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Mechanical
Autofrettage pressure for a target plastic fraction
Peak internal pressure that pushes the yield front to radius c in a thick cylinder, found from radial equilibrium in the plastic zone and the Lamé field in the elastic ring outside it.
- Ref
- MEC-71
- Rev
- v1
- Field
- Mechanical
- Result
- PA [MPa]
PA is the peak pressure that drives the yield front out to the radius c = a + f(b − a). Radial equilibrium in the yielded zone, with σθ − σr = σY, gives σr(r) = −PA + σY ln(r/a). At r = c the elastic outer ring (a Lamé cylinder just at yield at its inner surface) carries a contact pressure q = (σY/2)(1 − c²/b²), and PA = q + σY ln(c/a). The result lies between PY (f = 0) and PL (f = 1). A chosen f gives a candidate pressure, not an optimum.
Peak internal pressure that pushes the yield front to radius c in a thick cylinder, found from radial equilibrium in the plastic zone and the Lamé field in the elastic ring outside it. It lies between the first-yield pressure and the fully plastic limit.
Calculator
- Interface radius, c = a + f(b − a)70 mm
- First-yield pressure PY112.5 MPa
- Fully plastic limit PL207.9 MPa
- Position in the window, (PA − PY)/(PL − PY)0.6804
- Equivalent pressure in bar1,774 bar
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- For a 50 mm bore, 100 mm outer radius, σY = 300 MPa and f = 0.4, the pressure is 177.4 MPa, about 1,774 bar. It lies between first yield (112.5 MPa) and the fully plastic limit (207.9 MPa), 68 % of the way up.
- Plastifying more of the wall costs more pressure: raising f by 0.2 to 0.6 lifts the pressure to 195 MPa.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| σY Uniaxial yield strength | 100 to 1,500 MPa | 59.15 | 887.2 | 828.1 | +10% | +10 % |
| a Inner radius (bore) | 10 to 150 mm | 4.008 | 576.1 | 572.1 | +10% | −12.6 % |
| b Outer radius | 60 to 300 mm | 51.59 | 442.1 | 390.5 | +10% | +12.6 % |
| f Target plastic fraction of the wall | 0.05 to 0.95 | 123.3 | 207.8 | 84.46 | +10% | +2.36 % |
Sweep of Target plastic fraction of the wall
| Target plastic fraction of the wall | Autofrettage pressure (MPa) |
|---|---|
| 0.05 | 123.3 |
| 0.1625 | 144.5 |
| 0.275 | 161.9 |
| 0.3875 | 176.1 |
| 0.5 | 187.3 |
| 0.6125 | 195.8 |
| 0.725 | 202 |
| 0.8375 | 205.9 |
| 0.95 | 207.8 |
Sweep one variable
Variables and how each one contributes
Autofrettage pressure
What the equation gives you. Everything else feeds it.
Inner radius (bore)
The inner (bore) radius of the cylinder, in mm, in the same length unit as b. It must be smaller than b. Lowers PA: a larger bore leaves a thinner wall to resist the pressure. Raising it by 5 mm (to 55 mm) moves the result by −12.6 %, from 177.4 to 155 MPa.
Outer radius
The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Raises PA, because the elastic ring outside the plastic front becomes thicker and carries more. Raising it by 5 mm (to 105 mm) moves the result by +6.44 %, from 177.4 to 188.9 MPa.
Uniaxial yield strength
The uniaxial yield strength of the material at the autofrettage temperature, in MPa. The model treats the material as elastic and perfectly plastic and uses the Tresca criterion, so the wall yields where the hoop stress minus the radial stress reaches this value. Raises PA in direct proportion. Raising it by 25 MPa (to 325 MPa) moves the result by +8.33 %, from 177.4 to 192.2 MPa.
Target plastic fraction of the wall
The fraction of the wall thickness (b − a) that is yielded at peak pressure, a plain number between 0 and 1. A value of 0 means yielding only just begins and 1 means the whole wall has yielded. The slider runs from 0.05 to 0.95. Raises PA, increasingly steeply as the front nears the outside, because the pressure approaches the fully plastic limit. Raising it by 0.05 (to 0.45) moves the result by +2.92 %, from 177.4 to 182.6 MPa.
Worked example with the values above
- Interface radius: c = 50 + 0.4 × (100 − 50) = 70 mm.
- Logarithm and ring terms: ln(c/a) = 0.3365 and ½(1 − c²/b²) = 0.255, which add to 0.5915.
- PA = σY × that = 300 × 0.5915 = 177.4 MPa.
Assumptions and limits
- A long, uniform, single-piece thick cylinder with uniform internal pressure, no external pressure, small deformation and an initially stress-free isotropic material.
- Elastic, perfectly plastic behaviour with the Tresca criterion at constant temperature, and a hoop stress that is the largest principal stress (σθ ≥ σz ≥ σr). The axial stress must be checked separately: open ends, closed ends and axial restraint give different states.
- End regions, openings and attachments are ignored. This is a teaching and preliminary-analysis model, not a qualified manufacturing specification.
- Elastic unloading must follow the loading; see the reverse-yielding screening ratio before relying on residual stresses.
Common mistakes
- Using this pressure as an optimum: a chosen f is a target, not a best treatment.
- Entering f as a percentage (40) instead of a fraction (0.4).
- Using these cylinder formulas for a sphere, a non-uniform wall, nozzles, heads, welded or layered construction, significant axial restraint or large deformation. Those need an elastoplastic finite-element model of the real geometry and load path.
Statistics and uncertainty
PA is proportional to σY, so the pressure that produces a given front scatters with the yield strength: a ±5 % scatter gives about ±5 % in pressure. Equivalently, at a fixed pressure the plastic front is less well defined than its target, which is why practical treatments measure the result rather than rely on the formula alone.
Where it is used
- Setting the pressure for hydraulic autofrettage of gun barrels, injection bodies and high-pressure cylinders.
- Checking a quoted treatment pressure against the plastic zone it should give.
Maths you need
- Thick-walled cylinder: a wall too thick for the thin-wall formula
- Tresca criterion: yield when the largest stress difference reaches σY