Nonlinear thermal buckling and postbuckling analysis of bidirectional functionally graded tapered microbeams based on Reddy beam theory


In the present study, the nonlinear thermal buckling, postbuckling, and snap-through phenomenon of higher-order shear deformable tapered bidirectional functionally graded (BDFG) microbeams are comprehensively investigated under different types of thermal loading, for the first time. The thermomechanical properties of the BDFG microbeam are assumed to be functions of temperature, axial, and thickness directions. Reddy’s parabolic shear deformation beam theory with the von Kármán nonlinearity is employed to derive the variable coefficient governing nonlinear differential equations on the basis the physical neutral surface concept. Size-dependent effect is captured in the formulation employing the modified couple stress theory. The generalized differential quadrature method (GDQM) is used to discretize the motion equations considering different boundary conditions. The resulting system of nonlinear algebraic equations is solved iteratively using Newton’s method. Theoretical analysis and numerical results indicate that, depending on the shear deformation beam theory, boundary conditions, and the type of thermal load, the response of the BDFG microbeam may be of the bifurcation or snap-through buckling type of instability. Numerical parametric studies are conducted to explore the influences of thermal load type, material property gradient indexes, boundary conditions, material temperature dependency, taperness ratio, and microstructural length scale on critical thermal buckling load, thermal postbuckling, and equilibrium paths of the BDFG microbeam.

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  1. 1.

    Udupa G, Rao SS, Gangadharan K (2014) Functionally graded composite materials: an overview. Procedia Mater Sci 5:1291–1299

    Article  Google Scholar 

  2. 2.

    Ahankari SS, Kar KK (2017) Functionally graded composites: processing and applications. In: Composite materials. Springer, pp 119–168

  3. 3.

    Lam DC, Yang F, Chong A, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508

    MATH  Article  Google Scholar 

  4. 4.

    Lei J, He Y, Guo S, Li Z, Liu D (2016) Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity. AIP Adv 6(10):105202

    Article  Google Scholar 

  5. 5.

    Li Z, He Y, Lei J, Han S, Guo S, Liu D (2019) Experimental investigation on size-dependent higher-mode vibration of cantilever microbeams. Microsyst Technol 25(8):3005–3015

    Article  Google Scholar 

  6. 6.

    Li Z, He Y, Zhang B, Lei J, Guo S, Liu D (2019) Experimental investigation and theoretical modelling on nonlinear dynamics of cantilevered microbeams. Eur J Mech-A/Solids 78:103834

    MATH  Article  MathSciNet  Google Scholar 

  7. 7.

    Liebold C, Müller WH (2016) Comparison of gradient elasticity models for the bending of micromaterials. Comput Mater Sci 116:52–61

    Article  Google Scholar 

  8. 8.

    Nix WD (1989) Mechanical properties of thin films. Metall Trans A 20(11):2217

    Article  Google Scholar 

  9. 9.

    Son D, Jeong J-H, Kwon D (2003) Film-thickness considerations in microcantilever-beam test in measuring mechanical properties of metal thin film. Thin Solid Films 437(1):182–187

    Article  Google Scholar 

  10. 10.

    Tang C, Alici G (2011) Evaluation of length-scale effects for mechanical behaviour of micro-and nanocantilevers: I. Experimental determination of length-scale factors. J Phys D Appl Phys 44(33):335501

    Article  Google Scholar 

  11. 11.

    Tang C, Alici G (2011) Evaluation of length-scale effects for mechanical behaviour of micro-and nanocantilevers: II. Experimental verification of deflection models using atomic force microscopy. J Phys D Appl Phys 44(33):335502

    Article  Google Scholar 

  12. 12.

    Wi D, Sodemann A (2019) Investigation of the size effect on the resonant behavior of mesoscale cantilever beams. J Vib Control 25(23–24):2946–2955

    Article  Google Scholar 

  13. 13.

    Gopalakrishnan S, Narendar S (2013) Wave propagation in nanostructures: nonlocal continuum mechanics formulations. Springer Science and Business Media, New York

    Google Scholar 

  14. 14.

    Wang KF, Wang BL, Kitamura T (2016) A review on the application of modified continuum models in modeling and simulation of nanostructures. Acta Mech Sin 32(1):83–100

    MATH  Article  MathSciNet  Google Scholar 

  15. 15.

    Thai HT, Vo TP, Nguyen TK, Kim SE (2017) A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos Struct 177:196–219

    Article  Google Scholar 

  16. 16.

