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Carbon nanotube/polymer nanocomposite plate- and shell-like structures will be the next generation lightweight structures in advanced applications due to the superior multifunctional properties combined with lightness. Here material optimization of carbon nanotube/polymer nanocomposite beams and shells is tackled via ad hoc nonlinear finite element schemes so as to control the loss of stability and overall nonlinear response. Three types of optimizations are considered: variable through-the-thickness volume fraction of random carbon nanotubes (CNTs) distributions, variable volume fraction of randomly oriented CNTs within the mid-surface, aligned CNTs with variable orientation with respect to the mid-surface. The collapse load, which includes both limit points and deformation thresholds, is chosen as the objective/cost function. An efficient computation of the cost function is carried out using the Koiter reduced order model obtained starting from an isogeometric solid-shell model to accurately describe the point-wise material distribution. The sensitivity to geometrical imperfections is also investigated. The optimization is carried out making use of the Global Convergent Method of Moving Asymptotes. The extensive numerical analyses show that varying the volume fraction distribution as well as the CNTs orientation can lead to significantly enhanced performances towards the loss of elastic stability making these lightweight structures more stable. The most striking result is that for curved shells, the unstable postbuckling response of the baseline material can be turned into a globally stable response maintaining the same amount of nanostructural reinforcement but simply tailoring strategically its distribution.

This is a fundamental study on the buckling temperature and post-buckling analysis of functionally graded graphene nanoplatelet-reinforced composite (FG-GPLRC) disk covered with a piezoelectric actuator and surrounded by the nonlinear elastic foundation. The matrix material is reinforced with graphene nanoplatelets (GPLs) at the nanoscale. The displacement–strain of thermal post-buckling of the FG-GPLRC disk via third-order shear deformation theory and using Von Karman nonlinear plate theory is obtained. The equations of the model are derived from Hamilton’s principle and solved by the generalized differential quadrature method. The direct iterative approach is presented for solving the set of equations that includes highly nonlinear parameters. Finally, the results show that the radius ratio of outer to the inner (Ro/Ri), the geometrical parameter of GPLs, nonlinear elastic foundation, externally applied voltage, and piezoelectric thickness play an essential impact on the thermal post-buckling response of the piezoelectrically FG-GPLRC disk surrounded by the nonlinear elastic foundation. Another important consequence is that, when the effect of the elastic foundation is considered, there is a sinusoidal effect from the Ro/Ri parameter on the thermal post-buckling of the disk and this matter is true for both boundary conditions.

This paper aims to investigate the size scale effect on the buckling and post-buckling of single-walled carbon nanotube (SWCNT) rested on nonlinear elastic foundations using energy-equivalent model (EEM). CNTs are modelled as a beam with higher order shear deformation to consider a shear effect and eliminate the shear correction factor, which appeared in Timoshenko and missed in Euler–Bernoulli beam theories. Energy-equivalent model is proposed to bridge the chemical energy between atoms with mechanical strain energy of beam structure. Therefore, Young’s and shear moduli and Poisson’s ratio for zigzag (n, 0), and armchair (n, n) carbon nanotubes (CNTs) are presented as functions of orientation and force constants. Conservation energy principle is exploited to derive governing equations of motion in terms of primary displacement variable. The differential–integral quadrature method (DIQM) is exploited to discretize the problem in spatial domain and transformed the integro-differential equilibrium equations to algebraic equations. The static problem is solved for critical buckling loads and the post-buckling deformation as a function of applied axial load, CNT length, orientations and elastic foundation parameters. Numerical results show that effects of chirality angle, boundary conditions, tube length and elastic foundation constants on buckling and post-buckling behaviors of armchair and zigzag CNTs are significant. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.

This study systematically investigates stiffness and buckling response of finite structures consisting of repeated unit cells of a reference truss lattice microstructure and a topology optimized microstructure with enhanced buckling strength. Structural stability is evaluated using linear buckling, nonlinear pre-buckling, and post-buckling analyses, subjected to two benchmark loading cases representing uniaxial compression and shear loading. Numerical results indicate that geometric and material nonlinearities play a surprisingly small role in uniaxial loading, whereas strong effects are seen for the shear loading case for which the microstructure was not optimized.

In the present research, finite element solutions of thermal post-buckling load-bearing strength of functionally graded (FG) sandwich shell structures are reported by adopting a higher-order shear deformation type kinematics. For the numerical calculation, nine nodes are considered for each element. A specialized MATLAB code is developed incorporating the present mathematical model to evaluate the numerical buckling temperature. The Green–Lagrange nonlinear strain is adopted for the formulation of the sandwich structure. The eigenvalue equation of the FG sandwich structure is solved to predict the post-buckling temperature values of the structure. Moreover, three kinds of temperature distributions across the panel thickness are assumed, viz., uniform, linear and nonlinear. In addition, the properties are described using the power law distributions. The numerical solutions are first validated and, subsequently, the impact of alterations of structural parameters, viz., the curvature ratios, core–face thickness ratios, support conditions and power law index (nZ) including the amplitude ratio on the thermal post-buckling response of FG sandwich curved panels have been studied in details. The investigation reveals different interesting outcomes, which may help for future references for the analysis and design of the graded sandwich structure.

