An efficient numerical method for the quasi-static behaviour of micropolar viscoelastic Timoshenko beams for couple stress problems

A viscoelastic Timoshenko micro-beam is investigated using micropolar viscoelasticity theory. The governing equations are derived by applying the principle of virtual displacements, and can be used in a variety of problems. The equations are then reduced to the special case of a homogeneous uniform beam and solved for quasi-static elastic and viscoelastic cases. The viscoelastic material is modeled via standard linear solid model. For the elastic case, a closed-form solution is reached, but for the viscoelastic case, the inverse Laplace transform is calculated numerically by utilizing De Hoog's algorithm. It is shown that the proposed model can capture the size effects, and converges to the classical Timoshenko beam if the beam is thick enough. The model also converges to the micropolar Euler-Bernoulli model if the thickness of the beam is small. The role of distributed couple stress is also investigated in this paper. The results indicate that in a cantilever beam, the distributed couple stress has a high influence on the deflection of the beam, micro-rotation, stress components and couple stress. However, in a beam resting on three supports, it only affects the micro-rotation and the shear stress component in transverse direction (������������������). Further investigation shows an extremum in some of the results over time, illustrating the importance of taking viscoelastic behaviour into account.