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Nonlinear Stability of Curved Multi-phase Composite Panels: Influence of Agglomeration
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in Randomly Distributed Carbon Nanotubes with Non-uniform In-plane Loads
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Abstract
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The nonlinear stability characteristics of doubly curved panels made of three-phase composites with
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randomly dispersed carbon nanotubes (RD-CNTRFC) subjected to practically-relevant non-uniform
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in-plane loads are investigated in this study. Carbon nanotubes, when mixed with resin polymer, may
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give rise to bundles, termed as agglomerations, which can have a profound impact on the effective
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material properties. There exists a strong rationale to investigate the influence of such agglomeration
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on the nonlinear equilibrium path of panels, which can subsequently be included in the structural
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stability design process to enhance operational safety. A multi-stage bottom-up numerical framework
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is developed here to probe the nonlinear stability characteristics. The effective material properties of
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RD-CNTRFC panels are determined using the Eshelby-Mori-Tanaka approach and the Chamis
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method of homogenization. By considering von-Kármán non-linearity and Reddy's higher-order
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shear deformation theory, strain-displacement relations are established for the non-linear stability
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analysis. The governing partial differential equations are simplified into nonlinear algebraic relations
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using Galerkin's method. Subsequently, by reducing the stiffness matrix neglecting the non-linear
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terms and solving the Eigenvalue problem, we obtain critical load and non-linear stability path of
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shell panels based on arc-length approach. In the present study, various shell geometries such as
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cylindrical, elliptical, spherical and hyperbolic shapes are modeled along with the flat plate-like
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geometry to investigate the non-linear equilibrium paths, wherein a geometry-dependent
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programmable softening and hardening behavior emerges.
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Keywords: Doubly curved shells, Randomly distributed carbon nanotubes, Post-buckling analysis of
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composites, Three-phase composites, Programmable softening and hardening behaviour
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2
1. Introduction
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Shell panels are a critical structural component in the design of aircraft wings, forming the
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flaps and membranes. These panels are typically thin-walled structures and are often supported with
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stiffeners or stringers to enhance structural stiffness. However, this additional support increases the
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overall weight of the structure significantly. Carbon nanotubes (CNTs) offer a promising solution by
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improving the stiffness of composite panels without the need for additional stringers, providing high
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flexural resistance under both static and dynamic loads. Past experiences have shown that these thin-
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walled structures can become vulnerable under in-plane compression loads (Reddy, 2003),
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potentially leading to catastrophic failures during operation. Moreover, the in-plane loading applied
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to these components can be non-uniform, depending on the stiffness of neighboring elements,
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complexity of the adjacent structural configuration and surrounding loading conditions. Therefore,
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addressing the stability criteria of thin-walled structures becomes crucial. In this context, the present
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study delves into the stability performance of randomly distributed carbon nanotube-reinforced
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composite shell panels when subjected to non-uniform in-plane loadings. Investigating the influence
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of CNT aggregation on stability performance is of great significance for aircraft designers, as it aids
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in accurately predicting the stability performance of CNT-based composite panels considering real-
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world manufacturing complications. The current study focuses on investigating the non-linear
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stability of carbon nanotube (CNT) reinforced shell panels like Cylindrical, Spherical, Elliptical, and
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Hyperbolic in geometry, considering the random distribution of nanotubes in the polymer matrix.
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In general, thin-walled structures are susceptible to losing stability when subjected to a
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critical amount of in-plane compressive load (Garg et al. 2022; Dey et al. 2018b). This phenomenon
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is known as buckling and can result in the structure undergoing significant deformation during the
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post-buckling regime or even collapsing under further loads. As a result, researchers need to consider
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the post-buckling behavior when designing and analyzing thin-walled structures to ensure they can
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withstand the expected loading conditions and maintain stability. The members may lose one stable
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configuration and attain another stable configuration. The load at which it changes from one stable
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configuration to another is known as critical load or buckling load, while the member may withstand
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a higher amount of load and undergo large deformation. Such reserve strength is gained through a
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post-buckling path or non-linear equilibrium path (Reddy, 2003). The current study probes the post-
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buckling strength of different types of shell panels, including Cylindrical, Spherical, Elliptical, and
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Hyperbolic, where randomly distributed carbon nanotubes are investigated as reinforcement. Apart
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from aerospace applications, these thin-walled structures are widely used as structural members in
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the broader field of engineering, such as Automobile, Marine, Civil, and Mechanical Engineering,
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due to their combination of strength and lightweight characteristics. Despite their advantages, it is
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important to consider the post-buckling behavior of thin-walled structures to ensure they maintain
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stability under the expected loading conditions. Thus, enhancement of the buckling behavior has
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been a center of focus of the research community. To this end, different reinforcements have been
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used including CNTs and different forms of graphene derivatives (Chandra et al., 2022).
