Mechanical Analysis Of Timoshenko Micro-beam With Defects Based On MGE

Micro/nanostructures are widely used in micro/nano electromechanical systems.Due to the imperfect of manufacturing process and the factor of human,the existence of defects is inevitable.Only by studying the action mechanism of defects can we effectively avoid or prevent the adverse effects of defects in advance.At the same time,the micro-beam has a strong size effect,that is,its mechanical response increases with the decrease of size.If the size effect is ignored,it will also have an adverse impact on the micro/nano-structure.The main research work and conclusions of this paper are as follows:The thickness defect distribution function is introduced into the stiffness expression of Timoshenko beam.Then,based on the modified gradient elasticity(MGE)theory,the static control equation of micro-beam is obtained by using the variational principle.The static behavior of micro-beam with thickness defect is studied by solving the square process by six virtual node finite difference method.In order to further explore the influence of thickness defects on the mechanical behavior of micro-beam,the free vibration control equation of micro-beam is obtained based on Hamiltonian principle.Finally,the method of Newmark-β and the method of six virtual node finite difference are used to solve the equation,and the free vibration of micro-beam with thickness defects is analyzed.Finally,through numerical results,the effects of the depth of defect,the width of defect,the position of defect and internal characteristic length parameters on the deflection of microbeam and the free vibration of micro-beam are studied.The results show that the deflection of micro-beam will increase with the expansion of the defect,and the closer the location of defect is to the middle of the beam,the greater the deflection of micro-beam is.In the shear boundary layer,we can also capture the size effect.The effect of free vibration of micro-beam is mainly reflected in the amplitude and acceleration amplitude,but has little effect on the fundamental frequency;The smaller the internal characteristic length parameter of the micro-beam,the greater the variation of the amplitude of micro-beam with the width and depth of defect.If only the fundamental frequency of free vibration of Timoshenko micro-beam is considered,the influence of defects can be ignored;If its amplitude is to be considered,the influence of thickness defects cannot be ignored.