Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

doi:10.1007/s10483-020-2664-8

Abstract

Abstract: Advancements in manufacturing technology, including the rapid development of additive manufacturing (AM), allow the fabrication of complex functionally graded material (FGM) sectioned beams. Portions of these beams may be made from different materials with possibly different gradients of material properties. The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional (1D) finite element analysis. For this purpose, a sample beam is divided into discrete elements, and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory. Then, Hamilton's principle is used to derive the equations of motion for the beam. The effects of material properties and dimensions of FGM sections on the beam's natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model (TM). The presented model is validated by comparison with three-dimensional (3D) finite element simulations of the first three mode shapes of the beam.

Key words: finite element model (FEM), dynamics, functionally graded material (FGM), Timoshenko beam theory

2010 MSC Number:

N. V. VIET, W. ZAKI, Quan WANG. Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis. Applied Mathematics and Mechanics (English Edition), 2020, 41(12): 1787-1804.

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References

[1] OXMAN, N., KEATING, S., and TSAI, E. Functionally Graded Rapid Prototyping, Mediated Matter Group, MIT Media Lab, Cambridge (2004)
[2] SHAKIL, M., AHMAD, M., TARIQ, N. H., HASAN, B. A., AKHTER, J. I., AHMED, E., MEHMOOD, M., CHOUDHRY, M. A., and IQBAL, M. Microstructure and hardness studies of electron beam welded inconel 625 and stainless steel 304L. Vacuum, 110, 121-126(2014)
[3] SURESH, S. and MORTENSEN, A. Fundamentals of Functionally Graded Materials, IOM Communications Ltd, London (1998)
[4] BIRMAN, V. and BYRD, L. W. Modeling and analysis of functionally graded materials and structures. Applied Mechanics Reviews, 60(5), 195(2007)
[5] SUN, Z. and ION, J. C. Review laser welding of dissimilar metal combinations. Journal of Materials Science, 30, 4205-4214(1995)
[6] LIMA, D. D., MANTRI, S. A., MIKLER, C. V., CONTIERI, R., YANNETTA, C. J., CAMPO, K. N., LOPES, E. S., STYLES, M. J., BORKAR, T., CARAM, R., and BANERJEE, R. Laser additive processing of a functionally graded internal fracture fixation plate. Materials and Designs, 130, 8-15(1995)
[7] LOH, G. H., PEI, E., HARRISON, D., and MONZÓN, M. D. An overview of functionally graded additive manufacturing. Additive Manufacturing, 23, 34-44(2018)
[8] BOBBIO, L. D., OTIS, R. A., BORGONIA, J. P., DILLON, R. P., SHAPIRO, A. A., LIU, Z. K., and BEESE, A. M. Additive manufacturing of a functionally graded material from Ti-6Al-4V to invar:experimental characterization and thermodynamic calculations. Acta Materialia, 127, 133-142(2017)
[9] CARROLL, B. E., OTIS, R. A., PAUL, J., SUH, J. O., DILLON, R. P., SHAPIRO, A. A., HOFMANN, D., LIU, Z. K., and BEESE, A. M. Functionally graded material of 304L stainless steel and inconel 625 fabricated by directed energy deposition:characterization and thermodynamic modeling. Acta Materialia, 108, 46-54(2016)
[10] LIU, Y., LIU, C., LIU, W., MA, Y., ZHANG, C., CAI, Q., and LIU, B. Microstructure and properties of Ti/Al lightweight graded material by direct laser deposition. Materials Science and Technology, 34, 945-951(2017)
[11] BOBBIO, L. D., BOCKLUND, B., OTIS, R., BORGONIA, J. P., DILLON, R. P., SHAPIRO, A. A., MCENERNEY, B., LIU, Z. K., and BEESE, A. M. Characterization of a functionally graded material of Ti-6Al-4V to 304L stainless steel with an intermediate V section. Journal of Alloys and Compounds, 742, 1031-1036(2018)