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Following the discovery of carbon nanotubes (CNTs) by Sumio Lijima in 1991, it was
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observed that CNTs exhibit exceptional performance under mechanical load, thermal environment,
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and electric-magnetic loading (Fard, 2021; Fard and Pensky, 2023; Khaniki and Ghayesh, 2020; Roy
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et al., 2015, 2014). Numerous studies predicted CNT reinforced composites with functional graded
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distribution profiles perform well under bending but random distribution is considered to be a more
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practical distribution profile in terms of manufacturing of laminates. The present study examines the
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stability performance of shell panels reinforced with CNTs that are randomly distributed while
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considering the agglomeration impact of nanotubes in the polymer matrix. To improve the strength,
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fibers are added in the hybrid matrix (CNT and Polymer) that results in a multiphase composite. The
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behavior of CNT-reinforced flat panels under transverse and in-plane loads have been studied by
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several researchers and it can be observed that functional gradation with an X distribution pattern
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provides the maximum stiffness against bending (opposite for the O distribution pattern). This is
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because the density of nanotubes is highest at the surface and lowest near the neutral axis for the X
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distribution (Ansari et al., 2018; Chakraborty et al., 2021; Kiani and Mirzaei, 2018; Shen and Zhang,
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2010; Shen, 2012). Ansari et al. (2016) investigated the stability of functionally graded conical shell
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panels by using the Variational Differential Quadrature (VDQ) technique. First-order shear
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deformation theory (FSDT) was used to develop the strain-displacement model of the shell panel,
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and the Hamilton variational approach was used to construct the governing equations. The stability
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of Piezoelectric CNT-reinforced cylindrical panels was analyzed by Nasihatgozar et al. (2016). The
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effective material properties of functionally graded CNT composites were determined using the
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Eshelby-Mori-Tanaka approach and the Donnel shell theory was used to model the panel. The kp-
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Ritz meshless technique along with the arc-length method was used to trace the non-linear
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equilibrium path of CNT-reinforced cylindrical shell panels by Liew et al. (2014). It has been
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observed that the panel follows a non-linear equilibrium path with a softening phase initially and
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then hardening in the later stage. A finite element approach was presented by Gracia et al. (2017a,
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2017b) to obtain the stability characteristics of the FG-CNTRC cylindrical shell panels subjected to
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axial loadings and the effective material properties of composites were obtained using the Eshelby-
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Mori-Tanaka technique. Zghal et al. (2018) presented a finite element shell model considering the
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higher-order shear theory to investigate the FG-CNTRC curved panel stability characteristics. The
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authors observed significant improvement in buckling load due to the gradation of nanotubes and
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presented the stability response for various parametric properties. Chakraborty et al. (2019), using a
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semi-analytical solution, investigated the non-linear stability of a functional graded cylindrical shell
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panel subjected to non-uniform and localized loadings. It has been observed that the panel will
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deform continuously due to the tensile stresses developed in the transverse direction of the applied
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load. FG-CNTRC cylindrical shell panels’ buckling stresses under an in-plane load were estimated
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using the stable space approach by Liew and Alibeigloo (2021). The authors observed, that the panel
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with FG-X pattern exhibits minimum in-plane stresses and vice-versa for FG-O profile. The stability
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behavior of a cylindrical panel with an internal cutout was studied by Reddy and Ram (2021) using
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the finite element method. The finding reveals that the FG- X and FG-O pattern performs well as
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compared to other distributions and with an increase in cutout size the buckling load decreases.
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The investigations concerning the stability of CNT-reinforced composites is limited,
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particularly focusing on the effects of some of the practical and inevitable phenomenon like
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agglomeration. Shi et al. (2004) discussed the procedure to estimate the resulting material properties
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of randomly dispersed CNT-reinforced composites through the Eshelby-Mori-Tanaka
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micromechanical technique. Forougi et al. (2013) presented the stability performance of a randomly
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dispersed nanotube-reinforced polymer-fiber composite resting on an elastic foundation. The
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effective material properties were obtained using Mori-Tanaka micromechanical modeling and it was
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observed that with an increase in the volume fraction of nanotubes the non-dimensional buckling
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load increases. Arani et al. (2011) presented a finite element model to investigate the stability
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performance of randomly distributed CNT in a matrix considering the agglomeration effect. It was
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observed that an increase in agglomeration leads to a decrease in buckling strength when the panel is
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subjected to axial load. The influence of CNTs waviness and agglomeration with the inclusion of
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polymer matrix was considered in the design parameters of CNT-polymer composites by
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Georgantzinous et al. (2021) using FEM. Recent research has focused on the stability of RD-CNTRC
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plate and shell panels by considering sinusoidal shear deformation theory to model these panels
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(Daghigh et al., 2020; Georgantzinos et al., 2021). Dastjerdi and Malek-Mohammadi (2017)
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proposed a higher-order displacement theory along with Navier’s approach to model sandwich plates
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reinforced with RD-CNTRC face sheets subjected to bi-axial loadings. Sandwich CNTRC plate
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stability under a hygro-thermal environment was studied by Kiarasi et al. (2020) using Classical
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Laminated Plate Theory (CLPT). The influence of agglomeration was considered by obtaining the
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effective material properties using the Mori-Tanaka Approach. Safaei et al. (2019) modeled material
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properties of CNTRC sandwich plates by considering temperature sensitivity and obtained buckling
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response using a meshfree technique. The authors designed porosity in the core by considering
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gradation along the thickness of the plate. In a subsequent article (Moradi-Dastjerdi et al., 2020)
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stability of porous CNTRC sandwich plates was studied with two piezo-electric facesheets and
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subjected to thermo-mechanical loading.
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Fantuzzi et al. (2020) and Tornabene (2019) presented homogenization techniques to define
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the resulting material properties of three-phase randomly dispersed CNT-reinforced composite plate
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and shell panels. Following these homogenization techniques, Bacciocchi (2020) obtained the critical
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load of the multi-phase CNT-reinforced fiber composites. Based on experimentally obtained elastic
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properties of multi-walled CNT-reinforced fiber composites, Kamarian et al. (2020) investigated
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stability characteristics using the general differential quadrature (GDQ) approach. Thermal non-
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linear stability analysis was carried out for multi-phase shape memory alloy (SMA) fiber composites
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reinforced with CNTs by Mehar et al. (2021a) using the finite element approach. Some recent studies
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investigated the stability characteristics of randomly dispersed CNT-reinforced fiber composite
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structures like beam, plate, and shell panels using semi-analytical approaches (Chakraborty et al.,
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2022b, 2022a; Dash et al., 2022).
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The stability of flat and cylinder-shaped CNTRC panels was primarily covered in the above-
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mentioned studies, whereas the current study's focus is to investigate the nonlinear stability of
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various shell panels, followed by the possibility of exploiting geometry-dependent programmable
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features. In this section, a detailed review of the literature on doubly curved panels is framed to
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understand the behavioral pattern of doubly curved panels and broadly define the gap of research.
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Arefi and Amabili (2021) investigated the buckling characteristics of doubly curved nano-shell
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panels by incorporating the sinusoidal transverse shear effect under electro-magneto-elastic loadings.
