Analysis of Dynamics of Boundary Shape Perturbation of a Rotating Elastoplastic Radially Inhomogeneous Plane Circular Disk: Analytical Approach
Affiliation(s)
Cherkasy National Bohdan Khmelnytsky University, Cherkasy, Ukraine. Stability of Processes Department, S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.
Cherkasy National Bohdan Khmelnytsky University, Cherkasy, Ukraine. Stability of Processes Department, S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.
ABSTRACT
For a rotating inhomogeneous circular disk a way of calculating dynamics of boundary shape perturbation and failure of bearing capacity is proposed in terms of small parameter method. Characteristic equation of plastic zone critical radius is obtained as a first approximation. A formula of critical angular velocity is derived which determines the stability loss of the disc according to the self-balanced form. Efficiency of the proposed method is shown by an illustrative example considered in Section 7. Values of critical angular velocity of rotation are found numerically for different parameters of the disc.
For a rotating inhomogeneous circular disk a way of calculating dynamics of boundary shape perturbation and failure of bearing capacity is proposed in terms of small parameter method. Characteristic equation of plastic zone critical radius is obtained as a first approximation. A formula of critical angular velocity is derived which determines the stability loss of the disc according to the self-balanced form. Efficiency of the proposed method is shown by an illustrative example considered in Section 7. Values of critical angular velocity of rotation are found numerically for different parameters of the disc.
Cite this paper
D. Lila and А. Martynyuk, "Analysis of Dynamics of Boundary Shape Perturbation of a Rotating Elastoplastic Radially Inhomogeneous Plane Circular Disk: Analytical Approach," Applied Mathematics, Vol. 3 No. 5, 2012, pp. 451-456. doi: 10.4236/am.2012.35068.
D. Lila and А. Martynyuk, "Analysis of Dynamics of Boundary Shape Perturbation of a Rotating Elastoplastic Radially Inhomogeneous Plane Circular Disk: Analytical Approach," Applied Mathematics, Vol. 3 No. 5, 2012, pp. 451-456. doi: 10.4236/am.2012.35068.
References
[1] K. B. Bitseno and R. Grammel, “Technical Dynamics,” Vol. 2, Gosudarstvennoe Izdatelstvo Tekhniko-Teoreticheskoy Literatury, Moscow and Leningrad, 1952. [2] A. E. H. Love, “A Treatise on the Mathematical Theory of Elasticity,” Dover Publications, New York, 1927. [3] S. P. Timoshenko and J. N. Goodier, “Theory of Elasticity,” McGraw-Hill, New York, 1934. [4] А. N. Guz and Yu. N. Nemish, “Method of Boundary Form Perturbation in the Mechanics of Continua,” Vyshcha Shkola, Kiev, 1989. [5] V. V. Sokolovsky, “Plasticity Theory,” Vysshaya Shkola, Moscow, 1969. [6] A. N. Guz and I. Yu. Babich, “Three-Dimensional Stability Theory of Deformable Bodies,” Naukova Dumka, Kiev, 1985. [7] A. Nadai, “Plasticity and Fracture of Solid Bodies,” Vol. 1, Izdatelstvo Inostrannoy Literatury, Moscow, 1954. [8] D. D. Ivlev, “Continuum Mechanics,” Vol. 2, Phismatlit, Moscow, 2002. [9] D. D. Ivlev, “On the Loss of Bearing Capacity of Rotating Discs, Close to Circular Ones,” Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, No. 1, 1957, pp. 141-144. [10] D. V. Georgievskii, “Small Perturbations of an Undeformed State in Media with Yield Stress,” Doklady Physics, Vol. 48, No. 10, 2003, pp. 590-593. doi:10.1134/1.1623545 [11] D. D. Ivlev and L. V. Yershov, “Perturbation Method in the Theory of Elastoplastic Body,” Nauka, Moscow, 1978. [12] L. V. Yershov and D. D. Ivlev, “On the Stability Loss of Rotating Discs,” Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, No. 1, 1958, pp. 124-125. [13] D. M. Lila, “Eccentric Form of Stability Loss of a Rotating Elastoplastic Disc,” Reports of the National Academy of Sciences of Ukraine, No. 2, 2011, pp. 49-53. [14] D. M. Lila and А. А. Martynyuk, “About the Stability Loss of a Rotating Elastoplastic Circular Disc,” Reports of the National Academy of Sciences of Ukraine, No. 1, 2011, pp. 44-51. [15] D. M. Lila and А. A. Martynyuk, “Development of Instability in a Rotating Elastoplastic Annular Disk,” International