Turbine Blade Creep - 3D Spin Rig

Why a hot blade stretches

The blade of a high-pressure turbine spins at about 10,000 rpm and sits in gas hotter than the metal can stand without cooling air. Spinning loads it in tension: for a uniform blade between hub radius rh and tip radius rt = rh + l the stress at the root is s = integral(rho w^2 r dr) = rho w^2 (rt^2 - rh^2) / 2, about 330 MPa for a 100 mm nickel-alloy blade at 10,000 rpm. Cooling lowers the metal temperature to Tm = Tg - phi (Tg - Tc), around 1140 K here. At roughly 0.6 of the melting point the metal creeps: it deforms slowly and permanently under a stress well below its yield stress.

Norton law and the three stages

Dislocation creep follows de/dt = A s^n exp(-Q / R T) (Norton power law with an Arrhenius factor; here n = 6, Q = 500 kJ/mol). After a short primary stage the strain rate is nearly constant (secondary creep). Then voids and cracks grow and the load-bearing section shrinks; the Kachanov-Rabotnov damage variable w (0 = intact, 1 = ruptured) obeys dw/dt = B s^chi exp(-Q/RT) / (1 - w)^phi, and the effective stress is s / (1 - w), so de/dt = A (s/(1-w))^n exp(-Q/RT) accelerates: tertiary creep. With chi = n = phi the rupture time has a closed form, t_r = 1 / ((phi + 1) B s^chi exp(-Q/RT)), and the strain at rupture is e_r = A / B. Monkman and Grant noticed that e'min x t_r is roughly constant; in this model it is exactly e_r/(phi+1).

Larson-Miller master curve

Engineers extrapolate short hot tests to long lives with P = T (C + log10 t_r), C near 20 (T in kelvin, t_r in hours). Stress plotted against P falls on one curve for a given alloy. Because life depends on temperature through the exponential, 25 K more costs about a factor 3 in life here, and a 10 % overspeed (stress +21 %) costs a factor 3 as well (s^-6).

Tip clearance

The casing is cold-fixed, so the gap closes by alpha (Tm - 300) l (thermal growth), by s/E(T) l (elastic stretch) and by e l (creep). When the gap reaches zero the tip rubs. A rub often ends a blade’s life before rupture does; the model reports both times. Growth is drawn exaggerated (slider) so it is visible; the numbers are real.

What is simplified

The constants are tuned so that a single-crystal blade at 334 MPa and 1140 K lasts about 25,000 h, which is the right order of magnitude but not a datasheet. One uniform blade, one temperature, constant stress (the stress field of a real aerofoil and the temperature gradient are not resolved); no oxidation, thermal-mechanical fatigue, primary creep or coating failure; damage is a single scalar. Animation compresses the whole life into the playback time you choose.

• Hero blade (red band): the instrumented blade. Its middle section stretches and thins; its tip moves outward. Blade colour follows the metal temperature.
• Casing ring: fixed. The gap between the hero tip and the ring is the tip clearance, measured by the probe at the top.
• Board and monitor: the creep curve (dashed = prediction, green = this blade) and live readings.
• Graphs: creep strain, damage and clearance, Larson-Miller master curves, and rupture life against temperature.

• Run the nominal blade: slow, nearly linear creep, then a sudden acceleration just before rupture (tertiary creep).
• Pick "Overtemperature": weaker cooling raises the metal 85 K and the life collapses from years to weeks.
• Select "Overspeed 110 %" and compare with the nominal life: stress goes up 21 % and life divides by about 3.
• Choose "Tight clearance": the tip touches the casing long before the rupture, with sparks.
• Drag the "Life used" slider to scrub through the blade’s life, or watch the Larson-Miller graph for how stress, temperature and life trade off.