Buckling Prediction of MWCNT-reinforced Laminated Composite Structures under Hygro-Thermo-Mechanical Conditions

This study presents a detailed buckling analysis of laminated composites reinforced by multi-walled carbon nanotube (MWCNT) inclusions using a multiscale computational framework. It combines multiple analytical and computational techniques to assess the performance of these composites under varying hygro-thermo-mechanical conditions. The model incorporates nanoscopic MWCNT characteristics, estimates orthotropic constants, and investigates the impact of various factors on the critical buckling load of MWCNT-based laminates. Comparison with existing data validates our approach, marking the first usage of the multiscale finite element method for predicting the buckling behaviour of MWCNT-reinforced laminates. This research offers valuable design insights for various industries including aerospace and automotive.


Introduction
Laminated composites have been extensively used in various engineering applications due to their exceptional mechanical properties and design flexibility since the development of aerospace and automotive industries created an urgent demand for lightweight, high-strength and high-toughness materials [1,2].In recent years, the reinforcement of these composites with multi-walled carbon nanotubes (MWCNTs) has gained considerable attention due to the exceptional mechanical, thermal, and electrical properties of MWCNTs which has provided further potential for the development of enhanced laminated composites [3,4].MWCNT-reinforced composites exhibit enhanced mechanical performance, including increased stiffness, strength, and toughness [5,6,7].
However, the presence of MWCNTs in the composite system introduces complexities in predicting the structural behaviour of the composites, especially under hygro-thermo-mechanical conditions which result in the aging of the structures and can ultimately influence their structural integrity [8,9].The anisotropic nature of MWCNTs, their interaction with the matrix material, and their distribution and orientation within the composite can greatly affect the overall properties of the composite [10,11].Moreover, the behaviour of these composites under critical loading, such as buckling, is not yet fully understood [12].Recent studies have employed various analytical and computational techniques to predict the mechanical behaviour of MWCNT-reinforced composites [13,14].However, most of these studies have focused on the linear elastic and stable response of the composites and have not adequately addressed the complex response under critical loading conditions [15].Therefore, a more comprehensive approach that can accurately predict the buckling behaviour of MWCNT-reinforced composites under various conditions is highly desired.
To completely consider all involved factors and investigate the aforementioned challenges associated with MWCNT-reinforced composites A multi-scale analysis framework is proposed in the present study.The multi-scale framework serves as an interface that assimilates information across a spectrum of length scales enabling the extrapolation of macroscopic attributes from underlying microscopic phenomena [16].This approach effectively bridges the gap between the micro-structural behaviours and the overall mechanical properties, enhancing the predictive capabilities of the model.Despite their potential, multi-scale methods have not yet widely used in the study of MWCNTreinforced composites.Only a few recent studies have tried to use multi-scale methods to explore the behaviour of these advanced composites [17,18,19].However, these studies have primarily focused on general mechanical behaviour, leaving the specific domain of buckling behaviour -a critical aspect of structural stability -relatively unexplored.This gap in knowledge underlines the need for further research and development in this domain.
Therefore, this study aims to develop a multi-scale-based finite element method (FEM) to predict the buckling behaviour of MWCNT-reinforced laminated composites under various hygro-thermomechanical conditions.This approach will provide a more accurate and comprehensive understanding of the behaviour of these composites, which can contribute to the design and optimization of MWCNT-reinforced composites for various engineering applications [20,21].

Mechanical properties of the nanocomposite matrix
The Halpin-Tsai (H-T) equations are suitable for determining the elastic modulus of carbon nanotubes (CNT's) enhanced nanocomposites, assuming a uniform distribution of straight, perfectly aligned nanoparticles within the polymer matrix.Nonetheless, real-world experimental conditions often yield CNT agglomerates within the polymer matrix, making this a significant factor in most laboratory procedures [22].Consequently, current methodologies for integrating CNTs into polymer solutions require researchers to consider aspects like particle orientation and agglomeration during their theoretical computations to develop practically usable nanomaterials.
The H-T micromechanical framework is applied for computing Young's modulus of straightaligned CNT-based polymer nanocomposites [21].The equation is defined as follows: ). ( Here, Ecnt and Em signify Young's modulus of the CNT and polymer matrix, respectively.Vcnt stands for the CNT volume fraction, while Lcnt and dcnt symbolize CNT length and the exterior diameter, respectively.The   parameter is used to tie the random orientation of CNTs in the polymer matrix to the elastic modulus.Furthermore, the elastic modulus's association with CNT waviness is captured through the   parameter, defined as Here, W and A indicate the half-wavelength and amplitude of a wavy CNT, respectively.In order to factor in agglomeration effects in the polymer matrix, the H-T equation includes the   parameter: The shear modulus, considering the isotropic behavior of CNT-reinforced polymers, can be determined from the following formula [18]: The Poisson's ratio of the modified polymer matrix can be estimated using the subsequent approximation [18]: In this equation,  − and   represent the Poisson's ratio of the CNT and polymer matrix, correspondingly.

