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Bending and vibration of functionally graded material sandwich plates using an accurate theory
S. Natarajan, M. Ganapathi
TL;DR
The paper investigates bending and free flexural vibration in sandwich FGM plates, addressing through-thickness displacement variation and limitations of lower-order theories. It develops a higher-order QUAD-8 shear-flexible formulation for two sandwich configurations, validates it against available three-dimensional elasticity solutions, and studies gradient index and aspect-ratio effects. The HSDT13 model predicts accurate results for any sandwich construction, whereas other models depend on sandwich type and loading.
Problem
Existing sandwich FGM analyses mainly use lower-order theories, while theories capturing through-thickness displacement variation, interface slope discontinuity, and thickness stretch remain underused.
Method
The study uses a higher-order QUAD-8 shear-flexible element to analyze two sandwich FGM configurations, including in-plane and rotary inertia for vibration.
Results
HSDT13 predicts accurate results for any sandwich construction, while other models depend on sandwich type and loading situation.
Takeaways & Limitations
Higher-order models can represent the physical behavior and through-thickness variations of sandwich FGM plates across gradient indices, sandwich types, and thickness ratios.
Takeaways & Limitations
The numerical study considers only simply supported boundary conditions.
Abstract
from arXiv · showhide
In this paper, the bending and the free flexural vibration behaviour of sandwich functionally graded material (FGM) plates 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. The governing equations obtained here are solved for static analysis considering two types of sandwich FGM plates, viz., homogeneous face sheets with FGM core and FGM face sheets with homogeneous hard core. The in-plane and rotary inertia terms are considered for vibration studies. The accuracy of the present formulation is tested considering the problems for which three-dimensional elasticity solutions are available. A detailed numerical study is carried out based on various higher-order models to examine the influence of the gradient index and the plate aspect ratio on the global/local response of different sandwich FGM plates.
1 Introduction
The introduction motivates FGM sandwich plates as aerospace structures combining graded material properties with sandwich construction, while identifying limitations in commonly used theories.
- FGMs mix ceramics and metals with smooth, continuous property variation, helping avoid interface debonding associated with discrete material mismatches.
- Sandwich structures are widely used in aerospace vehicles for bending rigidity, low specific weight, vibration characteristics, and fatigue properties.
- Conventional sandwich FGM analyses have extensively used first-, third-order, and sinusoidal shear deformation theories.
- The paper addresses theories that account for through-thickness in-plane displacement variation, interface slope discontinuity, and thickness stretch affecting transverse deflection.
- The study presents an eight-noded shear-flexible quadrilateral element for static deflection and free vibration of thick and thin sandwich FGM plates.
2 Theoretical Formulation
The formulation models sandwich FGM plates with thickness-dependent material properties and higher-order displacement fields, then derives static and vibration equations using energy and finite-element procedures.
- 2.1 Functionally graded material plate: Material properties are graded through the thickness using a power-law distribution and homogenized with the rule of mixtures.
- Type A - FGM face sheet and homogeneous hard core: The model considers Type A plates with FGM face sheets, a homogeneous ceramic hard core, and metal-rich top and bottom surfaces.
- Type B - Homogeneous face sheet and FGM core: Type B plates contain homogeneous face sheets and an FGM core whose top surface is metal rich and bottom surface ceramic rich.
- 2.2 Plate formulation: Each sandwich plate comprises three discrete layers whose material properties follow the corresponding FGM power law.
- 2.2 Plate formulation: Higher-order expansions represent in-plane and transverse displacements through thickness, separating stretching and flexural contributions.
- 2.2 Plate formulation: A piecewise-linear zig-zag function captures interface slope discontinuities in in-plane displacements without a discrete-layer approach.
- 2.2 Plate formulation: The formulation defines membrane, bending, and transverse-shear strains, relates them through layer constitutive matrices, and derives motion equations from Lagrangian energy expressions.
- Free vibration: Finite-element integration uses higher-order Gaussian quadrature through thickness and 3 × 3 Gauss integration in-plane; generalized eigenanalysis yields frequencies and mode shapes.
3 Element description
The paper uses an eight-noded shear-flexible quadrilateral element based on higher-order kinematics for sandwich FGM plate analysis.
- The Q8-HSDT13 element is a serendipity quadrilateral shear-flexible plate element with 13 nodal degrees of freedom.Its kinematics use cubic variation through the plate thickness.
- The element is reported to avoid locking and spurious energy modes while passing the patch test and converging rapidly.
- Table 1 is identified as presenting alternate eight-noded finite element models.
4 Numerical results and discussion
Numerical analyses evaluate static and free-vibration responses of sandwich FGM plates with an eight-noded shear-flexible element across loading, geometry, material-gradation, and sandwich configurations. The formulation is compared with three-dimensional elasticity solutions and used to examine higher-order model behavior.
- Analysis setup: Simply supported analyses use 1-1-1, 1-2-1, and 2-2-1 sandwich thickness configurations, a/h values of 5, 10, and 100, and gradient indices n of 0, 0.5, 1, and 5.Both Type A and Type B sandwich FGM plates are considered, using Alumina–Aluminum constituent properties and constant Poisson’s ratio ν = 0.3.
- Static analysis: The static study considers mechanical and thermal loading for Type A FGM sandwich plates, with responses reported through nondimensionalized physical quantities.An 8 × 8 mesh is found adequate, and the formulation is validated against available three-dimensional elasticity solutions for an Al/SiC graded square plate.
- Static analysis: The present formulation gives displacements and stresses in good agreement with three-dimensional elasticity solutions for the mechanically loaded Al/SiC functionally graded square plate.The comparison uses effective properties based on the Mori–Tanaka homogenization scheme and HSDT11 results at gradient index n = 1.
- Static analysis: Higher- and lower-order models produce nearly similar mechanical-loading responses because smoothly varying FGM properties do not create visible differences in the evaluated through-thickness results.Their main distinctions concern slope discontinuities and polynomial variation in in-plane displacements, while thermal loading produces significantly different stress variations because the thermal-expansion coefficient varies through the thickness.
- Free flexural vibrations: The free-vibration formulation defines a nondimensionalized flexural frequency and is validated against available three-dimensional elasticity solutions using an 8 × 8 mesh.The convergence study reports frequency parameters for simply supported square Type A FGM sandwich plates.
- Free flexural vibrations: Fundamental frequencies decrease with increasing gradient index for Type A plates but decrease with decreasing gradient index for Type B plates.The paper attributes these trends to changes in metallic volume fraction and the resulting material rigidity; frequency studies cover thickness ratios, core thicknesses, and the first six modes.
5 Conclusion
The study evaluates higher-order models for bending and free vibration of sandwich FGM plates across material, construction, and thickness parameters. HSDT13 is reported as accurate across sandwich constructions, while other models depend on plate type and loading.
- HSDT13 predicts accurate results for any type of sandwich construction.
- Other higher-order models depend on the sandwich plate type and loading situation.
- The analyses vary the material gradient index, sandwich type, and thickness ratio.
- Figures 5 and 6 show six-mode deflected shapes for square, simply supported 1-2-1 Type A FGM plates with n = 1 and a/h = 5.