Research On Nonlinear Statics/Dynamics And Fatigue Life Analysis Of Piezoelastic Laminated Structures Including Damage Effects

In this thesis, considering piezoelectric plates and piezoelastic laminated plates/shells composed by the piezoelectric membranes and fibre-reinforced composite laminated plates as the subjects investigated, the nonlinear vibration, nonlinear dynamic response, postbuckling, bifurcation and chaos behaviors of piezoelectric/ piezoelastic plates including damage effects, as well as the fatigue life prediction of piezoelectric plates with damage are systematically studied, and the essential character of the mechanical property can be illustrated precisely. However, the establishment of the constitutive relations for piezoelectric materials with damage and corresponding damage evolution equations is very difficult. Moreover, the geometric nonlinearity, damage effects and piezoelectric effects will make the governing equations and boundary conditions more complicated, and the new analytical methods and computational methodology are necessary to solve this problem. Due to the above reasons, there is little research on this field. The problem is investigated in this thesis, and some achievements are obtained on several aspects such as the establishment of the constitutive relations with damage, the construction of the mechanical modeling and the sake of efficient computational method, etc. The research results are significant not only in the academic but also in the practical engineering. The main results contain as follows.Based on the Talreja's damage model with tensor valued internal state variables and the invariant theory, and considering the initial orthotropic symmetry of the material, the Helmhotlz free energy of piezoelectric materials with damage can be expressed as a polynomial in transformed strain, the electric field vector and tensorial damage variables using the integrity bases. Then using the thermodynamic relations, a constitutive model for piezoelectric material solids containing a substantive of distributed cracks is established. Based on the continuum damage mechanics and strain energy equivalence, a constitutive model is established for elastic composites with orthotropic damage. Suppose the damage variables remain constant throughout the thickness in each layer, then the damage constitutive model suited for the global analysis of the laminated structures is obtained and the Ladeveze's damage model of uni-ply for laminated structures is generalized.Based on the equivalent strain theory, the high-order shear deformation theory and the Hamilton variation principle, considering the damage effect in the composite laminated layers, the dynamic equations of the piezoelastic laminated plates with damage are derived. Then, an analytical solution is presented by using Galerkin method. Numerical results show, under the close-circuit condition, the influences of the damage, the side-to-thickness ratio and the ratio of the thickness of piezoelectric layer to the total thickness of laminated plates on the natural frequencies of the piezoelastic laminated plates with simply-supported condition.Based on the Von Kármán's nonlinear theory, and considering geometric nonlinearity, damage effects and piezoelectric effects, the nonlinear equilibrium equations of cross-ply piezoelastic laminated plates and the governing motion equations of the piezoelectric plate are established respectively. The damage evolution law is characterized by the Kachanov damage evolution equations. Through applying the finite difference method, Newmark method and iterative procedure, the above two governing equations are solved. The influences of damage effects, piezoelectric effects, load parameters and geometric parameters for the above two structures are discussed in detail.According to Von Kármán's nonlinear theory, the nonlinear equilibrium equations of the piezoelectric plate with damage are established under considering initial imperfection. The damage evolution equation proposed by Kachanov is still taken. In the numerical examples, the influence of the geometric nonlinearity on the postbuckling of the piezoelectric plate with damage is investigated, and the influence of various parameters including in-plane compressive load, external applied voltage and geometric parameter on the postbuckling behaviors of damaged piezoelectric plate and damage evolution property are discussed in detail.Based on the constitutive model for piezoelectric materials with damage and Von Kármán's nonlinear theory, the nonlinear dynamic equations of the piezoelectric plates with damage under cyclic impulsive loads are established. The Kachanvo damage evolution law under the once time impulsive load and the Chaboche fatigue damage development due to long time cyclic impulsive loads are employed to fatigue damage growth. The effect of cyclic impulsive loads and electrical loads on the fatigue damage and fatigue life prediction of the piezoelectric plate are determined and discussed.In the case of finite deformation, the bifurcation and chaos of the damaged piezoelectric plates are systematically studied. By applying the Galerkin method, introducing new variables, and employing the modern methods in nonlinear dynamics, the complex nonlinear dynamic system of the piezoelectric plate with damage are investigated qualitatively. The time history, phase plane portrait, Poincare map, bifurcation diagram and so on, for the nonlinear system are determined and the plenty and complex nonlinear dynamical behaviors are revealed. The influence of the damage and external applied voltage on the bifurcation and chaos of the piezoelectric plate with damage are considered in detail.