The Buckling And Resonance Behaviors Of Small-scale Slender Structure

The size-dependent nonlinear post-buckling resonance behaviors of small-scale beam-like structures are investigated in this thesis.The influences of the size effect,graded distribution of material,initial imperfection on vibration characteristics,instability regions,post-buckling behavior and nonlinear resonant response are discussed detailedly.On the basis of the modified couple stress theory,the small-scale effect is introduced into the constitutive relation of structures.Two kinds of dynamical models including transversely functionally graded material imperfect microbeam and bi-directional functionally graded material microbeam are derived by the method of Hamilton's principle,and the accuracy and universality of the presented models are verified by comparing with the previous models.A analytically study is presented for the buckling and post-buckling vibration of the imperfect heterogeneous microbeam.The exact analytical solutions of the critical buckling load and the static response are acquired for the buckling problem of imperfect microbeams with different boundary conditions.The difference of buckling behavior between the of perfect and imperfect microbeam is deeply analyzed.The numerical results are depicted to illustrate the influence of the dimensionless scale parameter and the imperfection amplitude on the circuital buckling load,static response and the dynamic properties of the imperfect microbeams.It is found that the imperfect microbeams buckle through a saddle-node bifurcation and the static responses of the upper and lower branch are asymmetrical in postbuckling domain.The initial imperfection and size effect have significant in-fluence on the postbuckling vibration of the imperfect mi-crobeams.Moreover,the internal resonances might be activated between the postbuckling vibration modes.The coupled longitudinal-transverse nonlinear resonant behaviors of the imperfect transversely functionally graded material microbeam are studied.The reduced-order model of the coupled nonlinear partial differential equations of motion is obtained by applying the Galerkin technique.Then,the pseudo-arclength continuation technique is employed to solve the reducedorder model of motion,and the frequency-and force-response curves are constructed using the acquired periodic solutions.The influences of the size effect,imperfection amplitude and gradient index on the resonant response are discussed.The results reveal that the initial imperfection and the asymmetry of material distribution in thickness of microbeam cause asymmetrical response.The cyclic-fold bifurcation of periodic solution is detected in resonant response of microbeam and the frequency-response curves may exhibit hardening-type,softening-type or softening-hardening-type nonlinearities,depending on the system parameters.The size dependent free vibration,post-buckling and dynamic stability of bi-directional functionally graded microbeam are investigated.The partial differential equations with variable coefficients and boundary conditions are discretized to algebra equations system using differential quadrature method.The linear free vibration,buckling and dynamic stability characteristics of microbeam are studied by solving the associated linear eigenvalue problem.The postbuckling problem is analyzed by performing bifurcation analysis using the pseudo-arclength continuation technique.The influences of the size effect,gradient index,type of material distribution and elastic foundation on the natural frequency,critical buckling load and instability region are analyzed.Particularly,the bifurcation behavior as the buckling of bi-directional functionally graded microbeam occurs,is carefully researched.Moreover,the post-buckling vibration of bi-directional functionally graded microbeam is investigated by introducing a small dynamic disturbance around the buckled configurations.It is demonstrated that size effect gives rise to additional rigidity and results in higher frequency,critical buckling load and shifts the instability region to higher excitation frequency.The buckling of bi-directional functionally graded microbeam occurs through transcritical bifurcation or pitchfork bifurcation depending on symmetry of material distribution in thickness of microbeam.The mode veering phenomenon is detected in post-buckling domain of microbeams.Based on the reduced-order model,the nonlinear resonance behaviors of bi-directional functionally graded microbeams are investigated by performing one-and two-parameter bifurcation analyses.In one-parameter bifurcation analysis,the frequency-and force-response curves are constructed by tracing the period motion of microbeam using the pseudo-arclength continuation technique.Cyclic-fold bifurcation which indicates jump phenomenon is detected in the period motion.The trajectories of cyclic-fold bifurcation points are achieved by implementing the two-parameter bifurcation analysis.The results indicate that the primary resonant response of microbeam exhibit hardening-type nonlinearities and cyclic-fold bifurcations are detected in periodic motion.The size effect and variation of material distribution will not change the type of nonlinearity and bifurcations,but give rise to the quantitative difference of results.The cusp bifurcation of periodic solution is detected in two-parameter bifurcation analysis.The bifurcation parameter values corresponding to cusp bifurcation are the critical values for the occurrence of “jump” phenomenon.The resonant responses of buckled microbeam exhibit softening-type nonlinearities.Finally,the research summary of this work are presented and a brief plan for further studies is given.