Schmidt-Ishlinskii Yield Criterion and a Rotating Cylinder with a ‎Rigid Inclusion

Document Type : Research Paper

Author

Institute of Machinery and Metallurgy, Khabarovsk Federal Research Center FEB RAS, Metallurgov, 1, Komsomolsk-on-Amur, 681005, Russian Federation

Abstract

An elastoplastic rotating cylinder with a rigid inclusion and fixed ends is investigated. The analysis is based on infinitesimal strain theory, Schmidt-Ishlinskii yield criterion, and its associated flow rule and perfectly plastic material behavior. Both loading and unloading stages are studied. The closed-formed solutions for all stages of deformation including secondary plastic flow are obtained. The results are illustrated by the distributions of the stresses and plastic strains in a cylinder rotating at different speeds.

Keywords

Main Subjects

[1] Conway, S.L., Shinbrot, T., Glasser, B.J., A Taylor vortex analogy in granular flows, Nature, 431(7007), 2004, 433–437.
[2] Neeleman, M., Prochaska, J.X., Kanekar, N., Rafelski, M., A cold, massive, rotating disk galaxy 1.5 billion years after the Big Bang, Nature, 581(7808), 2020, 269–272.
[3] Balbus, S.A., Spinning discs in the lab, Nature, 444(7117), 2006, 281–283.
[4] Zare, H.R., Darijani, H., A novel autofrettage method for strengthening and design of thick-walled cylinders, Materials & Design, 105, 2016, 366–374.
[5] Zare, H.R., Darijani, H., Strengthening and design of the linear hardening thick-walled cylinders using the new method of rotational autofrettage, International Journal of Mechanical Sciences, 124–125, 2017, 1–8.
[6] Kamal, S.M., Dixit, U.S., Roy, A., Liu, Q., Silberschmidt, V.V., Comparison of plane-stress, generalized-plane-strain and 3D FEM elastic–plastic analyses of thick-walled cylinders subjected to radial thermal gradient, International Journal of Mechanical Sciences, 131–132, 2017, 744–752.
[7] Gamer, U., Sayir, M., Elastic-plastic stress distribution in a rotating solid shaft, Zeitschrift für angewandte Mathematik und Physik, 35(5), 1984, 601–617.
[8] Gamer, U., Mack, W., Varga, I., Rotating elastic-plastic solid shaft with fixed ends, International Journal of Engineering Science, 35(3), 1997, 253–267.
[9] Lindner, T., Mack, W., Residual stresses in an elastic-plastic solid shaft with fixed ends after previous rotation, ZAMM - Journal of Applied Mathematics and Mechanics, 78(2), 1998, 75–86.
[10] Gamer, U., Lance, R.H., Stress distribution in a rotating elastic-plastic tube, Acta Mechanica, 50(1–2), 1983, 1–8.
[11] Prokudin, A.N., Firsov, S.V., Elastoplastic deformation of a rotating hollow cylinder with a rigid casing, PNRPU Mechanics Bulletin, 4, 2019, 120–135.
[12] Prokudin, A.N., Exact elastoplastic analysis of a rotating cylinder with a rigid inclusion under mechanical loading and unloading, ZAMM - Journal of Applied Mathematics and Mechanics, 100(3), 2020, e201900213.
[13] Mack, W., The rotating elastic–plastic solid shaft with free ends, Technische Mechanik, 12, 1991, 119–124.
[14] Mack, W., Rotating elastic-plastic tube with free ends, International Journal of Solids and Structures, 27(11), 1991, 1461–1476.
[15] Sofiyev, A.H., Schnack, E., Demir, F., Elasto-plastic stability of circular cylindrical shells subjected to axial load, varying as a power function of time, Structural Engineering and Mechanics, 24(5), 2006, 621–639.
[16] Eraslan, A.N., Arslan, E., Plane strain analytical solutions to rotating partially plastic graded hollow shafts, Turkish Journal of Engineering and Environmental Sciences, 31(5), 2007, 273–287.
[17] Akis, T., Eraslan, A.N., Exact solution of rotating FGM shaft problem in the elastoplastic state of stress, Archive of Applied Mechanics, 77(10), 2007, 745–765.
[18] Nejad, M.Z., Fatehi, P., Exact elasto-plastic analysis of rotating thick-walled cylindrical pressure vessels made of functionally graded materials, International Journal of Engineering Science, 86(Supplement C), 2015, 26–43.
[19] Eraslan, A.N., On the linearly hardening rotating solid shaft, European Journal of Mechanics - A/Solids, 22(2), 2003, 295–307.
[20] Gamer, U., Elastic-plastic deformation of the rotating solid disk, Ingenieur-Archiv, 54(5), 1984, 345–354.
[21] Gamer, U., The Elastic-Plastic Stress-Distribution in the Rotating Annulus and in the Annulus Under External-Pressure, ZAMM - Journal of Applied Mathematics and Mechanics, 64(4), 1984, T126–T128.
[22] Guven, U., The fully plastic rotating disk with rigid inclusion, ZAMM - Journal of Applied Mathematics and Mechanics, 77(9), 1997, 714–716.
[23] Güven, U., Elastic-Plastic Rotating Disk with Rigid Inclusion, Mechanics of Structures and Machines, 27(1), 1999, 117–128.
[24] Güven, U., Parmaksizoğlu, C., Altay, O., Elastic-Plastic Rotating Annular Disk with Rigid Casing, ZAMM - Journal of Applied Mathematics and Mechanics, 79(7), 1999, 499–503.
