A quasi-3D modified strain gradient formulation for static bending of functionally graded micro beams resting on Winkler-Pasternak elastic foundation

Document Type : Article


School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846, Iran


This paper presents the bending analysis of simply supported functionally graded (FG) size dependent beams based on modified strain gradient theory. The shear and normal deformations are considered in displacement field according to hyperbolic shear deformation theory. Governing equations and corresponding boundary conditions for FG micro beam are derived utilizing principle of minimum total potential energy. Mori–Tanaka homogenization scheme and the classical rule of mixture are used for prediction of material properties through the thickness. Effects of Winkler-Pasternak elastic foundation parameters are studied for different side to thickness ratios. Effects of different aspect ratios, elastic foundation parameters, power law gradient indexes and different loading conditions are investigated. The efficiency and accuracy of present model is demonstrated by comparing to the existing results in especial cases.


1. Patocka, F., Schneidhofer, C., Dorr, N., et al. "Novel resonant MEMS sensor for the detection of particles with dielectric properties in aged lubricating oils", Sens. Actuators A Phys., 315, p. 112290 (2020).
2. Ulkir, O. "Design and fabrication of an electrothermal MEMS micro-actuator with 3D printing technology", Mater. Res. Express, 7, p. 075015 (2020).
3. Moutlana, M.K. and Adali, S. "Fundamental frequencies of a nano beam used for atomic force microscopy (AFM) in tapping mode", MRS Adv., 3, pp. 2617-2626 (2018).
4. Li, X.-F., Wang, B.-L., and Lee, K.Y. "Size effect in the mechanical response of nanobeams", J. of Adv. Res. in Mech. Eng., 1(1), pp. 4-16 (2010).
5. Dell'Isola, F., Andreaus, U., and Placidi, L. "At the origins and in the vanguard of peridynamics, nonlocal and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola", Math. Mech. Solids, 20(8), pp. 887-928 (2015).
6. Cosserat, E. and Cosserat, F. "Theorie des corps deformables", Hermann, Paris, pp. 953-1173 (1909).
7. Thai, H-T., Vo, T.P., Nguyen, T.-K., et al. "A review of continuum mechanics models for size-dependent analysis of beams and plates", Compos. Struct., 177, pp. 196-219 (2017).
8. Mindlin, R.D. "Second gradient of strain and surfacetension in linear elasticity", Int. J. Solids Struct., 1(4), pp. 417-438 (1965).
9. Toupin, R. "Elastic materials with couple-stresses", Arch. Ration. Mech. Anal., 11(1), pp. 385-414 (1962).
10. Mindlin, R. and Tiersten, H. "Effects of couple-stresses in linear elasticity", Arch. Ration. Mech. Anal, 11, pp. 415-448 (1962).
11. Koiter, W.T. "Couple-stresses in the theory of elasticity, I & II", Philos. Trans. R. Soc. Lond., B, 67, pp. 17-44 (1969).
12. Yang, F., Chong, A., Lam, D.C.C., et al. "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), pp. 2731-2743 (2002).
13. Fleck, N. and Hutchinson, J. "Strain gradient plasticity", Adv. Appl. Mech., 33, pp. 295-361 (1997).
14. Fleck, N. and Hutchinson, J. "A reformulation of strain gradient plasticity", J. Mech. Phys. Solids, 49(10), pp. 2245-2271 (2001).
15. Lam, D.C., Yang, F., Chong, A., et al. "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids., 51(8), pp. 1477-1508 (2003).
16. Dal, H. "Analysis of gold micro-beams with modified strain gradient theory", Anadolu Universitesi Bilim Ve Teknoloji Dergisi A-Uygulamal Bilimler ve Muhendislik, 18(3), pp. 663-681 (2017).
17. Ashoori, A. and Mahmoodi, M. "A nonlinear thick plate formulation based on the modified strain gradient theory", Mech. Adv. Mater. Struct., 25(10), pp. 813-819 (2018).
18. Chu, L., Dui, G., and Ju, C. "Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory", Compos. Struct., 186, pp. 39- 49 (2018).
19. Thai, C.H., Ferreira, A., and Phung-Van, P. "Free vibration analysis of functionally graded anisotropic microplates using modified strain gradient theory", Eng. Anal. Bound. Elem., 117, pp. 284-298 (2020).
20. Farzam, A. and Hassani, B. "Size-dependent analysis of FG microplates with temperature-dependent material properties using modified strain gradient theory and isogeometric approach", Compos. B. Eng., 161, pp. 150-168 (2019).