Hygro-thermo-mechanical properties of the unidirectional composite lamina
Considering the impact of temperature and moisture on the design of polymer-based matrix composites, Chamis et al. [23] proposed the following empirical equations, which can be utilized to estimate the hygro-thermal effect on the epoxy matrix properties of a unidirectional composite lamina: where PwT denotes the matrix property at the use temperature T and moisture content M, and P0 denotes the matrix property at a reference temperature T0.F corresponds to the mechanical property retention ratio of the previous matrix properties.Additionally, Tgd and Tgw specify glass transition temperature, in the dry condition and the wet condition with moisture content of Mm, respectively.Equation ( 6) is valid for moisture content   ≤ 10% [24].

Finite element model and numerical analysis
In this study, an in-house FEM code is employed to predict the buckling behaviour of the MWCNTreinforced laminated composites.Within the Finite Element Analysis (FEA) context, mesh generation is an important pre-processing task to discretize the geometry of the structure into elements.In this work, the rectangular composite plates underwent discretization utilizing an 8-node stress/displacement, quadrilateral, shell element.This was equipped with reduced integration capabilities and exhibiting 5 degrees of freedom at each node (specifically the translations in the X, Y, and Z directions and rotations about the X and Y axes defined in the element's local coordinate system.The normal to the shell's middle surface usually defines the local Z-axis).Notably, the Kirchoff constraint [25] was numerically applied to maintain precision and accuracy in our simulations.
Once we had identified the applicable finite elements, a static analysis has been initiated preceding any buckling evaluation [26].This ensured the alignment with the imposed boundary conditions.To facilitate the linear elastic analysis, the equilibrium equations spanning across all elements are translated into a global Cartesian coordinate framework.Subsequently, these are assembled to conform to the prerequisites of nodal equilibrium and boundary conditions, thereby formulating a solvable structure that can be efficiently interpreted.
Proceeding into the domain of buckling analysis, we opted for the eigenvalue approach, and took advantage of the benefits of the Lanczos algorithm [27].A key aspect of this method was applying a reference load, onto the structure to simulate real-world stresses and assess the plate's resilience and deformation behaviour.
Figure 1 provides a comprehensive visualization of the representative model and the buckling analysis of a [0°/90°/0°] laminated composite plate, measuring 200 x 200 x 20 mm.In Figure 1(a), the front view showcases the partitioned model.This depiction highlights the applied uniaxial compressive load and the plate's boundary conditions, which adhere to a simply supported (SSSS) framework on all sides.Specifically, for the upper and down sides of the plate, the conditions U1=0 and U3=0 are imposed.This means that there is no translation allowed in the x-axis (represented by U1) and the z-axis (represented by U3).On the other hand, for the left and right sides of the plate, the conditions U2=0 and U3=0 are applied.This restricts any translation in the y-axis (represented by U2) and the z-axis.In Figure 1