    Yang F, Chong A, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743

    MATH  Article  Google Scholar 

  17. 17.

    Chong ACM, Yang F, Lam DC, Tong P (2001) Torsion and bending of micron-scaled structures. J Mater Res 16:1052–1058

    Article  Google Scholar 

  18. 18.

    Liu D, He Y, Dunstan DJ, Zhang B, Gan Z, Hu P, Ding H (2013) Toward a further understanding of size effects in the torsion of thin metal wires: an experimental and theoretical assessment. Int J Plast 41:30–52

    Article  Google Scholar 

  19. 19.

    Liu D, He Y, Tang X, Ding H, Hu P, Cao P (2012) Size effects in the torsion of microscale copper wires: experiment and analysis. Scr Mater 66:406–409

    Article  Google Scholar 

  20. 20.

    Park SK, Gao XL (2006) Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng 16(11):2355–2359

    Article  Google Scholar 

  21. 21.

    Dehrouyeh-Semnani AM (2014) A discussion on different non-classical constitutive models of microbeam. Int J Eng Sci 85:66–73

    Article  Google Scholar 

  22. 22.

    Dehrouyeh-Semnani AM, Nikkhah-Bahrami M (2015) A discussion on evaluation of material length scale parameter based on micro-cantilever test. Compos Struct 122:425–429

    Article  Google Scholar 

  23. 23.

    Dehrouyeh-Semnani AM, Nikkhah-Bahrami M (2015) A discussion on incorporating the Poisson effect in microbeam models based on modified couple stress theory. Int J Eng Sci 86:20–25

    Article  Google Scholar 

  24. 24.

    Aghazadeh R, Cigeroglu E, Dag S (2014) Static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories. Eur J Mech-A/Solids 46:1–11

    MATH  Article  MathSciNet  Google Scholar 

  25. 25.

    Al-Basyouni K, Tounsi A, Mahmoud S (2015) Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position. Compos Struct 125:621–630

    Article  Google Scholar 

  26. 26.

    Alimirzaei S, Mohammadimehr M, Tounsi A (2019) Nonlinear analysis of viscoelastic micro-composite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions. Struct Eng Mech 71(5):485–502

    Google Scholar 

  27. 27.

    Asghar S, Naeem MN, Hussain M, Taj M, Tounsi A (2020) Prediction and assessment of nonlocal natural frequencies of DWCNTs: vibration analysis. Comput Concr 25(2):133–144

    Google Scholar 

  28. 28.

    Attia MA, Emam SA (2018) Electrostatic nonlinear bending, buckling and free vibrations of viscoelastic microbeams based on the modified couple stress theory. Acta Mech 229(8):3235–3255

    MATH  Article  MathSciNet  Google Scholar 

  29. 29.

    Attia MA, Mohamed SA (2017) Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory. Appl Math Model 41:195–222

    MATH  Article  MathSciNet  Google Scholar 

  30. 30.

    Attia MA (2017) Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges. Meccanica 52(10):2391–2420

    MATH  Article  MathSciNet  Google Scholar 

  31. 31.

    Bellal M, Hebali H, Heireche H, Bousahla AA, Tounsi A, Bourada F et al (2020) Buckling behavior of a single-layered graphene sheet resting on viscoelastic medium via nonlocal four-unknown integral model. Steel Compos Struct 34(5):643–655

    Google Scholar 

  32. 32.

    Berghouti H, Adda Bedia E, Benkhedda A, Tounsi A (2019) Vibration analysis of nonlocal porous nanobeams made of functionally graded material. Adv Nano Res 7(5):351–364

    Google Scholar 

  33. 33.

    Bousahla AA, Bourada F, Mahmoud S, Tounsi A, Algarni A, Bedia E et al (2020) Buckling and dynamic behavior of the simply supported CNT-RC beams using an integral-first shear deformation theory. Comput Concr 25(2):155–166

    Google Scholar 

  34. 34.

    Boussoula A, Boucham B, Bourada M, Bourada F, Tounsi A, Bousahla AA et al (2020) A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates. Smart Struct Syst 25(2):197–218

    Google Scholar 

  35. 35.

    Dehrouyeh-Semnani AM, Dehrouyeh M, Torabi-Kafshgari M, Nikkhah-Bahrami M (2015) An investigation into size-dependent vibration damping characteristics of functionally graded viscoelastically damped sandwich microbeams. Int J Eng Sci 96:68–85

    MATH  Article  MathSciNet  Google Scholar 

  36. 36.

