Cette thèse traite du développement d’un élément continu de plaques orthotropes, sandwichs et multicouches. La démarche consiste dans un premier temps à établir la matrice de raideur dynamique de plaques orthotropes pour des conditions aux limites naturelles à partir d’une reformulation des éléments de plaques isotropes développés au laboratoire QUARTZ (EA7393). La démarche est basée d’une part sur la décomposition des conditions aux limites libres décrite par Gorman et d’autre part sur la résolution des équations de mouvement en se basant sur les développements en séries de Levy. La matrice de raideur dynamique est ensuite obtenue par projection des déplacements et des efforts de frontières sur des bases fonctionnelles compatibles avec les opérations d’assemblage. Dans un second temps, la formulation des éléments sandwichs et multicouches est décrite par superposition des plaques orthotropes précédemment développées.Les formulations présentées prennent en compte les vibrations de flexion et les vibrations dans le plan, dites vibrations de membrane. La validation de ces éléments est menée par une confrontation systématique de réponses harmoniques non amorties avec celles obtenues par diverses modélisations éléments finis. / This thesis deals with the development of a continuous element for orthotropic, sandwich and multilayer plates. This approach is based essentially on the construction of the dynamic stiffness matrix of orthotropic plates using natural boundary conditions from a reformulation of the isotropic plate elements developed in the QUARTZ laboratory (EA 7393). In order to develop the dynamic stiffness matrix of the studied element we resort on the first hand to the decomposition of free boundary conditions described by Gorman, on the second hand to the resolution of the equations of motion by using Levy series expansions. The dynamic stiffness matrix is then obtained by projecting movements and frontier efforts on functional bases compatible with assembly operations. Finally the continuous sandwich and multilayer plate element is described by superposition of continuous orthotropic plates element previously developed.The formulations presented takes into account the bending vibration and the vibration in the plane, called membrane vibration. The validation of all obtained results is conducted by a systematic comparison of undamped harmonic responses with those obtained by various finite element models.

A hybrid method is presented for the analysis of layers, plates, and multi-layered systems consisting of isotropic and linear elastic materials. The problem is formulated for the general case of a multi-layered system using a total potential energy formulation and employing the layerwise laminate theory of Reddy. A one-dimensional finite element model is used for the analysis of the multi-layered system through its thickness, and integral Fourier transforms are used to obtain the exact solution for the in-plane problem. Explicit expressions are obtained for the fundamental solution of the typical infinite layer, which are applied in the two-dimensional boundary integral equation model to produce the integral representation of the solution. The boundary integral equation model is two-dimensional, displacement-based and assumes piecewise continuous distribution of the displacement components through the system's thickness. The developed model describes the three-dimensional displacement field, the stress field, the strains and the interlaminar stresses over the entire domain of the problem as continuous functions of the position. This detailed three-dimensional analysis is achieved by incorporating only contour integrals. The boundary integral equations are discretized using the boundary element method and a numerical model is developed for the single numerical layer (element). This model is extended to the case of a multilayered system by introducing appropriate continuity conditions at the interfaces between the layers (firmly bonded layers, or separation, slip and friction between the layers). Assembly of the element matrices yields the global system of equations, which can be solved via iterative techniques. In addition, numerical techniques are developed for the evaluation of the boundary and domain integrals involved in the construction of the element matrices. The singular boundary integrals are computed using a special coordinate transformation, along with a subdivision of the boundary element and a transformation of the Gauss points. The domain integrals (regular, singular or near-singular) are transformed to regular definite integrals along the boundary through a semi-analytical approach. The proposed method provides a simple, efficient, and versatile model for a three-dimensional analysis of thick plates or multilayered systems. It can also be used to study plates resting on elastic foundations or plates with internal supports. The proposed method can be applied in an obvious manner to anisotropic materials and vibration problems. / Ph. D.

