Advances in materials Research

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Dynamic analysis of a porous microbeam model based on refined beam strain gradient theory via differential quadrature hierarchical finite element method

  • Ahmed Saimi (IS2M Laboratory, Faculty of Technology, University of Tlemcen) ;
  • Ismail Bensaid (IS2M Laboratory, Faculty of Technology, University of Tlemcen) ;
  • Ihab Eddine Houalef (IS2M Laboratory, Faculty of Technology, University of Tlemcen)
  • Received : 2022.04.15
  • Accepted : 2023.01.02
  • Published : 2023.06.25
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Abstract

In this paper, a size-dependent dynamic investigation of a porous metal foams microbeamsis presented. The novelty of this study is to use a metal foam microbeam that contain porosities based on the refined high order shear deformation beam model, with sinusoidal shear strain function, and the modified strain gradient theory (MSGT) for the first time. The Lagrange's principle combined with differential quadrature hierarchicalfinite element method (DQHFEM) are used to obtain the porous microbeam governing equations. The solutions are presented for the natural frequencies of the porous and homogeneoustype microbeam. The obtained results are validated with the analytical methods found in the literature, in order to confirm the accuracy of the presented resolution method. The influences of the shape of porosity distribution, slenderness ratio, microbeam thickness, and porosity coefficient on the free vibration of the porous microbeams are explored in detail. The results of this paper can be used in various design formetallic foammicro-structuresin engineering.

Keywords

Acknowledgement

We acknowledge with grateful thanks the support by the laboratory of mechanical and material systems engineering (IS2M) in university of Tlemcen, as well as the General Directorate of Scientific Research and Technological Development (DGRSDT) of the Ministry of Higher Education of Algeria.