    Dehrouyeh-Semnani AM, Mostafaei H, Nikkhah-Bahrami M (2016) Free flexural vibration of geometrically imperfect functionally graded microbeams. Int J Eng Sci 105:56–79

    MATH  Article  MathSciNet  Google Scholar 

  37. 37.

    Ghayesh MH, Farajpour A (2019) Vibrations of shear deformable FG viscoelastic microbeams. Microsyst Technol 25(4):1387–1400

    Article  Google Scholar 

  38. 38.

    Ghayesh MH (2019) Vibration characterisation of AFG microcantilevers in nonlinear regime. Microsyst Technol 25(8):3061–3069

    Article  MathSciNet  Google Scholar 

  39. 39.

    Ghayesh MH (2019) Viscoelastic mechanics of Timoshenko functionally graded imperfect microbeams. Compos Struct 225:110974

    Article  Google Scholar 

  40. 40.

    Hussain M, Naeem MN, Tounsi A, Taj M (2019) Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity. Adv Nano Res 7(6):431

    Google Scholar 

  41. 41.

    Jouneghani FZ, Babamoradi H, Dimitri R, Tornabene F (2020) A modified couple stress elasticity for non-uniform composite laminated beams based on the Ritz formulation. Molecules 25(6):1404

    Article  Google Scholar 

  42. 42.

    Mollamahmutoğlu Ç, Mercan A (2019) A novel functional and mixed finite element analysis of functionally graded micro-beams based on modified couple stress theory. Compos Struct 223:110950

    Article  Google Scholar 

  43. 43.

    Reddy JN (2011) Microstructure-dependent couple stress theories of functionally graded beams. J Mech Phys Solids 59(11):2382–2399

    MATH  Article  MathSciNet  Google Scholar 

  44. 44.

    Salamat-talab M, Nateghi A, Torabi J (2012) Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. Int J Mech Sci 57(1):63–73

    Article  Google Scholar 

  45. 45.

    Tlidji Y, Zidour M, Draiche K, Safa A, Bourada M, Tounsi A et al (2019) Vibration analysis of different material distributions of functionally graded microbeam. Struct Eng Mech 69(6):637–649

    Google Scholar 

  46. 46.

    Ebrahimi F, Rastgo A (2008) An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Struct 46(12):1402–1408

    Article  Google Scholar 

  47. 47.

    Zhao F-Q, Wang Z-M, Liu H-Z (2007) Thermal post-bunkling analyses of functionally graded material rod. Appl Math Mech 28(1):59–67

    MATH  Article  Google Scholar 

  48. 48.

    Nateghi A, Salamat-talab M (2013) Thermal effect on size dependent behavior of functionally graded microbeams based on modified couple stress theory. Compos Struct 96:97–110

    Article  Google Scholar 

  49. 49.

    Komijani M, Esfahani S, Reddy J, Liu Y, Eslami M (2014) Nonlinear thermal stability and vibration of pre/post-buckled temperature-and microstructure-dependent functionally graded beams resting on elastic foundation. Compos Struct 112:292–307

    Article  Google Scholar 

  50. 50.

    Akgöz B, Civalek Ö (2014) Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium. Int J Eng Sci 85:90–104

    Article  Google Scholar 

  51. 51.

    Akgöz B, Civalek Ö (2017) Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams. Compos B Eng 129:77–87

    Article  Google Scholar 

  52. 52.

    Levyakov S (2015) Thermal elastic of shear-deformable beam fabricated of functionally graded material. Acta Mech 226(3):723–733

    MATH  Article  MathSciNet  Google Scholar 

  53. 53.

    Ebrahimi F, Salari E (2015) Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments. Compos Struct 128:363–380

    Article  Google Scholar 

  54. 54.

    Ebrahimi F, Salari E (2015) Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions. Compos B Eng 78:272–290

    Article  Google Scholar 

  55. 55.

    Sun Y, Li S-R, Batra RC (2016) Thermal buckling and post-buckling of FGM Timoshenko beams on nonlinear elastic foundation. J Therm Stress 39(1):11–26

    Article  Google Scholar 

  56. 56.

    Mouffoki A, Bedia E, Houari MSA, Tounsi A, Mahmoud S (2017) Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory. Smart Struct Syst 20(3):369–383

    Google Scholar 

  57. 57.

