Journal of Computational Applied Mechanics

Stability Analysis of a Laminated Composite Micro Scaled Beam Embedded in Elastic Medium using Modified Coupled Stress Theory

Document Type : Research Paper

Author

Civil Engineering, Engineering Fac., Bursa Technical University, Bursa,Turkey

Abstract

This paper presents size dependent stability analysis a cantilever micro laminated beam embedded in elastic medium by using the modified coupled stress theory which includes the length scale parameter. The micro beam subjected to compressive load is considered as three composite laminas and embedded in elastic medium which is modelled in the Winkler foundation model. In the obtaining of the governing equations, the energy principle is used. In the solution of the buckling problem, the energy based Ritz method is implemented with algebraic polynomials. In order to accuracy obtained expressions and used method, a comparative study is performed. Many parametric studies are presented in order to investigate the buckling of laminated micro beams. For this purpose, effects of stacking sequence of laminas, geometric parameters, length scale parameter, fiber orientation angle, the parameter of elastic medium on critical buckling loads of laminated micro beams are investigated.

Keywords

Main Subjects

  1. D. Mindlin and H.F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, Vol.11, No.1, pp. 415–48, 1962.
  2. D. Mindlin, Influence of couple-stresses on stress concentrations, Experimental mechanics, Vol. 3, No.1, pp. 1–7, 1963.
  3. C. Eringen, Nonlocal polar elastic continua, International Journal of Engineering Science, Vol. 10, No.1, pp. 1-16, 1972.
  4. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, Vol. 54, pp. 4703–10, 1983.
  5. A. Toupin, Elastic materials with couple stresses, Archive for Rational Mechanics and Analysis, Vol.11, No.1: 385–414, 1962.
  6. C.C. Lam, F. Yang, A.C.M. Chong, J. Wang and P. Tong, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, Vol. 51, No.8, pp. 1477–508, 2003.
  7. Yang, A. Chong, D. Lam and P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, Vol. 39, No.10, pp. 2731-2743, 2002.
  8. K. Park and X.L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, Vol. 16, No.11, pp. 2355-2359, 2006.
  9. Liu and J.N. Reddy,  A Nonlocal curved beam model based on a modified couple stress theory, International Journal of Structural Stability and Dynamics, Vol. 11, No.3, pp. 495-512, 2011.
  10. Ansari, R. Gholami and S. Sahmani, Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Composite Structures, Vol. 94, No.1, pp. 221-228, 2011.
  11. N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science, Vol. 48, No.11, pp. 1507-1518, 2010.
  12. N. Reddy, Microstructure-dependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids, Vol. 59, No.11, pp. 2382-2399, 2011.
  13. Akgöz and Ö. Civalek, Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory, Composite Structures, Vol. 98, pp.314-322, 2013.
  14. Asghari, M.T. Ahmadian, M.H. Kahrobaiyan and M. Rahaeifard, On the size-dependent behavior of functionally graded micro-beams, Materials and Design, Vol. 31, No.5, pp. 2324-2329, 2010.
  15. Karličić, M. Cajić, T. Murmu and S. Adhikari, Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems, European Journal of Mechanics-A/Solids, Vol. 49, pp.183-196, 2015.
  16. L. Chaht, A. Kaci, M.S.A. A. Houari, Tounsi, O.A. Bég and S.R. Mahmoud, Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect, Steel and Composite Structures, Vol. 18, No.2, pp. 425-442, 2015.
  17. L. Ke, Y.S. Wang, J. Yang and S. Kitipornchai, Nonlinear free vibration of size-dependent functionally graded microbeams, International International Journal of Engineering Science, Vol. 50, No.1, pp. 256-267, 2012.
  18. M. Wang, Y. Xiang, J. Yang and S. Kitipornchai, Buckling of nano-rings/arches based on nonlocal elasticity, International Journal of Applied Mechanics, Vol. 4, No. 3, pp.1250025, 2012.
