Analysis of Membrane Wrinkling (Rimpling)
Theory and Physics
What is Membrane Wrinkling?
Professor, what kind of phenomenon is wrinkling (rinkuring) in membrane structures?
Membrane structures have almost zero bending stiffness. When compressive stress occurs, wrinkles form. This becomes a problem in space solar panels, airbags, and tent structures.
Mechanics of Wrinkling
When compressive stress occurs in a membrane:
1. The membrane cannot withstand compression — buckling = wrinkling due to lack of bending stiffness
2. Wrinkle direction — wrinkles form orthogonal to the compression direction
3. Wrinkle wavelength — depends on membrane tension, plate thickness, and curvature
Modeling in FEM
Two approaches:
1. Shell elements (thin thickness) — Directly simulate wrinkle shape. NLGEOM=YES + initial imperfection for buckling → wrinkle
2. Membrane elements (zero bending stiffness) + wrinkle model — "Tension field theory" that sets compressive stress to zero
Can't membrane elements directly represent wrinkles?
Membrane elements have no bending stiffness, so the "shape" of wrinkles does not appear under compression. Instead, compressive stress is set to zero to obtain the "stress field in the wrinkled state." Abaqus's *NO COMPRESSION or membrane wrinkle algorithms.
Summary
Origin of Tank Liquid Sloshing and Membrane Wrinkling Theory
Membrane wrinkling theory originated from the difference between tension and compression. The distinction between "tension membranes" where wrinkles do not occur and "wrinkled membranes" with compressive stress was made by Stein & Hedgepeth (1961, NASA). They established the "relaxed principal stress theory" where stress in the principal compression direction is set to 0 in wrinkled regions. The theoretical foundation of current FEM wrinkle analysis lies in this 1961 paper.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "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 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 $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire contents" (body force), 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 ending up with "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. Because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—deliberately absorbing vibration 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, stress-strain relationship is linear
- Isotropic material (unless specified otherwise): Material properties are direction-independent (anisotropic materials require separate tensor definition)
- Quasi-static assumption (for static analysis): Ignores inertial/damping forces, considers only equilibrium between external and internal forces
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and creep, constitutive law extension is needed
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads/elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note 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 as N in mm system, N in m system |
Numerical Methods and Implementation
Shell Element Approach (Obtaining Wrinkle Shape)
```
*SHELL SECTION
0.025, 5 $ Plate thickness 0.025mm (membrane)
*STEP, NLGEOM=YES
*STATIC, RIKS $ Wrinkling is a type of buckling
```
Give an initial imperfection (first buckling mode shape) to manifest the wrinkle pattern.
Membrane Element Approach (Tension Field)
```
*MEMBRANE SECTION
0.025
*NO COMPRESSION $ Set compressive stress to zero
```
Wrinkle shape does not appear, but the stress field in wrinkled regions is obtained.
Summary
Wrinkle Finite Element Method: Modified Material Property Method
Methods for handling membrane wrinkles in FEM include: ① "Relaxed stiffness method" that clips principal stress to zero, ② "Analytical tracking method" that calculates only non-wrinkled regions, ③ "Buckling analysis method" that reproduces actual geometric wrinkles with fine meshes. In practice, the modified material property method (relaxed stiffness method) is the most robust, and the combination of Abaqus membrane element M3D4R and Wrinkle determination subroutine is widely used in industry.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with 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 2-3x. Recommendation: 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. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix with initial value or every few iterations. Cost per iteration is low, but convergence is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$〜$10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Apply total load not all at once, but in small increments. The arc-length method (Riks method) can track beyond extremum points in load-displacement relationships.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—initially a rough answer, but accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: opening to an estimated location and adjusting forward/backward (iterative) is more efficient than searching sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like "flexible curves"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Practical Wrinkle Analysis
Most important in space structures (solar panels, membrane structure antennas). Wrinkles degrade optical surface accuracy.
Practical Checklist
Deployed Membrane Analysis for Space Solar Power Satellites
Thin-film solar panels (thickness 0.01mm) for Space Solar Power Satellites (SSPS) may experience thermal deformation and wrinkles after deployment. Since the 2010s, JAXA has conducted thermal-structural coupled wrinkle analysis of polyimide films (thickness 12.5μm), identifying that wrinkles with 3-5mm wavelength caused by the combination of solar radiation pressure and thermal expansion affect power generation efficiency, and reflected this in membrane tension design.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate (visualize in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's "prep work". If mesh quality is poor, results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "calculation ran = results are correct"? This is actually the most common trap for CAE beginners. The solver always returns "some answer" for the given mesh. But if the mesh is too coarse, that answer is far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct."
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most important step in the entire analysis.
Software Comparison
Wrinkle Analysis Tools
Special Settings for Thin Film Analysis in Ansys Mechanical
Ansys Mechanical's Shell181 or Membrane elements (SHELL181, KEYOPT(1)=1) function as pure membrane elements with zero out-of-plane stiffness. Wrinkle determination is typically implemented by combining material input that only permits "principal stress ≥ 0" (USERFLD + relaxed stiffness subroutine). ESA used this method for designing Europe's large membrane antenna (diameter 15m) for spacecraft, predicting on-orbit wrinkle shape with ±3% accuracy.
Three Most Important Questions for Selection
- "What to solve?": Does it support the physical models/element types needed for membrane wrinkle (rinkling) analysis? For example, presence of LES support for fluids, contact/large deformation capability for structures make a difference.
- "Who will use it?": Tools with rich GUI for beginner teams, flexible script-driven tools for experienced users. Similar to the difference between automatic (GUI) and manual (script) transmission cars.
- "How far to expand?": Selection considering future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced Wrinkle Research
Singular Field at Wrinkle Tip and Effective Membrane Thickness
In membranes where wrinkles spread, the actual effective thickness becomes "membrane thickness ÷ number of wrinkles", significantly reducing equivalent bending stiffness. Molecular dynamics simulation of a 0.1mm thick aluminum foil wrinkle tip in the 2020s revealed that local stress reaches 3-5 times the bulk yield stress. This high local stress is a key parameter determining crack initiation life in folded space deployable structure lifespan design.
Related Topics
Coupled AnalysisFlag/Membrane FSI AnalysisStructuralNonlinear Analysis of Cables and RopesStructuralAnalysis of Sandwich Panels用語集Bending StiffnessCoupled Analysis熱座屈解析StructuralGeometric Stiffening Effect (Spin Stiffening)
この記事の評価ご回答ありがとうございます！参考に
なったもっと
詳しく誤りを
報告
なった
詳しく
報告