Applied Mechanics, Vol. 48, No. 2, 2012, pp. 224233. [16] D. M. Lila, “On the Instability of Rotating Elastoplastic Stepped Annular Disc, Loaded over the Boundary in the Middle Plane,” International Applied Mechanics (to be published). [17] D. M. Lila and А. A. Martynyuk, “Stability Loss of Rotating Elastoplastic Discs of the Specific Form,” Applied Mathematics, Vol. 2, No. 5, 2011, pp. 579-585. doi:10.4236/am.2011.25077 [18] I. V. Demianushko and I. A. Birger, “Stress Calculation of Rotating Discs,” Mashinostroyeniye, Moscow, 1978. [19] M. Mazière, J. Besson, S. Forest, B. Tanguy, H. Chalons and F. Vogel, “Overspeed Burst of Elastoviscoplastic Rotating Disks—Part I: Analytical and Numerical Stability Analyses,” European Journal of Mechanics A/Solids, Vol. 28, No. 1, 2009, pp. 36-44. doi:10.1016/j.euromechsol.2008.07.008 [20] M. Mazière, J. Besson, S. Forest, B. Tanguy, H. Chalons and F. Vogel, “Overspeed Burst of Elastoviscoplastic Rotating Disks: Part II—Burst of a Superalloy Turbine Disk,” European Journal of Mechanics A/Solids, Vol. 28, No. 3, 2009, pp. 428-432. doi:10.1016/j.euromechsol.2008.10.002
[1] K. B. Bitseno and R. Grammel, “Technical Dynamics,” Vol. 2, Gosudarstvennoe Izdatelstvo Tekhniko-Teoreticheskoy Literatury, Moscow and Leningrad, 1952. [2] A. E. H. Love, “A Treatise on the Mathematical Theory of Elasticity,” Dover Publications, New York, 1927. [3] S. P. Timoshenko and J. N. Goodier, “Theory of Elasticity,” McGraw-Hill, New York, 1934. [4] А. N. Guz and Yu. N. Nemish, “Method of Boundary Form Perturbation in the Mechanics of Continua,” Vyshcha Shkola, Kiev, 1989. [5] V. V. Sokolovsky, “Plasticity Theory,” Vysshaya Shkola, Moscow, 1969. [6] A. N. Guz and I. Yu. Babich, “Three-Dimensional Stability Theory of Deformable Bodies,” Naukova Dumka, Kiev, 1985. [7] A. Nadai, “Plasticity and Fracture of Solid Bodies,” Vol. 1, Izdatelstvo Inostrannoy Literatury, Moscow, 1954. [8] D. D. Ivlev, “Continuum Mechanics,” Vol. 2, Phismatlit, Moscow, 2002. [9] D. D. Ivlev, “On the Loss of Bearing Capacity of Rotating Discs, Close to Circular Ones,” Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, No. 1, 1957, pp. 141-144. [10] D. V. Georgievskii, “Small Perturbations of an Undeformed State in Media with Yield Stress,” Doklady Physics, Vol. 48, No. 10, 2003, pp. 590-593. doi:10.1134/1.1623545 [11] D. D. Ivlev and L. V. Yershov, “Perturbation Method in the Theory of Elastoplastic Body,” Nauka, Moscow, 1978. [12] L. V. Yershov and D. D. Ivlev, “On the Stability Loss of Rotating Discs,” Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, No. 1, 1958, pp. 124-125. [13] D. M. Lila, “Eccentric Form of Stability Loss of a Rotating Elastoplastic Disc,” Reports of the National Academy of Sciences of Ukraine, No. 2, 2011, pp. 49-53. [14] D. M. Lila and А. А. Martynyuk, “About the Stability Loss of a Rotating Elastoplastic Circular Disc,” Reports of the National Academy of Sciences of Ukraine, No. 1, 2011, pp. 44-51. [15] D. M. Lila and А. A. Martynyuk, “Development of Instability in a Rotating Elastoplastic Annular Disk,” International Applied Mechanics, Vol. 48, No. 2, 2012, pp. 224233. [16] D. M. Lila, “On the Instability of Rotating Elastoplastic Stepped Annular Disc, Loaded over the Boundary in the Middle Plane,” International Applied Mechanics (to be published). [17] D. M. Lila and А. A. Martynyuk, “Stability Loss of Rotating Elastoplastic Discs of the Specific Form,” Applied Mathematics, Vol. 2, No. 5, 2011, pp. 579-585. doi:10.4236/am.2011.25077 [18] I. V. Demianushko and I. A. Birger, “Stress Calculation of Rotating Discs,” Mashinostroyeniye, Moscow, 1978. [19] M. Mazière, J. Besson, S. Forest, B. Tanguy, H. Chalons and F. Vogel, “Overspeed Burst of Elastoviscoplastic Rotating Disks—Part I: Analytical and Numerical Stability Analyses,” European Journal of Mechanics A/Solids, Vol. 28, No. 1, 2009, pp. 36-44. doi:10.1016/j.euromechsol.2008.07.008 [20] M. Mazière, J. Besson, S. Forest, B. Tanguy, H. Chalons and F. Vogel, “Overspeed Burst of Elastoviscoplastic Rotating Disks: Part II—Burst of a Superalloy Turbine Disk,” European Journal of Mechanics A/Solids, Vol. 28, No. 3, 2009, pp. 428-432. doi:10.1016/j.euromechsol.2008.10.002