Analysis On The Timoshenko Micro-beam Model Based On The Modified Gradient Elasticity Theory

When it is describing the mechanics behaviors of micro-structures,the size effect can’t be captured by the mechanics and math models which are based on the classic elasticity theory;the shearing effect can’t be captured by the Bernoulli-Euler beam model.Based on the modified gradient elasticity theory(MGE),a new Timoshenko micro-beam model,which is affected by the shearing effect and strain gradient,is proposed by using the variational principle.The new model can be simplified to the MGE Bernoulli-Euler beam model when the shearing effect is neglected,also can be simplified to the classical Timoshenko beam model when two internal character length scales vanish.When the shearing effect and the size effect are neglected,the new beam model can be simplified to the classic Bernoulli-Euler beam model.To solve this model,a new central finite difference method with 6 virtual nodes is developed.Three numerical examples of the homogeneous MGE beams and axially functionally graded(AFG)MGE beams are considered in this thesis.The size effect and shearing effect are captured from all the examples,i.e.the deflection,the rotation angle and the shearing angle decrease with the internal length scales;the size effect vanishes when the beam size is large enough.With the increment of the internal character length scales,the shearing effect plays a more and more important role on the deformation of MGE Timoshenko beam.The influence of l3(the internal length scale along the beam thickness direction)on deflection and on rotation is greater than that of l1(the internal length scale along the beam length direction),but the influence caused by l1 can’t be ignored.The influence of l1(the internal length scale along the beam thickness direction)on shearing angle is greater than that of l3(the internal length scale along the beam length direction).There is a linear relationship between the shear boundary layer and l1;there is a linear relationship between the maximal shearing angle and l1.The shearing angle affected by the Poisson’s effect.