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Post-buckling strength of nanocomposite doubly curved FG-CNTRC shell panels lying on an elastic
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foundation in a thermal environment was examined by Duc et al. (2019). The Galerkin approach was
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used along with von-Kármán type nonlinearity. The influence of functionally graded doubly curved
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shallow shell panels were studied by Dung and Dong (2016) for post-buckling analysis. The shell
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panels were modeled using third-order shear deformation theory, which takes geometrical non-
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linearities into account. Imperfection in the panel was added for a more realistic portrayal of the shell
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panel’s non-linear equilibrium path. Huang et al. (2021) modeled micro doubly curved FG-CNTRC
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shell panels by considering the transverse shear effect with a higher-order model. It was observed by
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the authors that the spherical panels predict maximum buckling strength compared to other panels
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and the hyperbolic panel possesses the lowest strength. Investigation of doubly curved shell panel’s
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stability under a hygro-thermal environment was studied by Karimiasl et al. (2019) using a multi-
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scale perturbation technique. The authors carried out the post-buckling analysis for CNT-based,
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Graphene-based and Shape memory alloy-based three-phase composites. Panda and co-authors (Kar
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et al., 2016; Kar and Panda, 2016; Mahapatra et al., 2016; Mehar et al., 2021b; Panda and Singh,
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2013) have made an effort to analyze the non-linear stability and vibration of curved panels in
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thermal environment. Kumar et al. (2003) studied the buckling and parametric instability of doubly
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curved panels including tensile patch loading. They observed that spherical panels have high
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buckling strength compared to other curved panels. The non-linear buckling of doubly curved
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shallow shell panels under hygrothermal loadings was studied using the finite element approach by
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Kundu and Han (2009a). The authors obtained the large deformation in panels with temperature
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increment and multiple deformation modes was observed due to change in moisture content. In a
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subsequent article (Kundu and Han, 2009b), the authors also studied the vibration and stability
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characteristics of doubly curved panels under in-plane loads. To trace the non-linear equilibrium path
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of the shell panel Kundu and Sinha (2007) opted for the incremental iterative arc-length approach.
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The authors compared the approach with the published works and found a good correlation for both
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snap-through and snapback behavior. The authors (Kundu et al., 2007) modelled piezo composite
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doubly curved panels and studied the non-linear equilibrium path using a finite element approach.
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The non-linear static instability of anisotropic laminated doubly curved panels was studied by Li et
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al. (Li et al., 2021; Liu et al., 2021). A closed-form solution was presented to obtain the non-linear
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equilibrium path of doubly curved shell panels under out-of-plane loading.
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The stability of thin-walled structures has attracted many researchers' interest as discussed in
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the concise review presented here. The problem gets more difficult for composite materials because
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of their stacking sequences, complex reinforcement and matrix elements, specific properties of the
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reinforcements (including irregularities such as agglomeration) along different loading circumstances
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and boundary conditions. Most of the published works on non-linear stability for curved panels are
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presented using the finite element approach which normally takes a large computation time. For
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different structural analyses where multiple realizations of the deterministic model are necessary
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(such as uncertainty quantification and reliability analysis, sensitivity analysis and optimization), it
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becomes practically impossible to deal with intensive simulation models (Dey et al., 2018a; Dey et
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al., 2019; Kumar et al., 2019; Trinh et al., 2020). To reduce such computational time a semi-
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analytical approach may be an efficient alternative with adequate physical insights (Haldar et al.,
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2018). Further, the nonlinear stability of doubly curved three-phase composite shells panels with
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randomly distributed carbon nanotubes has not been investigated considering the effect of
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agglomeration and non-uniform in-plane loading. Based on these considerations and identified
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research gaps, we aim to address the following issues in this article: (a) doubly curvature geometry-
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dependent (considering spherical, hyperbolic, elliptical and cylindrical panels) modulation of the
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softening and hardening behavior in the nonlinear stability analysis of three-phase randomly
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distributed CNT reinforced composite panel, (b) influence of different non-uniform in-plane linear
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and non-linear loads on non-linear stability of RD-CNTRFC doubly curved panels, (c) influence of
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CNTs agglomeration on non-linear stability of doubly curved panels, (d) coupled influence of
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transverse shear stresses (for moderately thick panels) and double curvatures on developing strain-
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displacement relations, and subsequently analysing the stability behavior. In the context of broader
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applications, the insights provided in this article, particularly regarding the softening and hardening
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behaviour of shell panels, would hold significant implications for engineering systems across the
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length scales. For example, in the realm of energy harvesting, understanding how the structural
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characteristics of shell panels change under various conditions can be instrumental in optimizing the
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design of energy harvesting devices. By harnessing these insights, it would become possible to
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generate optimized voltage output that can be utilized to power sensors and actuators efficiently (Yan
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et al., 2020), making this knowledge highly valuable for practical applications in the field.
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For developing the semi-analytical computational framework of nonlinear stability analysis,
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the strain-displacement relations would be developed here considering the geometrical non-
227
linearities of von-Kármán type. The governing equations would subsequently be obtained by
228
minimizing the variation of total energy of the system and the equations would be simplified into
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algebraic relations using Galerkin’s approach. The non-linear equilibrium path of the curved panels
230
would be obtained using Riks-Arc length approach to trace the actual path for load and deflection.
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The numerical results would explore the aims outlined in the preceding paragraph based on the
232
developed semi-analytical framework.
233
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234
(a)
(b)
Fig. 1: (a) Coordinate system of doubly curved panels; (b) Type of agglomerations.
235
236
2. Mathematical Formulation
237
Fig. 1(a) shows the doubly curved panel under consideration having radius of curvature R1
238
and R2 along y and x axes, respectively, with plan length of a, width of b, and thickness of h. The
239
panel is constructed with layers of lamina having an identical thickness and bonded together with
240
polymer matrix. The origin of coordinate system is located at the edge of panel mid-layer. Four
241
different geometries of panel are modelled for the present investigation; Spherical Panel
242
( )
12
,R R R R==
, Cylindrical Panel
( )
21
,R R R= =
, Elliptical Panel
( )
21
,2R R R R==
, and
243
Hyperbolic Panel
( )
21
,R R R R= =−
. In the present study, three-phase Fiber/CNT/Polymer
244
composite panel is modelled by assuming the dispersion of CNTs to be random within polymer
245
matrix, resulting in a composition of hybrid matrix where the polymer is further reinforced with
246
straight fibers. Due to improper mixing of nanotubes in matrix polymer, the nanoparticles may form
247
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bundles which are known as agglomerations (Shown in Fig. 1(b)). These can be categorized into
248
three types: Null agglomeration
( )
1; 1
==
, Partial agglomeration
( )
0.5; 0.75
==
, and
249
Complete agglomeration
( )
0.5; 1
==
. Here,
and
are the agglomeration coefficients.
250
2.1 Determination of effective elastic properties of RD-CNTRFC
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Effective material properties of CNT-reinforced doubly curved composite panels are
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estimated in two stages. Initially, hybrid matrix effective material properties are determined through
253
Eshelby-Mori-Tanaka approach, and further Chamis approach is utilized to assess the material
254
characteristics of three-phase composites.
