High-Temperature Stress-Strain Behaviour
The stress-strain response of materials at elevated temperature — instantaneous elastic-plastic response, time-dependent creep strain, the combined total strain, and the interaction between plasticity and creep.
The total strain decomposition
At elevated temperature, the total strain in a component under load is the sum of several components: the elastic strain, the thermal strain, the instantaneous plastic strain and the time-dependent creep strain. Each component has a different physical origin and a different dependence on stress, temperature and time. The structural analysis must decompose the total strain correctly and track each component. The stress depends on the elastic strain (through Hooke's law), so the redistribution of strain among the components affects the stress state.
Total strain decomposition: eps_total = eps_elastic + eps_thermal + eps_plastic + eps_creep where: eps_elastic = sigma / E(T) [recoverable] eps_thermal = integral(alpha(T) dT) [recoverable] eps_plastic = from yield surface flow rule [permanent] eps_creep = from creep constitutive law [permanent] sigma = stress [Pa] E(T) = temperature-dependent elastic modulus [Pa] alpha(T) = temperature-dependent CTE [1/K]
Instantaneous elastic-plastic response
When a load is applied at elevated temperature, the material responds instantaneously with an elastic strain (proportional to the stress and inversely proportional to the temperature-dependent modulus) and, if the stress exceeds the temperature-dependent yield, a plastic strain. The stress-strain curve at elevated temperature is softer than at room temperature — lower modulus, lower yield, lower hardening. The instantaneous response is the starting point for the subsequent creep. If the load is high enough to cause instantaneous plasticity, the plastic strain provides a starting offset for the creep strain accumulation.
Time-dependent creep strain
After the instantaneous response, the creep strain begins to accumulate. The creep strain rate depends on the current stress, the temperature and the accumulated creep strain. At a constant stress, the creep strain follows the classic three-stage curve: primary (decelerating), secondary (steady) and tertiary (accelerating). The total strain at any time is the instantaneous strain plus the accumulated creep strain. The creep strain is permanent — it is not recovered when the load is removed (though there is a small recoverable anelastic component in some materials).
Plasticity-creep interaction
The interaction between instantaneous plasticity and creep is an important and sometimes neglected aspect of high-temperature behaviour. If the component yields under the applied load, the plastic strain changes the material state (the dislocation density, the hardness) and may affect the subsequent creep rate. Some constitutive models treat plasticity and creep as separate mechanisms (superimposed strain rates); others treat them as a single inelastic strain (unified models). The choice of model affects the predicted deformation, particularly in the transition region where both plasticity and creep are significant. The model should be validated against test data that exercises both mechanisms.
Cyclic loading at high temperature
Under cyclic loading at high temperature, each cycle produces instantaneous plastic strain and creep strain during the hold periods. The creep strain accumulates cycle by cycle, and the plastic strain may produce fatigue damage. The interaction between creep and fatigue is a critical life-limiting mechanism in many high-temperature components. The analysis must track the creep strain per cycle, the plastic strain range per cycle, and the combined damage accumulation. This is the domain of creep-fatigue interaction assessment.
Recovery and reverse creep
When the load is removed at elevated temperature, the elastic strain is recovered instantaneously. The creep strain is not recovered (it is permanent), but a small anelastic strain may recover over time. If the load is reversed (compression after tension), the creep may continue in the new direction, though the prior creep history affects the rate. Recovery and reverse creep are important in components with cyclic or variable loads. Some constitutive models capture recovery; others do not. The model should be selected based on whether the loading history includes significant unloading or reversal.
The total strain at high temperature is the sum of elastic, thermal, plastic and creep components. An analysis that omits any component gives an incorrect strain and hence an incorrect stress distribution. Always decompose the strain correctly and use a constitutive model that captures all relevant components.
Separating instantaneous and time-dependent inelasticity
At elevated temperature the analyst must distinguish instantaneous plastic strain from creep strain even though both are permanent. Plasticity is driven primarily by the current stress state relative to the temperature-dependent yield surface, whereas creep evolves with time under sustained stress. During a thermal or mechanical transient the structure may yield first and then creep during the subsequent dwell. On unloading, the residual stress field produced by plasticity changes the stresses available to drive creep in the next stage. Constitutive models that simply add an elastic-plastic law and a creep law can be effective, but their interaction should be checked against the material behaviour expected for the alloy and temperature range.
Multiaxial stress and structural interpretation
Real components rarely experience the uniaxial conditions used to generate material curves. Pressure boundaries, welds, notches and rotating hardware develop multiaxial membrane, bending and local stresses. The FEA therefore requires an appropriate multiaxial flow formulation and a sensible interpretation of equivalent stress, principal stress and hydrostatic stress. Equivalent stress may control deformation rate in many metal creep models, while rupture and cavity growth can be sensitive to stress triaxiality. A good assessment does not report only a single von Mises contour; it checks the stress components and their evolution to understand whether the constitutive assumptions remain appropriate in the locations that govern life.