References

  1. Abdelrahman, A. A., Esen, I., & Eltaher, M. A. (2021). Vibration response of Timoshenko perforated microbeams under accelerating load and thermal environment. Applied Mathematics and Computation, 407, 126307. https://doi.org/https://doi.org/10.1016/j.amc.2021.126307
  2. Ahmed, S., Abdelhamid, H., Ismail, B., & Ahmed, F. (2020). An Differential Quadrature Finite Element and the Differential Quadrature Hierarchical Finite Element Methods for the Dynamics Analysis of on Board Shaft. European Journal of Computational Mechanics, 29(4-6), 303-344. https://doi.org/10.13052/ejcm1779-7179.29461
  3. Akgoz, B., & Civalek, O. (2013). A size-dependent shear deformation beam model based on the strain gradient elasticity theory. International Journal of Engineering Science, 70, 1-14. https://doi.org/https://doi.org/10.1016/j.ijengsci.2013.04.004
  4. Assem, H., Hadjoui, A., & Saimi, A. (2022). Numerical analysis on the dynamics behavior of FGM rotor in thermal environment using h-p finite element method. Mechanics Based Design of Structures and Machines, 50(11), 3925-3948. https://doi.org/10.1080/15397734.2020.1824791
  5. Aydogdu, M. (2009). A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Physica E: Low-dimensional Systems and Nanostructures, 41(9), 1651-1655. https://doi.org/https://doi.org/10.1016/j.physe.2009.05.014
  6. Barati, M. R., & Zenkour, A. M. (2017). Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions. Composite Structures, 182, 91-98. https://doi.org/https://doi.org/10.1016/j.compstruct.2017.09.008
  7. Bardell, N. S. (1996). An engineering application of the h-p Version of the finite element method to the static analysis of a Euler-Bernoulli beam. Computers & Structures, 59(2), 195-211. https://doi.org/https://doi.org/10.1016/0045-7949(95)00252-9
  8. Bensaid, I., & Guenanou, A. (2017). Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities. Advances in materials Research, 6(1), 45-63. https://doi.org/10.12989/AMR.2017.6.1.045
  9. Bensaid, I., & Saimi, A. (2022). Dynamic investigation of functionally graded porous beams resting on viscoelastic foundation using generalised differential quadrature method. Australian Journal of Mechanical Engineering, 1-20. https://doi.org/10.1080/14484846.2021.2017115
  10. Bensaid, I., Saimi, A., & Civalek, O. (2022). Effect of two-dimensional material distribution on dynamic and buckling responses of graded ceramic-metal higher order beams with stretch effect. Mechanics of Advanced Materials and Structures, 1-17. https://doi.org/10.1080/15376494.2022.2142342
  11. Chai, Q., & Wang, Y. Q. (2022). Traveling wave vibration of graphene platelet reinforced porous joined conical-cylindrical shells in a spinning motion. Engineering Structures, 252, 113718. https://doi.org/https://doi.org/10.1016/j.engstruct.2021.113718
  12. Chen, D., Kitipornchai, S., & Yang, J. (2016). Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core. Thin-Walled Structures, 107, 39-48. https://doi.org/https://doi.org/10.1016/j.tws.2016.05.025
  13. Chen, D., Yang, J., & Kitipornchai, S. (2015). Elastic buckling and static bending of shear deformable functionally graded porous beam. Composite Structures, 133, 54-61. https://doi.org/https://doi.org/10.1016/j.compstruct.2015.07.052
  14. Ebrahimi, F., & Jafari, A. (2018). A four-variable refined shear-deformation beam theory for thermomechanical vibration analysis of temperature-dependent FGM beams with porosities. Mechanics of Advanced Materials and Structures, 25(3), 212-224. https://doi.org/10.1080/15376494.2016.1255820
  15. Ebrahimi, F., Mahmoodi, F., & Barati, M. R. (2017). Thermo-mechanical vibration analysis of functionally graded micro/nanoscale beams with porosities based on modified couple stress theory. Advances in materials Research, 6(3), 279.
  16. Fleck, N. A., Muller, G. M., Ashby, M. F., & Hutchinson, J. W. (1994). Strain gradient plasticity: Theory and experiment. Acta Metallurgica et Materialia, 42(2), 475-487. https://doi.org/https://doi.org/10.1016/0956-7151(94)90502-9
  17. Ihab Eddine, H., Ismail, B., Ahmed, S., & Abdelmadjid, C. (2023). Free Vibration Analysis of Functionally Graded Carbon Nanotube-Reinforced Higher Order Refined Composite Beams Using Differential Quadrature Finite Element Method. European Journal of Computational Mechanics, 31(4), 1-34. https://doi.org/10.13052/ejcm2642-2085.3143
  18. Jabbari, M., Mojahedin, A., Khorshidvand, A. R., & Eslami, M. R. (2014). Buckling Analysis of a Functionally Graded Thin Circular Plate Made of Saturated Porous Materials. Journal of Engineering Mechanics, 140(2), 287-295. https://doi.org/doi:10.1061/(ASCE)EM.1943-7889.0000663