255
If
cnt
V
is the CNT fraction in terms of volume present in the entire hybrid matrix volume (
V
),
256
then the nanotubes can exist both inside and outside of aggregated form of CNT-matrix sphere,
257
termed as inclusion. Consequently, the overall CNT volume percent of the hybrid matrix can be
258
calculated as (Shi et al., 2004),
259
in m
cnt cnt cnt
V V V=+
(1)
260
Here,
in
cnt
V
and
m
cnt
V
denote the volume of CNTs inside the inclusion, and the volume of CNTs outside
261
the inclusion, respectively. If
in
V
is the total volume of inclusion in the hybrid matrix then the
262
agglomeration coefficients (
and
) can be expressed as (Shi et al., 2004),
263
;in
in cnt
cnt
V
V
VV
==
(2)
264
2.1.1 Eshelby-Mori-Tanaka approach for hybrid matrix
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Using the Eshelby-Mori-Tanaka (E-M-T) method, the elastic moduli of a hybrid matrix can
266
be calculated (Tornabene et al., 2019) as
267
1
1
1
1 (1 ) 1
in
out
hm out
in
out
K
K
KK K
K
−
=+
+ − −
(3)
268
11
2
1
1
1 (1 ) 1
in
out
hm out
in
out
G
G
GG G
G
−
=+
+ − −
(4)
269
32
62
hm hm
hm
hm hm
KG
KG
−
=+
(5)
270
9
3hm hm
hm
hm hm
KG
EKG
=+
(6)
271
Here,
1
1
3(1 )
out
out
+
=−
,
2
2(4 5 )
15(1 )
out
out
−
=−
. The parameters
,,
hm hm hm
EG
, and
hm
K
represent the hybrid
272
matrix elastic properties like Elastic modulus, Poisson's ratio, Shear modulus, and Bulk modulus,
273
respectively. Shear moduli (
in
G
and
out
G
) and Bulk moduli (
in
K
and
out
K
) of inclusion and matrix
274
are presented in Appendix-A.
275
2.1.2 Chamis method for estimation of resulting material properties of randomly distributed CNT
276
reinforced three-phase composites
277
The obtained hybrid matrix is further fortified with straight fibers, resulting in a three-phase
278
fiber polymer composites. The effective material properties are estimated in the present study using
279
Chamis technique. The volume proportion of fibres and hybrid matrix can be crucial variables to
280
estimate the effective properties. Fibers volume fraction is assumed as
0.8
f
V=
(Lee, 2018) and the
281
volume proportion of hybrid matrix can be determined using rule of mixture
( )
1
hm f
VV+=
. Fibers
282
are the reinforcing agent in polymer matrix and they improve the material stiffness. It has been
283
observed in the published study (Lee, 2018) that the in-plane shear modulus can play an important
284
role for the stability of CNT reinforced composite panels. Therefore for accurate non-linear stability
285
analysis of CNT reinforced fiber composites, the volume fraction of fibers should be calibrated for a
286
higher percentage. Considering the influence of in-plane shear modulus in CNT reinforced
287
composites proposed by Lee, 2018, we have considered a higher volume fraction of fibers (
0.8
f
V=
288
) in the present study. The resulting expressions of three-phase composites’ elastic constants are
289
12
discussed in Appendix-A.
290
2.2 Kinematic relations
291
Incorporating the transverse shear effect of higher order, proposed by Reddy (HSDT),
292
displacement variables can be expressed as (Reddy, 1984),
293
00
,1
1
( , , , ) 1 ( , , ) ( , , ) ( ) ( , , )
x
z
u x y z t u x y t zw x y t z x y t
R
= + − +
(7)
294
00
,2
2
( , , , ) 1 ( , , ) ( , , ) ( ) ( , , )
y
z
v x y z t v x y t zw x y t z x y t
R
= + − +
(8)
295
0
( , , , ) ( , , )w x y z t w x y t=
(9)
296
Here, mid-plane displacements are represented as
00
,,uv
and
0
w
. The total rotational components can
297
be expressed as,
00
1 , 2 ,
;
x x y y
ww
= + = +
.
2
4
( ) 1 3
z
zz h
= −
represents the function of
298
transverse shear.
x
and
y
are rotation components of cross-section referring to y and x-directions,
299
respectively. Subsequently, the strain-displacement relationship for doubly curved panels can be
300
expressed as (Reddy, 1984),
301
0
,
0, 1,
1
()
x
xx xx xx x
u
z w z
R
= − − +
(10)
302
0
,
00
, 2,
2
()
y
yy yy yy y
v
z w z
R
= − − +
(11)
303
00
,,
0 0 0
, , 1, 2,
12
2 2 ( ) ( )
yx
xy xy xy xy y x
uv
z w z w z z
RR
= − − − − + +
(12)
304
'1
()
xz z
=
(13)
305
'2
()
yz z
=
(14)
306
Here
0
xx
,
0
yy
, and
0
xy
are the mid-plane strains and expressed by considering von-kármán
307
geometrical non-linearities as (Reddy, 1984),
308
13
2
00
00
,,
11
02
00
0 0 0 0
,,
22
00
0 0 0 0
, , , ,
12
1
2
1
2
+ + −
= = + + −
+ + − −
xx
xx
yy y y
xy
y x x y
wu
uw
RR
wv
vw
RR
uv
u v w w
RR
(15)
309
The curved panels are composed of multi-layered laminates where local coordinate of each
310
lamina is transformed to global system. The resulting constitutive relations can be presented as
311
(Reddy, 1984),
312
k k k
Q=
(16)
313
k
and
k
are the vector representation of kth layer stress and strain vectors which can be illustrated
314
as
11 22 23 13 12
, , , ,
T
kk
=
,
11 22 23 13 12
, , , ,
T
kk
=
.
k
Q
is the reduced stiffness matrix for kth
315
layer of panel and each elements of the matrix can be represented as
1,2,6
ij
Q (i, j= )
and along the
316
thickness direction it can be represented as
4,5
ij
Q (i, j= )
(Reddy, 1984).
317
11 22 12 22
11 12 22
12 21 12 21 12 21
44 23 55 13 66 12
; ; ;
111
;;
c c c c
c c c c
c c c
E E E
Q Q Q
Q G Q G Q G
= = =
−−−
= = =
(17)
318
The constitutive equations are then represented in terms of resultant force/moment-strain relations as
319
(Reddy, 1984),
320
A B C
0
= + +
(18)
321
B C D
0
= + +M
(19)
322
D E
0
= + + F
(20)
323
V H =
(21)
324
,
M
,
, and
V
are the resultant forces, Moments, Additional moments, and Shear force. Here A
325
matrix refers to in-plane stiffness, B is the coupling of in-plane and flexure stiffness, D matrix
326
represents the flexure stiffness, C , E , F are the stiffness contribution due to higher order terms and
327
14
H is the stiffness representing the influence of transverse shear stress (defined in Appendix-B).
