On the Importance of the Recovery Procedure in the Semi-Analytical Solution for the Static Analysis of Curved Laminated Panels: Comparison with 3D Finite Elements.

The manuscript presents an efficient semi-analytical solution with three-dimensional capabilities for the evaluation of the static response of laminated curved structures subjected to general external loads. A two-dimensional model is presented based on the Equivalent Single Layer (ESL) approach, where the displacement field components are described with a generalized formulation based on a higher-order expansion along the thickness direction. The fundamental equations are derived from the Hamiltonian principle, and the solution is found by means of Navier's approach. Then, an efficient recovery procedure, derived from the three-dimensional elasticity equations and based on the Generalized Differential Quadrature (GDQ) method, is adopted for the derivation of the three-dimensional solution. Some examples of investigation are presented, where the numerical predictions of refined three-dimensional Finite-Element-based models are matched with a high level of accuracy. The model is validated for both straight and curved panels, taking into account different lamination schemes and load shapes. Furthermore, it is shown that the numerical solution to the elasticity problem in the recovery procedure is determining and accurately predicting the three-dimensional static response of the doubly-curved shell solid.

Numerical Study on the Buckling Behavior of FG Porous Spherical Caps Reinforced by Graphene Platelets.

The buckling response of functionally graded (FG) porous spherical caps reinforced by graphene platelets (GPLs) is assessed here, including both symmetric and uniform porosity patterns in the metal matrix, together with five different GPL distributions. The Halpin-Tsai model is here applied, together with an extended rule of mixture to determine the elastic properties and mass density of the selected shells, respectively. The equilibrium equations of the pre-buckling state are here determined according to a linear three-dimensional (3D) elasticity basics and principle of virtual work, whose solution is determined from classical finite elements. The buckling load is, thus, obtained based on the nonlinear Green strain field and generalized geometric stiffness concept. A large parametric investigation studies the sensitivity of the natural frequencies of FG porous spherical caps reinforced by GPLs to different parameters, namely, the porosity coefficients and distributions, together with different polar angles and stiffness coefficients of the elastic foundation, but also different GPL patterns and weight fractions of graphene nanofillers. Results denote that the maximum and minimum buckling loads are reached for GPL-X and GPL-O distributions, respectively. Additionally, the difference between the maximum and minimum critical buckling loads for different porosity distributions is approximately equal to 90%, which belong to symmetric distributions. It is also found that a high weight fraction of GPLs and a high porosity coefficient yield the highest and lowest effects of the structure on the buckling loads of the structure for an amount of 100% and 12.5%, respectively.

Special Issue on Advanced Mechanical Modeling of Nanomaterials and Nanostructures.

The increased requirements in design and manufacturing nanotechnology have favored the development of enhanced composite materials with tailored properties, such as functionally graded (FG) and carbon-based materials, primarily carbon nanotubes (CNTs), and graphene sheets or nanoplatelets, because of their remarkable mechanical properties, electrical conductivity, and high permeability [...].

Thermomechanical Buckling Analysis of the E&P-FGM Beams Integrated by Nanocomposite Supports Immersed in a Hygrothermal Environment.

Due to the widespread use of sandwich structures in many industries and the importance of understanding their mechanical behavior, this paper studies the thermomechanical buckling behavior of sandwich beams with a functionally graded material (FGM) middle layer and two composite external layers. Both composite skins are made of Poly(methyl methacrylate) (PMMA) reinforced by carbon-nano-tubes (CNTs). The properties of the FGM core are predicted through an exponential-law and power-law theory (E&P), whereas an Eshelby-Mori-Tanaka (EMT) formulation is applied to capture the mechanical properties of the external layers. Moreover, different high-order displacement fields are combined with a virtual displacement approach to derive the governing equations of the problem, here solved analytically based on a Navier-type approximation. A parametric study is performed to check for the impact of different core materials and CNT concentrations inside the PMMA on the overall response of beams resting on a Pasternak substrate and subjected to a hygrothermal loading. This means that the sensitivity analysis accounts for different displacement fields, hygrothermal environments, and FGM theories, as a novel aspect of the present work. Our results could be replicated in a computational sense, and could be useful for design purposes in aerospace industries to increase the tolerance of target productions, such as aircraft bodies.

