Creep Constitutive Models
An overview of creep constitutive models — from simple Norton secondary creep through Norton-Bailey primary-plus-secondary, time-hardening and strain-hardening forms, to damage-coupled and unified viscoplastic models.
The role of the constitutive model
The creep constitutive model defines the relationship between the creep strain rate and the current state — stress, temperature, accumulated strain and damage. It is the core of the high-temperature structural analysis: the model determines the predicted deformation, the stress redistribution and the life. The choice of model affects every result. A model that omits primary creep will under-predict early deformation. A model that omits tertiary creep will not predict rupture. A model that uses the wrong hardening rule will give incorrect results under variable loading. The model must be selected, calibrated and validated for the specific material and application.
Classification of models
- Secondary-creep-only (Norton): constant creep rate, no primary or tertiary — simplest, good for long-duration steady-state
- Primary-plus-secondary (Norton-Bailey): includes decelerating primary creep and steady secondary — good for moderate durations
- Time-hardening: creep rate depends on time — simple but incorrect for variable loads
- Strain-hardening: creep rate depends on accumulated creep strain — correct for variable loads with stress changes
- Damage-coupled (Kachanov-Rabotnov): includes tertiary creep and rupture — necessary for life prediction
- Unified viscoplastic: combines plasticity and creep into a single inelastic strain — most general, most complex
The Norton secondary creep model
The Norton model is the simplest creep constitutive relation: the creep rate is a power law of stress, with an Arrhenius temperature term. It represents only the secondary (steady-state) creep. It has two or three material constants (A, n, Q) that must be calibrated from secondary creep data. The Norton model is widely used because it is simple, it captures the dominant stress and temperature dependence, and it is available in all FEA codes. Its limitations are that it ignores primary creep (under-predicting early strain) and tertiary creep (not predicting rupture). It is adequate for long-duration steady-state operation where the secondary stage dominates.
The Norton-Bailey model
The Norton-Bailey model extends the Norton law to include primary creep by adding a time-dependence (or strain-dependence) term. The creep rate decreases with time (or accumulated strain) during primary creep and approaches the Norton secondary rate as the primary stage ends. The Norton-Bailey model has additional constants for the primary creep exponent. It is the standard model for primary-plus-secondary creep in most FEA codes. It is adequate for moderate-duration operation where the primary stage is significant but the tertiary stage has not begun.
Damage-coupled and unified models
For life prediction, the constitutive model must include tertiary creep or a damage variable. The Kachanov-Rabotnov model introduces a damage parameter that evolves with time and stress, amplifying the effective stress and accelerating the creep rate in the tertiary stage. Unified viscoplastic models (e.g. Bodner-Partom, Miller, Chaboche) combine plasticity and creep into a single inelastic strain with internal state variables for hardening and recovery. These models are the most general but require extensive calibration data and are computationally more expensive. They are used for complex loading histories where the interaction between plasticity and creep is important.
State variables and path dependence
A creep model is more than a curve-fitting equation: it defines what the material is assumed to remember about its prior history. A simple secondary-creep law may depend only on the current stress and temperature, whereas hardening or unified models introduce internal variables representing accumulated creep strain, hardening, recovery or damage. This distinction becomes important whenever the stress, temperature or constraint changes with time. Two components can arrive at the same instantaneous stress and temperature with different prior histories and therefore different subsequent creep rates. For FEA, the analyst should understand which state variables the solver stores, how they are initialised, and whether they are carried correctly through load steps, restarts and temperature changes.
Constitutive integration and numerical checks
The material equation must also be integrated accurately over each time increment. Strong temperature sensitivity, high stress exponents and rapid early redistribution can make the creep response numerically stiff, so apparently smooth results may still be time-step dependent. A practical verification is to repeat representative periods with smaller increments and compare accumulated creep strain, stress relaxation and reaction load. Constitutive checks on a single element under constant stress, constant strain and a simple load change are also valuable before the law is used in a complex assembly. These tests separate material-model behaviour from mesh, contact and boundary-condition effects and are one of the quickest ways to identify an incorrectly parameterised creep model.
No single creep constitutive model is right for all applications. The model must be selected based on the material, the loading history, the duration and the life-limiting mechanism. Using an overly simple model may miss critical behaviour; using an overly complex model may require calibration data that is not available.