Nonlinear forced vibration analysis of higher order shear-deformable functionally graded microbeam resting on nonlinear elastic foundation based on modified couple stress theory

Abstract

Geometrically nonlinear forced vibration analysis of higher order shear-deformable functionally graded microbeam is presented, where the beam is supported on a three-parameter Winkler–Pasternak-type nonlinear elastic foundation and subjected to a harmonically varying distributed load. The modified couple stress theory of elasticity is employed in the formulation to address the size-dependent effect. Hamilton’s principle is used to derive the displacement-based governing equations considering Reddy’s third-order shear deformation theory. Ritz method is followed to convert the governing equations to nonlinear algebraic form in the frequency domain by approximating the displacement fields. A mixed algorithm for nonlinear equations based on the iterative substitution method with successive relaxation and Broyden’s method is successfully employed to solve the stable regions of the frequency-response curves. The results are presented for hinged and clamped beams, and the effects of different parameters such as size-dependent thickness, load amplitude, foundation parameters, and gradation-profile parameter are studied. The effect of thermal loading due to uniform temperature rise is also studied considering temperature-dependent material properties.

Keywords

Microbeam modified couple stress theory elastic foundation forced vibration geometric nonlinearity functionally graded

Get full access to this article

View all access options for this article.

References

Lam

DCC

Yang

Chong

ACM

, et al. Experiments and theory in strain gradient elasticity. J Mech Phys Solids 2003; 51: 1477–1508.

Liu

Pan

Liu

. Dimension effect on mechanical behavior of silicon micro-cantilever beams. Measurement 2008; 41: 885–895.

Yang

Chong

ACM

Lam

DCC

, et al. Couple stress based strain gradient theory for elasticity. Int J Solids Struct 2002; 39: 2731–2743.

Park

Gao

. Bernoulli-Euler beam model based on a modiﬁed couple stress theory. J Micromech Microeng 2006; 16: 2355–2359.

Fleck

Hutchinson

. Strain gradient plasticity. Adv Appl Mech 1997; 33: 296–358.

Kong

Zhou

Nie

, et al. Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci 2009; 47: 487–498.

Eringen

. On diﬀerential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 1983; 54: 4703–4710.

Wang

. Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J Appl Phys 2005; 98: 124301–124301.

Reddy

. Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 2007; 45: 288–307.

10.

Gurtin

Murdoch

. Surface stress in solids. Int J Solids Struct 1978; 14: 431–440.

11.

Lü

Chen

Lim

. Elastic mechanical behavior of nano-scaled FGM ﬁlms incorporating surface energies. Compos Sci Technol 2009; 69: 1124–1130.

12.

Wang

Feng

. Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl Phys Lett 2007; 90: 231904–231904.

13.

Yamanoushi M, Koizumi M, Hiraii T, et al. (eds). In: Proceedings of the first international symposium on functionally gradient materials, Japan, 1990.

14.

Reddy

Chin

. Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stresses 1998; 21: 593–626.

15.

Toupin

. Elastic materials with couple stresses. Arc Ration Mech Anal 1962; 11: 385–414.

16.

Mindlin

Tiersten

. Effects of couple-stresses in linear elasticity. Arc Ration Mech Anal 1962; 11: 415–448.

17.

Koiter

. Couple stresses in the theory of elasticity: I and II. Proc K Ned Akad Wet B 1964; 67: 17–44.

18.

Mindlin

. Micro-structure in linear elasticity. Arc Ration Mech Anal 1964; 16: 51–78.

19.

Kong

Zhou

Nie

, et al. The size-dependent natural frequency of Bernoulli-Euler micro-beams. Int J Eng Sci 2008; 46: 427–437.

20.

Gao

Mahmoud

. A new Bernoulli-Euler beam model incorporating microstructure and surface energy effects. Z Angew Math Phys 2014; 65: 393–404.

21.

Gao

Zhang

. A microstructure- and surface energy-dependent third-order shear deformation beam model. Z Angew Math Phys 2015; 66: 1871–1894.

22.

Gao

Reddy

. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 2008; 56: 3379–3391.

23.

Kahrobaiyan

Asghari

Ahmadian

. A Timoshenko beam element based on the modified couple stress theory. Int J Mech Sci 2014; 79: 75–83.

24.

Dehrouyeh-Semnani

Nikkhah-Bahrami

. The influence of size dependent shear deformation on mechanical behavior of microstructures-dependent beam based on modified couple stress theory. Compos Struct 2015; 123: 325–336.

25.

Bambill

Guerrero

Felix

. Natural vibrations of micro beams with nonrigid supports. J Vib Control 2017; 23: 3233–3246.

26.

Ghayesh

Farokhi

Amabili

. Nonlinear dynamics of a micro scale beam based on the modiﬁed couple stress theory. Compos Part B Eng 2013; 50: 318–324.

27.

Karparvarfard

SMH

Asghari

Vatankhah

. A geometrically nonlinear beam model based on the second strain gradient theory. Int J Eng Sci 2015; 91: 63–75.

28.

Rajabi

Ramezani

. A nonlinear microbeam model based on strain gradient elasticity theory with surface energy. Arch Appl Mech 2012; 82: 363–376.