Nonlocal Analysis of the Flexural-Torsional Stability for FG Tapered Thin-Walled Beam-Columns.

This paper addresses the flexural-torsional stability of functionally graded (FG) nonlocal thin-walled beam-columns with a tapered I-section. The material composition is assumed to vary continuously in the longitudinal direction based on a power-law distribution. Possible small-scale effects are included within the formulation according to the Eringen nonlocal elasticity assumptions. The stability equations of the problem and the associated boundary conditions are derived based on the Vlasov thin-walled beam theory and energy method, accounting for the coupled interaction between axial and bending forces. The coupled equilibrium equations are solved numerically by means of the differential quadrature method (DQM) to determine the flexural-torsional buckling loads associated to the selected structural system. A parametric study is performed to check for the influence of some meaningful input parameters, such as the power-law index, the nonlocal parameter, the axial load eccentricity, the mode number and the tapering ratio, on the flexural-torsional buckling load of tapered thin-walled FG nanobeam-columns, whose results could be used as valid benchmarks for further computational validations of similar nanosystems.

Free Vibration of Thin-Walled Composite Shell Structures Reinforced with Uniform and Linear Carbon Nanotubes: Effect of the Elastic Foundation and Nonlinearity.

In this work, we discuss the free vibration behavior of thin-walled composite shell structures reinforced with carbon nanotubes (CNTs) in a nonlinear setting and resting on a Winkler-Pasternak Foundation (WPF). The theoretical model and the differential equations associated with the problem account for different distributions of CNTs (with uniform or nonuniform linear patterns), together with the presence of an elastic foundation, and von-Karman type nonlinearities. The basic equations of the problem are solved by using the Galerkin and Grigolyuk methods, in order to determine the frequencies associated with linear and nonlinear free vibrations. The reliability of the proposed methodology is verified against further predictions from the literature. Then, we examine the model for the sensitivity of the vibration response to different input parameters, such as the mechanical properties of the soil, or the nonlinearities and distributions of the reinforcing CNT phase, as useful for design purposes and benchmark solutions for more complicated computational studies on the topic.

Quasi-3D Hyperbolic Shear Deformation Theory for the Free Vibration Study of Honeycomb Microplates with Graphene Nanoplatelets-Reinforced Epoxy Skins.

A novel quasi-3D hyperbolic shear deformation theory (QHSDT) with five unknowns is here employed, together with the Hamilton's principle and the modified couple stress theory (MCST) to analyze the vibrational behavior of rectangular micro-scale sandwich plates resting on a visco-Pasternak foundation. The sandwich structure features a Nomex or Glass phenolic honeycomb core, and two composite face sheets reinforced with graphene nanoplatelets (GPLs). The effective properties of both face sheets are evaluated by means of the Halpin-Tsai and extended rule of mixture (ERM) micromechanical schemes. The governing equations of the problem are derived by applying the Hamilton's principle, whose solutions are determined theoretically according to a classical Navier-type procedure. A parametric study checks for the effect of different material properties, length-scale parameters, foundation parameters and geometrical properties of the honeycomb cells, and the reinforcing GPLs, on the vibration response of the layered structure, which can be of great interest for many modern engineering applications and their optimization design.

A Modified Couple Stress Elasticity for Non-Uniform Composite Laminated Beams Based on the Ritz Formulation.