Composite structures are extremely useful for aerospace, automotive, marine and civil applications due to their very high specific structural properties. These structures are subjected to severe dynamic loading in their service life. Repeated exposure to these severe loading conditions can induce structural damage which ultimately may precipitate a catastrophic failure. Therefore, an interest in the continuous inspection and maintenance of engineering structures has grown tremendously in recent years. Sensitive aerospace applications can have small design margins and any inadequacy in knowledge of the system may cause design failure. Structures made from composite materials posses complicated failure mechanism as compared to those made from conventional metallic materials. In composite structural design, it is hence very important to properly model geometric intricacies and various imperfections such as delaminations and cracks. Two important issues are addressed in this thesis: (1) structural modeling of nonlinear delamination and uncertainty propagation in nonlinear characteristics of composite plate structures and (2) development of a model based damage detection system to handle uncertainty issues. An earlier proposed shear deformable C0 composite plate finite element is modified to alleviate modeling uncertainty issues associated with a damage detection problem. Parabolic variation of transverse shear stresses across the plate thickness is incorporated into the modified formulation using mixed shear interpolation technique. Validity of the proposed modification is established through available literature. Correction of the transverse shear stress term in the formulation results in about 2 percent higher solution accuracy than the earlier model. It is found that the transverse shear effect increases with higher modes of the plate deformation. Transverse shear effects are more prominent in sandwich plates. This refined composite plate finite element is used for large deformation dynamic analysis of delaminated composite plates. The inter-laminar contact at the delaminated region in composite plates is modeled with the augmented Lagrangian approach. Numerical simulations are carried out to investigate the effect of delamination on the nonlinear transient behavior of composite plates. Results obtained from these studies show that widely used unconditionally stable β-Newmark method presents numerical instability problems in the transient simulation of delaminated composite plate structures with large deformation. To overcome this instability issue, an energy and momentum conserving composite implicit time integration scheme presented by Bathe and Baig is used for the nonlinear dynamic analysis. It is also found that a proper selection of the penalty parameter is very crucial in the simulation of contact condition. It is shown that an improper selection of penalty parameter in the augmented Lagrangian formulation may lead to erroneous prediction of dynamic response of composite delaminated plates. Uncertainties associated with the mathematical characterization of a structure can lead to unreliable damage detection. Composite structures also show considerable scatter in their structural response due to large uncertainties associated with their material properties. Probabilistic analysis is carried out to estimate material uncertainty effects in the nonlinear frequencies of composite plates. Monte Carlo Simulation with Latin Hypercube Sampling technique is used to obtain the variance of linear and nonlinear natural frequencies of the plate due to randomness in its material properties. Numerical results are obtained for composite plates with different aspect ratio, stacking sequence and oscillation amplitude ratio. It is found that the nonlinear frequencies show increasing non-Gaussian probability density function with increasing amplitude of vibration and show dual peaks at high amplitude ratios. This chaotic nature of the dispersion of nonlinear eigenvalues is also revealed in eigenvalue sensitivity analysis. For fault isolation, variations in natural frequencies, modal curvatures and curvature damage factors due to damage are investigated. Effects of various physical uncertainties like, material and geometric uncertainties on the success of damage detection is studied. A robust structural damage detection system is developed based on the statistical information available from the probabilistic analysis carried out on beam type structures. A new fault isolation technique called sliding window defuzzifier is proposed to maximize the success rate of a Fuzzy Logic System (FLS) in damage detection. Using the changes in structural measurements between the damaged and undamaged state, a fuzzy system is generated and the rule-base and membership functions are generated using probabilistic informations. The FLS is demonstrated using frequency and mode shape based measurements for various beam type structures such as uniform cantilever beam, tapered beam in single as well as in multiple damage conditions. The robustness of the FLS is demonstrated with respect to the highly uncertain input information called measurement deltas (MDs). It is said, if uncertainty level is larger than or close to the changes in damage indicator due to damage, the true information would be submerged in the noise. Then the actual damaged members may not be identified accurately and/or the healthy members may be wrongly detected as damaged giving false warning. However, this being the case, the proposed FLS with new fault isolation technique tested with these noisy data having large variation and overlaps shows excellent robustness. It is observed that the FLS accurately predicts and isolates the damage levels up-to considerable uncertainty and noise levels in single as well as multiple damage conditions. The robustness of the FLS is also demonstrated for delamination detection in composite plates having very high material property uncertainty. Effects of epistemic uncertainty on damage detection in composite plates is addressed. The effectiveness of the proposed refined Reddy type shear deformable composite plate element is demonstrated for reducing the modeling or epistemic uncertainty in delamination detection.