<p>In this work, a Timoshenko beam model based on micropolar elasticity theory is formulated. In the micropolar theory, it is assumed that each infinitesimal macro-element contains a microstructure that can rigidly rotate independently of its surrounding medium. From a kinetic perspective, the microrotation is associated with a couple stress tensor, which renders the Cauchy stress tensor asymmetric. Starting from the assumed displacement and microrotation fields, the active components of the strain, wryness, stress, and couple stress tensors are derived. The principle of virtual work is then employed to obtain the governing equilibrium equations along with the corresponding boundary conditions. For the present formulation, three second-order differential equations are obtained for the lateral deflection, macro-rotation of the cross section, and the microrotation field. These equations are subsequently transformed into a system of first-order differential equations, from which exact analytical solutions can be derived for various loading and boundary conditions. Exact solutions to four sample problems are presented, and the effects of different parameters are investigated. In particular, it is shown that the proposed formulation can successfully capture the experimental data on microcantilevers reported in the literature. Furthermore, it is demonstrated that the size effect at the microscale can be effectively captured by the present beam model.</p>

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Exact analytical solution to the Timoshenko beam model based on micropolar elasticity

  • Soroush Eghbali,
  • Farzam Dadgar-Rad

摘要

In this work, a Timoshenko beam model based on micropolar elasticity theory is formulated. In the micropolar theory, it is assumed that each infinitesimal macro-element contains a microstructure that can rigidly rotate independently of its surrounding medium. From a kinetic perspective, the microrotation is associated with a couple stress tensor, which renders the Cauchy stress tensor asymmetric. Starting from the assumed displacement and microrotation fields, the active components of the strain, wryness, stress, and couple stress tensors are derived. The principle of virtual work is then employed to obtain the governing equilibrium equations along with the corresponding boundary conditions. For the present formulation, three second-order differential equations are obtained for the lateral deflection, macro-rotation of the cross section, and the microrotation field. These equations are subsequently transformed into a system of first-order differential equations, from which exact analytical solutions can be derived for various loading and boundary conditions. Exact solutions to four sample problems are presented, and the effects of different parameters are investigated. In particular, it is shown that the proposed formulation can successfully capture the experimental data on microcantilevers reported in the literature. Furthermore, it is demonstrated that the size effect at the microscale can be effectively captured by the present beam model.