Study On The Size Effect And The Numerical Method For Micro-Components

Attributed to the small mass and size, MEMS (Micro Electro Mechanical Systems, abbreviated as MEMS) come with high fundamental resonance frequencies, high quality factors and low power dissipation. MEMS have been widely used in the field of resonators, mixers, sensors and actuators. According to geometry and loaded forms of those devices and products, main structures can be simplified to be some typical structural forms, such as micro-beams, micro-plates, micro-membranes and micro-bars etc. In the micro scale range, the sub-micro scale range and the nano scale range, the mechanical properties of micro-component are very different from that in the macro scale range.At present, the size effects of the mechanical properties for the micro-component have been discovered in the mechanical properties experiments. The size effects of the micro-compoment cannot be described using classical continuum mechanics. Therefore, the creation of new theory which is able to explain and describe the effect of the micro-components is crucial. In order to describe and explain the size effect of the micro-components, the strain gradient is introduced into the strain gradient theory when the material intrinsic characteristic scale parameters are introduced into the constitutive relations in the new theory. At the same time, the strain gradient theory simplified as the classic elasticity theory when the material intrinsic characteristic scale parameters equal to zero. Therefore, the strain gradient theory is the expansion of the classical elasticity theory.In our paper, the micro-components in MEMS are mainly analyzed and the size effect models are established based on the strain gradient theory. The size effects of mechanical property are widely studied for the components in the micro scale range. Main content of our work are as follows:A nonlinear size effect model of micro-beam is established based on the strain gradient theory and the Hamiltonian variational principle. In the new model, the middle plane elongation and the residual stress are included while the three intrinsic characteristic scale paramenters are introduced. Therefore, the new model can predict not only the size effect of the micro-beams but also the effect of the middle plane elongation and the residual stress on the size effects. When the two material intrinsic characteristic scale parameters regarding as the dilatation gradient tensor and deviatoric stretch gradient tensor are zero, the new model can be simplified as the nonlinear model based on the couple stress theory. When the three material intrinsic characteristic scale parameters are zero, the new model can be simplified as the nonlinear model based on the classical theory. For the simply supported micro-beam and the clamped-clamped micro-beam, the size effects of the nonlinear static deformation, the nonlinear natural frequency and the nonlinear buckling load are analyzed. The size effect on the static deformation, the natural frequency and the buckling load of micro-beams are assessed when the feature size of the micro-beams approaches to the material intrinsic characteristic scale parameter. However, the size effect gradually reduced when the feature sizes of micro-beams are by far larger than the material intrinsic characteristic scale parameter. The effects of the middle plane elongation and the residual stress on the effect of micro-beams are significant. At the same time, the size effect phenomenon significantly reduced when the residual stress increases. The bending deformation, the buckling load and the nonlinear natural frequency based on the strain gradient theory are equal to the results based the classical theory when the feature size of the micro-beams are much larger than the material intrinsic characteristic length scales. Therefore, the new model can capture size effect on the nonlinear bending deformation, the nonlinear buckling load and the nonlinear natural frequency of micro-beams and can provide a theoretical basis for the structural design and experimental testing of the micro-beams.Based on the strain gradient theory and Hamiltonian variational principle, the size effect model for arbitrary boundary shape micro-plates is derived. The new model can captured the size effect of the mechanical properties of the micro-plates based on the introduction of three material intrinsic characteristic scale parameters. At the same time, the boundary conditions are derived at the arbitrary shape boundary and the coner of the plate. The new model not only includes the effect of the strain tensor and rotation gradient tentor, but also contains the stretch gradient skew tensor and dilatation strain gradient tensor, can fully descrie the size effects of the static deformation and the natural frequency of the micro-plate. For the simply supported micro-plate and the clamped micro-plate, the size effects of the static and the natural frequency are analyzed. The numerical results are compared with that of the couple stress micro-plates and the classical micro-plates. The results show that the normalized static deformation and the normalized nature frequency of circular micro-plates have significant size effects. The size effect is different when the constraint of the circular plate is different.The size effect model of the electrostatically acturated nonlinear micro-beams is established based on the strain gradient theory and Hamiltonian variational principle. Taking into account size effect on mechanical propreties of materials and the inherent nonlinear property of electrostatic force and the middle plane elongation, approximate expression of the pull-in voltage of electrostatically actuated micro-beams are obtained by using the Rayleigh-Ritz method. The results show that the dimensionless pull-in voltage increases significantly with the dimensionless micro-beam thickness decreasing, showing a significant size effect. When the material intrinsic characteristic length increases, the size effect of the dimensionless pull-in voltage is more significantly, indicating that the effect of the strain gradient on the micro-beam pull-in voltage is remarkable. When the dimensionless micro-beam thickness decreases, the effect of the residual stress on the dimensionless pull-in voltage is significant. The size effect of the normalized pull-in voltage is weakened when the residual stress increases, indicating the residual stress can decrease the effect of the strain gradient on the pull-in voltage. The effect of the middle plane elongation on the pull-in voltage is reduced markedly when the strain gradient is considered. The results can prove the reference in the design of micro structures in MEMS.Based on the strain gradient theory and the Hamiltonian variational principle, the size effect model and the numerical simulation model of the micro-beam in the plane bar system are derived. The size effect model and the numerical simulation model can analyze the size effect of micro-beam in the plane bar system due to the introducation of the material intrinsic characteristic scale parameters. At the same time, the size effects of the static deformation and the electrostatic pull-in voltage for the nonlinear micro-beam can be analyzed based on the numerical model. Due to a large number of micro-springs are used in MEMS, the size effects of the stiffness of the plane micro-spring are analyzed. The results show that the stiffnesses of micro-spring have significant size effect and provide a theoretical basis for the analysis and design of complex plane micro-spring.The above mechanical models can capture size effect on the static deformation, the natural frequency, the buckling load and the pull-in voltage of the micro components. The research results can provide a theoretical guideline for the optimal design and experimental tests of the micro-components in MEMS devices.