Size-dependent nonlinear post-buckling analysis of functionally graded porous Timoshenko microbeam with nonlocal integral models. (English) Zbl 1502.74043
Summary: Strain-driven (\(\varepsilon\)D) and stress-driven (\(\sigma\)D) two-phase local/nonlocal integral models (TPNIM) are applied to study the size-dependent nonlinear post-buckling behaviors of functionally graded porous Timoshenko microbeams. The differential governing equations and boundary conditions of motion are derived on the basis of the principle of minimum potential energy and expressed in nominal form. The integral relation between local and nonlocal stresses is transformed unitedly into equivalent differential form with constitutive constraints for \(\varepsilon\)D- and \(\sigma\)D-TPNIMs. Bending deflection and cross-sectional rotation are derived explicitly with Laplace transformation for linear buckling. Taking into account boundary conditions and constitutive constraints, a general eigenvalue problem is obtained to determine linear buckling mode shape (LBMS) and nominal buckling load (NBL). Local and nonlocal LBMS based Ritz-Galerkin methods and general differential quadrature method (GDQM) based Newton-Raphson’s method are utilized to obtain the numerical solutions for linear and nonlinear NBLs. Numerical results show that nonlocal LBMS based Ritz-Galerkin method and GDQM would lead to same prediction for linear NBLs as analytical method, and nonlocal LBMSs based Ritz-Galerkin and GDQM based Newton-Raphson’s method would lead to exact same prediction for nonlinear post-buckling loads. However, local LBMS based Ritz-Galerkin method is failed to provide accurate prediction for both linear NBLs and nonlinear post-buckling loads, through the difference between local and nonlocal LBMSs is not significant. The influence of nonlocal parameters, porous distribution patterns and boundary conditions on the linear buckling and nonlinear post-buckling behaviors are investigated numerically.
MSC:
| 74G60 | Bifurcation and buckling |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
Keywords:
Laplace transform; Ritz-Galerkin method; general differential quadrature method; Newton-Raphson methodReferences:
| [1] | Wang, B.; Zhou, S.; Zhao, J.; Chen, X., Size-dependent pull-in instability of electrostatically actuated microbeam-based MEMS, J Micromech Microeng, 21 (2011) |
| [2] | Gorgani, H. H.; Adeli, M. M.; Hosseini, M., Pull-in behavior of functionally graded micro/nano-beams for MEMS and NEMS switches, Microsyst Technol-Micro- Nanosyst-Inform Storage Process Syst, 25, 3165-3173 (2019) |
| [3] | Sun, Y. H.; Cheng, J. H.; Wang, Z. G.; Yu, Y. P.; Tian, L. Y.; Lu, J. A., Analytical approximate solution for nonlinear behavior of cantilever FGM MEMS beam with thermal and size dependency, Math Probl Eng, 2019 (2019) |
| [4] | Mao, J-J.; Zhang, W., Buckling and post-buckling analyses of functionally graded graphene reinforced piezoelectric plate subjected to electric potential and axial forces, Compos Struct, 216, 392-405 (2019) |
| [5] | Wang, Y.; Xie, K.; Fu, T.; Zhang, W., A third order shear deformable model and its applications for nonlinear dynamic response of graphene oxides reinforced curved beams resting on visco-elastic foundation and subjected to moving loads, Eng Comput, 38, 2805-2819 (2022) |
