Creep Analysis by Norton's Law
Creep Analysis by Norton's Law: Theoretical Foundations
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.
Computational Methods for Creep Analysis by Norton's Law
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.
Creep Analysis by Norton's Law in Practice
Practical Applications of Creep
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.
Creep Analysis by Norton's Law: Software & Solver 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.
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.