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NURBS-based finite element analysis of functionally graded plates: static bending, vibration, buckling and flutter
Navid Valizadeh, Sundararajan Natarajan, Octavio A Gonzalez-Estrada, Timon Rabczuk, Tinh Quoc Bui, Stephane PA Bordas
TL;DR
The paper addresses static and dynamic analysis of FGM plates while controlling shear locking in thin-plate finite elements. It applies a NURBS-based isogeometric formulation with FSDT and a modified shear correction, finding that increasing the gradient index increases static deflection but decreases vibration, buckling, and flutter measures.
Problem
Thin plates analyzed with lower-order NURBS functions suffer from shear locking, motivating a formulation for FGM plate response across multiple static and dynamic problems.
Method
The paper uses NURBS-based Bubnov–Galerkin isogeometric finite elements with FSDT and an artificial shear correction factor for lower-order NURBS functions.
Results
Increasing gradient index n increases static deflection while decreasing free flexural vibration, critical buckling load, and flutter frequency through reduced structural stiffness.
Takeaways & Limitations
The formulation studies static bending, mechanical and thermal buckling, vibration, and supersonic flutter of FGM plates within one NURBS-based framework.
Takeaways & Limitations
The modified shear correction factor is problem dependent, and the present study considers only simple geometries.
Abstract
from arXiv · showhide
In this paper, a non-uniform rational B-spline based iso-geometric finite element method is used to study the static and dynamic characteristics of functionally graded material (FGM) plates. The material properties are assumed to be graded only in the thickness direction and the effective properties are computed either using the rule of mixtures or by Mori-Tanaka homogenization scheme. The plate kinematics is based on the first order shear deformation plate theory (FSDT). The shear correction factors are evaluated employing the energy equivalence principle and a simple modification to the shear correction factor is presented to alleviate shear locking. Static bending, mechanical and thermal buckling, linear free flexural vibration and supersonic flutter analysis of FGM plates are numerically studied. The accuracy of the present formulation is validated against available three-dimensional solutions. A detailed numerical study is carried out to examine the influence of the gradient index, the plate aspect ratio and the plate thickness on the global response of functionally graded material plates.
1. Introduction
The paper develops a NURBS-based isogeometric finite element formulation for static and dynamic analysis of FGM plates, addressing shear locking in thin-plate applications. It examines bending, vibration, buckling, and flutter while relating the response to material gradation and geometry.
- FGMs provide a smooth material transition that can reduce inter-laminar shear stresses and delamination relative to laminated composites.
- Existing FGM plate analyses use finite-element and meshfree methods with several plate theories, including FSDT and higher-order shear-deformation theories.
- Thin plates remain vulnerable to shear locking, motivating techniques based on selective integration, mixed interpolation, shear-gap methods, strain smoothing, and p-adaptivity.
- Prior work covers FGM vibration, thermal response, buckling, postbuckling, and supersonic flutter using analytical, finite-element, meshfree, and enriched methods.
- This study investigates NURBS-based isogeometric finite elements for static and dynamic Reissner–Mindlin plates and introduces an artificial shear correction factor for lower-order NURBS functions.
- The proposed correction is problem dependent, while higher-order NURBS functions have also been reported as a route to suppress shear locking.
2. Theoretical Formulation
The formulation models ceramic–metal FGM plates with thickness-wise gradation, Reissner–Mindlin kinematics, and NURBS-based finite-element discretization. It derives stiffness, inertia, thermal, buckling, vibration, and aerodynamic quantities for the targeted analyses.
- 2.1. Functionally graded material: The ceramic-rich top surface grades through the thickness to a metal-rich bottom surface according to a power-law distribution.
- 2.1. Functionally graded material: FGM effective properties are evaluated using either the rule of mixtures or Mori–Tanaka homogenization, with density computed by the rule of mixtures.
- 2.1. Functionally graded material: The gradient index n controls composition: n = 0 represents a fully ceramic plate, while n = 1 gives linear ceramic–metal variation.
- Temperature distribution through the thickness: The temperature field varies only through the thickness and is obtained from a steady-state heat-transfer problem with prescribed ceramic- and metal-surface temperatures.
- 2.2. Reissner-Mindlin Plates: Reissner–Mindlin kinematics uses mid-plane displacements and independent rotations, with membrane, bending, and transverse-shear strains entering the constitutive relations.
- 2.2. Reissner-Mindlin Plates: Thickness-integrated material and shear distributions define extensional, coupling, bending, thermal, mass, and transverse-shear stiffness quantities.
- 2.2. Reissner-Mindlin Plates: The finite-element equations are obtained from strain and kinetic energies, Lagrange equations, and Galerkin discretization using linear and geometric stiffness matrices.