    Abderrahmane M, Bessaim A, Ahmed HMS, Kaci A, Abdelouahed T, Bedia EAA (2019) Thermo-mechanical vibration analysis of non-local refined trigonometric shear deformable FG beams. Int J Hydromech 2(1):54–62

    Article  Google Scholar 

  58. 58.

    She G-L, Yuan F-G, Ren Y-R (2017) Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory. Appl Math Model 47:340–357

    MATH  Article  MathSciNet  Google Scholar 

  59. 59.

    She G-L, Jiang X, Karami B (2019) On thermal snap-buckling of FG curved nanobeams. Mater Res Express 6(11):115008

    Article  Google Scholar 

  60. 60.

    Dehrouyeh-Semnani AM, Mostafaei H, Dehrouyeh M, Nikkhah-Bahrami M (2017) Thermal pre-and post-snap-through buckling of a geometrically imperfect doubly-clamped microbeam made of temperature-dependent functionally graded materials. Compos Struct 170:122–134

    MATH  Article  Google Scholar 

  61. 61.

    Jia X, Ke L, Zhong X, Sun Y, Yang J, Kitipornchai S (2018) Thermal-mechanical-electrical buckling behavior of functionally graded micro-beams based on modified couple stress theory. Compos Struct 202:625–634

    Article  Google Scholar 

  62. 62.

    Sahmani S, Fattahi A, Ahmed N (2019) Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Eng Comput 35(4):1173–1189

    Article  Google Scholar 

  63. 63.

    Dehrouyeh-Semnani AM (2018) On the thermally induced non-linear response of functionally graded beams. Int J Eng Sci 125:53–74

    MATH  Article  MathSciNet  Google Scholar 

  64. 64.

    Dehrouyeh-Semnani AM, Jafarpour S (2019) Nonlinear thermal stability of temperature-dependent metal matrix composite shallow arches with functionally graded fiber reinforcements. Int J Mech Sci 161:105075

    Article  Google Scholar 

  65. 65.

    Dehrouyeh-Semnani AM, Dehdashti E, Yazdi MRH, Nikkhah-Bahrami M (2019) Nonlinear thermo-resonant behavior of fluid-conveying FG pipes. Int J Eng Sci 144:103141

    MATH  Article  MathSciNet  Google Scholar 

  66. 66.

    Salari E, Vanini SS, Ashoori A, Akbarzadeh A (2020) Nonlinear thermal behavior of shear deformable FG porous nanobeams with geometrical imperfection: snap-through and postbuckling analysis. Int J Mech Sci, 105615

  67. 67.

    Ebrahimi F, Hosseini SHS (2019) Nonlinear vibration and dynamic instability analysis nanobeams under thermo-magneto-mechanical loads: a parametric excitation study. Eng Comput, 1–14

  68. 68.

    Zhang Z, Zhou D, Xu X, Li X (2020) Analysis of thick beams with temperature-dependent material properties under thermomechanical loads. Adv Struct Eng, 1369433220901810

  69. 69.

    Wu H, Liu H (2020) Nonlinear thermo-mechanical response of temperature-dependent FG sandwich nanobeams with geometric imperfection. Eng Comput, 1–21

  70. 70.

    Shafiei N, Mirjavadi SS, Afshari BM, Rabby S, Hamouda A (2017) Nonlinear thermal buckling of axially functionally graded micro and nanobeams. Compos Struct 168:428–439

    Article  Google Scholar 

  71. 71.

    Dehrouyeh-Semnani AM (2017) A comment on “Nonlinear thermal buckling of axially functionally graded micro and nanobeams” [Composite Structures 168 (2017) 428–439]. Compos Struct 178:308–310

    Article  Google Scholar 

  72. 72.

    Wang Y, Xie K, Shi C, Fu T (2019) Nonlinear bending of axially functionally graded microbeams reinforced by graphene nanoplatelets in thermal environments. Mater Res Express 6(8):085615

    Article  Google Scholar 

  73. 73.

    Akgoz B (2019) Static stability analysis of axially functionally graded tapered micro columns with different boundary conditions. Steel Compos Struct 33(1):133–142

    Google Scholar 

  74. 74.

    Wang Y, Ren H, Fu T, Shi C (2020) Hygrothermal mechanical behaviors of axially functionally graded microbeams using a refined first order shear deformation theory. Acta Astronaut 166:306–316

    Article  Google Scholar 

  75. 75.

