Dynamic Characteristics Of Plates With Side-cracks And Plane-cracks

Plate structures are the most common components in engineering applications,and have been widely used in civil,aircraft,and aerospace structures.In some circumstances,cracks are generated in a plate due to overloading,degradation of material or impact of foreign objects.The existence of cracks not only reduces the local stiffness of the structure,but also can significantly change the dynamic characteristics,which reduces the reliability of the structure and may even cause damage accidents.In this thesis,the modeling of plate structures with side crack and plane crack is investigated to reveal the mechanism of influences of cracks on the dynamic characteristics of the plate structure.The research includes frequencies,modes of free vibration as well as nonlinear dynamic response of cracked rectangular plate structures under various boundary conditions.The main contents of this research are as follows:A semi-analytical model of free vibration of a stiffened plate with side crack is established,and the influence of crack parameters and stiffener positions on the vibration characteristics of the stiffened plate is studied.First,a set of crack functions are introduced and incorporated into the orthogonal polynomial to construct the admissible function for the displacement field of the stiffened plate under different distributions of stiffener.Such admissible functions can not only describe the singularity in stress at the tip of the crack and the discontinuity in displacement across the crack,but also satisfy the basic boundary conditions of the stiffened plate.Then,based on the classical theories of plate and beam,the plate and its stiffener are modeled separately and jointed by implementing the condition for compatibility of displacement.The Ritz method with special admissible function was used to investigate the change trend of the vibration characteristics of the stiffened plate under the coupling effect of different crack parameters and stiffener positions.It can be found that the cracks on the surface of the plate have a certain effect on the natural frequency of the structure;as the length of the crack increases and the stiffener breaks,the natural frequency will drop sharply;the effect of cracks on the frequency of the stiffened plate not only depends on the parameters of the crack,but also determined by the mode of the structure;the crack will change the distribution of the nodal lines and contour lines on the mode,and the mode of the crack plate can be used to identify the length,position and direction of the crack.The natural frequencies and modes of cracked Mindlin plates under in-plane initial stress are investigated.The Ritz method with a special admissible function is used to determine the initial stress distribution of the cracked plate under in-plane load,and the additional strain energy caused by the in-plane initial stress is obtained,and then the total potential energy and kinetic energy of the plate considering the in-plane prestressed crack are derived.Then use the Ritz method to determine the buckling load,natural frequency and corresponding mode of the plate.By analyzing the vibration characteristics of the plate under different load factors(inplane compressive or tensile load)and crack parameters,it is found out that the buckling loads and natural frequencies decrease with the development of crack.The buckling load is particularly sensitive to the crack length.The natural frequencies of the cracked plate under inplane compressive load decrease with growing crack length.For the cracked plate subjected to a tensile in-plane load,it is observed that a growing crack length may lead to bigger low-order natural frequencies.The nonlinear dynamic behavior of the plate with in-plane initial stress cracks under lateral external excitation is studied.Based on the Mindlin plate theory and von Karman large deflection theory,the strain energy and kinetic energy of the cracked plate under in-plane initial stress are derived.The nonlinear dynamic equations of cracked plate with in-plane initial stress are obtained by the Hamilton's principle,and then truncated into a set of ordinary differential equations by Galerkin method.The effects of various crack parameters and different in-plane initial stresses on the dynamical behavior of the cracked rectangular plate are investigated and presented in bifurcation diagram,waveforms,phase-space,Poincare maps and Maximum Lyapunov exponents.The Lyapunov exponents are used as quantitative tools to analysis the regular and irregular motions of the cracked plate with different in-plane tensile and pressure preload.Different dynamical behaviors are observed by slowly changing the amplitude of transverse external excitation.The results show that the appearance of cracks makes the nonlinear dynamic response of the plate complicated;and the in-plane initial pressure will change the dynamic response of the cracked plate from double-cycle motion to irregular motion such as pseudo-period or chaos,which leads to the parameter range of irregular motion becomes larger and exacerbates the complexity of the dynamic response of the cracked plate;on the contrary,the initial tensile load in the plane will make the dynamic response of the plate simple.A semi-analytical method was used to establish a nonlinear vibration model of a functionally graded material plate with plane-crack for the first time.Firstly,the plate is divided into four sub-regions according to the plane crack region.Then,based on the classical plate theory and von Karman large deflection theory,the kinetic energy and potential energy of each region are derived.According to the boundary conditions and the compatibility of displacement for each sub-region,the admissible function of the displacement field is constructed.The natural frequencies and modes of FGM plates with plane cracks can be obtained by using Ritz method under the assumption of small deformation.Then,the Lagrange dynamic equation is used to derive the nonlinear dynamic equation of the cracked FGM plate considering the contact effect.In this chapter,the proposed method is used to analyze the effects of different crack parameters and material gradient parameters on the vibration characteristics of exponential functionally graded material plates,and the nonlinear dynamic response of cracked FGM plates is studied.The study found that:the plane crack reduces the natural frequency of the FGM plate,and weakens the influence of the of gradient of the material parameter on the natural frequency;under the action of lateral external load,the contact effect of the plane-crack will lead to complex nonlinear dynamics response.