Due to the large application of tapered beams in smart devices, such as scanning tunneling microscopes (STM), nano/micro electromechanical systems (NEMS/MEMS), atomic force microscopes (AFM), as well as in military aircraft applications, this study deals with the vibration behavior of laminated composite non-uniform nanobeams subjected to different boundary conditions. The micro-structural size-dependent free vibration response of composite laminated Euler-Bernoulli beams is here analyzed based on a modified couple stress elasticity, which accounts for the presence of a length scale parameter. The governing equations and boundary conditions of the problem are developed using the Hamilton's principle, and solved by means of the Rayleigh-Ritz method. The accuracy and stability of the proposed formulation is checked through a convergence and comparative study with respect to the available literature. A large parametric study is conducted to investigate the effect of the length-scale parameter, non-uniformity parameter, size dimension and boundary conditions on the natural frequencies of laminated composite tapered beams, as useful for design and optimization purposes of small-scale devices, due to their structural tailoring capabilities, damage tolerance, and their potential for creating reduction in weight.

Buckling Behavior of FG-CNT Reinforced Composite Conical Shells Subjected to a Combined Loading.

The buckling behavior of functionally graded carbon nanotube reinforced composite conical shells (FG-CNTRC-CSs) is here investigated by means of the first order shear deformation theory (FSDT), under a combined axial/lateral or axial/hydrostatic loading condition. Two types of CNTRC-CSs are considered herein, namely, a uniform distribution or a functionally graded (FG) distribution of reinforcement, with a linear variation of the mechanical properties throughout the thickness. The basic equations of the problem are here derived and solved in a closed form, using the Galerkin procedure, to determine the critical combined loading for the selected structure. First, we check for the reliability of the proposed formulation and the accuracy of results with respect to the available literature. It follows a systematic investigation aimed at checking the sensitivity of the structural response to the geometry, the proportional loading parameter, the type of distribution, and volume fraction of CNTs. .

Nonlocal Buckling Analysis of Composite Curved Beams Reinforced with Functionally Graded Carbon Nanotubes.

This work deals with the size-dependent buckling response of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) (FG-CNTRC) curved beams based on a higher-order shear deformation beam theory in conjunction with the Eringen Nonlocal Differential Model (ENDM). The material properties were estimated using the rule of mixtures. The Hamiltonian principle was employed to derive the governing equations of the problem which were, in turn, solved via the Galerkin method to obtain the critical buckling load of FG-CNTRC curved beams with different boundary conditions. A detailed parametric study was carried out to investigate the influence of the nonlocal parameter, CNTs volume fraction, opening angle, slenderness ratio, and boundary conditions on the mechanical buckling characteristics of FG-CNTRC curved beams. A large parametric investigation was performed on the mechanical buckling behavior of FG-CNTRC curved beams, which included different CNT distribution schemes, as useful for design purposes in many practical engineering applications.

Higher-Order Thermo-Elastic Analysis of FG-CNTRC Cylindrical Vessels Surrounded by a Pasternak Foundation.

This study analyses the two-dimensional thermo-elastic response of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) cylindrical pressure vessels, by applying the third-order shear deformation theory (TSDT). The effective properties of FG-CNTRC cylindrical pressure vessels are computed for different patterns of reinforcement, according to the rule of mixture. The governing equations of the problem are derived from the principle of virtual works and are solved as a classical eigenproblem under the assumption of clamped supported boundary conditions. A large parametric investigation aims at showing the influence of some meaningful parameters on the thermo-elastic response, such as the type of pattern, the volume fraction of CNTs, and the Pasternak coefficients related to the elastic foundation.

Wave Propagation of Porous Nanoshells.

This study aims at investigating the wave propagation of porous nanoshells. The Bi-Helmholtz non-local strain gradient theory is employed in conjunction with a higher-order shear deformation shell theory, in order to include the size-dependent effects. The nanoshells are made of a porous functionally graded material (P-FGM), whose properties vary continuously along the thickness direction. A variational approach is here applied to handle the governing equations of the problem, which are solved analytically to compute the wave frequencies and phase velocities as function of the wave numbers. The sensitivity of the wave response is analyzed for a varying porosity volume fraction, material properties, non-local parameters, strain gradient length scales, temperature, humidity, and wave numbers. Based on the results, it is verified that the size-dependence of the response is almost the same to the one of plates, beams and tubes.