    Nemat-Alla M (2003) Reduction of thermal stresses by developing two-dimensional functionally graded materials. Int J Solids Struct 40(26):7339–7356

    MATH  Article  Google Scholar 

  76. 76.

    Nejad MZ, Hadi A, Rastgoo A (2016) Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory. Int J Eng Sci 103:1–10

    MATH  Article  MathSciNet  Google Scholar 

  77. 77.

    Karamanlı A, Vo TP (2018) Size dependent bending analysis of two directional functionally graded microbeams via a quasi-3D theory and finite element method. Compos B Eng 144:171–183

    Article  Google Scholar 

  78. 78.

    Shafiei N, Mirjavadi SS, MohaselAfshari B, Rabby S, Kazemi M (2017) Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput Methods Appl Mech Eng 322:615–632

    MATH  Article  MathSciNet  Google Scholar 

  79. 79.

    Trinh LC, Vo TP, Thai H-T, Nguyen T-K (2018) Size-dependent vibration of bi-directional functionally graded microbeams with arbitrary boundary conditions. Compos B Eng 134:225–245

    Article  Google Scholar 

  80. 80.

    Li L, Li X, Hu Y (2018) Nonlinear bending of a two-dimensionally functionally graded beam. Compos Struct 184:1049–1061

    Article  Google Scholar 

  81. 81.

    Yang T, Tang Y, Li Q, Yang X-D (2018) Nonlinear bending, buckling and vibration of bi-directional functionally graded nanobeams. Compos Struct 204:313–319

    Article  Google Scholar 

  82. 82.

    Rajasekaran S, Khaniki HB (2019) Size-dependent forced vibration of non-uniform bi-directional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass. Appl Math Model 72:129–154

    MATH  Article  MathSciNet  Google Scholar 

  83. 83.

    Yu T, Hu H, Zhang J, Bui TQ (2019) Isogeometric analysis of size-dependent effects for functionally graded microbeams by a non-classical quasi-3D theory. Thin-Walled Struct 138:1–14

    Article  Google Scholar 

  84. 84.

    Chen X, Zhang X, Lu Y, Li Y (2019) Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams. Int J Mech Sci 151:424–443

    Article  Google Scholar 

  85. 85.

    Chen X, Lu Y, Zhu B, Zhang X, Li Y (2019) Nonlinear resonant behaviors of bi-directional functionally graded material microbeams: one-/two-parameter bifurcation analyses. Compos Struct 223:110896

    Article  Google Scholar 

  86. 86.

    Sahmani S, Safaei B (2019) Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects. Thin-Walled Struct 140:342–356

    Article  Google Scholar 

  87. 87.

    Sahmani S, Safaei B (2019) Nonlocal strain gradient nonlinear resonance of bi-directional functionally graded composite micro/nano-beams under periodic soft excitation. Thin-Walled Struct 143:106226

    Article  Google Scholar 

  88. 88.

    Sahmani S, Safaei B (2020) Influence of homogenization models on size-dependent nonlinear bending and postbuckling of bi-directional functionally graded micro/nano-beams. Appl Math Model 82:336–358

    MATH  Article  MathSciNet  Google Scholar 

  89. 89.

    Mirjavadi SS, Afshari BM, Shafiei N, Hamouda A, Kazemi M (2017) Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams. Steel Compos Struct 25(4):415–426

    Google Scholar 

  90. 90.

    Shafiei N, She G-L (2018) On vibration of functionally graded nano-tubes in the thermal environment. Int J Eng Sci 133:84–98

    MATH  Article  MathSciNet  Google Scholar 

  91. 91.

    Tang Y, Ding Q (2019) Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads. Compos Struct 225:111076

    Article  Google Scholar 

  92. 92.

    Lal R, Dangi C (2019) Thermomechanical vibration of bi-directional functionally graded non-uniform Timoshenko nanobeam using nonlocal elasticity theory. Compos B Eng 172:724–742

    Article  Google Scholar 

  93. 93.

    Ebrahimi-Nejad S, Shaghaghi GR, Miraskari F, Kheybari M (2019) Size-dependent vibration in two-directional functionally graded porous nanobeams under hygro-thermo-mechanical loading. Eur Phys J Plus 134(9):465

    Article  Google Scholar 

  94. 94.

    Karami B, Janghorban M, Rabczuk T (2020) Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory. Compos B Eng 182:107622

    Article  Google Scholar 

  95. 95.

