Non-linear behavior of functionally graded elastoplastic beam under torsion

Year 2024, Issue: 057, 27 - 50, 30.06.2024

Murat Karaca , Bahadır Alyavuz

https://doi.org/10.59313/jsr-a.1415211

Abstract

The torsional behavior of beams graded in one and two directions under large displacements and angular deformations was analyzed using the power law and sinusoidal functions. Functionally graded material is elastoplastic, consisting of ceramic and metal. A nonlinear finite element method with isoparametric hexahedral elements was used. The finite element formulation was developed by using the updated Lagrangian formulation based on the virtual displacement principle. An iterative solution using Newton-Raphson and updated Newton-Raphson methods was used to solve the nonlinear equation system. The propagation of the plastic region was calculated based on the flow theory of plasticity. Elastoplastic behavior and effective material properties were determined according to the TTO model. Numerical investigations have shown that functionally graded beams behave quite differently from homogeneous beams under torsion. Yielding of the material starts at the outer boundaries of the section of the homogeneous beams, and the plastic region propagates symmetrically. On the other hand, yielding and propagation of plastic regions tend to shift to regions with more ceramic volume with higher effective Young modulus in functionally graded beams. Beams graded in the axial direction have a non-linear variation of rotation angle along the axial direction, unlike beams graded in section and pure metal beams. The amount of non-linearity increases with increasing volume of the ceramic material, which has higher torsional stiffness. Unlike homogeneous beams, the largest shear stresses can occur within the section rather than at the outer boundaries of the section. In beams graded from ceramic to metal using the power law, the section moves along the transverse direction in addition to the rotation. This transverse displacement occurs in the grading direction, and its magnitude is about 3% of the thickness at 12.5° rotation angle. Also, the shear stresses are not zero in the section's midpoint. The effects of material distribution on displacements, stresses, and plastic region propagation were examined, and essential points were reported.