  19. Jalaei, M. H., Thai, H. T., & Civalek, Ӧ. (2022). On viscoelastic transient response of magnetically imperfect functionally graded nanobeams. International Journal of Engineering Science, 172, 103629. https://doi.org/https://doi.org/10.1016/j.ijengsci.2022.103629
  20. Karamanli, A., & Aydogdu, M. (2019). On the vibration of size dependent rotating laminated composite and sandwich microbeams via a transverse shear-normal deformation theory. Composite Structures, 216, 290-300. https://doi.org/https://doi.org/10.1016/j.compstruct.2019.02.044
  21. Karamanli, A., & Aydogdu, M. (2020). Vibration of functionally graded shear and normal deformable porous microplates via finite element method. Composite Structures, 237, 111934. https://doi.org/https://doi.org/10.1016/j.compstruct.2020.111934
  22. Karamanli, A., Aydogdu, M., & Vo, T. P. (2021). A comprehensive study on the size-dependent analysis of strain gradient multi-directional functionally graded microplates via finite element model. Aerospace Science and Technology, 111, 106550. https://doi.org/https://doi.org/10.1016/j.ast.2021.106550
  23. Karamanli, A., & Vo, T. P. (2021a). Bending, vibration, buckling analysis of bi-directional FG porous microbeams with a variable material length scale parameter. Applied Mathematical Modelling, 91, 723-748. https://doi.org/https://doi.org/10.1016/j.apm.2020.09.058
  24. Karamanli, A., & Vo, T. P. (2021b). A quasi-3D theory for functionally graded porous microbeams based on the modified strain gradient theory. Composite Structures, 257, 113066. https://doi.org/https://doi.org/10.1016/j.compstruct.2020.113066
  25. Karamanli, A., Vo, T. P., & Civalek, O. (2022). Finite element formulation of metal foam microbeams via modified strain gradient theory. Engineering with Computers. https://doi.org/10.1007/s00366-022-01666-x
  26. Ke, L.-L., Wang, Y.-S., & Wang, Z.-D. (2011). Thermal effect on free vibration and buckling of sizedependent microbeams. Physica E: Low-dimensional Systems and Nanostructures, 43(7), 1387-1393. https://doi.org/https://doi.org/10.1016/j.physe.2011.03.009
  27. Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., & Tong, P. (2003). Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8), 1477-1508. https://doi.org/https://doi.org/10.1016/S0022-5096(03)00053-X
  28. Li, Y. S., Feng, W. J., & Cai, Z. Y. (2014). Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory. Composite Structures, 115, 41-50. https://doi.org/https://doi.org/10.1016/j.compstruct.2014.04.005
  29. Lim, C. W., Zhang, G., & Reddy, J. N. (2015). A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298-313. https://doi.org/https://doi.org/10.1016/j.jmps.2015.02.001
  30. Liu, C., Liu, B., Zhao, L., Xing, Y., Ma, C., & Li, H. (2017). A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains. International Journal for Numerical Methods in Engineering, 109(2), 174-197. https://doi.org/https://doi.org/10.1002/nme.5277
  31. Mindlin, R. D. (1965). Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1(4), 417-438. https://doi.org/https://doi.org/10.1016/0020-7683(65)90006-5
  32. Mirsalehi, M., Azhari, M., & Amoushahi, H. (2017). Buckling and free vibration of the FGM thin microplate based on the modified strain gradient theory and the spline finite strip method. European Journal of Mechanics - A/Solids, 61, 1-13. https://doi.org/https://doi.org/10.1016/j.euromechsol.2016.08.008
  33. Nguyen, N.-D., Nguyen, T.-N., Nguyen, T.-K., & Vo, T. P. (2022). A new two-variable shear deformation theory for bending, free vibration and buckling analysis of functionally graded porous beams. Composite Structures, 282, 115095. https://doi.org/https://doi.org/10.1016/j.compstruct.2021.115095
  34. Peano, A. (1976). Hierarchies of conforming finite elements for plane elasticity and plate bending. Computers & Mathematics with Applications, 2(3), 211-224. https://doi.org/https://doi.org/10.1016/0898-1221(76)90014-6
  35. Rezaei, A. S., & Saidi, A. R. (2016). Application of Carrera Unified Formulation to study the effect of porosity on natural frequencies of thick porous-cellular plates. Composites Part B: Engineering, 91, 361-370. https://doi.org/https://doi.org/10.1016/j.compositesb.2015.12.050
  36. Saimi, A., Fellah, A., & Hadjoui, A. (2021). Nonlinear dynamic analysis of the response to the excitation forces of a symmetrical on-board rotor mounted on hydrodynamic bearings using h-p finite elements method. Mechanics Based Design of Structures and Machines, 1-32. https://doi.org/10.1080/15397734.2021.1991806
