The Elasticity Theory With Strain Gradient Effect And Its Applications

Micro-components are the basic components in Micro-Electro-Mechanical-Systems (MEMS). Mastering the load behaviors of micro-components accurately is the basis to make accurate control of MEMS. When the size of micro-component reduces to mciro-scale, not only its mechanical property but also the multi-field coupling property show obvious size effect phenomena. All these phenomena are difficult to be explained by the classical theory without including any material lenth scale parameter. Preliminary researches have shown that strain gradient theory can explain these size effect phenomena. However, there are some problems in the existing theories. In some theories, the higher-order constants are not independent; in some theories, a wrong equilibrium condition is introduced; and some theories are approximated theories. All of these theories cannot describe the size effect in micro-components accurately. Whether it is from the point of the development of strain gradient theory or describing the mechanical behaviors of micro-components accurately, it is essential to present a correct and effective theory and further study the mechanical properties including the electromechanical coupling properties of micro-components. The main results are shown as follows:The symmetry/antisymmetry and hydrostatic/deviatoric decompositions of the strain gradient tensor are proposed, and the strain gradient is split into independent components. The constitutive relations are reformulated based on these independent components, and the corresponding higher-order material elastic tensors are constructed. Based on the higher-order tensor theory and the relations among higher-order material elastic tensors, the number of independent material length scale parameters is proved to be three for isotropic strain gradient elasticity theory. So the basic theoretical problem of constitutive relationship is solved and a strain gradient theory with independent material length scale parameters is presented. The variational principle of strain gradient theory in the form of independent components is developed and the governing equations and boundary conditions are derived. Furthermore, general formulations of the present strain gradient elasticity theory in orthogonal curvilinear coordinate systems are derived, and are then specified for the cases of cylindrical coordinates and spherical coordinates.On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. To facilitate the derivation, the bending stress resultants are first defined in terms of the stress measures in the strain gradient theory. Then, the bending equilibrium equations and force-prescribed boundary conditions are derived. The bending problems of plane-strain cantilever and Bernoulli-Euller cantilever are solved. Comparisons between the theoretical and experimental results of cantilever bending for epoxy and silicon demonstrate the validity of the new strain gradient elasticity theory. Further, the size effects of torsion of cylindrical bars, shearing of fixed-end layers, pure bending of thin beams and expansion of solid sphere are analyzed in current theory and Aifantis theory with only one material length scale parameter, respectively. Comparison between the results in these two theories show that the current theory can explain the size effect phenomena in all kinds of deformation problems effectively, while single-parameter theory is limited.For the laminated composite structures used widely in MEMS, a displacement field is proposed for the bilayered beams and plates to apply the displacement variational method. The size-dependent bilayered beam and plate models are proposed based on the new theory and the principle of minimum potential, respectively. The static bending problems of simply supported bilayered beam and plate are solved, respevtively. The size effects of deflections for bilayered beam and plate are captured, and the influence of the thickness ratio of the upper layer to the lower layer on the deflection is revealed.A size-dependent flexoelectric theory is proposed for isotropic centrosymmetric dielectrics based on the new present strain gradient elasticity theory, in which the size effect, polarization gradient effect, flexoelectric effect are considered and only three material length-scale parameters and two flexocoupling coefficients are involved. The constitutive relations are given in the form of independent components. The variational priciple of the new flexoelectric theory in the form of indepedent components is developed, and the governing equations and boundary conditions, which include the effect of the electrostatic force, are derived. Just applying the mathmatical reformulations, the coupling of strain gradient to polarization is proved to occur at its dilatation gradient and the antisymmetric part of the deviatoric rotation gradient only. The deviatoric stretch gradient and the symmetric part of the deviatoric rotation gradient cannot induce polarization.Based on the new present flexoelectric theory, a flexoelectric Bernoulli-Euler beam model and a flexoelectric Kirchhoff circular plate model are developed, respectively. For the flexoelectric Bernoulli-Euller beam model, a static bending of cantilever is solved in two cases, of which one is subjected to a concentrated foce at its free end and the other is subjected to a voltage across the beam thickness. For the flexoelectric Kirchhoff circular plate model, a static bending of simply supported axisymmetric circular micro-plate is solved in two cases, of which one is subjected to a distributed load and the other is subjected to a voltage across the plate thickness. For the bending of both cantilever beam and simply supported axisymmetric circular plate, the size-dependent direct and inverse flexoelectric effects are captured. In addition, the direct and inverse flexoelectric effects decrease with the decrease of the flexocoupling coefficients. Especially, the direct and inverse flexoelectric effects disappear when the flexocoupling coefficients equal zero.The present strain gradient elasticity theory and flexoelectric theory can explain the size effect of mechanical property and the flexoelectric effect of electromechanical coupling property effectively. The research results will be meanfulling for the design calculation, performance prediction and experimental research of products in MEMS.