  19. Kocatürk and Ş.D. Akbaş, Wave propagation in a microbeam based on the modified couple stress theory, Structural Engineering and Mechanics, Vol. 46, No. 3, pp. 417-431, 2013.
  20. Aissani, M.B. Bouiadjra, M. Ahouel and A. Tounsi, A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium, Structural Engineering and Mechanics, Vol. 55, No. 4, pp. 743-763, 2015.
  21. Ş.D. Akbaş, Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium, Smart Structures and Systems,  18, No.6, pp. 1125-1143, 2016.
  22. Ş.D. Akbaş, Forced vibration analysis of functionally graded nanobeams, International Journal of Applied Mechanics,  9, No.07, pp. 1750100, 2017.
  23. Ş.D. Akbaş, Static analysis of a nano plate by using generalized differential quadrature method, International Journal of Engineering & Applied Sciences,  8, No.2, pp. 30-39, 2016.
  24. Ş.D. Akbaş, Analytical solutions for static bending of edge cracked micro beams, Structural Engineering and Mechanics,  59, pp. 579-599, 2016.
  25. Ş.D. Akbaş, Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory, International Journal of Structural Stability and Dynamics,  17, pp. 1750033, 2017.
  26. Ş.D. Akbaş, Forced vibration analysis of cracked nanobeams, Journal of the Brazilian Society of Mechanical Sciences and Engineering,  40, No.8, pp. 392, 2018.
  27. Ş.D. Akbaş, Forced vibration analysis of cracked functionally graded microbeams, Advances in Nano Research,  6, No.1, pp.39-55, 2018.
  28. Ş.D. Akbaş, Longitudinal forced vibration analysis of porous a nanorod, Journal of Engineering Sciences and Design,  7, No.4, pp.736-743, 2019.
  29. Ş.D. Akbaş, Axially forced vibration analysis of cracked a nanorod, Journal of Computational Applied Mechanics, 50, No.1, pp. 63-68, 2019.
  30. Ş.D. Akbaş, Modal analysis of viscoelastic nanorods under an axially harmonic load, Advances in nano research,  8, No.4, pp. 277-282, 2020.
  31. Romano and J.N. Reddy, Experimental validation of the modified couple stress Timoshenko beam theory for web-core sandwich panels, Composite Structures, Vol. 111, , pp.130–137, 2014.
  32. Ö. Yaylı, S.Y. Kandemir, and A.E. Çerçevik, A practical method for calculating eigenfrequencies of a cantilever microbeam with the attached tip mass, Journal of Vibroengineering, Vol. 18, No.5, pp. 3070-3077, 2016.
  33. H. Ghayesh, H. Farokhi and M. Amabili, Nonlinear dynamics of a microscale beam based on the modified couple stress theory, Composites Part B: Engineering, Vol. 50, pp. 318-324, 2013.
  34. H. Ghayesh, M. Amabili and H. Farokhi, Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory, International Journal of Engineering Science, Vol. 63, pp. 52-60, 2013.
  35. Alimoradzadeh and Ş.D. Akbaş, Superharmonic and subharmonic resonances of atomic force microscope subjected to crack failure mode based on the modified couple stress theory, The European Physical Journal Plus, Vol. 136, No.5, pp. 1-20, 2021.
  36. Akgöz and Ö. Civalek, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science, Vol. 49, No.11, pp. 1268-1280, 2011.
  37. Alimoradzadeh and Ş.D. Akbaş, Superharmonic and subharmonic resonances of a carbon nanotube-reinforced composite beam, Advances in Nano Research, Vol. 12, No.4, pp. 353-363, 2022.
  38. Alimoradzadeh and Ş.D. Akbaş, Nonlinear dynamic responses of cracked atomic force microscopes, Structural Engineering and Mechanics, Vol. 82, No.6, pp. 747-756, 2022.
  39. Alimoradzadeh and Ş.D. Akbaş, Nonlinear vibration analysis of carbon nanotube-reinforced composite beams resting on nonlinear viscoelastic foundation, Geomechanics and Engineering, Vol. 32, No.2, pp.125-135, 2023.
  40. Alimoradzadeh and Ş.D. Akbaş (2023). Thermal nonlinear dynamic and stability of carbon nanotube-reinforced composite beams, Steel and Composite Structures, Vol. 46, No.5, pp.637-647.