,
,
328
and
are the strains due to bending, additional bending and shear, respectively, as
329
,,
,,
0 0 0
, , , 1, 2, 1, 2,
1 2 1 2
21
21
, , 2 ; , , ;
,
yy
xx
TT
xx yy xy x y y x
T
vu
uv
w w w
R R R R
vu
RR
= − + − + − + + = +
−−
=
(22)
330
In terms of stresses, the resultant forces, and moments for kth layer of laminates can be evaluated as
331
(Reddy, 1984),
332
1
/2
/2 1
/2
/2
, , (1, , ( )) (1, , ( ))
()
k
k
xx xx
xx xx xx z
n
h
yy yy yy yy yy
hkz
xy xy xy xy xy
h
xz xz xz
h
yz yz yz
z z dz z z dz
M
Vz dz
V
−
−=
−
= =
= =
1
1
()
k
k
z
n
kz
z dz
−
=
(23)
333
2.3 Governing partial differential equations
334
The governing partial differential equations can be derived by minimizing the variation of
335
energy of the system (Soldatos, 1991; Xiao-Ping, 1996) referring to displacement variables. The
336
obtained partial differential equations for doubly curved RD-CNT reinforced composite panels are
337
,,
,,
ˆˆ
:xx x xy y
xx x xy y
1
MM
u0
R
+
+ + =
(24)
338
339
(25)
340
, , , , , ,
12
,,
,
ˆ
ˆˆˆ
: 2 ( ) ( )
ˆˆ
( ) ( ) 0
yy
xx
xx xx xy xy yy yy xx x xy y x
xy x yy y y
w M M M w w
RR
ww
+ + − − + +
+ + =
(26)
341
1 , ,
: 0
xx x xy y xz
V
+ − =
(27)
342
2 , ,
: 0
xy x yy y yz
V
+ − =
(28)
343
,,
,.
()
ˆˆ
:yy y xy x
yy x xy x
2
MM
v0
R
+
+ + =
15
Here,
ˆ ˆ ˆ
( ), ( ), ( )
xx xx xx yy yy yy xy xy xy
N n N n N n
= − = − = −
and
,,
xx yy xy
n n and n
are the in-plane
344
membrane stresses that the panel may experience due to axial non-uniform compression loads. These
345
membrane stresses can be determined by satisfying compatibility and edge conditions (the detailed
346
determination procedure is discussed in Appendix – C). The present study would investigate non-
347
linear stability of doubly curved panels under linear and non-linear in-plane compression loads as
348
shown in Fig. 2. The load distribution profile for linearly varying in-plane loads is expressed as:
349
01
xx
y
NN b
=−
, and Parabolic in-plane load can be expressed as (Kumar et al., 2016):
350
0
41
xx
yy
NN
bb
=−
. Here
is the load factor that represents the linear distribution of in-plane
351
loads. It may be noted for linear in-plane loads that the panel may not experience pre-buckling
352
stresses as linear in-plane loads match with the neutral axis layer. Thus the investigation of non-
353
uniform loading conditions may be critical for stability analysis.
354
355
Fig. 2: Linear in-plane loads (a) Triangular
( )
1.0
=
; (b) Trapezoidal
( )
0.5
=
; (c) Partial Tension
356
( )
1.5
=
; and (d) Parabolic in-plane load.
357
2.4 Boundary conditions
358
The present methodology can solve for different boundary conditions like all edges are
359
clamped (CCCC), alternate clamped (SCSC) and simply-supported edges (CSCS) (Chakraborty et
360
al., 2019). For numerical investigation, here the non-linear stability analysis of doubly curved panels
361
is performed considering all of the edges as simply supported, and it can be expressed as
362
(Chakraborty et al., 2019a),
363
For x = 0, a:
00
;0
xx xx xx xx xx y
n N v w M
− =− = = = = =
(29)
364
For y = 0, b:
00
0; 0
yy yy yy yy x
n u w M
− = = = = = =
(30)
365
16
The assumed displacement field variables can be obtained by satisfying the above boundary
366
conditions (Chakraborty et al., 2019a),
367
11
cos sin
Q
P
pq
pq
p x q y
uU ab
==
=
(31)
368
11
sin cos
Q
P
pq
pq
p x q y
vV ab
==
=
(32)
369
11
sin sin
Q
P
pq
pq
p x q y
wW ab
==
=
(33)
370
11
cos sin
Q
P
xx pq
pq
p x q y
Kab
==
=
(34)
371
11
sin cos
Q
P
yy pq
pq
p x q y
Lab
==
=
(35)
372
Here ( ) represents the approximated displacement variables which depend on finite terms p and q
373
along x and y directions respectively.
374
The above displacement field variables [equations (31) – (35)] are substituted in the
375
governing relations [equations (24) - (28)] and further these partial differential equations (PDEs) are
376
simplified into non-linear algebraic equations using Galerkin’s approach. Next, the algebraic
377
relations are rearranged in the form of
0K=
. Here,
is the displacement vector, and K is the
378
stiffness matrix. Further K can be represented in terms of linear stiffness matrix Kl by neglecting
379
non-linear displacement terms, in-plane axial load, and the geometrical stiffness matrix Kg
380
containing participation of in-plane load. To obtain these critical buckling loads, the non-linear terms
381
are omitted, and a standard Eigenvalue solution is employed. Furthermore, for the evaluation of non-
382
linear equilibrium paths, the Riks arc-length approach (Vasios, 2015) is utilized, taking into account
383
the non-linear terms in the strain-displacement relationship. It can be noted that neglecting the non-
384
linear terms may not yield very accurate results in non-linear stability analysis for doubly curved
385
panels, while the important behavioral trends can be predicted.
386
387
388
389
17
3. Result and Discussion
390
3.1 Comparative analysis and validation
391
Geometry-dependent non-linear stability of doubly curved panels such as spherical, elliptical,
392
cylindrical, and hyperbolic panels is the focus of this study. In order to trace non-linear equilibrium
393
path of the panel accurately, geometrical non-linearities are introduced when designing the strain-
394
displacement model and transverse shear deformation theory of higher order is taken into account.
