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Isogeometric analysis for functionally graded microplates based on modified couple stress theory
Hoang X. Nguyen, Tuan N. Nguyen, M. Abdel-Wahab, S. P. A. Bordas, H. Nguyen-Xuan, Thuc P. Vo
TL;DR
The paper addresses the need to model size-dependent bending, vibration, and buckling of functionally graded microplates with suitable shear and thickness-deformation descriptions. It combines modified couple stress theory, four-variable refined and quasi-3D theories, and NURBS-based isogeometric analysis. Convergence and verification studies support the validity and efficiency of the approach, which is applied across plate geometries and boundary conditions.
Problem
Small-scale functionally graded microplates require modeling that captures size-dependent effects, shear deformation, and thickness stretching while handling higher-order continuity requirements.
Method
The approach combines one-parameter modified couple stress theory, novel seventh-order four-variable refined and quasi-3D theories, and NURBS-based isogeometric analysis.
Results
Convergence and verification studies show the proposed computational approach is valid and efficient compared with existing literature results.
Takeaways & Limitations
The framework is applied to static bending, free vibration, and buckling of rectangular and circular functionally graded microplates with varied boundary conditions.
Takeaways & Limitations
The functionally graded material model acknowledges that the rule of mixtures fails to describe interactions between material phases, motivating the Mori–Tanaka scheme.
Abstract
from arXiv · showhide
Analysis of static bending, free vibration and buckling behaviours of functionally graded microplates is investigated in this study. The main idea is to use the isogeometric analysis in associated with novel four-variable refined plate theory and quasi-3D theory. More importantly, the modified couple stress theory with only one material length scale parameter is employed to effectively capture the size-dependent effects within the microplates. Meanwhile, the quasi-3D theory which is constructed from a novel seventh-order shear deformation refined plate theory with four unknowns is able to consider both shear deformations and thickness stretching effect without requiring shear correction factors. The NURBS-based isogeometric analysis is integrated to exactly describe the geometry and approximately calculate the unknown fields with higher-order derivative and continuity requirements. The convergence and verification show the validity and efficiency of this proposed computational approach in comparison with those existing in the literature. It is further applied to study the static bending, free vibration and buckling responses of rectangular and circular functionally graded microplates with various types of boundary conditions. A number of investigations are also conducted to illustrate the effects of the material length scale, material index, and length-to-thickness ratios on the responses of the microplates.
1. Introduction
The paper addresses computational modeling of functionally graded microplates, where small-scale applications require accurate yet tractable predictions. It combines modified couple stress theory, refined and quasi-3D plate theories, and NURBS-based isogeometric analysis to study bending, vibration, and buckling.
- Motivation: Micro- and nano-scale applications motivate models for predicting microstructure behaviour, but experiments are difficult and simulations can be computationally expensive.The paper identifies the need for mathematical and mechanical models suited to small-scale structures.
- Limitations of existing plate theories: Classical plate theory neglects shear deformation and is mainly suitable for thin plates, whereas first-order shear deformation theory also applies to moderately thick plates.FSDT nevertheless produces inaccurate shear stresses and violates traction-free conditions without shear correction factors.
- Computational approach: Higher-order and refined plate theories avoid shear correction factors, while continuity challenges motivate isogeometric analysis.IGA uses the same basis functions to describe CAD geometry and approximate unknown fields, addressing C1-continuity requirements without extra variables.
- Paper contribution: The study investigates bending, free vibration, and buckling of functionally graded microplates using modified couple stress theory with four-variable refined and quasi-3D theories.NURBS functions describe geometry boundaries and construct the approximation basis.
- Study scope: Numerical examples cover rectangular and circular functionally graded microplates with varied boundary conditions.The paper reports these analyses as part of its numerical study and conclusion structure.
2. A novel theory for FG microplates
The paper develops a four-variable quasi-3D and refined-plate framework for functionally graded microplates, combining higher-order thickness functions with modified couple stress theory. NURBS-based isogeometric analysis then provides the computational formulation for static, vibration, and buckling problems.
- Modified couple stress theory: Modified couple stress theory represents size-dependent effects using one material length scale parameter and a symmetric deviatoric couple stress tensor.The strain energy formulation uses classical strain and symmetric curvature measures.
- Functionally graded material: Functionally graded material properties are modeled from metal and ceramic phases whose volume fractions vary through the thickness according to material index n.The rule of mixtures and Mori–Tanaka scheme are used to estimate effective properties.
- Refined plate theory: The refined plate displacement field separates bending and shear transverse displacements and imposes zero tangential shear values at the plate surfaces.This construction satisfies traction-free surface conditions without a shear correction factor.
- Quasi-3D theory: The quasi-3D theory adds thickness stretching to transverse shear deformation through a thickness-dependent transverse displacement.The refined theory is recovered by replacing the quasi-3D thickness functions with g(z) and 1.
- Seventh-order plate theory: A novel seventh-order thickness function is proposed for the four-variable refined plate and quasi-3D theories.The proposed functions are presented alongside existing distributions for higher-order and refined plate formulations.
- Governing formulation: The constitutive and weak formulations derive static bending, vibration, and buckling equations from the displacement, strain, curvature, and material matrices.Hamilton’s principle and weak formulation are used for the governing problem forms.
3. FG microplate formulation based on NURBS basis functions
The formulation uses NURBS basis functions for geometry and field approximation, supporting the higher-order continuity needed for refined microplate analysis. The resulting approximation is assembled into global equations for static bending, free vibration, and buckling.
- 3. FG microplate formulation based on NURBS basis functions: NURBS basis functions are introduced as the basis for isogeometric analysis and the formulation of couple-stress microplate responses.The formulation addresses static bending, free vibration, and buckling using refined plate and quasi-3D theories.
