Influence of material damage on propagation of a longitudinal magnetoelastic wave in a rod
DOI:
https://doi.org/10.7242/1999-6691/2018.11.4.30Keywords:
longitudinal deformation, nonlinear elastic rod, material damage, magnetic field, evolution equation, generalized Burgers equation, asymptotic solutionAbstract
At present, continuum damage mechanics is in good progress, studying both the stress-strain state of structures and the accumulation of damages in their materials. Treatment of a number of problems is associated with the necessity of taking into account the fact that structural elements operate under the action of external magnetic field, which affects the formation and propagation of elastic waves. In this work, for an electrically conducting rod performing longitudinal oscillations, we formulate a self-consistent system, which includes the equation of rod dynamics, the equation of variation of the external magnetic field strength, and the kinetic equation of accumulation of damage in the material of the rod. Here we assume that damage is uniformly distributed in the rod material, the magnetic field is stationary and use the classical model of the homogeneous Bernoulli rod as the rod model. The linearized system and the system of equations, including geometrical and physical elastic nonlinearities, are considered sequentially. In the first case, it is shown that the characteristic features of the waves described by this system are the dispersion and frequency-dependent damping due to the presence of two types of dissipation, one of which is caused by damage of the material and the other by a magnetic field. In the second case, an evolution equation for the function of longitudinal deformation is derived as a generalization to the Burgers equation. Its approximate solution is found and analyzed. Depending on the ratio of the damage to conductivity parameters it allows us to estimate the possibility of the existence of stationary waves that retain their shape and velocity during propagation in space. Moreover, the limiting cases of the evolution equation are considered, in particular, in the absence of conductivity of the electromagnetic field and damage in the material. For these cases exact solutions of the stationary profile are obtained.
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References
Kachanov L.M. Introduction to Continuum Damage Mechanics. Springer, 1986. 140 p.
Rabotnov Yu.N. Creep Problems in Structural Members. Amsterdam, North-Holland Publishing Company, 1969. 836
Maugin G.A. The thermomechanics of plasticity and fracture. Cambridge University Press, 1992. 369
Zuev L.B., Murav’ev V.V., Danilova Yu.S. Criterion for fatigue failure in steels. Technical Physics Letters, 1999, vol. 25, no. 5, pp. 352-353. DOI
Hirao M., Ogi H., Suzuki N., Ohtani T. Ultrasonic Attenuation Peak During Fatigue of Polycristalline Copper. Acta Mater., 2000, vol. 48, pp. 517-524. DOI
Wang J., Fang Q.F., Zhu Z.G. Sensitivity of ultrasonic attenuation and velocity change to ciclic deformation in pure aluminum. Status Solidi, 1998, vol. 169, pp. 43-48. DOI
Klepko V.V., Lebedev E.V., Kolupaev B.B., Kolupaev B.S. Energy dissipation and modulus defect in heterogeneous systems based on flexible-chain linear polymers. Sci. Ser. B, 2007, vol. 49, no. 1-2, pp. 18-21. DOI
Volkov V.M. Structures the loosening of metals and fracture of machine. Vestnik VGAVT – Bulletin of VSAWT, 2003, no. 4, pp. 50-69.
Collins J.A. Failure of Materials in Mechanical Design: Analysis, Prediction, Prevention. 2nd Edition, John Wiley & Sons, 1993. 654 p.
Makhutov N.A. Deformatsionnyye kriterii razrusheniya i raschet elementov konstruktsiy na prochnost’ [Deformation Criteria of Fracture and Calculation of Construction Elements for Strength]. M.: Mashinostroyeniye, 1981. 272 p.
Romanov A.N. Razrusheniye pri malotsiklovom nagruzhenii [Fracture under low-cycle loading]. M.: Nauka, 1988. 278 p.
Berezina T.G., Mints I.I. Vliyaniye struktury na razvitiye tret’yey stadii polzuchesti khromomolibdenovanadiyevykh staley [The influence of the structure on the development of the third stage of creep of chromomolybdenum vanadium steels] // Zharoprochnost’ i zharostoykost’ metallicheskikh materialov [Heat-strength and heat-resistant of metallic materials]. M.: Nauka, 1976. Pp. 149-152.
Uglov A.L., Erofeyev V.I., Smirnov A.N. Akusticheskiy kontrol’ oborudovaniya pri izgotovlenii i ekspluatatsii [Acoustic control of equipment during its manufacture and operation]. M.: Nauka, 2009. 280 p.
Erofeev V.I., Nikitina E.A. The self-consistent dynamic problem of estimating the damage of a material by an acoustic method. Phys., 2010, vol. 56, no. 4, pp. 584-587. DOI
Erofeev V.I., Nikitina E.A., Sharabanova A.V. Wave propagation in damaged materials using a new generalized continuum. Mechanics of generalized continua. One hundred years after the Cosserats. Series: Advances in Mechanics and Mathematics, vol. 21, ed. G.A. Maugin, A.V. Metrikine. Springer, 2010. P. 143-148. DOI
Stulov A., Erofeev V. Frequency-dependent attenuation and phase velocity dispersion of an acoustic wave propagating in the media with damages. Generalized Continua as Models for Classical and Advanced Materials. Series: Advances Structured Materials, vol. 42, ed. H. Altenbach, S. Forest, Springer, 2016, P. 413-423. DOI
Erofeev V.I., Mal’khanov A.O. Magnetic field effect on strain wave localization. Mach. Manuf. Reliab., 2010, vol. 39, no. 1, pp. 78-82. DOI
Erofeyev V.I., Zemlyanukhin A.I., Catson V.M., Malkhanov A.O. Nonlinear localized longitudinal waves in a plate under magnetic field. mekh. splosh. sred – Computational Continuum Mechanics, 2010, vol. 3, no. 4, pp. 5-15. DOI
Erofeyev V.I., Malkhanov A.O. Nonlinear longitudinal localized magnetoelastic waves in a plate in an arbitrarily oriented magnetic field. mekh. splosh. sred – Computational Continuum Mechanics, 2012, vol. 5, no. 1, pp. 79-84. DOI
Erofeev V.I., Mal’khanovA.O. Localized strain waves in a nonlinearity elastic conducting medium interacting with a magnetic field. Solids, 2017, vol. 52, no. 2, pp. 224-231. DOI
Erofeyev I., Kazhayev V.V., Semerikova N.P. Volny v sterzhnyakh. Dispersiya. Dissipatsiya. Nelineynost’ [Waves in the Rods: Dispersion. Dissipation. Nonlinearity]. M.: Fizmatlit, 2002. 208 p.
Korsunskii S.V. Rasprostraneniye zvukovykh puchkov konechnoy amplitudy v elektroprovodyashchikh sredakh [Propagation of large amplitude acoustic beams through electrically conducting medium]. Akusticheskiy zhurnal – Soviet Physics. Acoustics, 1990, vol. 36, no. 1, pp. 48-
Ryskin M., Trubetskov D.I. Nelineynyye volny [Nonlinear waves]. M.: Lenand, 2017. 312 p.
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