Creep Strain Accumulation
How creep strain accumulates over time under constant and variable loading — the integration of the creep rate, the effect of stress and temperature history, and the assessment of accumulated strain against design limits.
Accumulation under constant load
Under a constant load at constant temperature, the creep strain accumulates according to the creep curve. The total creep strain at any time is the integral of the creep rate from zero to that time. The creep rate is given by the constitutive model (e.g. Norton-Bailey for primary and secondary, or a damage model for tertiary). For a secondary-creep-only model, the accumulated creep strain is simply the minimum creep rate multiplied by the time. For models that include primary creep, the early-time accumulation is faster. For models that include tertiary creep, the late-time accumulation accelerates towards rupture.
Accumulation under variable loading
Under variable loading (varying stress or temperature), the creep strain accumulates by integrating the creep rate over the loading history. At each instant, the creep rate is evaluated at the current stress and temperature. The accumulated strain is the time integral of the instantaneous creep rate. The constitutive model must define how the creep rate depends on the current state (stress, temperature, accumulated strain, damage). The treatment of the loading history — whether the creep rate depends on time (time hardening) or on the accumulated strain (strain hardening) — has a significant effect on the predicted accumulation under variable loads.
The strain limit criterion
A common design criterion limits the total accumulated creep strain to a specified value (e.g. 1% or 2%). This prevents excessive deformation, loss of clearance or functional failure. The strain limit is typically applied to the total inelastic strain (creep plus plasticity) over the design life. The analysis must integrate the creep strain over the full service history and compare the accumulated strain to the limit. In regions of stress concentration, the local strain may exceed the limit even if the nominal strain is acceptable — the assessment must be made at the critical locations.
Stress and temperature history effects
The creep strain accumulation depends on the full stress and temperature history, not just the current values. A component that experiences a high-temperature excursion early in life may accumulate significant primary creep strain that persists. A component that experiences cyclic loading may accumulate creep strain during each hold period. The order of the loading matters if the creep rate depends on the accumulated strain (strain hardening) — a high-stress period followed by a low-stress period gives a different result from the reverse. The analysis must use the actual or representative service history.
Assessment against design limits
The accumulated creep strain must be assessed against the applicable design limits. These may be specified in design codes (e.g. ASME, EN, API) or in project-specific criteria. The limits may include: a maximum total strain, a maximum creep strain per unit time (creep rate limit), a maximum strain at a specific location (e.g. a stress concentration), or a strain-based damage fraction for creep-fatigue interaction. The assessment should consider the uncertainty in the creep properties and the loading history — a single deterministic calculation may not be adequate for safety-critical components.
Creep strain accumulation is path-dependent — the order and duration of stress and temperature excursions matters. Always use the actual or representative service history in the analysis, and integrate the creep rate over the full history. Do not assess creep strain from a single nominal condition.
History dependence and numerical integration
Accumulated creep strain is the time integral of the instantaneous creep strain rate, so any change in stress or temperature changes the future accumulation rate. In a real structure, stress itself evolves because creep changes the deformation and redistributes load. The problem is therefore coupled and history-dependent: calculating strain from the initial stress and multiplying by total time is generally valid only for very simple constant-stress conditions. Transient FEA updates the stress, temperature and creep increment sequentially, which is why the time discretisation must resolve periods of rapid change such as start-up, initial relaxation or load transfer.
Engineering measures of accumulated deformation
The most useful strain measure depends on the failure mode. Equivalent creep strain is convenient for mapping general deformation, but axial strain may govern elongation, hoop creep strain may govern diameter growth, and through-thickness or local principal strains may be more relevant to cracking or clearance. For bolted assemblies, loss of preload can be a more meaningful output than the peak creep strain itself. Results should therefore be reduced to quantities tied to function and integrity, with histories plotted at critical locations so that the analyst can see whether deformation is stabilising, continuing approximately linearly or accelerating towards an unacceptable state.