Interlaminar stresses are estimated from plate theories by equilibrium. The elasticity equations of equilibrium are integrated with respect to the thickness coordinate z using the linear distribution in z of the in-plane stresses. This procedure, for example, requires fourth order derivatives of the out-of-plane displacement w with respect to the in-plane coordinates x and y to compute the interlaminar normal stress. Since compatible elements for the plate bending problem at most require the displacement and its first derivatives to be continuous across element boundaries, low degree interpolation polynomials are used. Thus, fourth order derivatives of the finite element polynomials are either meaningless, or at least inaccurate. In order to compute high order derivatives, an approximate polynomial solution of high degree to the governing partial differential equation for w(x,y) is determined using the finite element solution as a first approximation. A rectangular subdomain that may consist of several elements is selected from the finite element model. The displacement w(,y) over the subdomain is expanded in a Chebyshev series. Then collocation is used to determine the unknown Chebyshev coefficients such that the Chebyshev series matches displacement w and its normal derivative from the finite element solution at discrete points on the boundary of the subdomain, and the partial differential equation is enforced at discrete points within the subdomain. Interlaminar shear and normal stresses are computed from the third and fourth derivatives, respectively, of the Chebyshev series at the collocation points. / Master of Science

Three dimensional contact problems of square orthotropic laminates indented by a rigid spherical indenter are solved. Simplified problems of indentations of beam and isotropic square plate are studied first to develop an efficient numerical technique and to gather the knowledge of the shape of the contact area in order to solve for the three dimensional orthotropic cases. The approach combines an exact solution method in conjunction with a simple discretization numerical scheme. Numerical sensitivity due to the ill-posed nature of the problem was experienced but was cured by enhancing the numerical approach with a least square spirit. Well agreement is obtained by comparing the results of these simplified studies with available published solutions. For isotropic plate, contact area is found to be either a circle or a hypotrochoid of four lobes featured with a shorter length of contact along the through-the- corner directions of the plate. Hertz's theory fails earlier than assuming the contact area to be a circle. In-plane dependence of the contact stress is presented to illustrate the difference of contact behavior between a square plate and a circular plate. Load-indentation relation reveals indenting a square plate is harder than indenting a circular plate of a diameter equal to the side length of the square plate. Solutions of multi-layered orthotropic cases are achieved by employing a modified analytical approach with the same numerical method. Three different configurations of plate are implemented for the orthotropic case, namely, a single layered magnesium (Mg) plate, which is slightly orthotropic, and a single and double layered plates of graphite-epoxy (G-E), which are highly orthotropic. Results for the (Mg) plate agrees with the previous isotropic case. Concept of modifying the previous hypotrochoids is introduced to seek for the contact stresses for comparatively large indentation conditions. Single-layered (G-E) plate was implemented for small indentations. The result supports the validity of Hertz's theory for small indentation and shows a relatively longer contact length in the direction of less stiffness. Two layered (G-E) plate illustrates similar distributions for the contact stresses along both of the in-plane directions with a smaller range of validity of Hertzian type behavior than the previous cases. The boundary effect prevails at the initial stage of indentation but is overcome by the effect of material orthotropy as the indentation proceeds. Thus, the contact area for small indentation appears to be the same kind of hypotrochoids as located in the isotropic case but changes to be the other type of hypotrochoids as the indentation advances. / Ph. D.

Free-edge effects in composite laminates were studied experimentally. Strains were determined and compared on a ply-by-ply basis for the curved edges of a hole in thick composite panels and along the straight free-edge of the panels. The experimental technique of moire interferometry was extended to take measurements of in-plane deformations on singly-curved surfaces. A replication scheme was developed to produce high-frequency diffraction gratings on singly-curved surfaces. Two different techniques were developed to interrogate specimen gratings on 25.4 mm (1 in.) diameter holes. Eight thick composite laminates from three material systems were tested in uniaxial compression on a screw-driven testing machine. Interlaminar deformations were measured at the straight free-edge on four of the specimens. Strain distributions on the straight free-edge were compared with FEM results for two specimens. Good agreement was obtained for one specimen while poor agreement was obtained for the other. / Ph. D.

In this study, a computational model for accurate analysis of composite laminates and laminates with including delaminated interfaces is developed. An accurate prediction of stress distributions, including interlaminar stresses, is obtained by using the Generalized Laminate Plate Theory of Reddy in which layer-wise linear approximation of the displacements through the thickness is used. Analytical, as well as finite-element solutions of the theory, are developed for bending and vibrations of laminated composite plates for the linear theory. Geometrical nonlinearity, including buckling and post-buckling are included and used to perform stress analysis of laminated plates. A general two-dimensional theory of laminated cylindrical shells is also developed in this study. Geometrical nonlinearity and transverse compressibility are included. Delaminations between layers of composite plates are modeled by jump discontinuity conditions at the interfaces. The theory includes multiple delaminations through the thickness. Geometric nonlinearity is included to capture layer buckling. The strain energy release rate distribution along the boundary of delaminations is computed by a novel algorithm. The computational models presented herein are accurate for global behavior and particularly appropriate for the study of local effects. / Ph. D.

In this work, a finite element formulation and associated computer program is developed for the transient large deformation analysis of laminated composite plate/shell structures. In order to satisfy the plate/shell surface traction boundary conditions and to have accurate stress description while maintaining the low cost of the analysis, a newly assumed displacement field theory is formulated by adding higher-order terms to the transverse displacement component of the first-order shear deformation theory. The laminated shell theory is formulated using the Updated Lagrangian description of a general continuum-based theory with assumptions on thickness deformation. The transverse deflection is approximated through the thickness by a quartic polynomial of the thickness coordinate. As a result both the plate/shell surface tractions (including nonzero tangential tractions and nonzero normal pressure) and the interlaminar shear stress continuity conditions at interfaces are satisfied simultaneously. Furthermore, the rotational degree of freedoms become layer dependent quantities and the laminate possesses a transverse deformation capability (i.e. the normal strain is no longer zero). Analytical integration through the thickness direction is performed for both the linear analysis and the nonlinear analysis. Resultants of the stress integrations are expressed in terms of the laminate stacking sequence. Consequently, the laminate characteristics in the normal direction can be evaluated precisely and the cost of the overall analysis is reduced. The standard Newmark method and the modified Newton Raphson method are used for the solution of the nonlinear dynamic equilibrium equations. Finally, a variety of numerical examples are presented to demonstrate the validity and efficiency of the finite element program developed herein. / Ph. D.

Laminated composite plates have become an important and proven structural material in aerospace and ocean vehicles. However, because of the inherent orthotropy of laminated composite materials the analysis of these structures is complex and usually cannot be adequately performed using classical methods. In this dissertation the formulation of the fully coupled fluid-structure interaction of a laminated composite plate and its surrounding fluid medium is presented. The solution technique involves the finite element method for modeling the structural response and the boundary element method for modeling the acoustic field. The model incorporates the Mindlin plate theory which includes five degrees of freedom. An improved integration technique is demonstrated which significantly reduces the approximation error. Storage requirements are reduced by grouping complex numbers. Finally the fully coupled fluid-structure interaction involving laminated composite plates is modeled using the combined FEM-BEM approach demonstrating the usefulness and the significance of the method. / Ph. D.