29.

Ramezani

. A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory. Int J Non-Lin Mech 2012; 47: 863–873.

30.

Asghari

Ahmadia

Kahrobaiyan

, et al. On the size-dependent behavior of functionally graded micro-beams. Mater Des 2010; 31: 2324–2329.

31.

Salamat-talab

Nateghi

Torabi

. Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. Int J Mech Sci 2012; 57: 63–73.

32.

Reddy

. Microstructure-dependent couple stress theories of functionally graded beams. J Mech Phys Solids 2011; 59: 2382–2399.

33.

Şimşek

Reddy

. Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int J Eng Sci 2013; 64: 37–53.

34.

Al-Basyouni

Tounsi

Mahmoud

. Size dependent bending and vibration analysis of functionally graded micro beams based on modiﬁed couple stress theory and neutral surface position. Compos Struct 2015; 125: 621–630.

35.

Ebrahimi

Barati

. A modified nonlocal couple stress-based beam model for vibration analysis of higher-order FG nanobeams. Mech Adv Mater Struct 2018; 25: 1121–1132.

36.

Lei

Guo

, et al. Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM. J Therm Stresses 2017; 40: 665–689.

37.

Mirjavadi

Afshari

Shafiei

, et al. Effect of temperature and porosity on the vibration behavior of two dimensional functionally graded micro-scale Timoshenko beam. J Vib Control. 2018; 24: 4211–4225.

38.

Arbind

Reddy

. Nonlinear analysis of functionally graded microstructure-dependent beams. Compos Struct 2013; 98: 272–281.

39.

Komijani

Esfahani

Reddy

, et al. Nonlinear thermal stability and vibration of pre/post-buckled temperature- and microstructure-dependent functionally graded beams resting on elastic foundation. Compos Struct 2014; 112: 292–307.

40.

Shaﬁei

Mousavi

Ghadiri

. On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams. Int J Eng Sci 2016; 106: 42–56.

41.

Ansari

Pourashraf

Gholami

. An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory. Thin-Walled Struct 2015; 93: 169–176.

42.

Ansari

Gholami

Shojaei

, et al. 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 2016; 40: 9872–9891.

43.

Wisnom

. Size effects in the testing of fibre-composite materials. Compos Sci Technol 1999; 59: 1937–1957.

44.

Asl

Niezrecki

Sherwood

, et al. Vibration prediction of thin-walled composite I-beams using scaled models. Thin-Walled Struct 2017; 113: 151–161.

45.

Asl ME, Niezrecki C, Sherwood J, et al. Similitude analysis of the frequency response function for scaled structures. In: Barthorpe R, Platz R, Lopez I, Moaveni B, Papadimitriou C (eds) Model validation and uncertainty quantification. Vol. 3. Cham, Switzerland: Springer, 2017, pp.209–217.

46.

Wang

. Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 2011; 93: 342–350.

47.

Barati

Zenkour

. Forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments. Mech Adv Mater Struct 2018; 25: 669–680.

48.

Lü

Lim

Chen

. Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. Int J Solids Struct 2009; 46: 1176–1185.

49.

Bhushan

Takashima

, et al. Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscopy 2003; 97: 481–494.

50.

Yang

Lau

Tan

, et al. An electro-thermal micro-actuator based on polymer composite for application to dual-stage positioning systems of hard disk drives. Sens Actuat A Phys 2012; 187: 98–104.

51.

Niknam

Aghdam

. A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Compos Struct 2015; 119: 452–462.

52.

El-Borgi

Fernandes

Reddy

. Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation. Int J Non-Lin Mech 2015; 77: 348–363.

53.

Reddy

Kim

. A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos Struct 2012; 94: 1128–1143.

54.

Shen

. Functionally graded materials nonlinear analysis of plates and shells, Florida, USA: CRC Press, 2009.

55.

Zhang

. Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory. Compos Struct 2013; 100: 121–126.

56.

Fallah

Aghdam

. Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation. Euro J Mech A/Solids 2011; 30: 571–583.

57.

Majumdar

Das

. A study on thermal buckling load of clamped functionally graded beams under linear and nonlinear thermal gradient across thickness. Proc IMechE, Part L: J Materials: Design and Applications. 2018; 232: 769–784.

58.

Paul

Das

. Free vibration behavior of a thermally post-buckled FG Timoshenko beam under large deflection using a tangent stiffness-based method. Mech Adv Mater Struct 2018; 25: 982–994.

59.

Das

Sahoo

Saha

. A numerical analysis of large amplitude beam vibration under different boundary conditions and excitation patterns. J Vib Control 2011; 18: 1900–1915.

60.

Das

Sahoo

Saha

. Nonlinear vibration analysis of clamped skew plates by a variational method. J Vib Control 2009; 15: 985–1017.

61.

Press

Teukolsky

Vetterling

, et al. Numerical recipes in Fortran 77: the art of scientific computing, 2nd ed. New York: Cambridge University Press, 1992.

62.

Ribeiro

. Nonlinear forced vibrations of thin/thick beams and plates by the finite element and shooting methods. Comput Struct 2004; 82: 1413–1423.