Isogeometric analysis for functionally graded microplates based on modified couple stress theory (1604.00547v1)

Abstract: 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.

• The paper introduces an isogeometric analysis framework combining NURBS, modified couple stress theory, and refined plate theories to analyze functionally graded microplates.
• Numerical results confirm the framework's accuracy and rapid convergence, demonstrating that increasing the material length scale enhances microplate stiffness and mechanical properties.
• This approach provides accurate and efficient simulation for analyzing size-dependent behavior crucial for designing micro-scale FGM components like sensors and actuators.

Isogeometric Analysis of Functionally Graded Microplates with Modified Couple Stress Theory

The paper explores the computational analysis of functionally graded microplates by integrating isogeometric analysis with modified couple stress theory (MCST) and refined plate theories. The authors aim to address the challenges in analyzing the bending, vibration, and buckling behaviors of microplates, particularly how size-dependent effects, shear deformations, and thickness stretching influence these behaviors. To achieve this, the paper employs a novel isogeometric approach that combines Non-Uniform Rational B-Splines (NURBS)-based computational mechanics with an enhanced theoretical framework for functionally graded materials (FGMs).

Methodology and Theoretical Framework

The work introduces a versatile computational framework that employs MCST, known for utilizing a single material length scale parameter, facilitating the capture of microscale effects more effectively than other higher-order theories. This theory is integrated with a novel seventh-order displacement field formulation based on quasi-3D refined plate theories. This formulation considers both shear deformations and thickness stretching without needing shear correction factors, a distinct advantage over classical and first-order shear deformation theories.

The isogeometric analysis approach leverages NURBS to define both geometry and field approximations, which is particularly advantageous in satisfying C$^1$ continuity conditions required by higher-order shear deformation theories. This technique allows for accurate geometric representation alongside efficient computation, as evidenced by the included numerical examples.

Results and Comparisons

The paper provides extensive numerical verifications against established literature, confirming the validity of the proposed approach. The results demonstrate that the proposed method achieves rapid convergence and aligns closely with benchmark solutions from previous studies. By varying parameters such as material length scale ratio, material index, and geometrical dimensions, the authors show significant agreement across a broad spectrum of scenarios, including isotropic and FGMs with both rectangular and circular geometries.

The detailed results also illustrate that increasing the material length scale improves structural stiffness, leading to lower deflections and higher critical buckling loads and natural frequencies. This finding underpins the importance of size-dependent effects in the design and analysis of micro-scale structures.

Practical and Theoretical Implications

The implications of this research are extensive for the design and analysis of micro-scale engineering applications, particularly where FGMs are utilized due to their beneficial gradient properties. The insights gained could be pivotal for small-scale elements used in electronics, sensors, and actuator systems that require careful consideration of size-dependent mechanical behavior.

Additionally, from a computational standpoint, the integration of NURBS within an isogeometric framework offers a promising direction for future research, particularly when paired with modified couple stress theory. This combination allows for more accurate simulations while maintaining computational efficiency, potentially setting a new standard for such analyses.

Future Directions

The paper opens avenues for further refinement and extension of isogeometric methods in the analysis of complex materials and geometries. Future research may focus on expanding this approach to nonlinear dynamic problems, exploring other enriching degrees of freedom, and employing different types of basis functions beyond NURBS for enriched isogeometric analysis.

In conclusion, this paper introduces a robust computational framework combining isogeometric analysis with modified couple stress and advanced plate theories, yielding comprehensive insights into the mechanical behaviors of functionally graded microplates. These findings not only validate the utility of MCST in capturing micro-scale effects but also unveil the practical enhancements made possible through the isogeometric approach.