Size Effect On Mechanical Properties Of Micro-beams

With the advances of Micro-electro-mechanical systems (MEMS), a wide range of MEMS devices and products have been fabricated. According to geometry and loaded forms of those devices and products, main structures can be simplified to be some typical structural forms. Micro-beam is one of most used structural forms in MEMS devices and products. Due to small size and special processing technology, mechanical properties of micro-beams are very different from that in the macro-scale range. Therefore, thoroughly understanding and accurate characterization of mechanical properties of micro-beams is critical for optimal design and reliability of MEMS devices and products. Nowadays, many micro-scale experiments have verified that mechanical properties of some materials at the micron scale are size dependent. No material length scale parameter is introduced into the constitutive relation of conventional theory, so it cannot characterize and interpret size effect on the mechanical properties of materials and structures at the micron scale. Strain gradient theories, which include effects of strain gradient tensor and possess material length scale parameter, can characterize and interpret size effect on the mechanical properties of micro-structures at the micron scale.Based on strain gradient theories (including couple stress theory and strain gradient elasticity theory), size effect on the mechanical properties of micro-beams are studied in this work. Theoretical models are established and influence of size effect on mechanical properties of micro-beams is analyzed thoroughly. Conclusions can render basic guidelines for optimal design and experimental tests of MEMS devices and products. Main contents of this work are as follows:In order to analyze influence of size effect of materials on the dynamic property of micro-beams, a theoretical model of dynamic property of micro-beams is established on the basis of a modified couple stress theory. It can be seen that in the new model, only one additional material length scale parameter is included besides two classical material constants. In the light of feature sizes of simply supported micro-beams and cantilever micro-beams, the governing equations of micro-beams are solved and size effect on natural frequencies of micro-beams are assessed. It is found that natural frequencies predicted by the newly established model are 2.6 and 3 times as that calculated from classical beam model when feature sizes of micro-beams (for rectangular and circular micro-beams, feature sizes are thickness and diameter, respectively) equal to material internal length scale. However, Natural frequencies predicted by the newly established model are approximate to that calculated from classical beam model when feature sizes of micro-beams are by far larger than material internal length scale parameter. Therefore, the new model can capture size effect on natural frequencies of micro-beams and can provide the theoretical basis for optimal design and experimental test of dynamic properties of MEMS products and devices.For buckling problems of slender columns under the action of axial compressive loads, a theoretical model for buckling characteristics of slender columns is established by a combination of the basic equations of the modified couple stress theory and the energy variational method. It can be seen that in the new model, only one additional material length scale parameter is included besides two classical material constants. Solutions of corresponding boundary value problems for buckling are presented and size effect on normalized buckling loads of slender columns with retangular and circular cross sectional shapes are analyzed. It is shown that the buckling load of slender columns predicted by the newly established model are 7 and 9 times as that calculated from classical beam model when feature sizes (for rectangular and circular micro-beams, feature sizes are thickness and diameter, respectively) of slender columns equal to material internal length scale parameter. However, the buckling load of slender columns predicted by the newly established model conforms to that calculated from classical beam model when feature sizes of micro-beams are by far larger than material internal length scale parameter. Therefore, the new model can capture size effect on buckling loads of slender columns and can provide a theoretical basis for design and control of similar buckling MEMS devices and products.For pull-in characteristics of electrostatically actuated micro-structures commonly used in practical engineerings, a theoretical model for pull-in characteristics of micro-beams is established based on a modified couple stress theory. It should be emphasized that, in the new model, only one additional material length scale parameter is included besides two classical material constants.Taking into account size effect on mechanical properties of materials and the inherent nonlinear property of electrostatic force, approximate analytical solutions of the pull-in voltage and pull-in displacement of electrostatically actuated micro-beams are obtained by using the Rayleigh-Ritz method. In the light of feature size of cantilever micro-beams and fixed-fixed micro-beams, size effect on pull-in characteristics of micro-beams is assessed. It is shown that the normalized pull-in voltage of the micro-beams increases by a factor on 2.7 as the thickness of microbeams equals to the material internal length scale parameter and exhibits size dependent. However, size effect is almost diminishing as the thickness of microbeams is far greater than material internal length scale parameter. Moreover, the normalized pull-in displacement of microbeams is size independence and equals to 0.448 and 0.398 for cantilever beams and clamped-clamped beams, respectively. Therefore, the new model can capture size effect on pull-in characteristics of electrically actuated micro-beams and can give a theoretical guide for optimal design and experimental tests of electrostatically actuated MEMS structures.The modified couple stress proposed by Yang et al. only includes the effect of strain tensor and rotation gradient tensor. Afterwards, Lam et al. presented a new strain gradient elasticity theory, which takes into account the effects of strain tensor, rotation gradient tensor and stretch gradient tensor. In order to analyze the bending behaviour of micro-beams, a bending model for micro-beams is established based on the strain gradient elasticity theory and governing equation and boundary conditions are derived. It can be seen that only 3 additional material length scale parameters are introduced besides 2 classical material constants in the new model. Moreover, boundary conditions consist of 2 classical boundary conditions and 1 additional higher-order boundary condition. In the light of geometries and boundary conditions of micro-beams, size effect on normalized bending rigidities of micro-beams and boundary layer effect are assessed. It is shown that bending rigidities of micro-beams based on strain gradient elasticity theory are about 12 times as that from classical theory model, when the thickness of retangular micro-beams become comparable to material internal length scale parameter and exhibit size dependent (h=20μm, l=17.6μm). However, bending rigidities of micro-beams based on starin gradient elasticity theory conform to that calculated from classical theory model when the feature sizes of micro-beams are far greater than material internal length scale parameters. Moreover, the higher-order boundary conditions results in boundary layer effect of deformed cantilever micro-beams at the fixed end. Therefore, the newly established model includes influences of rotation gradient tensor, deviatoric stretch gradient tensor and dilatation gradient tensor besides strain tensor, and can capture size effect on normalized bending rigidities and boundary layer effect of cantilever micro-beams at the fixed end.Taking into account the effect of rotation gradient tensor and stretch gradient tensor on dynamic property of micro-beams, a theoretical model for flexural micro-beams is established based on the strain gradient elasticity theory and by using Hamilton principle. Governing equation, initial conditions and boundary conditions of flexural micro-beams motion are derived. It can be seen that only 3 additional material length scale parameters are introduced in the new model besides 2 classical material constants. Moreover, boundary conditions consist of 2 classical boundary conditions and 1 additional higher-order boundary condition. In the light of geometry and boundary conditions of cantilever and simply supported micro-beams, size effect on natural frequencis of micro-beams is assessed. It is shown that natural frequencies of micro-beams based on strain gradient elasticity theory are about 12 times as that from classical theory model, as the thickness of retangular micro-beams become comparable to material internal length scale parameter and exhibit size dependent (h=20μm, l=17.6μm). However, natural frequencies of beams based on starin gradient elasticity theory conform to that calculated from classical theory model when the feature sizes of micro-beams are far greater than material internal length scale parameters. Therefore, the newly established model includes influences of rotation gradient tensor, deviatoric stretch gradient tensor and dilatation gradient tensor besides strain tensor, and can capture size effect on normalized natural frequencies of micro-beams.The above mechanical models can capture size effect on mechanical properties of micro-beams and conclusions can render theoretical guidelines for optimal design and experimental tests of MEMS devices.