- 2.2. Reissner-Mindlin Plates: Flutter analysis combines the stiffness, consistent mass, and aerodynamic matrices; the critical parameter is defined by the first eigenvalue coalescence.
3. Non-Uniform Rational B-Splines
The finite-element approximation uses NURBS basis functions that represent both geometry and displacement fields, preserving exact geometry and allowing adjustable continuity. NURBS curves and surfaces are built from knot vectors, control points, degrees, and weights.
- NURBS basis functions are used for finite-element approximation, while the same functions represent geometry exactly.
- The illustrated non-uniform rational B-spline curve has order 3 and uses the knot vector Ξ = {0, 0, 0, 0, 1/3, 1/3, 1/3, 1/2, 2/3, 1, 1, 1, 1}.
- NURBS continuity can be tailored to the problem, and surfaces are formed by tensor products over parametric dimensions ξ and η.
- The construction uses knot vectors, control points, curve degree, weights, and B-spline basis functions over a control-point net.
- A NURBS surface combines the bidirectional control net and tensor-product basis functions through a weighting function.
- The displacement field is approximated over the control mesh using nodal displacements and rotations multiplied by NURBS basis functions.
4. Shear Locking
Mindlin theory permits transverse shear deformation, but thin plates require vanishing shear strain as thickness approaches zero to avoid locking.
- 4. Shear Locking: Mindlin theory allows plate normals to remain straight while becoming non-normal after deformation.This relaxes continuity requirements on the assumed displacement fields.
- 4. Shear Locking: Thin-plate formulations must ensure that transverse shear strain vanishes as thickness approaches zero.The thin-plate limit imposes this condition on the displacement field.
Artificial shear correction factor
The formulation modifies the shear correction factor to suppress locking in lower-order NURBS elements and evaluates the base factor through energy equivalence.
- Artificial shear correction factor: Lower-order NURBS basis functions suffer from shear locking when applied to thin plates.The problem motivates stabilization of the shear response.
- Artificial shear correction factor: The paper applies Kikuchi and Ishii’s artificial shear correction technique to lower-order NURBS basis functions.The modified factor is introduced specifically to suppress the shear-locking syndrome.
- Artificial shear correction factor: The modified factor depends on positive integers n and β, element diameter le, element thickness h, and shear correction factor υ.Here le is the maximum or diagonal element length.
- Artificial shear correction factor: The underlying shear correction factor is obtained using the energy equivalence principle.This factor is used together with the locking-alleviation modification.
- Artificial shear correction factor: The numerical study examines material gradient index, plate skewness, aspect ratio, thickness, and boundary conditions through the global response.The FGM plate has a ceramic-rich top surface and metal-rich bottom surface.
Skew boundary transformation
The study transforms skew-boundary degrees of freedom into local coordinates for boundary-condition enforcement and evaluates NURBS-based FGM plate responses across meshes and parameters.
- Skew boundary transformation: Boundary conditions on skew edges are specified in a local coordinate system because skew-edge boundaries are not parallel to global axes.The generalized displacement vector is transformed between global and local coordinates.
- Skew boundary transformation: The nodal transformation matrix provides the mapping needed to impose boundary conditions on skew boundaries.The relation applies to generalized displacement vectors at boundary nodes.
- Skew boundary transformation: The convergence of quadratic, cubic, and quartic NURBS results is quite fast for simply supported and clamped plates.For cubic and quartic elements, convergence is almost achieved with 16 control points per side.
- Skew boundary transformation: The present formulation shows very good agreement with other approaches for normalized center deflection.The comparison uses results available in the literature.
- Skew boundary transformation: Standard IGA experiences shear locking for thin plates with a/h > 100, whereas S-IGA results are nearly independent of length-to-thickness ratio.S-IGA results agree very well with the NS-DSG3 element.
- Skew boundary transformation: For thermo-mechanical loading, normalized stresses agree well with published results, while central deflection increases linearly with load and with gradient index.Metallic plates have the largest deflection and ceramic plates the lowest.
- Skew boundary transformation: For isotropic plates, axial stress varies linearly through thickness; for FGM plates, it is nonlinear and smaller at the bottom than at the top.The maximum compressive top-surface stress occurs for the FGM plate with n = 2.
FGM plates subjected to thermal loading
Thermal loading produces FGM plate deflections that differ from purely mechanical responses and from isotropic plates, while the IGA results agree well with published solutions. The response varies systematically with gradient index, temperature, and load.
- Validation: The thermal-loading results agree very well with available results from element-free kp-Ritz and NS-DSG3 methods.The paper reports matching results for the thermal deflection problem.
- Thermal loading: The metal plate gives the maximum thermal deflection because of its high thermal conductivity.The cited study varies the top-surface temperature from 0°C to 500°C while maintaining the bottom surface at 20°C.
