Size-dependent postbuckling for microbeams: analytical solutions using a reformulated strain gradient elasticity theory

This paper presents closed-form solutions for exploring size-dependent postbuckling of Euler–Bernoulli and Timoshenko microbeams using a novel strain gradient elasticity theory. To capture the size effects of microbeams, a reformulated strain gradient elasticity theory incorporating strain gradient and couple stress effects simultaneously with only one material parameter for each effect is employed. The governing equations and boundary conditions of both Euler–Bernoulli and Timoshenko microbeams are derived using the principle of minimum potential energy and von Kármán geometric nonlinearity theory. These models can be degenerated into modified couple stress theory if the strain gradient material length-scale parameter is taken to be zero. The analytical solutions of the critical buckling load are derived for both Euler–Bernoulli and Timoshenko microbeams. The numerical results are depicted to illustrate the effects of material length-scale parameter, length-to-thickness ratio and beam thickness on the postbuckling response.