Thick Shell Theory (Degenerate Solid)
Theory and Physics
What is a Degenerated Solid Shell?
Professor, what is a "degenerated solid"? Is it a hybrid of shell and solid elements?
Exactly. The Degenerated Solid Shell Element is created by degenerating (reducing) the degrees of freedom in the thickness direction of a 3D solid element into a shell element. It was proposed by Ahmad-Irons-Zienkiewicz (1970).
The concept is simple:
1. Start with a 3D solid element having nodes on its top and bottom surfaces.
2. Use the displacement of the mid-surface and the rotation angle about the normal direction as degrees of freedom.
3. Assume a linear displacement distribution in the thickness direction (Mindlin assumption).
Starting from 3D and adding assumptions to make it 2D. That's the opposite approach of conventional shell theory.
Yes. Conventional shell theory starts from 2D equations, but the degenerated solid approach starts from 3D and "degenerates the unused degrees of freedom." The result converges to the same Mindlin shell, but the implementation is simpler because it's based on 3D.
Handling Thick Shells
What is meant by a "thick shell"?
Shells with moderate thickness where $R/t$ is around 10 to 30. It's the intermediate region, neither thin ($R/t > 30$) nor thick ($R/t < 10$, essentially solid).
For thick shells:
- Shear deformation cannot be ignored.
- The through-thickness stress $\sigma_z$ is not exactly zero.
- Membrane-bending coupling is strong.
Can Mindlin shell elements handle this?
They can handle shear deformation, but $\sigma_z \neq 0$ cannot be treated by conventional shell elements. To handle this, you need solid shell elements (shell elements with displacement degrees of freedom in the thickness direction) or solid elements.
Solid Shell Elements
What kind of elements are "solid shells"?
They look like solid elements (HEX8 or HEX20), but their internal formulation is optimized for shells.
| Element | Solver | Features |
|---|---|---|
| SC8R | Abaqus | 8-node solid shell. Reduced integration + anti-locking measures |
| SOLSH190 | Ansys | Solid shell. Bending representation with one element in thickness direction |
| CHEXA(solid-shell) | LS-DYNA | LSDYNA implementation of solid shell |
What are the advantages of solid shells?
Summary
Let me organize the thick shell theory.
Key points:
- Degenerated solid — Creating a shell by degenerating the thickness direction from a 3D solid
- Intermediate region $R/t = 10 \sim 30$ — Neither thin nor solid
- Solid shell elements — Look like solids, behave like shells. Handle contact surfaces and $\sigma_z$
- SC8R (Abaqus), SOLSH190 (Ansys) — Representative elements
- Bending representation with one element in thickness direction — Efficient
So we choose: thin → Mindlin shell, intermediate → solid shell, thick → solid element.
Judgment based on $R/t$ is fundamental. If in doubt, solve with both and compare the results.
Mindlin-Reissner Thick Shell Theory
The Mindlin-Reissner theory, which forms the basis for thick shells, was formulated independently by Raymond Mindlin and Eric Reissner between 1945 and 1951. Unlike the Kirchhoff assumption, it allows "normals to tilt due to shear deformation," explicitly treating transverse shear strains εxz and εyz as degrees of freedom. This extends the applicable range to plates with thickness/span ratios up to about 1/5, making it an essential theory for analyzing interlaminar shear in composite laminates.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? 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 enough that acceleration is negligible." It 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's Hooke's law $F=kx$, the essence of the stiffness term. So, a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously, the rubber band. 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"—different concepts.
- External force term (load term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire interior" (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 applying "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated 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—deliberately absorbing vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep swaying 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 the 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: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
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 system 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) | Unify to N in mm system, N in m system. |
Numerical Methods and Implementation
Implementation of Solid Shells
Please tell me about implementation considerations for solid shells.
Solid shells are "thin" solid elements, so the aspect ratio in the thickness direction becomes very large. For regular solid elements, accuracy degrades at aspect ratio > 5, but solid shells internally correct for this.
Locking Countermeasures
Locking issues in solid shells:
1. Shear locking — Occurs in thin plate bending. Countered by ANS method.
2. Volumetric locking — Occurs with incompressible materials. Countered by EAS method or B-bar method.
3. Trapezoidal locking — When the element tapers in the thickness direction. Unique to solid shells.
4. Curvature thickness locking — When elements become trapezoidal in the thickness direction on curved surfaces.
Is trapezoidal locking unique to solid shells?
Yes. When there is taper (trapezoidal shape) in the thickness direction, regular solid elements cannot correctly represent bending. Solid shell elements eliminate trapezoidal locking using EAS (Enhanced Assumed Strain).
Usage Tips
Points to note when using solid shell elements:
- One element in the thickness direction — Two or more are unnecessary (design philosophy of solid shells).
- Correctly specify the element's "thickness direction" — Specify stack direction in Abaqus's *SOLID SECTION.
- Meshing curved surfaces — Mesh the top and bottom CAD surfaces separately, then connect them in the thickness direction.
It's efficient that one element in the thickness direction is enough. With regular solid HEX8, you needed at least 4 elements in the thickness direction.
That's precisely the biggest advantage of solid shells. They achieve bending accuracy equivalent to regular shell elements with just one HEX8-equivalent element in the thickness direction. The DOF count is similar to shell elements, but they offer benefits in contact and thickness variation.
Summary
Let me organize the numerical methods for solid shells.
Key points:
- Represent bending with one element in the thickness direction — Efficient.
- Locking countermeasures are essential — Combination of ANS + EAS.
- Trapezoidal locking is unique to solid shells — Countered by EAS method.
- Specifying stack direction is important — Correctly define the thickness direction.
- Optimal for problems with contact surfaces on both sides — An advantage not found in regular shell elements.
MITC Element's Shear Locking Countermeasure
The MITC (Mixed Interpolation of Tensorial Components) method is a thick shell locking countermeasure technique developed by Bathe and Dvorkin at MIT in 1986. By independently interpolating shear strains, it ensures uniform accuracy from thin to thick plates. MITC4 corresponds to 4-node shells, MITC9 to 9-node shells, demonstrating high performance with displacement errors within 5% even for thickness/span ratios of 1/1000.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommendation: Use when stress evaluation is important.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix every iteration. Shows quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial values or every few iterations. Cost per iteration is low, but convergence speed is
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