Mindlin–Reissner Plate Theory
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly so acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That is Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External force term (load term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | In mm system: N, in m system: N (unified). |
Numerical Methods and Implementation
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.
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