| [6] | Yang, F.; Chong, A. C.M.; Lam, D. C.C.; Tong, P., Couple stress based strain gradient theory for elasticity, Int J Solids Struct, 39, 2731-2743 (2002) · Zbl 1037.74006 |
| [7] | Wang, Y. G.; Lin, W. H.; Liu, N., Nonlinear bending and post-buckling of extensible microscale beams based on modified couple stress theory, Appl Math Model, 39, 117-127 (2015) · Zbl 1428.74130 |
| [8] | Borchani, W.; Jiao, P. C.; Borcheni, I.; Lajnef, N., Post-buckling analysis of microscale non-prismatic beams subjected to bilateral walls, Extrem Mech Lett, 21, 82-89 (2018) |
| [9] | Chen, X. C.; Li, Y. H., Size-dependent post-buckling behaviors of geometrically imperfect microbeams, Mech Res Commun, 88, 25-33 (2018) |
| [10] | Taati, E., On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment, Internat J Engrg Sci, 128, 63-78 (2018) · Zbl 1423.74355 |
| [11] | Zhao B, Long CY, Peng XL, Chen J, Liu T, Zhang ZH et al. Size effect and geometrically nonlinear effect on thermal post-buckling of micro-beams: a new theoretical analysis. Contin Mech Thermodyn. · Zbl 1516.74044 |
| [12] | Hosseini, S. M.H.; Arvin, H., Thermo-rotational buckling and post-buckling analyses of rotating functionally graded microbeams, Int J Mech Mater Des, 17, 55-72 (2021) |
| [13] | Ansari, R.; Gholami, R.; Shojaei, M. F.; Mohammadi, V.; Darabi, M. A., Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin’s strain gradient theory, Appl Math Model, 40, 9872-9891 (2016) · Zbl 1443.74014 |
| [14] | Shenas, A. G.; Ziaee, S.; Malekzadeh, P., Post-buckling and vibration of post-buckled rotating pre-twisted FG microbeams in thermal environment, Thin-Walled Struct, 138, 335-360 (2019) |
| [15] | Lam, D. C.C.; Yang, F.; Chong, A. C.M.; Wang, J.; Tong, P., Experiments and theory in strain gradient elasticity, J Mech Phys Solids, 51, 1477-1508 (2003) · Zbl 1077.74517 |
| [16] | Li, L.; Hu, Y. J., Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects, Int J Mech Sci, 120, 159-170 (2017) |
| [17] | Barati, M. R.; Zenkour, A. M., Thermal post-buckling analysis of closed circuit flexoelectric nanobeams with surface effects and geometrical imperfection, Mech Adv Mater Struct., 26, 1482-1490 (2019) |
| [18] | Fedorchenko, A. I.; Wang, A.-B.; Cheng, H. H., Thickness dependence of nanofilm elastic modulus, Appl Phys Lett, 94 (2009) |
| [19] | McDowell, M. T.; Leach, A. M.; Gaill, K., On the elastic modulus of metallic nanowires, Nano Lett, 8, 3613-3618 (2008) |
| [20] | Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J Appl Phys, 54, 4703-4710 (1983) |
| [21] | Liu, C.; Ke, L. L.; Wang, Y. S.; Yang, J.; Kitipornchai, S., Buckling and post-buckling of size-dependent piezoelectric Timoshenko nanobeams subject to thermo-electro-mechanical loadings, Int J Struct Stab Dyn, 14 (2014) · Zbl 1359.74099 |
| [22] | Zhong, J.; Fu, Y. M.; Tao, C., Linear free vibration in pre/post-buckled states and nonlinear dynamic stability of lipid tubules based on nonlocal beam model, Meccanica, 51, 1481-1489 (2016) · Zbl 1341.74083 |
| [23] | Dai, H. L.; Ceballes, S.; Abdelkefi, A.; Hong, Y. Z.; Wang, L., Exact modes for post-buckling characteristics of nonlocal nanobeams in a longitudinal magnetic field, Appl Math Model, 55, 758-775 (2018) · Zbl 1480.74081 |
| [24] | Nguyen, T. B.; Reddy, J. N.; Rungamornrat, J.; Lawongkerd, J.; Senjuntichai, T.; Luong, V. H., Nonlinear analysis for bending, buckling and post-buckling of nano-beams with nonlocal and surface energy effects, Int J Struct Stab Dyn, 19 (2019) · Zbl 1535.74128 |