This review provides a comprehensive look at buckling failures in curved shell structures, unifying theoretical advancements, design guidelines, and practical applications to bridge gaps in existing literature. It critically analyses how geometric, material, and load-boundary imperfections influence buckling behaviour, providing insights into discrepancies between theoretical and experimental results. The paper’s key novelty lies in this comprehensive and unified approach, which uniquely evaluates modern design codes such as NASA SP-8007, ECCS, and PD 5500, highlighting their limitations for composite materials. The review also addresses the increasing use of lightweight composites like carbon fibre-reinforced plastic (CFRP) and examines real-world failure cases to highlight key design and manufacturing challenges. It provides a critical analysis of advanced numerical and experimental techniques like geometrically and materially nonlinear analysis with imperfections (GMNIA) and digital image correlation (DIC), while also proposing future research directions. The interdisciplinary approach is relevant to aerospace, civil, and marine engineering, aiming to promote knowledge transfer and contribute to the development of safer and more efficient structural designs.

The present study investigates how to apply continuous tow shearing (CTS) in a manufacturable design parameterization to obtain reduced imperfection sensitivity in lightweight, cylindrical shell designs. The asymptotic nonlinear method developed by Koiter is applied to predict the post-buckled stiffness, whose index is constrained to be positive in the optimal design, together with a minimum design load. The performance of three machine learning methods, namely, Support Vector Machine, Kriging, and Random Forest, are compared as drivers to the optimization towards lightweight designs. The new methodology consists of contributions in the areas of problem modeling, the selection of machine learning strategies, and an optimization formulation that results in optimal designs around the compromise frontier between mass and stiffness. The proposed ML-based framework proved to be able to solve the inverse problem for which a target design load is given as input, returning as output lightweight designs with reduced imperfection sensitivity. The results obtained are compatible with the existing literature where hoop-oriented reinforcements were added to obtain reduced imperfection sensitivity in composite cylinders.

Present paper undergoes with the analysis of the post-buckling behaviors of multi-scale hybrid nanocomposite beam-type structures manufactured from both macro- and nanoscale reinforcing elements, namely carbon fibers and carbon nanotubes (CNTs), respectively, in addition to the host polymeric matrix. The equivalent material properties of the hybrid nanocomposite will be gathered utilizing a two-step micromechanical scheme while the influences of the CNTs’ agglomeration phenomenon are covered. Continued by using the concept of the virtual work’s principle, the nonlinear governing equation of the motion will be derived on the basis of the combination of the von Karman hypothesis with the well-known Euler–Bernoulli beam theory while the beam is rested on a three-parameter nonlinear foundation. It is noteworthy that the impact of the existence of an initial deflection in the continuous system is included in the present study, too. At the end of the manuscript, the obtained governing equation will be solved analytically within the framework of the Galerkin’s method once both simply supported–simply supported (S–S) and clamped–clamped (C–C) boundary conditions are considered. It is shown that the stability response of the NC structure can be deeply influenced tailoring the agglomeration parameters.

In this paper, a multi-objective probabilistic design optimisation approach is presented for reliability and robustness analysis of composite structures and demonstrated on a mono-omega-stringer stiffened panel. The proposed approach utilises a global surrogate model of the composite structure while accounting for uncertainties in material properties as well as geometry. Unlike the multi-level optimisation approach which freezes some parameters at each level, the proposed approach allows for all parameters to change at the same time and hence ensures global optimum solutions in the given parameter design space (for both probabilistic and deterministic optimisations) within a certain degree of accuracy. The proposed approach is used in this study to conduct extensive multi-objective probabilistic and deterministic optimisations (without considering safety factors) on a mono-stringer stiffened panel. In particular, a global surrogate model is developed utilising the computational power of a high-performance computing facility. The inputs of the surrogate model are the omega-stringer geometry and the mechanical properties of the composite material, while the outputs are the fundamental linear buckling load (LBL) and the nonlinear post-buckling strength (NPS). LBL and NPS are obtained via detailed parametric finite element models of the mono-stringer stiffened panel; in the nonlinear model, the interface between the skin and the omega-stringer is modelled via cohesive elements to allow for debonding in the post-buckled regime. Extensive multi-objective optimisations are conducted on the surrogate model using deterministic and probabilistic approaches to examine the omega-stringer geometric parameters mostly affecting the system robustness and reliability. The differences between deterministic and probabilistic designs are highlighted as well.

Lightweight thin-walled structures are crucial for many engineering applications. Advanced manufacturing methods are enabling the realization of composite materials with spatially varying material properties. Variable angle tow fibre composites are a representative example, but also nanocomposites are opening new interesting possibilities. Taking advantage of these tunable materials requires the development of computational design methods. The failure of such structures is often dominated by buckling and can be very sensitive to material configuration and geometrical imperfections. This work is a review of the recent computational developments concerning the optimisation of the response of composite thin-walled structures prone to buckling, showing how baseline products with unstable behaviour can be transformed in stable ones operating safely in the post-buckling range. Four main aspects are discussed: mechanical and discrete models for composite shells, material parametrization and objective function definition, solution methods for tracing the load-displacement path and assessing the imperfection sensitivity, structural optimisation algorithms. A numerical example of optimal material design for a curved panel is also illustrated.