- Thermal loading: FGM plate deflections are much lower than those of isotropic plates, indicating high-temperature resistance behaviour.This comparison is explicitly reported for the thermal-loading response.
- Thermal loading: Thermal loading produces center deflections that differ completely from those under purely mechanical loading.The comparison is made for the same FGM plate response under thermal and mechanical cases.
- Thermo-mechanical loading: Under thermo-mechanical loading, center deflection increases linearly with load, with the metallic phase showing the largest change and the ceramic phase the least.The response is reported for a plate with a ceramic-rich top surface at 300°C and a metal-rich bottom surface at 20°C.
- Mechanical loading: For mechanical loading, the maximum compressive stress at the top surface occurs at gradient index n = 1.The metallic or ceramic plate shows the minimum tensile stress at the bottom surface.
Skew plates
For skew FGM plates, axial stress increases as the skew angle decreases, and the same trend appears across multiple gradient indices. The isogeometric results agree with element-free and discrete-shear-gap formulations.
- Mechanical response: Axial stresses increase as the skew angle decreases for a gradient index n = 0.5.The stresses are reported through the plate thickness under a uniform mechanical load.
- Validation: The isogeometric analysis agrees well with element-free kp-Ritz, NS-DSG3, and ES-DSG3 results.The comparison concerns the axial-stress distributions of skew FGM plates.
- Validation: The formulation is validated against available analytical and numerical solutions before the detailed vibration study.The paper presents non-dimensionalized free-flexural frequencies for this comparison.
Square plates
For square FGM plates, the NURBS-based formulation accurately reproduces normalized vibration results across aspect ratios and boundary conditions. Frequencies decrease with gradient index, while clamping raises them relative to simple supports.
- Frequency comparison: IGA results are superior to other methods irrespective of the material gradient index.The comparison uses quadratic NURBS elements with 14 control points per side for a/h ratios of 5, 10, and 20.
- Boundary conditions: IGA gives higher frequencies than the element-free kp-Ritz method for simply supported and clamped boundary conditions.The comparison concerns the first two non-dimensionalized frequencies.
- Material and geometry effects: Increasing the gradient index or decreasing the skew angle decreases the natural frequency.The paper attributes the reduction to stiffness degradation, with increased metallic volume fraction contributing to the degradation.
Mechanical Buckling
Mechanical and thermal buckling, together with flutter, are analyzed for FGM plates using the isogeometric formulation. Increasing the gradient index generally reduces stability and aerodynamic pressure, while boundary conditions and temperature profiles also affect critical values.
- Mechanical buckling: Increasing the gradient index decreases the critical mechanical buckling load, with the reduction significant for n ≤ 2 and smaller thereafter.The trend is reported for mechanical loading across skew angles and plate configurations.
- Thermal buckling: Thermal stability increases with skew angle, while increasing gradient index decreases the critical buckling load through stiffness degradation.The degradation is attributed to increased metallic volume fraction.
- Thermal buckling: Nonlinear temperature variation through the thickness yields higher critical values than a linear temperature distribution.Both linear and nonlinear temperature rises are considered for simply supported FGM skew plates.
- Flutter validation: The formulation is validated against available results for critical aerodynamic pressure and critical frequency, including isotropic plates with and without cracks.The validation includes multiple boundary conditions.
- Flutter: Clamped plates have higher critical aerodynamic pressure than simply supported plates.The comparison is made for square Aluminum-Alumina FGM plates with a/h = 100.
- Flutter: Critical aerodynamic pressure decreases as the gradient index increases, with the decrease fastest at low gradient-index values.The paper relates this trend to the higher stiffness of ceramic plates and lower stiffness of metallic plates.
6. Conclusions
The paper applies a NURBS-based iso-geometric finite element formulation with FSDT to static and dynamic FGM plate responses. Numerical studies examine gradient index, aspect ratio, and thickness effects, while modified shear correction alleviates shear locking in thin plates.
- 6. Conclusions: The formulation uses NURBS-based Bubnov-Galerkin iso-geometric finite elements with first-order shear deformation plate kinematics.
- 6. Conclusions: NURBS basis functions exactly represent geometry and can extend the formulation from simple to complex geometries by changing geometry information.
- 6. Conclusions: A modified shear correction factor alleviates shear locking when the formulation is applied to thin plates.
- 6. Conclusions: Numerical experiments examine the influence of gradient index, plate aspect ratio, and plate thickness on the global response of FGM plates.
- 6. Conclusions: With increasing gradient index n, static deflection increases, whereas free flexural vibration, critical buckling load, and flutter frequency decrease.The decrease is attributed to reduced structural stiffness from increased metallic volume fraction.