Large-Deformation (Geometric Nonlinearity) Analysis
Large Deformation (Geometric Nonlinear) Analysis: Theoretical Foundations
What is Large Deformation?
Professor, how is "Large Deformation" analysis different from regular FEM?
Regular linear FEM assumes infinitesimal deformation: the shape before and after deformation are "almost the same". Large deformation analysis handles problems where the shape changes significantly due to deformation. Equilibrium is evaluated on the deformed shape.
Sources of Geometric Nonlinearity
Three nonlinear effects:
1. Large strain โ Strain is not infinitesimal ($\varepsilon << 1$). Rubber, etc.
2. Large rotation โ Element rotation is not small. Large deformation of beams or shells.
3. Follower force โ Load direction follows the deformation. Pressure loads, etc.
All of these are ignored in linear analysis, right?
Linear analysis assumptions: $\varepsilon << 1$, rotation $\theta << 1$, loads act on the initial shape. If any of these do not hold, NLGEOM=YES (large deformation option) is required.
When is NLGEOM Needed?
| Condition | NLGEOM Required? |
|---|---|
| Strain > 5% | Mandatory |
| Displacement/Dimension Ratio > 10% | Mandatory |
| Rotation Angle > 10ยฐ | Mandatory |
| Pressure Load (large area change) | Required |
| Post-buckling behavior | Mandatory |
| Rubber/Hyperelasticity | Mandatory |
So if displacement is more than 10% of the dimension, large deformation is needed.
For a 1 mm thick plate, if it deflects more than 0.1 mm, large deformation. It's needed more often than you might think.
NLGEOM Settings
Summary
Key Points:
- Evaluate equilibrium on the deformed shape โ Linear analysis stays on the initial shape.
- Large strain + Large rotation + Follower force โ Three nonlinear effects.
- Displacement/Dimension Ratio > 10% is mandatory โ Needed more often than you think.
- NLGEOM=YES (Abaqus), SOL 106/400 (Nastran), NLGEOM ON (Ansys)
Green and Almansi Finite Strain
Finite deformation theory requires two configurations: "current configuration" and "reference configuration". Green-Lagrange strain (reference configuration basis) and Almansi strain (current configuration basis) coincide under infinitesimal deformation, but when the stretch ratio exceeds 1.2, a difference of over 10% arises. The distinction between these two types of strain, independently proposed by Green and Almansi in the 1900s, is directly linked to the difference between Total Lagrangian (reference configuration) and Updated Lagrangian (current configuration) FEM formulations.
Computational Methods for Large Deformation (Geometric Nonlinear) Analysis
Newton-Raphson Method
What is the basic algorithm for large deformation analysis?
Newton-Raphson Method: Apply load in increments, iteratively satisfying equilibrium at each increment.
1. Load increment โ Apply total load divided into $n$ steps.
2. Equilibrium iteration โ Newton-Raphson iteration at each increment until internal and external forces match.
3. Tangent stiffness matrix update โ Recalculate stiffness based on deformed shape.
So it solves simultaneous equations repeatedly at each increment. Much heavier than linear analysis.
Linear analysis solves simultaneous equations once. Large deformation analysis solves them $n$ increments ร $m$ iterations. Computational cost is 10 to 100 times higher.
Total Lagrangian Method vs. Updated Lagrangian Method
Abaqus's NLGEOM=YES uses UL. Nastran's SOL 106 is TL-based.
Summary
Arc-Length Method and Snap-Through Tracking in Large Deformation Analysis
When the load-displacement curve shows "snap-back", it cannot be tracked with normal load control. The Riks method (arc-length method), proposed by Kemper and Riks in 1972, simultaneously increments load and displacement, enabling tracking up to unstable equilibrium paths. Applications to industrial analysis, such as shell snap-through and buckling deformation of rubber seals, have been standardized since the 1980s as the RIKS step in Abaqus.
Large Deformation (Geometric Nonlinear) Analysis in Practice
Large Deformation in Practice
Typical problems requiring large deformation analysis:
| Problem | Reason for Large Deformation |
|---|---|
| Rubber components | Strain > 100% |
| Sheet metal forming | Large strain + large rotation |
| Cables/Ropes | Geometric stiffness change |
| Membrane structures | Initial shape is "flat", undergoes large deformation in use |
| Post-buckling | Deformed shape is important |
| Medical devices (stents) | Large deformation during expansion |
Practical Checklist
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