Keywords

Functionally graded material, torsion, hexahedral finite element

References

• [1] M. Koizumi, “FGM activities in Japan,” Composites Part B, vol. 28, no. 1, pp. 1-4, 1997, doi: 10.1016/S1359-8368(96)00016-9.
• [2] T. Hirai, “Functional gradient materials,” in. Materials Science and Technology, 3 b., vol. 17B, R. J. Brook, Ed., Weinheim, VCH Verlagsgesellschaft mbH, 1996, pp. 293-341.
• [3] S. H. Chi and Y. L. Chung, “Mechanical behavior of functionally graded material plates under transverse load-Part I: Analysis,” International Journal of Solids and Structures, vol. 43, no. 13, pp. 3657-3674, June 2006, doi: 10.1016/j.ijsolstr.2005.04.011.
• [4] F. Tornabene and E. Viola, “Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution,” European Journal of Mechanics A/Solids, vol. 28, no. 5, pp. 991-1013, Sept. 2009, doi: 10.1016/j.euromechsol.2009.04.005.
• [5] V. Boggarapu, R. Gujjala, S. Ojha, S. Acharya, P. V. Babu, S. Chowdary and D. k. Gara, “State of the art in functionally graded materials,” Composite Structures, vol. 262, p. 113596, April 2021, doi: 10.1016/j.compstruct.2021.113596.
• [6] R. Hill, “A self-consistent mechanics of composite materials”, J. Mech. Phys. Solids, vol. 13, no. 4, pp. 213-222, Aug. 1965, doi: 10.1016/0022-5096(65)90010-4.
• [7] T. Mori and K. Tanaka, “Average stress in matrix and average elastic energy of materials with misfitting inclusions,” Acta Metallaugica, vol. 21, no. 5, pp. 571-574, May 1973, doi: 10.1016/0001-6160(73)90064-3.
• [8] M. M. Gasik and K. R. Lilius, “Evaluation of properties of W-Cu functional gradient materials by micromechanical model,” Computational Materials Science, vol. 3, no. 1, pp. 41-49, Sept. 1994, doi: 10.1016/0927-0256(94)90151-1.
• [9] T. Reiter and G. J. Dvorak, “Micromechanical models for graded composite materials: II. thermomechanical loading,” J. Mech. Phys. Solids, vol. 46, no. 9, pp. 1655-1673, Sept. 1998, doi: 10.1016/S0022-5096(97)00039-2.
• [10] I. Tamura, Y. Tomota and H. Ozawa, “Strenght and ductility of Fe-Ni-C alloys composed of austenite and martensite with various strenght,” Proceeding of the Third International Conference on Strenght of Metals and Alloys, Cambridge Institute of Metals, vol. 1, pp. 611-615, 1973.
• [11] R. L. Williamson, B. H. Rabin and J. T. Drake, “Finite element analysis of thermal residual stresses at graded ceramic metal interfaces. Part I. Model description and geometrical effects,” Journal of Applied Physics, vol. 74, pp. 1310-1320, July 1993, doi: 10.1063/1.354910.
• [12] H. Huang and Q. Han, “Elastoplastic buckling of axially loaded functionally graded material cylindrical shells,” Composite Structures, vol. 117, pp. 135-142, Nov. 2014, doi: 10.1016/j.compstruct.2014.06.018.
• [13] Y. Zhang, H. Huang and Q. Han, “Buckling of elastoplastic functionally graded cylindrical shells under combined compression and pressure,” Composites: Part B, vol. 69, pp. 120-126, Feb. 2015, doi: 10.1016/j.compositesb.2014.09.024.
• [14] H. Zafarmand and M. Kadkhodayan, “Nonlinear material and geometric analysis of thick functionally graded plates with nonlinear strain hardening using nonlinear finite element method,” Aerospace Science and Technology, vol. 92, pp. 930-944, Sept. 2019, doi: 10.1016/j.ast.2019.07.015.
• [15] F. J. Rooney and M. Ferrari, “Torsion and flexure of inhomogeneous elements,” Composites Engineering, vol. 5, no. 7, pp. 901-911, 1995, doi: 10.1016/0961-9526(95)00043-M.
• [16] C. O. Horgan and A. M. Chan, “Torsion of functionally graded isotropic linearly elastic bars,” Journal of Elasticity, vol. 52, pp. 181-199, Aug. 1998, doi: 10.1023/A:1007544011803.
• [17] C. O. Horgan, “On the torsion of functionally graded anisotropic linearly elastic bars,” IMA Journal of Applied Mathematics, vol. 72, pp. 556-562, Oct. 2007, doi: 10.1093/imamat/hxm027.
• [18] R. C. Batra, “Torsion of a functionally graded cylinder,” AIAA Journal, vol. 44, no. No.6, pp. 1363-1365, June 2006, doi: 10.2514/1.19555.
• [19] U. Anita, “Implementation of the method of fundamental solutions and homotopy analysis method for solving a torsion problem of a rod made of functionally graded material,” Advanced Materials Research, vols. 123-125, pp. 551-554, Aug. 2010, doi: 10.4028/www.scientific.net/AMR.123-125.551.
• [20] M. R. Hematiyan and E. Estakhrian, “Torsion of functionally graded open-section members,” International Journal of Applied Mechanics, vol. 4, no. 2, p. 1250020, June 2012, doi: 10.1142/S1758825112500202.