  37. Saimi, A., & Hadjoui, A. (2016). An engineering application of the h-p version of the finite elements method to the dynamics analysis of a symmetrical on-board rotor. European Journal of Computational Mechanics, 25(5), 388-416. https://doi.org/10.1080/17797179.2016.1245597
  38. Simsek, M., & Reddy, J. N. (2013). Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. International Journal of Engineering Science, 64, 37-53. https://doi.org/https://doi.org/10.1016/j.ijengsci.2012.12.002
  39. Stolken, J. S., & Evans, A. G. (1998). A microbend test method for measuring the plasticity length scale. Acta Materialia, 46(14), 5109-5115. https://doi.org/https://doi.org/10.1016/S1359-6454(98)00153-0
  40. Teng, M. W., & Wang, Y. Q. (2021). Nonlinear forced vibration of simply supported functionally graded porous nanocomposite thin plates reinforced with graphene platelets. Thin-Walled Structures, 164, 107799. https://doi.org/https://doi.org/10.1016/j.tws.2021.107799
  41. Thai, H.-T., & Vo, T. P. (2012). Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. International Journal of Mechanical Sciences, 62(1), 57-66. https://doi.org/https://doi.org/10.1016/j.ijmecsci.2012.05.014
  42. Thai, S., Thai, H.-T., Vo, T. P., & Patel, V. I. (2017). Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis. Computers & Structures, 190, 219-241. https://doi.org/https://doi.org/10.1016/j.compstruc.2017.05.014
  43. Thang, P. T., Nguyen-Thoi, T., Lee, D., Kang, J., & Lee, J. (2018). Elastic buckling and free vibration analyses of porous-cellular plates with uniform and non-uniform porosity distributions. Aerospace Science and Technology, 79, 278-287. https://doi.org/https://doi.org/10.1016/j.ast.2018.06.010
  44. Touratier, M. (1991). An efficient standard plate theory. International Journal of Engineering Science, 29(8), 901-916. https://doi.org/10.1016/0020-7225(91)90165-Y
  45. Wang, C. M., Kitipornchai, S., Lim, C. W., & Eisenberger, M. (2008). Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory. Journal of Engineering Mechanics, 134(6), 475-481. https://doi.org/doi:10.1061/(ASCE)0733-9399(2008)134:6(475)
  46. Wang, Y. Q. (2018). Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronautica, 143, 263-271. https://doi.org/https://doi.org/10.1016/j.actaastro.2017.12.004
  47. Wang, Y. Q., Ye, C., & Zu, J. W. (2019). Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets. Aerospace Science and Technology, 85, 359-370. https://doi.org/https://doi.org/10.1016/j.ast.2018.12.022
  48. Wang, Y. Q., Zhao, H. L., Ye, C., & Zu, J. W. (2018). A Porous Microbeam Model for Bending and Vibration Analysis Based on the Sinusoidal Beam Theory and Modified Strain Gradient Theory. International Journal of Applied Mechanics, 10(05), 1850059. https://doi.org/10.1142/s175882511850059x
  49. Wang, Y. Q., Zhao, H. L., & Zu, C. Y. a. J. W. (2018). A Porous Microbeam Model for Bending and Vibration Analysis Based on the Sinusoidal Beam Theory and Modified Strain Gradient Theory. International Journal of Applied Mechanics, 10(5), 1850059.
  50. Wang, Y. Q., & Zu, J. W. (2017). Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerospace Science and Technology, 69, 550-562. https://doi.org/https://doi.org/10.1016/j.ast.2017.07.023
  51. Wattanasakulpong, N., Gangadhara Prusty, B., Kelly, D. W., & Hoffman, M. (2012). Free vibration analysis of layered functionally graded beams with experimental validation. Materials & Design (1980-2015), 36, 182-190. https://doi.org/https://doi.org/10.1016/j.matdes.2011.10.049
  52. Wattanasakulpong, N., & Ungbhakorn, V. (2014). Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerospace Science and Technology, 32(1), 111-120. https://doi.org/https://doi.org/10.1016/j.ast.2013.12.002
  53. Xing, Y., & Liu, B. (2009). High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain. International Journal for Numerical Methods in Engineering, 80(13), 1718-1742. https://doi.org/https://doi.org/10.1002/nme.2685
  54. Xu, H., Wang, Y. Q., & Zhang, Y. (2021). Free vibration of functionally graded graphene platelet-reinforced porous beams with spinning movement via differential transformation method. Archive of Applied Mechanics, 91(12), 4817-4834. https://doi.org/10.1007/s00419-021-02036-7
  55. Yang, F., Chong, A. C. M., Lam, D. C. C., & Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), 2731-2743. https://doi.org/https://doi.org/10.1016/S0020-7683(02)00152-X
  56. Ye, C., & Wang, Y. Q. (2021). Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells: internal resonances. Nonlinear Dynamics, 104(3), 2051-2069. https://doi.org/10.1007/s11071-021-06401-7