This document summarizes research on the isogeometric analysis of functionally graded microplates. It introduces the use of modified strain gradient theory to capture size-dependent effects in microplates, which are modeled using non-uniform rational B-splines (NURBS) in isogeometric analysis. Static bending, nonlinear bending, free vibration, and dynamic analyses are performed to study the effects of material gradation and size scales on microplate stiffness and vibrational characteristics. The results demonstrate increased plate stiffness and more significant small scale effects with decreasing thickness comparable to length scale parameters.

1. ISOGEOMETRIC ANALYSES OF FUNCTIONALLY GRADED MICROPLATES
INTRODUCTIONS
• Functionally graded (FG) microplates are commonly
used in small-scale structures and systems, the
mechanical responses of which are size-dependent as
observed in experimental studies.
• Size effect can be captured using either molecular
dynamics (MD) simulation or higher-order continuum
mechanics. The MD is too computationally expensive.
• The Modified Strain gradient Theory (MST) is
therefore used to capture the size effects in this study
with three length scale parameters.
THEORETICAL FORMULATIONS
• The strain energy based on the MST is given as:
• The third-order shear deformation theory gives
• Material properties varies based on the mixture rule
- Rule of mixture
• - Volume fraction
IGA-BASED FORMULATIONS
• Isogeometric Analysis (IGA) is considered as an
advanced numerical method in computational
mechanics.
• The distinguish feature of IGA is that it uses the tools
of CAD technology as basic components in numerical
analyses.
NUMERICAL RESULTS
• Static bending analysis
CONCLUSIONS
Son Thai, Huu-Tai Thai
School Engineering and Mathematical Sciences, La Trobe University
Micro-photography of the mechanical microstructure elements (Gusev et al., 2010)
   
 1 11
2
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1 2
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2 2
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4
3
4
3
x x
y y
z w
u u z
h x
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u v z
h y
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 
 
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
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• 2D-NURBS basis functions
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,
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• Governing equation of motion
     KK dKK Md f
• Nonlinear bending analysis
• Free-vibration analysis
• Dynamic analysis
• The inclusion of the size effect results in an increase
in the plate stiffness. The small scale effect becomes
significant when the plate thickness is such small
that it is comparable to length-scale parameters.
• The stiffness of FG plates decrease with the increase
of gradient index n.
Publications
S. Thai, H.-T. Thai, T. P. Vo, and V. I. Patel, “Size-dependant behaviour of functionally graded
microplates based on the modified strain gradient elasticity theory and isogeometric analysis,”
Comput. Struct., vol. 190, pp. 219–241, Oct. 2017.
S. Thai, H.-T. Thai, T. P. Vo, and H. Nguyen-Xuan, “Nonlinear static and transient isogeometric
analysis of functionally graded microplates based on the modified strain gradient theory,” Eng.
Struct., vol. 153, pp. 598–612, Dec. 2017.
NURBS basis functions and a complicated geometry is represented by NURBS
Meshes of square and circular plates
Results of displacements at centre point and deformed geometries
Results of natural frequencies and deformed geometries of mode 1 and 2