  41. Alimoradzadeh and Ş.D. Akbaş (2023). Nonlinear free vibration analysis of a composite beam reinforced by carbon nanotubes, Steel and Composite Structures, Vol. 46, No.3, pp.335-344.
  42. Alimoradzadeh and Ş.D. Akbaş, Nonlinear oscillations of a composite microbeam reinforced with carbon nanotube based on the modified couple stress theory, Coupled Systems Mechanics, Vol. 11, No.6, pp. 485-504, 2022.
  43. J. Chen and X. P. Li, Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory, Archive of Applied Mechanics, Vol. 83, No.3, pp. 431-444, 2013.
  44. M.C. Roque, D.S. Fidalgo, A.J.M. Ferreira and J.N. Reddy, A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method, Composite Structures, Vol. 96, pp. 532-537, 2013.
  45. M. Abadi and A.R. Daneshmehr, An investigation of modified couple stress theory in buckling analysis of micro composite laminated Euler–Bernoulli and Timoshenko beams, International Journal of Engineering Science, Vol. 75, pp. 40-53, 2014.
  46. Mohammadabadi, A.R. Daneshmehr and M. Homayounfard, Size-dependent thermal buckling analysis of micro composite laminated beams using modified couple stress theory, International Journal of Engineering Science, Vol. 92, pp. 47-62, 2015.
  47. Hosseini Hashemi and H. Bakhshi Khaniki, Free vibration analysis of nonuniform microbeams based on modified couple stress theory: an analytical solution, International Journal of Engineering, Vol. 30, No.2, pp. 311-320, 2017.
  48. Feng, S. Kitipornchai and J. Yang, Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs), Composites Part B: Engineering, Vol. 110, pp. 132-140, 2017.
  49. H. Dong, Y.F. Zhang and Y.H. Li, An analytical formulation for postbuckling and buckling vibration of micro-scale laminated composite beams considering hygrothermal effect, Composite Structures, Vol. 170, pp. 11-25, 2017.
  50. D. Nguyen, T.K. Nguyen, H.T. Thai and T.P. Vo, A Ritz type solution with exponential trial functions for laminated composite beams based on the modified couple stress theory, Composite Structures, Vol. 191, pp. 154-167, 2018.
  51. Z. Jouneghani, P.M. Dashtaki, R. Dimitri, M. Bacciocchi and F. Tornabene, First-order shear deformation theory for orthotropic doubly-curved shells based on a modified couple stress elasticity, Aerospace Science and Technology, Vol. 73, pp.129-147, 2018.
  52. B. Khaniki, S. Hosseini-Hashemi and A. Nezamabadi, Buckling analysis of nonuniform nonlocal strain gradient beams using generalized differential quadrature method, Alexandria Engineering Journal, Vol. 57, No.3, pp.1361-1368, 2018.
  53. Lal and C. Dangi, Thermomechanical vibration of bi-directional functionally graded non-uniform timoshenko nanobeam using nonlocal elasticity theory, Composites Part B: Engineering, Vol. 172, pp. 724-742, 2019.
  54. Bhattacharya and D. Das, Free vibration analysis of bidirectional-functionally graded and double-tapered rotating micro-beam in thermal environment using modified couple stress theory, Composite Structures, Vol. 215, pp. 471-492, 2019.
  55. Z. Jouneghani, H. Babamoradi, R. Dimitri and F. Tornabene, A modified couple stress elasticity for non-uniform composite laminated beams based on the Ritz formulation, Molecules, Vol. 25, No.6, pp. 1404, 2020.
  56. Priyanka and J. Pitchaimani, Static stability and free vibration characteristics of a micro laminated beam under varying axial load using modified couple stress theory and Ritz method, Composite Structures, pp. 115028, 2021.
  57. Ş.D. Akbaş, Size dependent vibration of laminated micro beams under moving load, Steel and Composite Structures,  46, No.2, pp. 253, 2023.
  58. Ş.D. Akbaş, Moving Load Analysis of Laminated Porous Micro Beams Resting on Elastic Foundation, International Journal of Applied Mechanics,  15, No.8, pp. 2350066, 2023.
Journal of Computational Applied Mechanics
Volume 55, Issue 1
January 2024
Pages 26-38
  • Receive Date: 03 January 2024
  • Revise Date: 06 February 2024
  • Accept Date: 06 February 2024