Shape Memory Alloy (SMA) Model
Theory and Physics
What are Shape Memory Alloys?
Professor, shape memory alloys (SMAs) are materials that return to their original shape even after deformation, right?
SMA (Shape Memory Alloy) is represented by NiTi (Nitinol). It exhibits two special behaviors:
1. Superelasticity (Superelasticity) — Even under large deformation (6-8% strain), it fully recovers upon unloading. Temperature is constant.
2. Shape Memory Effect — After deformation, heating causes recovery to the original shape. Phase transformation occurs with temperature change.
Phase Transformation
SMA characteristics originate from the martensite → austenite phase transformation:
- High Temperature (Austenite) — Hard. Exhibits superelasticity.
- Low Temperature (Martensite) — Soft. Heating returns it to austenite → shape recovery.
Modeling in FEM
Abaqus's *SUPERELASTIC (Superelastic SMA model). Models stress-induced martensitic transformation.
$\varepsilon^{tr}$ is the transformation strain. Stress-induced austenite → martensite transformation.
Summary
Discovery of the Shape Memory Effect
The shape memory effect of NiTi (Nitinol) alloy was accidentally discovered in 1963 by William Buehler and Frederick Wang at the US Naval Ordnance Laboratory (NOL). The alloy name "Nitinol" is derived from the initials of Nickel Titanium Naval Ordnance Laboratory. The phase transformation between martensite (low-temperature phase) and austenite (high-temperature phase) is the physical basis for shape memory and superelasticity.
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 enough that 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", 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. 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 force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). 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 typical mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — 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 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 for a smoother ride. 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 inhomogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear.
- Isotropic material (unless specified otherwise): 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 equilibrium between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For plasticity, creep, and other nonlinear material behaviors, 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 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 as N in mm system, N in m system. |
Numerical Methods and Implementation
FEM Settings for SMA
```
*MATERIAL, NAME=NiTi
*DEPVAR
24,
*USER MATERIAL, CONSTANTS=14
$ Auricchio model parameters
```
Or:
```
*SUPERELASTIC
sigma_SL, sigma_EL, sigma_SU, sigma_EU, epsilon_L, ...
```
So the superelastic hysteresis loop (different loading-unloading paths) is reproduced in FEM, right?
Yes. Loading causes stress-induced martensitic transformation → unloading causes reverse transformation to austenite. Hysteresis dissipates energy.
Summary
Identification Experiments for the Brinson Constitutive Law
The representative SMA constitutive law by Brinson (1993) requires 5-6 parameters: phase transformation start/finish stresses (σsAs, σfAs, σsMs, σfMs) and maximum transformation strain εL. The standard procedure is to identify transformation temperatures using DSC (Differential Scanning Calorimetry) and perform isothermal tensile tests at multiple temperatures to read transformation stresses from σ-ε curves.
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 2-3 times. Recommended 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 for the situation.
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 each iteration. Exhibits quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using 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 $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load in small increments rather than all at once. The arc-length method (Riks method) can trace beyond extremum points in the load-displacement relationship.
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" — starts with a rough answer but accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
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 a "flexible curve" — 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
SMA in Practice
Practical Checklist
Design Analysis of Intravascular Stents
In designing coronary stents (diameter 3-4mm) made of Nitinol, FEA predicts the behavior where they transform to austenite at body temperature 37°C and expand the vessel wall with a force of about 0.3-0.5N. Combining Abaqus's *SMATERIAL keyword (Superelastic) with the Auricchio-Taylor model has been used as computational evidence for FDA 510(k) submissions since the 2010s.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy ingredients (prepare CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question — which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Easily Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "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 the same as "writing the exam question". If the question 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
SMA Tools
Selection Guide
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