Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams

Abstract

A microstructure-dependent nonlinear third-order beam theory which accounts for through-thickness power-law variation of a two-constituent material is developed using Hamilton's principle. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Kármán nonlinear strains. The modified couple stress theory contains a material length scale parameter that can capture the size effect in a functionally graded material beam. The influence of the material length scale parameter on linear bending is investigated. The finite element models are also developed to determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on linear and nonlinear response. Key wordsModified couple stress theory, equations of motion, general thirdorder beam theory, von Kármán nonlinearity, functionally graded beam, nonlinear finite element model.

A unified beam theory with strain gradient effect and the von Kármán nonlinearity

“…The unified beam theory presented in this paper differs from the general third‐order beam theory proposed by Arbind, Reddy, and Srinivasa in several aspects. …”

“… 1.The micro‐structure dependent properties in the paper by Arbind, Reddy, and Srinivasa are provided through modified couple stress theory, whereas, in this paper strain gradient theory is employed. In modified couple stress theory the nonlocal terms present in the strain energy depend on the product of the deviatoric part of the symmetric couple stress tensor and the components of the symmetric curvature tensor.…”

“…Modified couple stress theory can be considered as a special case of the strain gradient theory . 2.The generalized third‐order beam theory presented by Arbind, Reddy, and Srinivasa assumes a displacement field with 7 primary variables with C 0 continuity for all variables in the case where the beam is studied with no nonlocal features. When modified couple stress theory is considered, some of the primary variables will require C 1 continuity.…”

Study of squeeze film damping in a micro-beam resonator based on micro-polar theory

“…Latin By adding a term representing the force acting on the micro-beam owing to the pressure of the squeeze gas film, the equation of motion governing the transverse deflection of the beam is as follow (Li et al, 2007;Asghar et al, 2010;Arbind et al, 2014, Sedighi et al, 2014 (EI) eq 4 ( , )…”

A Theoretical Analysis of Functionally Graded Beam under Thermal Loading

“…Conjunction, the temperature field was obtained by exactly solving Fourier's heat conduction equation and, as classically done by a staggered solution approach, it was considered as an external load within the mechanical analysis. Arbind et al [7] developed the governing equations for a general third order theory for functionally graded beam considering the modified couple stress theory and von Kármán nonlinearity using Hamilton's principle.…”