• [21] T. T. Nguyen, N. I. Kim and J. Lee, “Analysis of thin-walled open-section beams with functionally graded materials,” Composite Structures, vol. 138, pp. 75-83, March 2016, doi: 10.1016/j.compstruct.2015.11.052.
• [22] I. Ecdesi and A. Baksa, “Torsion of Functionally Graded Anisotropic Linearly Elastic Circular Cylinder,” Engineering Transactions, vol. 66, no. 4, pp. 413-426, 2018, doi: 10.24423/EngTrans.923.20181003.
• [23] G. J. Nie, A. Pydah and R. C. Batra, “Torsion of bi-directional functionally graded truncated conical cylinders,” Composite Structures, vol. 210, pp. 831-839, Feb. 2019, doi: 10.1016/j.compstruct.2018.11.081.
• [24] E. T. Akinlabi, M. N. Mikhin and E. V. Murashkin, “Functionally graded prismatic triangular rod under torsion,” Journal of Physics: Conf. Series, vol. 1474, p. 012003, 2020, doi: 10.1088/1742-6596/1474/1/012003.
• [25] I. Ecsedi, “Non-uniform torsion of functionally graded anisotropic bar of an elliptical cross section,” Acta Mech, vol. 231, pp. 2947-2953, July 2020, doi: 10.1007/s00707-020-02682-y.
• [26] G. C. Tsiatas and N. G. Babouskos, “Elastic-plastic analysis of functionally graded bars under torsional loading,” Composite Structures, vol. 176, pp. 254-267, Sept. 2017, doi: 10.1016/j.compstruct.2017.05.044.
• [27] M. Aminbaghai, J. Murin, V. Kutis, J. Hrabovsky, M. Kostolani and H. A. Mang, “Torsional warping elastostatic analysis of FGM beams with longitudinally varying material properties,” Engineering Structures, vol. 200, p. 109694, Dec. 2019, doi: 10.1016/j.engstruct.2019.109694.
• [28] P. K. Singh, M. Kumar and S. Mishra, “Finite element analysis of functionally graded bar under torsional load,” Materials Today: Proceedings, vol. 56, pp. 2960-2966, 2022, doi: 10.1016/j.matpr.2021.11.012.
• [29] X. Chen, G. Nie and Z. Wu, “Non-uniform torsion analysis of functionally graded beams with solid or thin-walled section using hierarchical Legendre expansion functions,” Mechanics of Advanced Materials and Structures, vol. 29, no. 14, pp. 2074-2097, 2022, doi: 10.1080/15376494.2020.1851828.
• [30] E. Mahmoodi and P. Malekzadeh, “Analytical solutions of multiple cracks and cavities in a rectangular cross-section bar coated by a functionally graded layer under torsion,” Arch Appl Mech, vol. 91, pp. 2189-2209, May 2021, doi: 10.1007/s00419-020-01877-y.
• [31] I. Ecdesi and A. Baksa, “Saint-Venant torsion of functional graded orthotropic piezoelectric hollow circular cylinder,” International Journal on Applied Physics and Engineering, vol. 2, pp.22-27, 2023, doi: 10.37394/232030.2023.2.4.
• [32] F. Mehralian and Y. T. Beni, “Size-dependent torsional buckling analysis of functionally graded cylindrical shell,” Composites, vol. Part B, no. 94, pp. 11-25, June 2016, doi: 10.1016/j.compositesb.2016.03.048.
• [33] T.-T. Nguyen, P. T. Thang and J. Lee, “Flexural-torsional stability of thin-walled functionally graded open-section beams,” Thin-Walled Structures, vol. 110, pp. 88-96, Jan. 2017, doi: 10.1016/j.tws.2016.09.021.
• [34] Y. Wang, C. Feng, Z. Zhao, F. Lu and J. Yang, “Torsional buckling of graphene platelets (GPLs) reinforced functionally graded cylindrical shell with cutout,” Composite Structures, vol. 197, pp. 72-79, Aug. 2018, doi.org/10.1016/j.compstruct.2018.05.056.
• [35] M. Bocciarelli, G.Bolzon and G.Maier, “A constitutive model of metal-ceramic functionally graded material behavior: Formulation and parameter identification,” Computational Materials Science, vol. 43, pp. 16-26, July 2008, doi: 10.1016/j.commatsci.2007.07.047.
• [36] D. Owen and E. Hinton, Finite Elements in Plasticity: Theory and practice. Swansea: Pineridge Press Limited, 1980.
• [37] K.-J. Bathe, Finite Element Procedures. New Jersey: Prentice-Hall Inc., 1996.
• [38] Z.-H. Jin, G. H. Paulino and R. H. D. Jr., “Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials,” Engineering Fracture Mechanics, vol. 70, no. 14, pp. 1885-1912, Sept. 2003, doi: 10.1016/S0013-7944(03)00130-9.
• [39] Oracle Corporation, JAVA SE. Version 18.0.0, Programming Language.
• [40] The Apache Software Foundation, NETBEANS IDE. Version 13, Integrated Development Environment.
• [41] The Math Works Inc., MATLAB. Version 2023a, Computer Software.
• [42] D. K. Nguyen, K. V. Nguyen, V. M. Dinh, B. S. Gan and S. Alexandrow, “Nonlinear bending of elastoplastic functionally graded ceramic-metal beams subjected to nonuniform distributed loads,” Applied Mathematics and Computation, vol. 333, pp. 443-459, Sept. 2018, doi: 10.1016/j.amc.2018.03.100.