Creep and rupture explained

Creep is a time-dependent deformation that occurs when a material is subjected to constant stress at a high temperature (usually above about $0.4 T_m$ times its melting temperature, in Kelvin).

Unlike elastic or plastic deformation, which occur almost instantaneously upon loading, creep deformation develops gradually over time. Creep is especially relevant for materials in high-temperature applications like turbines, boilers, and components in the aerospace and energy industries.

1. Stages of Creep

Creep generally occurs in three stages:

Primary (Transient) Creep

During the initial loading phase, the creep rate decreases over time. This is due to the material hardening as it deforms, which increases its resistance to further deformation. The strain $(\epsilon)$ as a function of time $(t)$ can be approximately described by: $\epsilon = \epsilon_0 + \epsilon_1 e^{-\alpha t}$ where:

$\epsilon_0$ : initial strain,
$\epsilon_1$ : strain increment,
$\alpha$ : a material constant.

Secondary (Steady-State) Creep

In this stage, the material undergoes a constant creep rate $(\dot{\epsilon})$ , balancing hardening and recovery mechanisms. The steady-state strain rate $(\dot{\epsilon}_{ss})$ is crucial for engineering calculations, and it depends on stress $(\sigma)$ and temperature $(T)$ through an Arrhenius relationship:

$\dot{\epsilon}_{ss} = A \cdot \sigma^n \cdot e^{-Q / (R \cdot T)}$

where:

$A$ : material constant,
$\sigma$ : applied stress,
$n$ : stress exponent,
$Q$ : activation energy for creep,
$R$ : gas constant,
$T$ : absolute temperature.

This steady-state regime is often the longest and most predictable, so it’s essential for lifetime predictions.

Tertiary Creep

Eventually, microstructural degradation, such as void formation or micro-cracking, leads to an accelerated creep rate. This stage culminates in creep rupture or fracture. The strain rate accelerates due to the loss of load-bearing area and microstructural integrity: $\epsilon = \epsilon_0 + \epsilon_1 e^{\beta t} + \epsilon_2 e^{\gamma t}$ where $\beta$ and $\gamma$ are material-dependent constants that characterise tertiary acceleration.

2. Mechanisms of Creep

Creep can be attributed to several microscopic mechanisms, which vary depending on the material and temperature:

Dislocation Climb: At higher temperatures, dislocations can “climb” by diffusing atoms, allowing them to overcome obstacles. Dislocation climb is governed by vacancy diffusion and occurs more readily at higher temperatures.
Diffusional Creep (Nabarro-Herring): At high temperatures and low stresses, atoms diffuse through the grain boundaries or within the grains to accommodate strain. This process leads to elongation along the stress axis and shortens the perpendicular axes.
Grain Boundary Sliding: Under stress, grains can slide along their boundaries. At elevated temperatures, grain boundaries become more mobile, allowing this to happen more easily.

3. Creep Rupture

When materials reach the tertiary creep stage, they may eventually experience creep rupture due to the accumulation of microstructural damage like voids and cracks. The time to rupture $(t_r)$ under a constant stress can be empirically estimated by the Monkman-Grant equation: $t_r \dot{\epsilon}_{ss}^m = C$ where:

$t_r$ : time to rupture,
$\dot{\epsilon}_{ss}$ : steady-state creep rate,
$m$ : material constant, often close to $-1$ ,
$C$ : empirical constant.

Another useful model is the Larson-Miller Parameter (LMP), which correlates time to rupture with temperature and stress: $LMP = T (C + \log t_r)$ where $C$ is an empirically derived constant.

This approach allows for the extrapolation of rupture life under different temperatures and stresses, which is critical for designing components that will experience long-term, high-temperature loading.

4. Norton-Bailey Law for Creep

The creep strain over time for materials that follow the Norton-Bailey law can be expressed as: $\epsilon(t) = K \sigma^n t^m$ where:

$K$ : material constant,
$\sigma$ : applied stress,
$n$ : stress exponent,
$m$ : time exponent.

The Norton-Bailey law is useful for cases where both stress and time dependence are significant, often giving insight into the relationship between stress level, temperature, and expected creep deformation.

5. Creep Deformation Models and Life Prediction

Several phenomenological models and equations help predict the creep life of components. One widely used is Norton’s Law, which describes the steady-state creep rate as a function of stress and temperature. Another is the θ (theta) Projection Model, which uses parameters derived from experimental data to model creep strain over time, aiding long-term predictions.

Summary

Creep and rupture are critical considerations for materials under high-temperature stress. By understanding the stages, mechanisms, and predictive models (e.g., steady-state creep, Larson-Miller, and Monkman-Grant relationships), engineers can design components to withstand prolonged high-temperature service, minimising the risk of catastrophic failure.

Product Development Engineers Ltd

1. Stages of Creep

Primary (Transient) Creep

Secondary (Steady-State) Creep

Tertiary Creep

2. Mechanisms of Creep

3. Creep Rupture

4. Norton-Bailey Law for Creep

5. Creep Deformation Models and Life Prediction

Summary

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Creep and rupture explained

1. Stages of Creep

Primary (Transient) Creep

Secondary (Steady-State) Creep

Tertiary Creep

2. Mechanisms of Creep

3. Creep Rupture

4. Norton-Bailey Law for Creep

5. Creep Deformation Models and Life Prediction

Summary

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