[25] Güven, U., Elastic-plastic stress distribution in the rotating annular disk with variable thickness, Archive of Applied Mechanics, 61(8), 1991, 548–554.
[26] Güven, U., Elastic-plastic stress distribution in a rotating hyperbolic disk with rigid inclusion, International Journal of Mechanical Sciences, 40(1), 1998, 97–109.
[27] Eraslan, A.N., Orcan, Y., Elastic–plastic deformation of a rotating solid disk of exponentially varying thickness, Mechanics of Materials, 34(7), 2002, 423–432.
[28] Eraslan, A N., Orcan, Y., On the rotating elastic–plastic solid disks of variable thickness having concave profiles, International Journal of Mechanical Sciences, 44(7), 2002, 1445–1466.
[29] Orcan, Y., Eraslan, A.N., Elastic–plastic stresses in linearly hardening rotating solid disks of variable thickness, Mechanics Research Communications, 29(4), 2002, 269–281.
[30] Eraslan, A.N., Von Mises’ yield criterion and nonlinearly hardening rotating shafts, Acta Mechanica, 168(3–4), 2004, 129–144.
[31] Rees, D.W.A., Elastic-Plastic Stresses in Rotating Discs by von Mises and Tresca, ZAMM - Journal of Applied Mathematics and Mechanics, 79(4), 1999, 281–288.
[32] Eraslan, A.N., Von mises yield criterion and nonlinearly hardening variable thickness rotating annular disks with rigid inclusion, Mechanics Research Communications, 29(5), 2002, 339–350.
[33] Eraslan, A.N., Stress distributions in elastic-plastic rotating disks with elliptical thickness profiles using Tresca and von Mises criteria, ZAMM - Journal of Applied Mathematics and Mechanics, 85(4), 2005, 252–266.
[34] Eraslan, A.N., Mack, W., A computational procedure for estimating residual stresses and secondary plastic flow limits in nonlinearly strain hardening rotating shafts, Forschung im Ingenieurwesen, 69(2), 2005, 65–75.
[35] Alexandrova, N.N., Alexandrov, S., Vila Real, P.M.M., Displacement Field and Strain Distribution in a Rotating Annular Disk, Mechanics Based Design of Structures and Machines, 32(4), 2004, 441–457.
[36] Alexandrova, N.N., Alexandrov, S., Vila Real, P.M.M., Analysis of stress and strain in a rotating disk mounted on a rigid shaft, Theoretical and Applied Mechanics, 33(10), 2006, 65–90.
[37] Aleksandrova N.N., Application of Mises yield criterion to rotating solid disk problem, International Journal of Engineering Science, 51, 2012, 333–337.
[38] Aleksandrova, N.N., Exact deformation analysis of a solid rotating elastic–perfectly plastic disk, International Journal of Mechanical Sciences, 88, 2014, 55–60.
[39] Lomakin, E., Alexandrov, S., Jeng, Y.-R., Stress and strain fields in rotating elastic/plastic annular discs, Archive of Applied Mechanics, 86(1–2), 2016, 235–244.
[40] Schmidt, R., Über den Zusammenhang von Spannungen und Formänderungen im Verfestigungsgebiet, Ingenieur-Archiv, 3(3), 1932, 215–235.
[41] Ishlinsky, A.Yu., Hypothesis of strength of shape change (in Russian: Gipoteza prochnosti formoizmeneniya), Uchebnye Zapiski Moskovskogo Universiteta, Mekhanika, 46, 1940, 104–114.
[42] Haythornthwaite, R.M., Range of Yield Conditions in Ideal Plasticity, Journal of the Engineering Mechanics Division, 87(6), 1961, 117–134.
[43] Yu, M.-h., Twin shear stress yield criterion, International Journal of Mechanical Sciences, 25(1), 1983, 71–74.
[44] Ivlev, D.D., On the development of a theory of ideal plasticity, Journal of Applied Mathematics and Mechanics, 22(6), 1958, 1221–1230.
[45] Naghdi, P.M., Stress-strain relations in plasticity and thermoplasticity, Proc. 2nd Symp. Naval Structural Mechanics (Providence, RI 1959), Pergamon Press, 1960, 121–167.
[46] Yu, M.-h., Unified Strength Theory and Its Applications, Springer, Singapore, 2018.
[47] Sokolovsky, V.V., Theory of plasticity (in Russian: Teoriya plastichnosti), Vysshaya Shkola, Moscow, 1969.
[48] Kolupaev, V. A., Yu, M.-H., Altenbach, H., Fitting of the strength hypotheses, Acta Mechanica, 227(6), 2016, 1533–1556.
[49] Cai, Q., Pang, M., Zhang, Y.-Q., Liu, X., Elastic-plastic stress distribution of rotating annular disc based on twin-shear stress yield criterion, Journal of Zhejiang University (Engineering Science), 42(9), 2008, 1540–1544.
[50] Prokudin, A.N., Elastic-plastic analysis of rotating solid shaft by maximum reduced stress yield criterion, Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki, 24(1), 2020, 74–94.
[51] Ma, G., Hao, H., Miyamoto, Y., Limit angular velocity of rotating disc with unified yield criterion, International Journal of Mechanical Sciences, 43(5), 2001, 1137–1153.
[52] Xie, X., Wang, L., Zhang, Y. Unified solution of a borehole problem with size effect, Acta Mechanica, 225(6), 2014, 1769–1778.
[53] Liang, J., Li, Y. Analytical solutions for limit loads of simply supported conical shells under internal pressure with unified yield criterion, International Journal of Pressure Vessels and Piping, 172, 2019, 145–152.
[54] Koiter, W.T., Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface, Quarterly of Applied Mathematics, 11(3), 1953, 350–354.