Results and discussion
First, to validate the employed numerical model, a convergence and comparison studies have been performed, focused on the normalized critical buckling  ̄=    2 / 2 ℎ 3 , where b and h are the length of side and thickness of the plate, respectively, of a simply supported laminated plate as presented below in Table 1.This plate, under the influence of a uniaxial compressive load, maintains a distinct aspect ratio, b/h=10, and is analysed under various lamination schemes using Ε1/Ε2 = 40, G12 = G13 = 0.6E2, G23 = 0.5E2, υ12 = 0.25.The Table 1 compares the results derived from our present FEM simulations with those reported by Trinh et al. [28].It encompasses mesh sizes ranging from 6x6 to 16x16 results which emphasizes the significance of mesh refinement in obtaining accurate buckling predictions.The consistency between the present FEM results and those from a referenced study reaffirms the reliability and robustness of our computational approach.Next, our analysis is additionally validated by a comparison study on the normalized critical buckling of a simply supported laminated rectangular plate which is presented in Table 2. Here, we explored the implications of varying modulus ratios (E1/E2) on the buckling performance.Keeping the aspect ratio constant at b/h=10 and using the same material properties, this investigation encompasses modulus ratios spanning from 3 to 40.It's apparent that although all computational approaches show similar trends, there are slight differences in the precise buckling values.These variations could stem from differences in computational methodologies, boundary condition interpretations, or other subtle details intrinsic to each study.Nevertheless, the convergence in trends highlights the reliability of these computational strategies in predicting buckling behaviour under varying modulus ratios.
Figure 2 offers an insightful depiction of the critical buckling load response of a simply supported [0°/90°/±45°/0°/90°/±45°/0°/90°] nanocomposite plate.The study meticulously explores this behaviour as a function of Carbon Nanotube (CNT) volume fraction, while keeping parameters such as the fiber volume fraction, moisture content, and CNT aspect ratio, constant.The temperature spectrum is comprehensive, ranging from 200K to 400K, with increments of 50K.Notably, a consistent trend is observable across this spectrum: as the temperature increases, the critical buckling load for the nanocomposite plate decreases.This behaviour underscores the adverse impact of temperature elevation on the structural stability of the plate, emphasizing the temperature-sensitive nature of the material's mechanical properties.The findings underline the need for meticulous material optimization when designing structures that need to withstand varying thermal environments, especially if the objective is to harness the reinforcing capabilities of CNTs effectively.Further, the effect of moisture content on the critical buckling load of a simply supported [0°/90°/±45°/0°/90°/±45°/0°/90°] nanocomposite plate is illustrated in Figure 3.This comprehensive evaluation is conducted while keeping parameters such as temperature, fiber volume fraction, and CNT aspect ratio, the same.The analysed moisture spectrum is carefully chosen, spanning from 0% to 5% in progressive increments of 1.25%.A clear pattern has been found: as the moisture content within the plate increases, the critical buckling load experiences a clear reduction.This observation underpins the vulnerability of the plate's structural integrity when exposed to increasing moisture levels, suggesting that the internal cohesion of the material matrix is compromised, or the moisture-induced swelling leads to changes in material properties that affect buckling behaviour.On the other hand, the trend with respect to the CNT volume fraction remains consistent with the previous results.Regardless of the moisture levels, an augmentation in CNT volume fraction translates to a subsequent elevation in the critical buckling load.This suggests that, even under the influence of moisture, the presence of a higher CNT volume fraction consistently contributes to the structural robustness of the nanocomposite plate and an enhanced resistance against buckling.When compared against the Euler prediction, the FEM analysis shows a smaller deviation of just 1.06%.These findings emphasize the merits and precision of both theoretical and computational methodologies in assessing critical buckling loads.

Conclusions
Several key findings and further insights emerged from the present study of the critical buckling behaviour of laminated nanocomposite plates.The study emphasized the pivotal role of the carbon nanotubes (CNT's) volume fraction in enhancing the stability of the nanocomposite plates, with an increase in CNT volume consistently leading to improved resistance against buckling.Furthermore, environmental factors, particularly the temperature and moisture content, emerged as significant influencers, with both the rising temperature and moisture levels exhibiting a negative impact on the buckling load.
Within the present study, experimental, theoretical (Euler), and numerical (FEM) data have been employed.While the theoretical and FEM analyses were closely aligned, there were noticeable deviations as compared to the experimental results, which underlines the inherent complexities and limitations of each methodology.Additionally, our emphasis on mesh sensitivity in the FEM evaluation process underscored the critical nature of mesh generation for accurate results.Collectively, these findings have substantial implications for sectors that heavily rely on laminated nanocomposite materials.The insights provided can guide material scientists, structural engineers, and industry professionals in optimizing design choices concerning laminated nanocomposite plates, ensuring both performance and durability.

Figure 1 .
Figure 1.Representative model and buckling analysis of a [0 o /90 o /0 o ] laminated composite plate, with 200 x 200 x 20 dimensions (mm).(a) Front view of the partitioned model, with its applied uniaxial compressive load and the simply supported (SSSS) boundary conditions, and (b) Isometric view of the fundamental buckling mode.

Table 1 .
Convergence and comparison studies of normalized critical buckling of a simply supported rectangular plate, subjected to uniaxial compressive load with b/h=10 and various lamination schemes.

Table 2 .
Comparison study of normalized critical buckling of a simply supported laminated rectangular plate, subjected to uniaxial compressive load with b/h=10 and various modulus ratios.
[32][32], reports a critical buckling load of 3817 N for a 10-layered [0 o /90 o /±45 o /0 o /90 o /±45 o /0 o /90 o ] laminated composite plate, with 200 x 25 x 3.05 dimensions (mm).In contrast, the Euler-based theoretical estimation suggests a slightly diminished value of 3607 N.This deviation between empirical and theoretical values can arise from the inherent assumptions in Euler's calculations.Meanwhile, the employment of the Finite Element Method (FEM) yields an estimation close to the theoretical prediction, registering a critical buckling load of 3569 N. Comparatively, the FEM's output deviates from the experimental data by 6.49%, affirming the model's relative accuracy.
Finally, a comprehensive assessment of the critical buckling load based on the carbon nanotubes (CNT's) volume fraction by comparing experimental, theoretical (Euler), and FEA results is provided in Table3for clamped-clamped support conditions.For a CNT volume fraction set at 0.3 vol%, the experimental data

Table 3 .
Comparison study of the critical buckling load.