Analysis of Size-Dependent Linear Static Bending, Buckling, and Free Vibration Based on a Modified Couple Stress Theory

Abstract

The purposes of this paper are to study bending, buckling, and vibration by considering micro-scale effects using the Kirchhoff thin-plate theory and to consider small deflections, neglecting higher-order nonlinear terms. The governing equations for the bending, buckling, and vibration of the system are obtained using the equilibrium method coupled with the Kirchhoff thin-plate theory and a modified couple stress theory (MCST). The concept of the equivalent bending stiffness (EBS) of micro-thin plates is proposed to describe the scale effect. The Navier method is used to obtain analytical solutions for the bending, buckling, and free vibration of thin plates under simply supported boundary conditions with scale effects. The numerical results are presented to investigate the influence of scale effects on deflection, critical buckling load, buckling topography, and thin-plate natural frequency. The results show that the scale effect increases the equivalent stiffness of the thin plate, which leads to a decrease in deflection, a larger critical buckling load, and an increase in natural frequency, but does not affect the buckling topography. The MSCT is invalid when the thickness is greater than 10 times the scale effect parameter, thus defining the scope of application of the scale effect. This research study may contribute to the design of micro-scale devices such as MEMSs/NEMSs.

Size-dependent vibration analysis of the simply supported functionally graded porous material Al-Al2O3 rectangle microplates based on the modified couple stress theory with innovative consideration of neutral plane and scale distribution

“…Based on the MCST and scale distribution, the constitutive equation of the porous FGM plate can be denoted as follows (Tang et al ., 2022b):where σij, εij, χij and mij denote stress tensor, strain tensor, symmetric curvature tensor and couple stress tensor of the porous FGM plate, respectively, μ and λ are Lame constants, E is Young's modulus, υ is the Poisson ratio, normaleijk is the permutation symbol and θnormali is the porous FGM plate rotation vector and l is the scale parameter of porous FGM, which is related to the structure size, that is, power law index p and porosity.…”

“…Based on the MCST and scale distribution, the equation of motion of a porous FGM thin plate considering the effect of the physical neutral plane can be formulated as (Tang et al ., 2022b):where w is the deflection of the FGM thin plate, ρ* is the equivalent density at z, and p(x,y,t) is dynamic load. When considering the free vibration problem, the dynamic load p(x,y,t) = 0.…”

A Review on Nonlocal Theories in Fatigue Assessment of Solids

“…Size-dependent models can be classified into various frameworks [ 12 , 13 ] such as the nonlocal elasticity theory, which is intertwined with Eringen’s ground-breaking 1984 work [ 14 ] on evaluating the mechanical response of micro/nano-structures. This theory has been exploited for investigations of elastic waves lattice dispersion, wave propagation in nano-composites, dislocation mechanics, fracture mechanics, surface tension fluids, and other related topics.…”

A Size-Dependent Finite Element Method for the 3D Free Vibration Analysis of Functionally Graded Graphene Platelets-Reinforced Composite Cylindrical Microshells Based on the Consistent Couple Stress Theory

“…Based on the MCST, some two-dimensional (2D) microstructure-dependent shell theories have been developed for analyzing various mechanical behaviors of FG cylindrical and conical microshells. Tang et al [ 23 ] developed a microstructure-dependent Kirchhoff thin plate theory to study the microplate’s static bending, buckling, and free vibration behaviors. Incorporating the kinematic model of Love’s classical shell theory (LCST) into the MCST, Mehralian and Beni [ 24 ] developed a size-dependent LCST for the buckling analysis of single-walled CNTs (SWCNTs).…”