fractals; Forms, Chance and Dimensions (W.H. Freeman & Co, San Francisco, 1977). Google Scholar
4. M. F. Shlesinger, Asymptotic solutions of conti nuous-time random walks, J. Stat. Phys. 10(5) (1974) 421–434. Crossref, Web of Science, Google Scholar
5. S. P. Nasholm and S. Holm, On a fractional zener elastic wave equation, Fract. Calc. Appl. Anal. 16(1) (2013) 26–50. Crossref, Web of Science, Google Scholar
6. V. E. Tarasov, Continuous medium model for fractal media, Phys. Lett. A 336 (2005) 167–174. Crossref, Web of Science, Google Scholar
7. J. Li and M. Ostoja-Starzewski, Comment on “Hydrodynamics of fractal continuum flow” and “Map of fluid flow in fractal porous medium into fractal continuum flow”, Phys. Rev. E 88 (2013) 057001. Crossref, Web of Science, Google Scholar
8. J. Li and M. Ostoja-Starzewski, Micropolar continuum mechanics of fractal media, Int. J. Eng. Sci. 49 (2011) 1302–1310. Crossref, Web of Science, Google Scholar
9. M. Ostoja-Starzewski, J. Li, H. Joumaa and P. N. Demmie, From fractal media to continuum mechanics, Z. Angrew. Math. Mech. 94(5) (2013) 373–401. Crossref, Web of Science, Google Scholar
10. H. Joumaa and M. Ostoja-Starzewski, On the wave propagation in isotropic fractal media, Z. Angrew. Math. Phys. 62 (2011) 1117–1129. Crossref, Web of Science, Google Scholar
11. H. Joumaa, M. Ostoja-Starzewski and P. N. Demmie, Elastodynamics in micropolar fractal solids, Math. Mech. Solids 19(2) (2013) 117–134. Crossref, Web of Science, Google Scholar
12. A. S. Balankin, Stresses and strains in a deformable fractal medium and its fractal continuum model, Phys. Lett. A 377 (2013) 2535–2541. Crossref, Web of Science, Google Scholar
13. A. Balankin and B. Espinoza, Map of fluid flow in fractal porous medium into fractal continuum flow, Phys. Rev. E 85 (2012) 056314. Crossref, Web of Science, Google Scholar
14. A. S. Balankin, The concept of multifractal elasticity, Phys. Lett. 2(10) (1996) 51–59. Crossref, Web of Science, Google Scholar
15. A. Balankin and B. Espinoza, Reply to “Comment on ‘Hydrodynamics of fractal continuum flow’ and ‘Map of fluid flow in fractal porous medium into fractal continuum flow’”, Phys. Rev. E 88 (2013) 057002. Crossref, Web of Science, Google Scholar
16. G. Jumarie, On the representation of fractional brownian motion as an integral with respect to ${(d t)}^{α}$ ${(d t)}^{α}$ , Appl. Math. Lett. 18 (2005) 739–748. Crossref, Web of Science, Google Scholar
17. E. J. Marsden and M. J. R. Hughes, Mathematical Foundation of Elasticty (Prentice-Hall, New Jersey, 1983). Google Scholar
18. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, A. E. H., 1944). Google Scholar