395
The accuracy of the current model is established by comparing the critical load of CNTRC panels
396
(Khdeir et al., 1989; Lei et al., 2016; Shadmehri et al., 2012), and non-linear stability of laminated
397
composite shell panels (Girish and Ramachandra, 2006), as available in literature. The effective
398
material properties of hybrid matrix obtained using E-M-T approach are compared with Shi et al.
399
(2004).
400
In Table 1, the non-dimensional critical load of a flat polymer composite panel reinforced
401
with CNTs subjected to uniaxial and biaxial compressive in-plane loads are compared with the
402
literature. By assuming that nanoparticles are evenly distributed throughout the polymer matrix and
403
the edges are simply supported, the non-dimensional buckling parameter is obtained for various
404
volume fractions of CNTs and compared with the kp-Ritz solution approach based on transverse
405
shear theory of first order (Lei et al., 2016). One can notice that the current method closely matches
406
with the existing literature, wherein the critical load increases as the % CNTs fraction increases.
407
Here,
xcr
N
represents the instability load for CNT based composite flat panels.
408
Table 1. Non-dimensional critical load
2
3
xcr
m
Nb
Eh
of uniformly dispersed CNT flat panel with simply
409
supported edges
( / 1, / 10,[0 / 90 / 90 / 0])a b b h==
.
410
cnt
V
Uniaxial Compression
Biaxial Compression
Present
Lei et al. (2016)
Present
Lei et al. (2016)
0.11
20.12
19.84
10.06
9.93
0.14
23.21
22.64
11.60
11.36
0.17
31.29
30.95
15.64
15.49
411
18
Table 2. Non-dimensional buckling load
2
3
22
100xcr
Na
Eh
of simply supported laminated cylindrical
412
shell panel subjected to uniform in-plane compression loads (R/h = 10, R/a = 1, a = b =1).
413
Ply Layup
Present
Shadmehri et al.
(2012)
Khdeir et al.
(1989)
[0/90/0]
0.2754
0.2765
0.2813
[0/90]
0.1703
0.1525
0.1670
414
Table 2 shows the non-dimensional buckling load parameter for a laminated cylindrical shell
415
panel under in-plane compressive force with simply supported edges. With the help of the current
416
method, non-dimensional parameters are obtained for various ply layups and compared with the
417
solution obtained using Ritz method considering transverse shear of first order as presented by
418
Shadmehri et al. (2012) and higher order transverse shear effect as presented by Khdeir et al. (1989),
419
respectively. The current results are in good agreement with the results of published works.
420
Further, the non-linear stability of a laminated cylindrical composite shell is traced and
421
compared with the published literature of Girish and Ramachandra (2006), as shown in Fig. 3. It can
422
be noticed that the present non-linear stability results show close agreement with the available
423
literature. The cylindrical shell panel's non-linear equilibrium response exhibits softening behavior in
424
the early stages of deformation and hardening characteristics as the deformation progresses. Such a
425
trend is accurately captured by the current computational framework.
426
427
Fig. 3: Post-buckling path of simply supported cylindrical shell panel [0/90/90/0] subjected to in-
428
plane compression load
( / 1, / 0.38)a b a R==
.
429
19
430
Fig. 4: Comparison of effective elastic property of hybrid matrix with an increase in agglomeration
431
of CNTs
( )
1β=
.
432
433
After establishing the validations for stability behavior of the shell panels based on the
434
availability of literature (refer to Table 1 – 2 and Fig. 3), we concentrate on the accuracy of the
435
models for obtaining effective material properties of RD-CNT based polymer composites. Fig. 4
436
shows the comparative results for effective material properties relation given by Shi et al. (2004) for
437
RD-CNT based polymer composites. The material properties are modelled by considering the CNTs
438
agglomeration effect and the elastic modulus of CNT reinforced hybrid polymer composites are seen
439
to decrease with an increase in aggregation of CNTs in polymer, while they increase with % increase
440
of volume content of nanoparticles. To improve the effective elastic properties of CNT reinforced
441
composites, fiber reinforcement has been added to CNT reinforced hybrid matrix which can
442
significantly improve the stiffness property of the composite. In general, a good agreement of the
443
trends can be noticed with respect to published literature.
444
In this context, it can be noted that a random distribution of nanotubes leads to a
445
heterogeneous mixture, which was also predicted by Shi et al. (2004). In the discussed study by Shi
446
et al. (2004), the authors compared the characteristics of nanotube composites with isotropic and
447
anisotropic properties of nanotubes. They observed that the isotropic characteristics of CNTs may
448
overestimate the tensile modulus of elasticity, which is more pronounced when CNTs are uniformly
449
dispersed in the matrix. Building upon these conclusions, the authors of the present study considered
450
the material properties of nanotubes to be transversely isotropic. Effective material properties are
451
20
estimated here using the Mori-Tanaka approach for its simplicity and high accuracy, even at a high
452
volume fraction of inclusions, and also validated with the experimental results by Shi et al., (2004).
453
This approach also takes into account the heterogeneous nature of nanotubes in the matrix. To
454
demonstrate the presence of inhomogeneous behaviour of nanotubes in the polymer matrix, the
455
present micromechanical model considered the agglomeration and dispersion characteristics of
456
nanotubes in the polymer matrix, as shown in Figure 4.
457
Having validated the computational model from different aspects like the stability behavior
458
and evaluation of effective material properties, we present numerical results concerning the nonlinear
459
stability of doubly curved three-phase composite shells reinforced with randomly distributed carbon
460
nanotubes including the effect of agglomeration in the following subsections. The material properties
461
of CNTs, Polymer and Glass fibers used in the present investigation are mentioned in Table 3.
462
Table 3. Material properties of CNT, matrix, and fiber
463
SWCNT (10,10)
Hill’s Modulus
(Singh et al., 2022)
Glass Fiber properties
(Chamis, 1983)
Matrix Properties
(PmPV)
(Singh et al., 2022)
271
88
17
1089
442
cnt
cnt
cnt
cnt
cnt
k GPa
l GPa
m GPa
n GPa
p GPa
=
=
=
=
=
11 22 33
12 13 23
12 13 23
73.08
30.13
0.22
f f f
f f f
f f f
E E E GPa
G G G GPa
= = =
= = =
= = =
2.11
0.34
m
m
E Gpa
=
=
Elastic of SWCNT (10,10) (Singh et al., 2022)
11 12
5.646 0.175
cnt cnt
E TPa;
==
464
3.2 Influence of CNTs agglomeration
465
The current study examines the non-linear stability of composite shell panels with randomly
466
dispersed CNT reinforcement. The impact of CNT agglomeration on the non-linear stability of
467
several forms of shell panels, such as cylindrical, spherical, elliptical, and hyperbolic, are covered in
468
this section. In Fig. 5, numerical results are presented for panels subjected to uniform in-plane
469
compression loads with simply supported edges. The non-linear equilibrium path of each panel is
470
normalised with the critical load corresponding to null agglomeration of RD-CNTRFC cylindrical
471
21
(a)
(b)
(c)
(d)
Fig. 5: Influence of agglomeration on RD-CNTRFC panels
[0 / 90/ 0 / 90 / 0]
( / 1, / 100, 10 , 0.25, 1. 1)
cnt
a b b h R a w
= = = = = =
(a) Cylindrical; (b) Spherical; (c)
Elliptical; (d) Hyperbolic.