21. Cornacchia, F., Fantuzzi, N., Luciano, R., et al. "Solution for cross-and angle-ply laminated Kirchhoff nano plates in bending using strain gradient theory", Compos. B. Eng., 173, p. 107006 (2019).
22. Thai, C.H., Ferreira, A., and Phung-Van, P. "A nonlocal strain gradient isogeometric model for free vibration and bending analyses of functionally graded plates", Compos. Struct., 251, pp. 112634 (2020).
23. Phung-Van, P., Ferreira, A., Nguyen-Xuan, H., et al. "An isogeometric approach for size-dependent geometrically nonlinear transient analysis of functionally graded nanoplates", Compos. B. Eng., 118, pp. 125- 134 (2017).
24. Phung-Van, P., Thai, C.H., Nguyen-Xuan, H., et al. "Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis", Compos. B. Eng., 164, pp. 215-225 (2019).
25. Phung-Van, P., Thai, C.H., Nguyen-Xuan, H., et al. "An isogeometric approach of static and free vibration analyses for porous FG nanoplates", Eur J Mech A Solids, 78, p. 103851 (2019).
26. Farzam, A. and Hassani, B. "Isogeometric analysis of in-plane functionally graded porous microplates using modified couple stress theory", Aerosp Sci Technol.,91, pp. 508-524 (2019).
27. Farzam, A. and Hassani, B. "Thermal and mechanical buckling analysis of FG carbon nanotube reinforced composite plates using modified couple stress theory and isogeometric approach", Compos. Struct., 206, pp. 774-790 (2018).
28. Phung-Van, P., Thai, C.H., Wahab, M.A., et al. "Optimal design of FG sandwich nanoplates using sizedependent isogeometric analysis", Mech. Mater., 142, p. 103277 (2020).
29. Phung-Van, P., Ferreira, A., and Thai, C.H. "Computational optimization for porosity-dependent isogeometric analysis of functionally graded sandwich nanoplates", Compos. Struct., 239, pp. 112029 (2020).
30. Zhao, X., Zheng, S., and Li, Z. "Size-dependent nonlinear bending and vibration of  flexoelectric nanobeam based on strain gradient theory", Smart Mater. Struct., 28, p. 075027 (2019).
31. Zanoosi, A.A.P. "Size dependent thermo mechanical free vibration analysis of functionally graded porous microbeams based on modified strain gradient theory", J. Braz. Soc. Mech. Sci., 42(5), p. 236 (2020).
32. Timoshenko, S.P. "On the correction for shear of the differential equation for transverse vibrations of prismatic bars", Philos. Mag., 41(245), pp. 744-746 (1921).
33. Nam, V.H., Vinh, P.V., Chinh, N.V., et al. "A new beam model for simulation of the mechanical behaviour of variable thickness functionally graded material beams based on modified first order shear deformation theory", Materials, 12(3), p. 404 (2019).
34. Reddy, J. "A general non-linear third-order theory of plates with moderate thickness", Int. J. Nonlin. Mech., 25(6), pp. 677-686 (1990).
35. Ghugal, Y.M. and Sharma, R. "A hyperbolic shear deformation theory for  flexure and vibration of thick isotropic beams", Int. J. Comput. Methods, 6(04), pp. 585-604 (2009).
36. Soldatos, K. "A transverse shear deformation theory for homogeneous monoclinic plates", Acta Mech., 94(3-4), pp. 195-220 (1992).
37. Ninh, D.G. and Bich, D.H. "Nonlinear thermal vibration of eccentrically stiffened ceramic-FGM-metal layer toroidal shell segments surrounded by elastic foundation", Thin-Walled Struct., 104, pp. 198-210 (2016).
38. Ninh, D.G. and Bich, D.H. "Nonlinear buckling of centrically stiffened functionally graded toroidal shell segments under torsional load surrounded by elastic foundation in thermal environment", Mech. Res. Commun., 72, pp. 1-15 (2016).
39. Carrera, E. "Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking", Arch. Comput. Methods Eng. 10(3), pp. 215-296 (2003).
40. Demasi, L. "1 6 mixed plate theories based on the generalized unified formulation. Part I: Governing equations", Compos. Struct., 87(1), pp. 1-11 (2009).
41. Karamanl, A. and Vo, T.P. "Size dependent bending analysis of two directional functionally graded microbeams via a quasi-3D theory and finite element method", Compos. B. Eng., 144, pp. 171-183 (2018).
