Mindlin–Reissner Plate Theory
Mindlin: Theoretical Foundations
2D Version of Timoshenko Beam
Professor, is the Mindlin plate the Kirchhoff plate with shear deformation added?
Exactly. Just as the Timoshenko beam adds shear deformation to the Euler-Bernoulli beam, the Mindlin plate (Reissner-Mindlin plate) is a theory that adds shear deformation to the Kirchhoff plate.
Basic Assumptions
Assumptions of the Mindlin plate:
1. Straight lines in the thickness direction remain straight after deformation (but are not necessarily perpendicular to the mid-surface).
2. Strain in the thickness direction $\varepsilon_{zz} = 0$.
3. Shear deformation is considered — $\gamma_{xz} \neq 0, \gamma_{yz} \neq 0$.
The difference from the Kirchhoff plate is assumption 3, right? The shear strain is not zero.
Due to this difference, the rotation angles $\theta_x, \theta_y$ become independent of the derivative of the deflection $w$:
The degrees of freedom are not just $w$ but also independent $\theta_x, \theta_y$. So three degrees of freedom per node, right?
Yes. In the Kirchhoff plate, there was the constraint $\theta = -\partial w / \partial x$, but the Mindlin plate has no such constraint. Thanks to this independence, FEM implementation only requires $C^0$ continuity.
Governing Equations
The equilibrium equations for the Mindlin plate are three coupled partial differential equations:
Here, $Q_x, Q_y$ are the transverse shear forces, and $M_x, M_y, M_{xy}$ are the bending/twisting moments.
The Kirchhoff plate had a single fourth-order partial differential equation for $w$, but the Mindlin plate has coupled equations...
In return, each equation contains only second-order or lower derivatives. This is why it can be discretized with $C^0$ continuous FEM elements.
Shear Locking Problem
Mindlin plate elements also have shear locking, right?
The exact same problem as with the Timoshenko beam occurs. For thin plates ($b/t > 20$), shear deformation should theoretically be almost zero, but in standard FEM elements, the shear deformation does not vanish, causing the element to become overly stiff.
Many countermeasures have been developed:
- MITC Elements (Mixed Interpolation of Tensorial Components) — Bathe-Dvorkin method. Assumes independent interpolation for shear strain.
- DSG Method (Discrete Shear Gap) — Shear gap method.
- Reduced Integration — Applied to first-order elements.
- Assumed Natural Strain (ANS) — Assumes shear strain in the natural coordinate system.
I've heard of MITC elements.
MITC4 (4-node) and MITC9 (9-node) are elements developed by Professor Bathe, which eliminate shear locking while also passing the patch test. Many commercial solver shell elements are MITC-based. Abaqus's S4R, Nastran's CQUAD4 (shell), and Ansys's SHELL181 all incorporate MITC-based technology.
Summary
Let me organize the Mindlin plate theory.
Key points:
- Plate theory considering shear deformation — The 2D version of the Timoshenko beam.
- $w, \theta_x, \theta_y$ are independent variables — Can be discretized with $C^0$ FEM elements.
- Converges to Kirchhoff plate for thin plates — Agrees when $b/t > 20$.
- Shear locking is the biggest challenge — Countermeasures include MITC method, DSG method, reduced integration.
- Modern FEM shell elements are all Mindlin-based — The de facto standard theory.
So, if you use shell elements in FEM, Mindlin plate theory is fundamental knowledge you can't avoid, right?
Exactly. Questions like "Why is reduced integration recommended?" or "Why can shell elements be used even for thin plates?" can all be explained with knowledge of Mindlin plate theory and shear locking.
Establishment of Mindlin Plate Theory
Raymond D. Mindlin published a plate theory considering shear deformation in his 1951 paper "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates". E. Reissner also independently published a similar formulation in 1944, and it is now jointly referred to as "Mindlin-Reissner theory". This theory improves accuracy by 10-20% compared to Kirchhoff theory for plates with t/L > 0.05.
Computational Methods for Mindlin
Principle of MITC Elements
Please explain how MITC elements eliminate shear locking.
In standard Mindlin plate elements, $w$ and $\theta$ are interpolated with the same shape functions. For thin plates, $\gamma = \partial w / \partial x + \theta_y \approx 0$ should hold, but with same-order interpolation, it does not become numerically zero.
The MITC method interpolates shear strain independently. Instead of calculating shear strain from $w$ and $\theta$, it assumes shear strain at "tying points" on the element edges and interpolates it within the element.
What are "tying points"?
Shear strain is evaluated at specific points on the element edges, and these values are interpolated over the entire element. This eliminates parasitic shear (the cause of locking). They are called "tying points" because they "tie" (connect) points on the edges.
Shell/Plate Elements in Various Solvers
Please tell me about the Mindlin plate (shell) elements in various solvers.
| Element | Solver | Number of Nodes | Locking Countermeasure | Recommendation Level |
|---|---|---|---|---|
| S4R | Abaqus | 4 | Reduced integration + hourglass control | ○ General recommendation |
| S4 | Abaqus | 4 | Full integration + incompatible mode | ○ Precision analysis |
| S8R | Abaqus | 8 | Reduced integration | ◎ Highest accuracy |
| CQUAD4 | Nastran | 4 | MITC-based + incompatible | ◎ Industry standard |
| CQUAD8 | Nastran | 8 | Quadratic element | ○ |
| SHELL181 | Ansys | 4 | MITC-based | ○ General recommendation |
| SHELL281 | Ansys | 8 | MITC-based | ◎ Highest accuracy |
What's the difference between Abaqus's S4R and S4?
S4R uses reduced integration (1 integration point) + hourglass control. It's versatile and fast to compute, but its accuracy per element is lower than S4 (full integration + incompatible mode). S4 has neither locking nor hourglassing, but computational cost is somewhat higher.
Related Topics
Experience the theory firsthand with the interactive simulator for this field
All Simulators