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.

Modified couple stress-based free vibration behavior of pre-twisted tapered BFGM rotating micro beam considering spin-softening and Coriolis effects

“…For a simply supported beam, the lowest order functions for approximating the displacement and rotation fields are found in Das. 64 It is to be mentioned that the results of Figure 2 are generated with the 3-D stress–strain relationship, for which Ef is replaced by Ef(1-υf)/{(1+υf)(1-2υf)} in the first relation of equation (12) to calculate σxx . 44 The comparison shows that the present result is in good agreement.…”

“…For a simply supported beam, the lowest order functions for approximating the displacement and rotation fields are found in Das. 64 It is to be mentioned that the results of Figure 2 are generated with the 3-D stress-strain relationship, for which E f is replaced by…”

Size-dependent vibration analysis of fluid-infiltrated porous curved microbeams integrated with reinforced functionally graded graphene platelets face sheets considering thickness stretching effect

“…Based on the MCST, which suggests a material length-scale parameter, the strain energy of the structure includes two parts: the first one related to the classical elasticity theory (CET) and the second due to the deviator part of the couple stress tensor. So it can be written as 65 in which χ and m are the symmetric part of the curvature tensor and deviatoric part of the couple stress tensor, respectively, which can be obtained as where l m is the material length-scale parameter of MCST, and μ ( r ) is the Lame’s parameter. It is obvious that setting the material length-scale parameter l m to zero converts the equations of microstructure to macro one.…”

Strain gradient theory-based vibration analyses for functionally graded microbeams reinforced by GPL

“…Resonance behavior, bending and buckling of the GPL reinforced FG plates are studied in [22], and the effects of GPL distribution, geometric size, mass fraction of graphene are examined. The vibration, bending and buckling of the GPL reinforced microbeams on elastic foundations are analyzed in [5,[23][24][25][26][27]. It was found that the addition of graphene can improve the mechanical performance of the structures.…”