| [25] | Qing H, Cai YX. Semi-analytical and numerical post-buckling analysis of nanobeam using two-phase nonlocal integral models, Arch Appl Mech. |
| [26] | Benvenuti, E.; Simone, A., One-dimensional nonlocal and gradient elasticity: Closed-form solution and size effect, Mech Res Commun, 48, 46-51 (2013) |
| [27] | Pisano, A. A.; Fuschi, P., Closed form solution for a nonlocal elastic bar in tension, Int J Solids Struct, 40, 13-23 (2003) · Zbl 1083.74555 |
| [28] | C., Li; Yao, L. Q.; Chen, W. Q.; Li, S., Comments on nonlocal effects in nano-cantilever beams, Internat J Engrg Sci, 87, 47-57 (2015) |
| [29] | Reddy, J. N.; Pang, S. D., Nonlocal continuum theories of beams for the analysis of carbon nanotubes, J Appl Phys, 103, 16 (2008) |
| [30] | Zhang, J. Q.; Qing, H.; Gao, C. F., Exact and asymptotic bending analysis of microbeams under different boundary conditions using stress-derived nonlocal integral model, Zamm-Z Angew Math Mech, 99 (2019) |
| [31] | Zhang, P.; Qing, H., Exact solutions for size-dependent bending of timoshenko curved beams based on a modified nonlocal strain gradient model, Acta Mech, 231, 5251-5276 (2020) · Zbl 1457.74113 |
| [32] | Ren, Y. M.; Qing, H., On the consistency of two-phase local/nonlocal piezoelectric integral model, Appl Math Mech (English Ed), 42, 1581-1598 (2021) · Zbl 1505.74057 |
| [33] | Zhang P, Qing H. Well-posed two-phase nonlocal integral models for free vibration of nanobeams in context with higher-order refined shear deformation theory. J Vib Control. |
| [34] | Zhang, P.; Qing, H., On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams, Appl Math Mech (English Ed), 42, 931-950 (2021) · Zbl 1479.74081 |
| [35] | Qing, H., Well-posedness of two-phase local/nonlocal integral polar models for consistent axisymmetric bending of circular microplates, Appl Math Mech (English Ed), 43, 637-652 (2022) · Zbl 1505.74136 |
| [36] | Peddieson, J.; Buchanan, G. R.; McNitt, R. P., Application of nonlocal continuum models to nanotechnology, Internat J Engrg Sci, 41, 305-312 (2003) |
| [37] | Eringen, A. C., Theory of nonlocal elasticity and some applications, Res Mech, 21, 313-342 (1987) |
| [38] | Romano, G.; Luciano, R.; Barretta, R.; Diaco, M., Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours, Contin Mech Thermodyn, 30, 641-655 (2018) · Zbl 1392.74071 |
| [39] | Zhang, P.; Qing, H.; Gao, C., Theoretical analysis for static bending of circular Euler-Bernoulli beam using local and Eringen’s nonlocal integral mixed model, Zamm-Z Angew Math Mech, 99, Article e201800329 pp. (2019) · Zbl 07785951 |
| [40] | Zhang, P.; Qing, H.; Gao, C.-F., Analytical solutions of static bending of curved Timoshenko microbeams using Eringen’s two-phase local/nonlocal integral model, Zamm-Z Angew Math Mech, 100 (2020) · Zbl 07809732 |
| [41] | Wang, Y. B.; Zhu, X. W.; Dai, H. H., Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model, AIP Adv, 6, Article 085114 pp. (2016) |
| [42] | Romano, G.; Barretta, R.; Diaco, M.; de Sciarra, F. M., Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, Int J Mech Sci, 121, 151-156 (2017) |
| [43] | Zhu, X.; Wang, Y.; Dai, H.-H., Buckling analysis of Euler-Bernoulli beams using Eringen’s two-phase nonlocal model, Internat J Engrg Sci, 116, 130-140 (2017) · Zbl 1423.74360 |
