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Application of higher-order structural theory to bending and free vibration analysis of sandwich plates with CNT reinforced composite facesheets
Sundararajan Natarajan, Mohamed Haboussi, Ganapathi Manickam
TL;DR
The paper addresses limited higher-order modeling of CNT-reinforced sandwich plates, whose responses involve through-thickness displacement variation and interface slope discontinuities. It develops a shear-flexible higher-order finite-element formulation and finds that model choice, CNT fraction, distribution, and plate thickness substantially affect deflection, stresses, and frequencies.
Problem
Higher-order theories accounting for through-thickness displacement variation, interface slope discontinuity, and thickness stretch had not been exploited for CNT-reinforced sandwich structures, whose literature remained limited.
Method
A QUAD-8 shear-flexible finite element based on higher-order structural theory analyzes static deflection and free vibration of sandwich plates with homogeneous cores and CNT-reinforced facesheets.
Results
HSDT13 accurately evaluates global and local displacements and stresses; CNT increases generally decrease deflection, while natural frequencies vary significantly across plate theories.
Takeaways & Limitations
Higher-order model selection is important for representing global and local responses, particularly for thick plates and interface stress behavior.
Takeaways & Limitations
The numerical studies assume PMMA matrices, single-walled CNTs, specified efficiency parameters, and Ti-6Al-4V homogeneous cores unless otherwise stated.
Abstract
from arXiv · showhide
In this paper, the bending and free flexural vibration behaviour of sandwich plates with carbon nanotube (CNT) reinforced facesheets are investigated using QUAD-8 shear flexible element developed based on higher-order structural theory. This theory accounts for the realistic variation of the displacements through the thickness, and the possible discontinuity in slope at the interface, and the thickness stretch affecting the transverse deflection. The in-plane and rotary inertia terms are considered in the formulation. The governing equations obtained using Lagrange's equation of motions are solved for static and dynamic analyses considering a sandwich plate with homogeneous core and CNT reinforced face sheets. The accuracy of the present formulation is tested considering the problems for which solutions are available. A detailed numerical study is carried out based on various higher-order models deduced from the present theory to examine the influence of the volume fraction of the CNT, core-to-face sheet thickness and the plate thickness ratio on the global/local response of different sandwich plates.
1. Introduction
The paper addresses limited higher-order analysis of CNT-reinforced sandwich plates, developing and evaluating a shear-flexible finite-element formulation for static and vibration responses.
- Motivation: Sandwich structures combine low specific weight with high bending rigidity and favorable vibration and fatigue properties.Their response depends on bonding and the selected core and facesheet materials.
- Motivation: CNTs are promising reinforcements for sandwich facesheets because experiments report mechanical properties exceeding those of carbon fibers.
- Research gap: Prior studies widely used first-order shear deformation theory, while higher-order models accounting for through-thickness in-plane variation were less common for CNT-reinforced plates.
- Research gap: Available research on sandwich structures with CNT-reinforced facesheets was limited, especially for theories including slope discontinuity and thickness stretch.A layerwise theory could represent these effects but may become computationally expensive as the number of layers increases.
- Approach: The study employs a higher-order 8-noded quadrilateral plate element to analyze static deflection and free vibration in thick and thin CNT-reinforced sandwich plates.The formulation is assessed for mechanical and thermal static loading and free vibration using structural models derived from the higher-order theory.
- Paper organization: The paper computes CNT-reinforced composite properties, formulates higher-order kinematics, describes the finite element, and presents static and dynamic numerical results.
2. Theoretical Formulation
The formulation models a three-layer sandwich with a homogeneous core and CNT-reinforced facesheets, using graded CNT distributions and effective-property estimates for the constituent materials.
- Geometry and coordinates: A Cartesian coordinate system uses x and y in the plate plane and z through the thickness, with the origin at a corner on the middle plane.
- Material model: The CNT-reinforced layers use single-walled CNTs with a thickness-direction graded distribution in an isotropic matrix.
- Material model: Effective CNT-composite properties are estimated with a corrected rule of mixtures, using CNT efficiency parameters to represent imperfect CNT–matrix load transfer.The CNT and matrix volume fractions satisfy VCN + Vm = 1.
- Sandwich architecture: The sandwich consists of a homogeneous core of thickness hH between two CNT-reinforced facesheets of thickness hf.The core-to-facesheet thickness ratio is hH/hf.
- Thermal properties: The formulation includes CNT and matrix thermal expansion coefficients in the longitudinal and transverse directions.
- Geometry and coordinates: Figure 1 distinguishes single-layer and three-layer sandwich cross sections and indicates functionally graded CNTs in the facesheets.
3. Higher order accurate theory
The higher-order theory represents layerwise displacement fields with polynomial terms and a zigzag function, then derives static and free-vibration equations through Lagrangian mechanics and finite-element discretization.
- Displacement kinematics: Each layer uses higher-order in-plane and transverse displacement expansions containing polynomial terms and layer-dependent functions.The formulation includes separate displacement components for bending, membrane, shear, and stretching behavior.
