A micro scale Timoshenko beam model based on strain gradient elasticity theory

• Published in: European journal of mechanics, A, Solids Vol. 29; no. 4; pp. 591 - 599
• Main Authors: Wang, Binglei, Zhao, Junfeng, Zhou, Shenjie
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Masson SAS 01.07.2010; Elsevier
• Subjects: Beams (structural); Deflection; Elasticity; Exact sciences and technology; Free vibration; Fundamental areas of phenomenology (including applications); Mathematical models; Micro scale; Physics; Resonant frequency; Size effect; Solid mechanics; Static elasticity (thermoelasticity...); Strain; Strain gradient; Structural and continuum mechanics; Timoshenko beam; Timoshenko beams; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Micro scale; Timoshenko beam; Strain gradient; Size effect; Initial condition; Free vibration; Simple support; Bernoulli Euler model; Scale model; Modeling; Natural frequency; Scaling law; Static model; Hamiltonian mechanics; Reduced order systems; Bending vibration
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2009.12.005

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.