472
shell panels. The numerical results reveal that with an increase in CNTs agglomeration, effective
473
material properties decrease, resulting in a decrease of the panel stiffness. Therefore, with reference
474
to null agglomerations the panel exhibits maximum buckling strength, while the minimum value is
475
attained with respect to complete agglomerations. It can be observed that the cylindrical shell panel’s
476
non-linear equilibrium path starts with bifurcation point and follows a softening behaviour, and
477
further deformation leads to hardening behavior. Whereas Spherical, Elliptical and Hyperbolic shell
478
panels follow a continuous deformation non-linear equilibrium path due to the tensile stress
479
developed in the transverse direction of the panel with the application of in-plane compressive load.
480
Such non-unique mechanical characteristics give scope for programming (i.e. application-specific
481
22
(a)
(b)
(c)
(d)
Fig. 6: Non-linear stability path of the RD-CNTRFC shell panels
[0 / 90/ 0 / 90 / 0]
for
different mass fraction of CNTs
( / 1, / 10, / 2, 0.50, 0.75)a b b h R a
= = = = =
(a)
Cylindrical; (b) Spherical; (c) Elliptical; (d) Hyperbolic.
482
designing) the constitutive behaviour of softening and hardening based on the shell geometry for
483
application-specific demands.
484
3.3 Influence of CNTs mass fraction
485
The influence of CNTs mass fraction on non-linear stability analysis of different shell panels
486
is studied in Fig. 6. The panels are simply supported considering the CNTs to be partially
487
agglomerated in the hybrid matrix and subjected to uniform in-plane compressive load. The non-
488
linear equilibrium path of different panels is normalised with the critical load corresponding to mass
489
fraction (wcnt = 0.1) of the cylindrical shell panel. It can be observed from Fig. 6 that with an
490
increase in mass fraction of CNTs the stiffness of the panels improves, resulting in higher buckling
491
strength, but the rate of increase in post-buckling strength with an increase in mass fraction of
492
23
nanotubes decreases. This happens because the increment of nanotubes in a constant volume of
493
polymer matrix tends the nanotubes distribution to agglomerated stage which further reduces the
494
effective stiffness of composites and results in a decrease in the rate of enhancing buckling strength
495
with an increase of mass fractions of nanotubes. Similar observations are reported in published
496
literature (Thirugnanasambantham et al., 2021).
497
(a)
(b)
(c)
(d)
(e)
Fig 7: Influence of various in-plane compression loads on post-buckling path of different RD-
CNTRFC shell panels
[0 / 90/ 0 / 90 / 0]
( / 1, / 10, / 2, 1, 1, 0.25)
cnt
a b b h R a w
= = = = = =
(a)
Uniform; (b) Triangular; (c) Trapezoidal; (d) Partial Tension; (e) Parabolic.
24
3.4 Influence of non-uniform in-plane loads
498
In a practical scenario, it is often unlikely that a shell panel will always be subjected to uniform
499
compressive load. To investigate such effects, in this section, the panels are subjected to uniform and
500
non-uniform in-plane loads like triangular, trapezoidal, partial tension, and parabolic load variations
501
(refer to Fig. 7). For uniform and linear in-plane loads panel does not experience pre-buckling
502
stresses as the loads coincide with the panels’ neutral axis. But for a parabolic in-plane load, the
503
panel will experience the membrane stresses and therefore it has been incorporated for estimation of
504
exact stress distribution. The non-linear equilibrium path of each panel is normalised with the critical
505
load corresponding to cylindrical shell panel of uniform load distribution. The CNTs in the panel are
506
assumed to be properly mixed in the hybrid matrix. It can be noticed that for cylindrical shell panels
507
subjected to uniform and linear in-plane loads, the non-linear equilibrium path follows bifurcation
508
point. Whereas for cylindrical panels with parabolic in-plane loads, it follows the continuous
509
deformation similar to doubly curved panels. This is due to the nature of tensile stress developed in
510
the transverse direction to applied in-plane load. It can also be observed that panels subjected to
511
partial tensile in-plane load show maximum buckling strength, and the minimum corresponds to
512
uniform in-plane loads. It is also noticed that Elliptically curved panels exhibit maximum buckling
513
strength, followed by Spherical panel, Cylindrical panel, and minimum corresponds to Hyperbolic
514
shell panels (as shown in Fig. 7). In essence, we show in this section that the stability behaviour
515
including softening and hardening depends on the coupled effect of shell geometry and nature of in-
516
plane loading, which can be programmed based on application-specific demands for most optimal
517
performances.
518
3.5 Influence of Radius of curvature of shell panels
519
Fig. 8 represents the influence of radius of curvature of shell panels on non-linear stability
520
path of different shell panels. The non-dimensional ratio for equilibrium paths for the shell panel is
521
normalised with critical load corresponding to flat panel when subjected to uniform in-plane
522
compressive load. Thus the presented numerical results give a clear perspective on the stability
523
behaviour of different shell geometries with respect to the flat plate structure. CNTs are assumed to
524
25
(a)
(b)
(c)
(d)
Fig. 8: Influence of R/a ratio on non-linear stability path of different RD-CNTRFC shell
panels [0/90/0/90/0] configuration (a) Cylindrical; (b) Spherical; (c) Hyperbolic; (d) Elliptical
( / 1, / 10, 1, 1, 0.25)
cnt
a b b h w
= = = = =
.