- 3.1. B-splines and NURBS basis functions: The knot vector Ξ is non-decreasing, with n basis functions and polynomial order p defining the B-spline construction and element domains.Knot vectors may be uniform or open, with open vectors repeating the first and last knots p + 1 times.
- 3.1. B-splines and NURBS basis functions: For p ≥2, B-spline basis functions have C1 continuity across single knots, and two-dimensional functions are formed by tensor products of two knot vectors.The second knot vector H supplies the second parametric dimension.
- 3.1. B-splines and NURBS basis functions: The illustrated one- and two-dimensional B-splines use specified knot vectors, after which NURBS functions assign an additional weight ζA to each control point.The one-dimensional vector Ξ is combined with H for the two-dimensional construction.
- 3.1. B-splines and NURBS basis functions: B-splines are a special case of NURBS obtained when all control-point weights are assigned the same constant.Equal individual weights reduce the NURBS function to a B-spline function.
- 3.2. A novel NURBS-based formulation of modified couple stress theory: The displacement field is approximated by NURBS basis functions using control-point degrees of freedom qA = [u0A v0A wbA wsA]T.The number of basis functions is n × m, and each control point carries the corresponding displacement variables.
- 3.2. A novel NURBS-based formulation of modified couple stress theory: Substituting the displacement approximation into the strain-displacement and curvature relations yields strains and curvatures for global equilibrium equations.The global stiffness matrix combines classical and couple-stress contributions as K = Ks+Kc.
R1 R2 R3
The formulation defines geometric stiffness for buckling and uses the distribution function f(z) to avoid shear correction factors. Because second-order derivatives are required, the approximation must provide C1-continuity.
- R1 R2 R3: The geometric stiffness matrix is used in the formulation, while ω and λcr denote the natural frequency and critical buckling value, respectively.These quantities distinguish the vibration and buckling response measures.
- R1 R2 R3: Introducing f(z) allows the refined plate and quasi-3D theories to describe transverse shear stresses satisfying traction-free conditions without shear correction factors.This avoids the correction factors commonly required in first-order shear deformation theory.
- R1 R2 R3: The formulation uses second-order derivatives of NURBS approximation functions, requiring C1-continuity that is naturally supported by isogeometric analysis.In finite element analysis, satisfying this requirement can require more variables and increase computational cost.
4. Numerical examples and discussion
Numerical studies verify the proposed NURBS-based IGA approach and examine static bending, free vibration, and buckling of rectangular and circular FG microplates. Results generally agree with published or benchmark solutions, while material length scale, material index, geometry, thickness, and boundary conditions influence responses.
- Convergence and verification studies: An 11 × 11 cubic (p = 3) NURBS mesh is sufficient for the analyzed convergence cases.This mesh is used in subsequent examples unless otherwise specified.
- Convergence and verification studies: The proposed RPT and quasi-3D results show good agreement with published 2-D and quasi-3D solutions for Al/Al2O3 FG plates.The verification considers uniformly and sinusoidally distributed loads without couple-stress effects.
- Static bending analysis: Increasing material index n raises central deflection, whereas increasing material length scale ratio l/h reduces displacement by increasing plate stiffness.These trends are reported for CCCC square Al/Al2O3 microplates under sinusoidal and uniform loads.
- Free vibration analysis: Higher material length scale ratios increase natural frequencies, while RPT and quasi-3D predictions differ slightly because quasi-3D includes thickness stretching.RPT agrees especially well with TSDT for thinner plates; discrepancies grow mainly for thick plates as l/h approaches 1.
- Free vibration analysis: The circular-plate vibration results agree very well with existing theories and provide benchmark examples where modified-couple-stress results were previously unavailable.The study reports fundamental and first-six natural frequencies for simple and clamped supports.
- Buckling analysis: For buckling, shear-deformable RPT and quasi-3D predictions generally agree with reference theories, whereas CPT differs substantially for thick plates.The quasi-3D model produces results similar to FSDT and RPT while accounting for normal deformation.
- Buckling analysis: Quasi-3D and RPT buckling predictions are close for circular microplates in several boundary-condition cases, but can differ markedly for CCCC square plates.The square-plate discrepancy is attributed to the quasi-3D theories’ inclusion of normal deformation.
5. Conclusions
The study presents a unified computational approach for analyzing FG microplate bending, vibration, and buckling across geometric domains and boundary conditions. It combines modified couple stress theory, four-variable refined and quasi-3D theories, and NURBS-based IGA.
- Conclusions: The approach combines modified couple stress theory, four-variable refined and quasi-3D plate theories, and NURBS-based isogeometric analysis.It is applied to static bending, free vibration, and buckling of FG microplates with varied geometries and boundary conditions.
- Conclusions: The modified couple stress model uses one material length scale, while the quasi-3D theory captures shear deformation and thickness stretching with four unknowns.The quasi-3D formulation avoids shear correction factors.
- Conclusions: NURBS-based IGA exactly describes geometry and supports the higher-order derivative and continuity requirements of the formulation.
q0L4 w (a/2, a/2, 0) of
The paper presents tables and figures covering deflection, natural frequencies, buckling loads, geometry, meshes, and mode shapes for functionally graded microplates.
- Tables compare non-dimensional deflection for square microplates under different loading and boundary-condition configurations.
- Tables report comparisons and calculated values for non-dimensional natural frequencies of square and circular microplates.
- Tables present comparison and calculated values for non-dimensional critical buckling loads and the first six buckling loads.
- Figures depict effective-modulus schemes, basis functions, convergence, plate geometries, element meshes, deformed configurations, and vibration or buckling mode shapes.
- Figures examine how material index n and material length scale ratio l/h affect central deflection, natural frequency, and critical buckling load.