On The Linear Elastic Gradient Models And Their Applications

When the characteristic length of materials/structures scales down to micro-/nanometers, theirmechanical behaviors are found size-dependent. The physical origin of such size-dependent isdue to their spatial discrete, which results in the non-uniform deformation field. In fact, thesize-dependent phenomenon exists when the characteristic length of the materials/structures iscomparable to their internal length. To well depict their mechanical behaviors, use of thehigher order continuum theories which can take into account the microstructured effects hasbecome a topic of current interest. In the present thesis, we carry out a series of studiesfocusing on the establishment of the governing equations of beams, Kirchhoff plate, Mindlinplate and Flügge shell. At the meantime, we also derive the variational-consistent boundaryconditions of beams, Kirchhoff plate, Mindlin plate within the framework of, what we termedhere as, linear gradient elastic theory. The work of this thesis is expected to find theirapplications in practical engineering in Micro-/Nanoelectromechnical Systems(MEMS/NEMS). The main contents and conclusions are summarized as follows:1) By using the semi-inverse method, variational formulations for the buckling and vibrationof the strain gradient multi-walled carbon nanotubes (MWCNTs) are presented. Usingthese formulations, one can derive not only the governing equations of the MWCNTs, butalso the variational consistent sets of the coupled boundary conditions. Based on theRayleigh-Ritz method, the buckling stress, strain and vibration frequency of thesingle-walled carbon nanotubes (SWCNTs) and double-walled carbon nanotubes(DWCNTs) are obtained. These results are compared with those in published works. Twoelastic beam models are also compared to show that the strain gradient beam model ismore flexible than the nonlocal beam model when the small scale parameter is the same.In addition, the effects of gradient parameter, aspect ratio and boundary conditions on thebuckling and vibration of the SWCNTs and DWCNTs are investigated. The results alsoindicate that the coupled van der Waals (vdW) interaction between the adjacentnanotubes has prominent influence on the vibration of nanotubes.2) The prior one-dimensional beam problem is then extended to the two-dimensional plateand shell problems. According to the constitutive equations, strain-displacement relationsand equilibrium equations, the gradient Kirchhoff plate model is developed. Based on theHamilton’s variational principle, we also derive the variational-consistent boundaryconditions of gradient Kirchhoff plate model, and discuss the three typical boundary conditions encountered in engineering problems. The proposed method is capable ofdealing with size-dependent plate at micro/nanoscale dimension with complex geometriesand boundary conditions. Also, we find that the derived boundary conditions of Kirchhoffplate model reduce to that of Euler─Bernoulli beam model, provided that the spatialdimensions reduce to one-dimension. A simply supported plate is adopted to illustrate theproposed model, analytical solutions for the static bending subjected to the uniformloading and free vibration problems are obtained. The numerical results show that thedeflection predicted by the present model increases with the increase (decrease) ofgradient parameter l1(l2) compared with that predicted by conventional Kirchhoff platemodel, while the natural frequency exhibits the opposite behavior. These mean that theincorporation of l1softens the plate, while l2stiffens the plate.3) We develop the gradient Mindlin plate model from the constitutive equations,strain-displacement relations and equilibrium equations. Meanwhile, we also derive thevariational-consistent boundary conditions of gradient Mindlin plate model based on theHamilton’s variational principle, and discuss the three typical boundary conditionsencountered in engineering problems. The proposed method is capable of dealing withsize-dependent plate at micro/nanoscale dimension with complex geometries andboundary conditions. Also, we find that the derived boundary conditions of Mindlin platemodel reduces to that of Timoshenko beam model provided that the spatial dimensionreduces to one-dimension. To illustrate the proposed model, analytical solutions for thestatic bending subjected to the uniform loading and free vibration problems of a simplysupported plate are obtained in view of the governing equations and boundary conditions.The numerical results show that the deflection predicted by the present model increaseswith the increase (decrease) of l1(l2) compared with that predicted by conventionalMindlin plate model, while the natural frequency exhibits the opposite behavior. Inaddition, the effects of shear deformation and rotary inertia on the static bending and freevibration of the plate should be considered for plate with larger aspect ratios.4) Based on the gradient elastic theory, the governing equation of the gradient Kirchhoffcircular plate model is derived. Analytical solutions for the static bending subjected to thelinear loading and free vibration problems of a clamped circular plate are obtained basedon the boundary conditions discussed in Chapter3. The effects of the gradient parameterson the static bending and free vibration of circular plate are studied. Noteworthy, thenatural frequency is larger than that of the conventional Kirchhoff circular plate model,when the two non-zero gradient parameters are of the same value, i.e., l1=l2≠0. The reason of this phenomenon is that the strengthening non-classical boundary condition isused compared with the conventional circular plate model. Moreover, the effect ofgradient parameter l2on higher order frequency and mode shape of the plate is prominentthan that of l1.5) Based on the gradient elastic theory, the governing equations of motion are derived, andanalytical solutions for the free wave dispersion relations are obtained. The effects ofgradient parameters on the frequency and phase velocity of gradient Flügge shell areillustrated. By comparing with results of molecular dynamics (MD) simulations availablein the literature, the proposed shell model is found to have a better prediction ofdispersion relation than that of conventional shell model, especially for larger wavenumbers. The numerical results show that the effect of gradient parameters on thedispersion relation of SWCNT is insensitive to small wave numbers (i.e., ka<0.2), butbecome increasing prominent for larger wave numbers. The number of cut-offfrequencies is a function of circumferential mode number. Moreover, the predictedangular frequency and phase velocity increase with the increase of gradient parameter l2but decrease with the increase of l1.