525
be well mixed with the polymer matrix here. For cylindrical shell panels bifurcation point is
526
observed to be on the higher side with the least value of radius-to-span (R/a) ratio. With an increase
527
of R/a ratio, the minimum non-dimensional value is observed corresponding to flat panels. For lower
528
R/a value, cylindrical panels follow the softening behaviour for the initial non-dimensional
529
displacement parameter (w/h) and as deformation progresses hardening behaviour can be seen. But
530
with an increase in R/a value softening behaviour in cylindrical panel vanishes. It is also observed in
531
cylindrical panels that, with decreasing R/a value buckling strength also decreases with an increase in
532
deformation. Similar behaviour can be noticed for Spherical and Hyperbolic panels, whereas
533
Elliptical panels follow the same characteristic for initial w/h value and in a later stage reverse
534
behaviour can be observed. Interestingly, in Spherical panels, the difference in non-linear
535
equilibrium path for different R/a values is found to be insignificant at higher values of w/h.
536
26
Whereas, a significant difference can be observed for Elliptical and Hyperbolic shell panels. At the
537
transition point of softening and hardening behavior of shell panels, stiffness is minimum and
538
corresponding to this stiffness, displacement amplitude can be maximum when a panel is subjected
539
to static and dynamic loads. As an application of the knowledge presented in this article, such
540
softening and hardening behavior can be of significant importance for obtaining the resulting voltage
541
output in energy harvesting devices to operate sensors and actuators (Yan et al., 2020).
542
In essence, along with the discussions of the preceding subsections, we establish here that a
543
non-linear programming of hardening and softening constitutive behavior is achievable as a function
544
of shell geometry including the radius of curvature and nature of non-uniform loading to address
545
application-specific functional demands optimally. The distribution of CNTs and agglomeration
546
further influence the stability behaviour, albeit not affecting the general trends significantly.
547
(a)
(b)
(c)
(d)
Fig. 9: Influence of thickness of shell panels [0/90/0/90/0] subjected to uniform compressive
load (a) Cylindrical Panel; (b) Elliptical Panel; (c) Spherical Panel; (d) Hyperbolic Panel
( / 1, / 2, 1, 1, 0.25)
cnt
a b R a w
= = = = =
.
548
27
3.6 Influence of thickness of shell panels
549
Due to the influence of transverse shear effect, thickness of panels is an important parameter
550
to explore the non-linear stability of shell panels. Therefore, the present model is developed to
551
include the higher-order shear theory. The width-to-thickness ratio (b/h) of the shell panels is taken
552
into account while modelling them by assuming that the CNT distribution is evenly mixed with the
553
polymer matrix. The non-dimensional parameter is then obtained by normalising the non-linear
554
equilibrium route with the critical load. It can be observed from Fig. 9 that with an increase in width-
555
to-thickness ratio (b/h) non-linear equilibrium path of shell panels decreases and this is due to the
556
reduction of panels’ stiffness with the decrease of shell panel thickness.
557
4. Conclusions
558
In the present work, the non-linear equilibrium path is investigated for different doubly
559
curved panel geometries reinforced with randomly distributed CNTs in the polymer matrix. Due to
560
random distribution, the effective material properties and the stability behaviour are significantly
561
influenced which have been investigated including the effect of agglomeration. Along with the
562
parameters involved with CNT distribution and shell geometry, the effect of non-uniform edge
563
loading is explored in this study to understand their coupled influences on the non-linear equilibrium
564
path. Some of the salient conclusions drawn from the numerical analysis are mentioned here.
565
• Panels with complete agglomeration of CNTs show minimum buckling strength compared to null
566
agglomeration. This is due to the reduction of effective material properties of CNT-reinforced
567
fiber composites that result in a reduction of stiffness of composite with an increase of CNTs
568
agglomeration in the polymer matrix.
569
• Higher stability of the panel is achieved with an increase in mass fraction of CNTs due to an
570
improvement in the stiffness of shell panels. The non-linear equilibrium path of elliptical shell
571
panels exhibits a higher load-deformation path compared to hyperbolic panels.
572
• It is observed that with an increase of mass fraction of nanotubes, keeping the volume of polymer
573
constant, the non-linear stability of panels improves. However, the rate of increase in non-linear
574
28
stability decreases subsequently due to the change of state from null agglomerated to
575
agglomerated phase with an increase in mass fraction of CNTs in polymer matrix resulting in
576
deterioration of materials properties.
577
• Cylindrical panels follow softening type of non-linear behaviour at the initial non-dimensional
578
deformation (w/h) and successive hardening characteristics can be observed at higher value of
579
w/h. However, the initial softening behaviour is dependent on the radius of curvature and it
580
vanishes with an increase of R/a value.
581
• For doubly curved panels non-linear equilibrium path follows continuous deformation with the
582
application of in-plane loads. This is due to the tensile stress developed in the transverse direction
583
with respect to the in-plane applied load.
584
• Thickness of shell panels significantly influences the non-linear stability characteristics, albeit
585
without altering the general trends significantly. With an increase of shell panel thickness, the
586
panel stiffness also improves resulting in higher stability compared to thin panels.
587
In summary, multiple shell geometries such as cylindrical, elliptical, spherical and
588
hyperbolic shapes are modelled here along with the flat plate-like geometry, wherein a geometry-
589
dependent programmable softening and hardening behavior emerges with a quantifiable coupled
590
dependence on the non-uniform nonlinear loading condition, shell thickness, radius of curvature and
591
the CNT parameters. Such control on the nonlinear stability behavior of thin-walled structures would
592
lead to enhancing the performance of a multitude of critical engineering applications including
593
energy harvesting, soft robotics, lightweight structural systems and the design of sensors and
594
actuators. Future work in the field of post-buckling for doubly curved panels reinforced with
595
randomly distributed Carbon Nanotubes (CNTs) in the polymer matrix could involve developing
596
new analytical and numerical models, conducting experimental studies to validate theoretical
597
predictions, exploring the potential of using other types of nanoparticles or hybrid reinforcements,
598
investigating the impact of various boundary conditions and loading scenarios, accessing the
599
influence of different defects and progressive damages such as layer-wise debonding (including
600
quantification of the subsequent effect of reduced stiffness), exploring the effect of operating
601
29
environment and evaluating the potential applications of these panels in various critical engineering
602
fields.
603
Appendix
604
Appendix-A, Appendix-B and Appendix-C are provided in a supplementary document.
605
Acknowledgments
606
SN and TM would like to acknowledge the initiation grant received from the University of
607
Southampton during the period of this research work.
608
Conflict of interest
609
The authors declare that they have no known competing financial interests or personal relationships
610
that could have appeared to influence the work reported in this paper.
611
References
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