42. Benahmed, A., Houari, M.S.A., Benyoucef, S., et al. "A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foundation", Geomech. Eng., 12(1), pp. 9-34 (2017).
43. Nguyen, T.-K., Vo, T.P., Nguyen, B.-D., et al. "An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi- 3D shear deformation theory", Compos. Struct., 156, pp. 238-252 (2016).
44. Farzam, A. and Hassani, B. "A new efficient shear deformation theory for FG plates with in-plane and through-thickness stiffness variations using isogeometric approach", Mech. Adv. Mater. Struct., 26, pp. 512- 525 (2019).
45. Farzam-Rad, S.A., Hassani, B., and Karamodin, A. "Isogeometric analysis of functionally graded plates using a new quasi-3D shear deformation theory based on physical neutral surface", Compos. B. Eng., 108, pp. 174-189 (2017).
46. Thai, C.H., Ferreira, A., Tran, T., et al. "A sizedependent quasi-3D isogeometric model for functionally graded graphene platelet-reinforced composite microplates based on the modified couple stress theory", Compos. Struct., 234, p. 111695 (2020).
47. Winkler, E. "Theory of elasticity and strength", Dominicus Prague, Czechoslovakia (1867).
48. Pasternak, P. "On a new method of analysis of an elastic foundation by means of two foundation constants" (in Russian), Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture, USSR (1954).
49. Atmane, H.A., Tounsi, A., and Bernard, F. "Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations", Int. J. Mech. Mater. Des., 13(1), pp. 71-84 (2017).
50. Lee, W.-H., Han, S.-C., Park, W.-T. "A refined higher order shear and normal deformation theory for E-, P-, and S-FGM plates on Pasternak elastic foundation", Compos. Struct., 122, pp. 330-342 (2015).
51. Li, Q., Wu, D., Gao, W., et al. "Size-dependent instability of organic solar cell resting on Winkler- Pasternak elastic foundation based on the modified strain gradient theory", Int. J. Mech. Sci., 177, p. 105306 (2020).
52. Ninh, D.G., Tien, N.D., Hoang, V.N.V., et al. "Vibration of cylindrical shells made of three layers W-Cu composite containing heavy water using Flugge-Lur'e-Bryrne theory", Thin-Walled Struct., 146, p. 106414(2020).
53. Zeighampour, H., Beni, Y.T., and Dehkordi, M.B."Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory", Thin-Walled Struct., 122, pp. 378-386 (2018).
54. Cao, C.-Y. and Zhong, Y. "Dynamic response of a beam on a Pasternak foundation and under a moving load", J. of Chongqing Univ., 7(4), pp. 311-316 (2008).
55. Kural, S. and Ozkaya, E. "Size-dependent vibrations of a micro beam conveying  fluid and resting on an elastic foundation", J. Vib. Control, 23(7), pp. 1106- 1114 (2017).
56. Eyebe, G., Betchewe, G., Mohamadou, A., et al. "Nonlinear vibration of a nonlocal nanobeam resting on fractional-order viscoelastic Pasternak foundations", Fractal Fract., 2(3), p. 21 (2018).
57. Bich, D.H. and Ninh, D.G. "Research on dynamical buckling of imperfect stiffened three-layered toroidal shell segments containing  fluid under mechanical loads", Acta Mech., 228, pp. 711-730 (2017).
58. Bich, D.H. Ninh, D.G. "An analytical approach: Nonlinear vibration of imperfect stiffened FGM sandwich toroidal shell segments containing  fluid under external thermo-mechanical loads", Compos. Struct., 162, pp. 164-181 (2017).
59. Bich, D.H. and Ninh, D.G. "Post-buckling of sigmoidfunctionally graded material toroidal shell segment surrounded by an elastic foundation under thermomechanical loads", Compos. Struct., 138, pp. 253-263 (2016).
60. El Meiche, N., Tounsi, A., Ziane, N., et al. "A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate", Int. J. Mech. Sci., 53(4), pp. 237-247 (2011).
61. Mori, T. and Tanaka, K. "Average stress in matrix and average elastic energy of materials with misfitting inclusions", Acta Metall., 21(5), pp. 571-574 (1973).
62. Trinh, L.C., Nguyen, H.X., Vo, T.P., et al. "Sizedependent behavior of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory", Compos. Struct., 154, pp. 556-572 (2016).
63. Ke, L.-L. and Wang, Y.-S. "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory", Compos. Struct., 93(2), pp. 342-350 (2011).