Asymptotically Accurate and Shear-Locking-Free Finite Element Implementation of First-Order Shear Deformation Theory for Plates

Core Concepts

The authors present an asymptotically accurate and shear-locking-free finite element implementation of the first-order shear deformation theory (FSDT) for linear-elastic homogeneous plates.

Abstract

The key highlights and insights of the content are:

The authors formulate the FSDT for plates in rescaled coordinates and rotation angles, which makes the problem independent of the plate thickness and inherently shear-locking-free.

They develop a finite element implementation of the rescaled FSDT using isogeometric analysis with NURBS shape functions, which ensures C1-continuity of the solution and achieves asymptotic accuracy.

Numerical simulations of circular and rectangular plates are performed, showing complete agreement between the analytical solution, the numerical solution based on the 2D FSDT, and the numerical solution of the 3D elasticity theory.

The rescaled formulation and the use of isogeometric elements eliminate the need for high-order interpolation schemes and sophisticated integration techniques, significantly improving the computational efficiency.

The proposed approach provides a simple and effective way to avoid the shear-locking effect, which is a common issue in the numerical implementation of FSDT for plates.

Stats

The authors provide the following key figures and equations to support their analysis:

Equation (9): The rescaled variational formulation of the FSDT for plates.
Equation (12): The expression for the true average displacement of the plate.
Equations (48)-(51): The analytical solution for the bending of a rectangular plate with one clamped edge and three free edges.
Equations (52)-(53): The solution based on the classical Kirchhoff plate theory.

Quotes

"The goal of this paper is therefore twofold. First, we give the formulation of the FSDT for plates in the rescaled coordinates and rotation angles. This formulation occurs naturally when the coordinates in the mid-plane are scaled by the plate thickness h (thus becoming dimensionless), while the rotation angles are multiplied by h, resulting in equal and finite orders of the bending and shear stiffnesses as well as the scaled rotation angles and bending measures."
"Since this formulation is independent of the plate thickness and inherently shear-locking-free, no high-order interpolation scheme and/or sophisticated integration technique is required for the discretization and FE-implementation, so the computational efficiency can be significantly improved."

Key Insights Distilled From

Asymptotically accurate and locking-free finite element implementation of first order shear deformation theory for plates

by Khanh Chau L... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2310.19443.pdf

Deeper Inquiries

How can the proposed approach be extended to handle more complex plate geometries and boundary conditions

The proposed approach can be extended to handle more complex plate geometries and boundary conditions by incorporating higher-order NURBS discretization, multipatch modeling, and advanced coupling techniques.

Higher-order NURBS Discretization: By increasing the order of the NURBS basis functions, the accuracy of the solution can be improved, allowing for more complex geometries to be accurately represented. This higher-order discretization enables finer control over the shape functions and can capture intricate plate geometries more effectively.

Multipatch Modeling: For geometries that cannot be represented by a single NURBS patch, multipatch modeling can be employed. This involves connecting multiple NURBS patches at their interfaces using techniques like the penalty method or Nitsche method to ensure continuity and smooth transitions between patches.

Advanced Coupling Techniques: To handle complex boundary conditions, advanced coupling techniques can be utilized. This includes methods like the bending strip method or the reduced integration approach to ensure compatibility and stability at the boundaries of the plate.

By incorporating these extensions, the proposed approach can effectively handle a wide range of complex plate geometries and boundary conditions.

What are the potential limitations or drawbacks of the rescaled FSDT formulation, and how can they be addressed

The rescaled FSDT formulation, while offering advantages such as shear-locking-free behavior and computational efficiency, may have some limitations and drawbacks that need to be addressed:

Limited Applicability: The rescaled formulation may not be suitable for all types of plate structures, especially those with highly irregular geometries or material properties. In such cases, the assumptions made in the rescaled formulation may not hold, leading to inaccuracies in the results.

Boundary Effects: For plates with complex boundary conditions, the rescaled formulation may struggle to accurately capture the behavior near the edges. This can result in discrepancies between the numerical solution and the actual physical response of the plate.

Convergence Issues: In some cases, the rescaled formulation may exhibit convergence issues, especially when dealing with highly non-linear or large deformation problems. This can affect the accuracy and stability of the numerical solution.

To address these limitations, further research can focus on refining the rescaled formulation, incorporating adaptive meshing techniques, and exploring hybrid approaches that combine the rescaled FSDT with other structural theories for improved accuracy and robustness.

Can the variational-asymptotic method used in this work be applied to other structural theories beyond the FSDT for plates

The variational-asymptotic method used in this work can indeed be applied to other structural theories beyond the FSDT for plates. The method is a powerful tool for deriving accurate and efficient finite element formulations for a wide range of structural problems. Here are some ways in which the variational-asymptotic method can be extended to other structural theories:

Shell Structures: The variational-asymptotic method can be applied to develop accurate finite element formulations for shell structures, including higher-order shell theories and specialized shell models for specific applications.

Composite Materials: By incorporating the variational-asymptotic method, it is possible to derive finite element formulations for composite materials, considering the anisotropic behavior and complex material properties of composites.

Nonlinear Analysis: The method can be extended to handle nonlinear structural analysis, including large deformations, material nonlinearity, and contact problems. By incorporating nonlinear terms into the energy functional, accurate numerical solutions can be obtained for a wide range of nonlinear structural problems.

By applying the variational-asymptotic method to these and other structural theories, researchers can develop efficient and accurate finite element formulations for a diverse set of structural engineering applications.