| [44] | Wang, Y.; Huang, K.; Zhu, X.; Lou, Z., Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model, Math Mech Solids, 24, 559-572 (2019) · Zbl 1444.74034 |
| [45] | Zhu, X.; Li, L., A well-posed Euler-Bernoulli beam model incorporating nonlocality and surface energy effect, Appl Math Mech (English Ed), 40, 1561-1588 (2019) · Zbl 1430.74097 |
| [46] | Romano, G.; Barretta, R., Nonlocal elasticity in nanobeams: the stress-driven integral model, Internat J Engrg Sci, 115, 14-27 (2017) · Zbl 1423.74512 |
| [47] | Zhang, J.-Q.; Qing, H.; Gao, C.-F., Exact and asymptotic bending analysis of microbeams under different boundary conditions using stress-derived nonlocal integral model, Zamm-Z Angew Math Mech, 100 (2020) · Zbl 07794843 |
| [48] | Jiang, P.; Qing, H.; Gao, C. F., Theoretical analysis on elastic buckling of nanobeams based on stress-driven nonlocal integral model, Appl Math Mech (English Ed), 41, 207-232 (2020) · Zbl 1462.74100 |
| [49] | He, Y.; Qing, H.; Gao, C., Theoretical analysis of free vibration of microbeams under different boundary conditions using stress-driven nonlocal integral model, Int J Struct Stab Dyn, 20, Article 2050040 pp. (2020) · Zbl 1535.74217 |
| [50] | Zhang, P.; Qing, H.; Gao, C. F., Exact solutions for bending of Timoshenko curved nanobeams made of functionally graded materials based on stress -driven nonlocal integral model, Compos Struct., 245 (2020) |
| [51] | Barretta, R.; Caporale, A.; Faghidian, S. A.; Luciano, R.; de Sciarra, F. M.; Medaglia, C. M., A stress-driven local-nonlocal mixture model for Timoshenko nano-beams, Compos Part B-Eng, 164, 590-598 (2019) |
| [52] | Bian, P-L.; Qing, H.; Gao, C.-F., One-dimensional stress-driven nonlocal integral model with bi-Helmholtz kernel: Close form solution and consistent size effect, Appl Math Model, 89, 400-412 (2021) · Zbl 1485.74054 |
| [53] | Vaccaro, M. S.; Barretta, R.; de Sciarra, F. M.; Reddy, J. N., Nonlocal integral elasticity for third-order small-scale beams, Acta Mech, 233, 2393-2403 (2022) · Zbl 1493.74061 |
| [54] | Vaccaro, M. S.; Pinnola, F. P.; de Sciarra, F. M.; Canadija, M.; Barretta, R., Stress-driven two-phase integral elasticity for timoshenko curved beams, Proc Inst Mech Eng Part N (J Nanoeng Nanosyst), 235, 52-63 (2021) |
| [55] | Vaccaro, M. S.; Pinnola, F. P.; Marotti de Sciarra, F.; Barretta, R., Dynamics of stress-driven two-phase elastic beams, Nanomaterials, 11 (2021) |
| [56] | Zhang P, Schiavone P, Qing H. Two-phase local/nonlocal mixture models for buckling analysis of higher-order refined shear deformation beams under thermal effect. Mech Adv Mater Struct. |
| [57] | Qing, H.; Wei, L., Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model, Commun Nonlinear Sci Numer Simul (2022) · Zbl 07840933 |
| [58] | Tang, Y.; Lv, X.; Yang, T., Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration, Compos Part B-Eng, 156, 319-331 (2019) |
| [59] | Fakher, M.; Hosseini-Hashemi, S., Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method, J Vib Control, 27, 378-391 (2021) |
| [60] | Wu, T. Y.; Liu, G. R., The generalized differential quadrature rule for fourth-order differential equations, Int J Numer Methods Biomed Eng, 50, 1907-1929 (2001) · Zbl 0999.74120 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