[12] BOBBIO, L. D., BOCKLUND, B., OTIS, R., BORGONIA, J. P., DILLON, R. P., SHAPIRO, A. A., MCENERNEY, B., LIU, Z. K., and BEESEA, A. M. Characterization of a functionally graded material of Ti-6Al-4V to 304L stainless steel with an intermediate V section. Journal of Materials Research, 33(11), 1642-1649(2018)
[13] ZHANG, X., CHEN, Y., and LIOU, F. Fabrication of SS316LIN625 functionally graded materials by powder-fed directed energy deposition. Science and Technology of Welding and Joining, 24, 504-516(2019)
[14] HU, S., GAGNOUD, A., FAUTRELLE, Y., MOREAU R., and LI, X. Fabrication of aluminum alloy functionally graded material using directional solidification under an axial static magnetic field. Scientific Report, 8, 7945(2018)
[15] REZAPOOR, M., RAZAVI, M., ZAKERI, M., RAHIMIPOUR, M. R., and NIKZAD, L. Fabrication of functionally graded Fe-TiC wear resistant coating on CK45 steel substrate by plasma spray and evaluation of mechanical properties. Ceramics International, 44, 22378-22386(2018)
[16] WANG, X. and LI, S. Free vibration analysis of functionally graded material beams based on Levinson beam theory. Applied Mathematics and Mechanics (English Edition), 37(7), 861-878(2016) https://doi.org/10.1007/s10483-016-2094-9
[17] LI, S. R., WAN, Z. Q., and ZHANG, J. H. Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories. Applied Mathematics and Mechanics (English Edition), 35, 591-606(2014) https://doi.org/10.1007/s10483-014-1815-6
[18] LI, S. R., SU, H. D., and CHENG, C. J. Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment. Applied Mathematics and Mechanics (English Edition), 30, 969-982(2009) https://doi.org/10.1007/s10483-009-0803-7
[19] HADJI, L., ATMANE, H. A., MECHAB, I., and ADDABEDIA, E. A. Free vibration of functionally graded sandwich plates using four-variable refined plate theory. Applied Mathematics and Mechanics (English Edition), 32, 925-942(2011) https://doi.org/10.1007/s10483-011-1470-9
[20] CAO, D. and GAO, Y. Free vibration of non-uniform axially functionally graded beams using the asymptotic development method. Applied Mathematics and Mechanics (English Edition), 40(1), 85-96(2019) https://doi.org/10.1007/s10483-019-2402-9
[21] LAL, A., SHEGOKAR, N. L., and SINGH, B. N. Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties. Applied Mathematical Modelling, 44, 274-295(2017)
[22] SIMSEK, M. Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Composite Structures, 133, 968-978(2015)
[23] SIMSEK, M. and KOCATURK, T. Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Composite Structures, 90, 465-473(2009)
[24] KAPURIA, S., BHATTACHARYYA, M., and KUMAR, A. N. Bending and free vibration response of layered functionally graded beams:a theoretical model and its experimental validation. Composite Structures, 82, 340-390(2008)
[25] LEE, J. K. and LEE, B. K. Free vibration and buckling of tapered columns made of axially functionally graded materials. Applied Mathematical Modelling, 75, 73-87(2019)
[26] ZHOU, Y. and ZHANG, X. Natural frequency analysis of functionally graded material beams with axially varying stochastic properties. Applied Mathematical Modelling, 67, 85-100(2009)
[27] VIET, N. V., ZAKI, W., and UMER, R. Analytical model of functionally graded material/shape memory alloy composite cantilever beam under bending. Composite Structures, 203, 764-776(2018)
[28] VIET, N. V. and ZAKI, W. Analytical investigation of the behavior of concrete beams reinforced with multiple circular superelastic shape memory alloy bars. Composite Structures, 210, 958-970(2019)
[29] VIET, N. V., ZAKI, W., and UMER, R. Bending models for superelastic shape memory alloy laminated composite cantilever beams with elastic core layer. Composites Part B, 147, 86-103(2018)
[30] TIMOSHENKO, S. P. On the transverse vibrations of bars of uniform cross-section. Philosophical Magazine, 125, 125-131(1922)
[31] ALSHORBAGY, A., ELTAHER, M., and MAHMOUD, F. Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling, 35, 412-425(2011)