    Reddy J, Chin C (1998) Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stress 21(6):593–626

    Article  Google Scholar 

  96. 96.

    Liu Y, Su S, Huang H, Liang Y (2019) Thermal-mechanical coupling buckling analysis of porous functionally graded sandwich beams based on physical neutral plane. Compos B Eng 168:236–242

    Article  Google Scholar 

  97. 97.

    Attia MA (2017) On the mechanics of functionally graded nanobeams with the account of surface elasticity. Int J Eng Sci 115:73–101

    MATH  Article  MathSciNet  Google Scholar 

  98. 98.

    Attia MA, Rahman AAA (2018) On vibrations of functionally graded viscoelastic nanobeams with surface effects. Int J Eng Sci 127:1–32

    MATH  Article  MathSciNet  Google Scholar 

  99. 99.

    Attia MA, Mohamed SA (2018) Pull-in instability of functionally graded cantilever nanoactuators incorporating effects of microstructure, surface energy and intermolecular forces. Int J Appl Mech 10(08):1850091

    Article  Google Scholar 

  100. 100.

    Attia MA, Mohamed SA (2019) Coupling effect of surface energy and dispersion forces on nonlinear size-dependent pull-in instability of functionally graded micro-/nanoswitches. Acta Mech 230(3):1181–1216

    MATH  Article  MathSciNet  Google Scholar 

  101. 101.

    Reddy JN (1984) A simple higher-order theory for laminated composite plates. ASME J Appl Mech 51:745–752

    MATH  Article  Google Scholar 

  102. 102.

    Eslami MR, Hetnarski RB, Ignaczak J, Noda N, Sumi N, Tanigawa Y (2013) Theory of elasticity and thermal stresses, vol 197. Springer, Dordrecht

    Google Scholar 

  103. 103.

    Attia MA, Mahmoud FF (2016) Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories. Int J Mech Sci 105:126–134

    Article  Google Scholar 

  104. 104.

    Attia MA, Mahmoud FF (2017) Analysis of viscoelastic Bernoulli-Euler nanobeams incorporating nonlocal and microstructure effects. Int J Mech Mater Des 13(3):385–406

    Article  Google Scholar 

  105. 105.

    Shu C (2012) Differential quadrature and its application in engineering. Springer Science and Business Media, New York

    Google Scholar 

  106. 106.

    Attia MA, Shanab RA, Mohamed SA, Mohamed NA (2019) Surface energy effects on the nonlinear free vibration of functionally graded Timoshenko nanobeams based on modified couple stress theory. Int J Struct Stab Dyn 19(11):1950127

    Article  MathSciNet  Google Scholar 

  107. 107.

    Shanab RA, Attia MA, Mohamed SA (2017) Nonlinear analysis of functionally graded nanoscale beams incorporating the surface energy and microstructure effects. Int J Mech Sci 131:908–923

    Article  Google Scholar 

  108. 108.

    Shanab RA, Mohamed SA, Mohamed NA, Attia MA (2020) Comprehensive investigation of vibration of sigmoid and power law FG nanobeams based on surface elasticity and modified couple stress theories. Acta Mech, 1–34

  109. 109.

    Mohamed N, Mohamed S, Eltaher M (2020) Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Eng Comput, 1–14

  110. 110.

    Emam SA (2009) A static and dynamic analysis of the postbuckling of geometrically imperfect composite beams. Compos Struct 90(2):247–253

    Article  MathSciNet  Google Scholar 

  111. 111.

    Dehrouyeh-Semnani AM (2017) On boundary conditions for thermally loaded FG beams. Int J Eng Sci 119:109–127

    MATH  Article  MathSciNet  Google Scholar 

  112. 112.

    Riks E (1979) An incremental approach to the solution of snapping and buckling problems. Int J Solids Struct 15(7):529–551

    MATH  Article  MathSciNet  Google Scholar 

  113. 113.

    Ma LS, Lee DW (2011) A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading. Compos Struct 93(2):831–842

    Article  Google Scholar 

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Attia, M.A., Mohamed, S.A. Nonlinear thermal buckling and postbuckling analysis of bidirectional functionally graded tapered microbeams based on Reddy beam theory. Engineering with Computers (2020).

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  • Bidirectional functionally graded microbeams
  • Modified couple stress theory
  • Temperature-dependent physical properties
  • Thermal pre- and postbuckling
  • Snap-through phenomena
  • Generalized differential quadrature method