Creep Analysis by Norton's Law
Theory and Physics
Norton Creep Law
Professor, the Norton law also appeared on the creep buckling page.
The Norton law (power law) is the most basic model for steady-state creep:
$A$ is the creep coefficient, $n$ is the stress exponent. When including temperature dependence:
Time Hardening Law and Strain Hardening Law
When also including transient creep (primary stage):
- Time Hardening Law: $\dot{\varepsilon}_{cr} = A \sigma^n t^m$ — a function of time $t$
- Strain Hardening Law: $\dot{\varepsilon}_{cr} = f(\sigma, \varepsilon_{cr})$ — a function of accumulated strain
Which one should I use?
For constant load, both are the same. If the load varies, the strain hardening law is more accurate.
FEM Settings
```
*CREEP, LAW=NORTON
A, n, m
```
Time integration with *VISCO step.
Summary
F.H. Norton, the Namesake of the Norton Law
The Norton law (power-law creep law) was proposed by F.H. Norton in his 1929 book "The Creep of Steel at High Temperatures." Norton was an engineer at GE and a pioneer who systematized the high-temperature deformation of steam turbine components. Nearly 100 years after its publication, it is still implemented in almost all FEM codes as the first approach to creep, and its universality stands out among material models.
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 "being carried away" feeling 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, which is the assumption that "acceleration can be ignored because the force is applied slowly." 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 $F=kx$, and it's 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" is not correct. Stiffness is "resistance to deformation," strength is "resistance to failure"—they are 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 contents" (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's actually "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 away. That's because the vibration energy is converted into heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they deliberately absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue 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 inhomogeneities
- 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: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and 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 load/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 Implementation of Creep
Abaqus *VISCO step:
```
*STEP, INC=10000
*VISCO, CETOL=0.005
0.01, 100000., 1e-8, 1000.
```
Automatic time stepping with CETOL (creep strain tolerance error).
Summary
Long-Term Records of Creep Tests
Creep tests to determine Norton law parameters (A, n) are extremely long-term. The Japanese Society of Mechanical Engineers' high-temperature material database (NIMS) contains test data for over 100,000 hours (about 11 years) for 316 stainless steel at 600°C and 100 MPa. This vast experimental data forms the basis for the 60-year design life of thermal and nuclear power plants.
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 critical.
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 every iteration. Achieves 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 speed 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 the full load not all at once, but in small increments. 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 get closer to the correct answer"—the initial answer is rough, but accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location 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
Creep in Practice
Creep evaluation for thermal power boiler tubes, turbine blades, nuclear vessels, high-temperature piping. Regulated by ASME NH.
Practical Checklist
Design Guidelines for Thermal Power Boiler Tubes
For main steam pipe design in supercritical pressure thermal boilers (steam temperature above 600°C, pressure above 25 MPa), creep analysis using the Norton law is mandatory. In Japan, JISB8201 based on the Electricity Business Act applies, setting the upper limit of allowable stress as 2/3 of the 100,000-hour creep rupture strength. This standard codifies a "design creep curve" extrapolated from the Norton law with a safety factor, and the basic concept has remained unchanged since the 1960s.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), put it on the heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—in cooking, which step is most prone to failure? Actually, it's the "prep work." If the 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 will be 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."
How to Think About Boundary Conditions
Setting boundary conditions is the same as "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
Tools for Creep
History of the Ansys CREEP Command
The Ansys command "TB,CREEP" for defining Norton law creep is one of the few legacy features whose basic syntax has remained almost unchanged for over 30 years since ANSYS 5.0 (released in 1993). It is now selectable as "Creep (Norton)" from Workbench/Mechanical, but internally the same CREEP constant table is used, and errors in parameter number correspondence during APDL migration are still reported today.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models/element types needed for Norton law creep analysis? For example, in fluids, the presence of LES support; in structures, the ability to handle contact/large deformation makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic transmission cars (GUI) and manual transmission cars (script).
- "How far will it expand?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced Topics in Creep
Multiaxial Creep: Extension to von Mises Equivalent Stress
When extending the uniaxial Norton law to multiaxial stress states, it is assumed that the direction of creep strain rate follows the Prandtl-Reuss rule and is proportional to the deviatoric stress. This isotropic creep assumption is also called the Norton-Bailey equation, named after Bailey who independently proposed the same equation in 1935. However, for heavily worked materials or welds, anisotropic creep along the principal axes becomes pronounced, and this assumption breaks down, as confirmed by experiments in the 1980s.
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