Obtaining the best mechanical properties of laminated composite materials has always been the core pursuit in automotive and aerospace industries. Functionally graded materials (FGMs) and variable stiffness composite (VSC) materials are the two typical kinds of advanced composite materials using different stiffness tailoring techniques. The former one comprises multiple phases of materials with different properties, and the volume fraction of each phase varies stepwise or continuously in a predetermined direction, e.g., thickness direction in a lamina. One of the promising FGM is the functionally graded carbon nanotube-reinforced composite (FG-CNTRC). While the latter one tailors the in-plane material properties by using curvilinear fibers instead of straight fibers that are used for traditional fiber-reinforced composites (FRCs). Both the through-the-thickness adjusting and the in-plane tailoring approaches may enhance the mechanical performances of the advanced composite structures when compared with the traditional uniformly distributed counterparts.

On account of the merits of these advanced materials, the dissertation conducted a systematic numerical investigation of the mechanical responses of these advanced composite material structures. The present dissertation is divided into two categories according to the material type. In the first part, we aimed to examine the effect of the matrix cracks in the FRC lamina on the global responses of hybrid composite structures containing FG-CNTRC laminae and damaged FRC laminae. In the second part, the buckling optimization, post-buckling behaviors, and failure and fracture analyses of VSCs and structures were performed. Specifically, the following research aspects were covered:

First, we investigated the vibration behaviors of rotating pre-twisted hybrid composite blades containing FG-CNTRC laminae and FRC laminae. The degraded stiffness of the cracked lamina was modeled through the self-consistent model (SCM). The blade was modeled as a shell structure that was formed by twisting a plate around its mean line. With the help of differential geometry theory, a novel shell model has been derived to describe the kinetics of the blade. The Coriolis effect and centrifugal force were both presented in the formulation, which resulted in a damped-like free vibration system governed by a system of second-order ordinary differential equations (ODEs). Furthermore, the system was reformulated to a system of first-order ODEs by utilizing the state space technique, and the improved moving least-squares Ritz (IMLS-Ritz) method was then used for discretizing the ODEs. After a carefully deliberate validating validation of the effectiveness of the presented model through a series of comparison studies, parametric studies including CNT distribution configuration, and rotating speed, the geometrical parameters on the vibration responses of cross-plied composite blades were systematically examined. The vibration characteristics of angle-plied composite blades were also investigated.

Second, a numerical framework for modeling the geometrically nonlinear large deformation behaviors of matrix cracked hybrid composite double-curved deep shell containing FG-CNTRC layers and FRC layers is proposed. The material properties of the FG-CNTRC laminae and damaged FRC laminae are obtained using the models proposed in the first part. To describe the geometrically nonlinear large deflection behaviors and account for deep and moderate thick shells, the von Kármán geometric nonlinearity assumptions and the term 1/(1+ϛ/R) were considered in the relationship between displacement and strain. The IMLS-Ritz method was engaged to discretize the non-linear partial differential equations, and the modified Newton-Raphson method in combination with the arc-length iteration technique was adopted to solve the discretized equations. Comparison studies indicated that the proposed predictive model could furnish very accurate results for the linear and nonlinear behaviors of thin to moderately thick and shallow to deep laminated doubly-curved shells. Moreover, parametric studies on the effect of CNT distribution, matrix crack density, load type, length-to-thickness ratio, radius-to-length ratio, aspect ratio, boundary condition, and fiber ply-angle on the geometrically nonlinear large deformation behaviors of spherical hybrid composite shells were investigated.

The optimization of a VSC panel is still a challenging problem, which can be attributed to the high non-convexity of the feasible design space and highly involved computation cost. To accelerate the design and optimization of VSC structures, in this dissertation, we proposed an effective and efficient adaptive surrogate-based harmony search algorithm (ASBHSA). The VSC was modeled using the fiber path methodology through which the fiber trajectory was generated by a mathematical formulation. In addition, the manufacturability and quality of the product were guaranteed by considering the curvature constraint. For the ASBHSA, a distance-based infill criterion in combination with a radial basis function (RBF) was proposed to guide the adaptive sampling. Instead of the genetic algorithm (GA), the global best harmony search (GHS) with a good ability to find the global minimum was integrated into the developed optimization framework. After solving some popular benchmark optimization problems and making comparisons of the optimal designs of VSCs with those in published works, the proposed optimization framework was demonstrated being able to effectively and efficiently find the global minimum. Subsequently, a series of designs and optimizations of laminated VSC plates under various boundary conditions and different loading conditions were executed systematically.

Learning about the fracture properties of a composite lamina at the mesoscale is of great help in the design and optimization of a composite structure. Hence, we performed the failure analyses of VSC laminas at mesoscale through which the VSC lamina can be modeled as an orthotropic and nonhomogenous brittle material. The orientation of the fiber path was generated by a periodic mathematical formulation. To predict the crack initiation and propagation in VSC laminas, a phase-field approach which was able to account for the orthotropic material properties was proposed. A staggered algorithm with iterations was utilized to decouple the coupled phase field and displacement problem. Additionally, the derived equations were discretized via the finite element method (FEM) in a fully vectorized manner by using array operations through MATLAB software, which was much more efficient than the classical algorithm. The proposed theoretical formulations and numerical implementations have been validated through several comparison studies. Thereafter, a systematic investigation of the fracture behaviors of pre-cracked and open-hole VSC laminae was performed, and some fundamental insights in the design and optimization of VSC laminates were provided on the basis of the reported results.

Last, the progressive failure analysis (PFA) of the open-hole VSC manufactured by the CTS technique under uniform edge shortening was also investigated. The linear variation of the fiber orientation along a reference axis was assumed. The PFA of a quasi-isotropic (QI) counterpart was also carried out for comparison. The puck failure criterion in combination with a material stiffness degradation model was implemented through ABAQUS user-defined subroutine UMAT to perform the PFA. Both the nonlinear buckling load and ultimate strength of the VSC panel were found much to be higher than those of the QI counterpart. Moreover, a considerable decrease in the stiffness was observed after entering into the post-buckling regime for QI, while it maintained a high level for the VSC panel. Another interesting finding was that the VSC behaved very notch-insensitive under the studied loading condition. The novel findings demonstrated the superior buckling, post-buckling, and failure-resistance properties of the VSC fabricated by the CTS technique.