- Displacement kinematics: Even powers in in-plane displacements and odd powers in transverse displacement represent stretching, whereas odd and even powers represent flexure.
- Interface behavior: The piecewise-linear zigzag function captures slope discontinuities in in-plane displacements at interfaces more economically than a discrete layer approach.Its alternating interface values represent the interface behavior observed in exact three-dimensional elasticity solutions for thick sandwich functionally graded materials.
- Strain and constitutive relations: The theory separates bending and membrane strains from transverse shear strains and applies constitutive relations to each layer.
- Governing equations: Lagrange’s equations produce governing equations for static deflection and free vibration, with inertia terms omitted for the static case.The external loading is represented by a distributed force q on the plate’s top surface.
- Numerical solution: The finite-element implementation uses higher-order Gaussian quadrature through the thickness and a 3×3 Gauss rule in the in-plane directions.Natural frequencies and mode shapes are obtained with a standard generalized eigenvalue algorithm.
4. Element description
The element description uses a continuous eight-noded serendipity quadrilateral shear-flexible element with 13 degrees of freedom and derives alternate models by removing selected higher-order variables.
- Primary element: The HSDT13 element is a continuous eight-noded serendipity quadrilateral shear-flexible plate element with 13 degrees of freedom per node.Its degrees of freedom include mid-surface displacements, rotations, stretching, higher-order, and zigzag variables.
- Alternate models: Five alternate discrete models are obtained from the original element by deleting appropriate degrees of freedom.These models are listed in Table 1 for comparison.
5. Numerical results and discussion
The numerical study evaluates static and free-vibration responses of simply supported sandwich plates with homogeneous cores and CNT-reinforced facesheets across structural models, geometry, loading, temperature, and CNT parameters. Higher-order theories capture through-thickness effects that lower-order models omit, while the formulation agrees well with available results.
- Study setup: The study considers simply supported plates with core-to-facesheet ratios hH/hf = 2, 6 and thickness ratios a/h = 5, 10, using PMMA/CNT facesheets and a Ti-6Al-4V core.Mechanical and thermal loading are examined for static response, together with free flexural vibration.
- Validation: The formulation is validated by convergence studies and literature comparisons, with an 8 × 8 mesh reported as adequate for the sandwich-plate frequency calculations.The reported static results are in excellent agreement with literature, and the frequency results are in very good agreement with available solutions.
- Static response: FG-V reinforcement produces the maximum center deflection, whereas FG-X produces the minimum center deflection.This comparison concerns the CNT distribution type in the sandwich plate static response.
- Static response: Increasing CNT volume fraction decreases nondimensionalized displacement, while the stress change depends on the thickness ratio.Increasing core thickness generally decreases both nondimensionalized displacements and stresses, attributed to increased flexural stiffness and CNT reinforcement.
- Structural-model effects: Higher-order models predict different neutral-surface responses because HSDT13, HSDT11A, and HSDT11B include through-thickness stretching effects absent from HSDT9, TSDT7, and FSDT5.Through-thickness displacement and stress profiles also differ substantially among plate theories, particularly when zig-zag functions represent interface slope discontinuities.
- Free vibration: The nondimensionalized natural frequency increases with CNT volume fraction and core-to-facesheet thickness, but decreases with increasing temperature and plate thickness.HSDT13 and HSDT11A show comparable performance in the frequency study, as in the static bending case.
- Free vibration: Flexural modes exhibit nonuniform transverse displacement through the thickness, indicating normal stresses in the thickness direction at the selected locations.The mode-shape study uses HSDT13 for square simply supported plates with hH/hf = 2 and a/h = 5.
6. Conclusions
The study finds that higher-order models capture both global and through-thickness responses of CNT-reinforced sandwich plates, with HSDT13 offering the broadest accuracy. CNT content, temperature, thickness, and distribution materially affect deflection, stresses, and natural frequencies.
- HSDT11A and HSDT11B approximate global responses under mechanical and thermal loading, but differ in through-thickness shear-stress predictions.The models therefore agree more closely on overall behavior than on local stress variation.
- HSDT13 accurately evaluates both global and local displacements and stresses, outperforming lower theories particularly for thick plates.Its advantage is most pronounced when through-thickness and local response characteristics matter.
- In-plane stresses vary nonlinearly and exhibit discontinuities at layer interfaces, while flexural and extensional mode types depend on structural location.Mode shapes can also demonstrate normal stress through the thickness.
- Increasing CNT volume fraction generally decreases deflection, whereas temperature variation noticeably changes stress distributions relative to mechanical loading.These trends show that reinforcement and loading environment influence different aspects of the structural response.
- Natural frequencies vary significantly among plate theories, and for thick plates changing CNT distribution from UD to FGX lowers nondimensionalized frequency in HSDT13 and HSDT11A.The frequency prediction is therefore sensitive to both the structural model and CNT distribution.