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Mechanical
Plastic zone radius from a prescribed autofrettage pressure
Radius of the yield front when the autofrettage pressure is given rather than chosen.
- Ref
- MEC-79
- Rev
- v1
- Field
- Mechanical
- Result
- c [mm]
The pressure relation PA = σY[ln(c/a) + ½(1 − c²/b²)] cannot be solved for c in closed form, so a prescribed pressure is turned into a root of g(c) = ln(c/a) + ½(1 − c²/b²) − PA/σY = 0 on the interval a < c < b, found here by bisection. The derivative g′(c) = 1/c − c/b² is positive on the interval (since c < b), so g increases steadily and there is exactly one root when PY < PA < PL. Pressures outside that window are rejected for a partial-plasticity calculation.
Radius of the yield front when the autofrettage pressure is given rather than chosen. It has no closed form, so it is found by bisection; there is exactly one root when the pressure lies between first yield and the fully plastic limit.
Calculator
- Plastic fraction of the wall, (c − a)/(b − a)0.4
- Plastic zone thickness, c − a20 mm
- First-yield pressure PY112.5 MPa
- Fully plastic limit PL207.9 MPa
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.
- A pressure of 177.4 MPa on a 50 to 100 mm cylinder in 300 MPa steel pushes the yield front to 70 mm, 40 % of the way through the wall.
- The front is sensitive to pressure: raising the pressure by 5 % to 186.3 MPa moves it to 74.46 mm, an extra 4.464 mm of plastic zone.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Inner radius (bore) | 10 to 150 mm | 11.02 | 76.14 | 65.12 | +10% | +31.6 % |
| PA Peak autofrettage pressure | 50 to 600 MPa | 52.51 | 81.28 | 28.77 | +10% | +14.5 % |
| b Outer radius | 60 to 300 mm | 55.74 | 79.85 | 24.11 | +10% | −6.61 % |
| σY Uniaxial yield strength | 100 to 1,500 MPa | 52.86 | 74.12 | 21.26 | +10% | −9.24 % |
Sweep of Peak autofrettage pressure
| Peak autofrettage pressure (MPa) | Elastic–plastic interface radius (mm) |
|---|---|
| 50 | — |
| 118.8 | 51.42 |
| 187.5 | 75.13 |
| 256.3 | — |
| 325 | — |
| 393.8 | — |
| 462.5 | — |
| 531.3 | — |
| 600 | — |
Sweep one variable
Variables and how each one contributes
Elastic–plastic interface radius
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. Raises the radius of the front, strongly: a larger bore lowers both the first-yield pressure and the fully plastic limit, so the same pressure sits nearer the limit. Raising it by 5 mm (to 55 mm) moves the result by +31.6 %, from 70 to 92.13 mm.
Outer radius
The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Lowers the radius of the front: a thicker wall resists the same pressure better, so less of it yields. Raising it by 5 mm (to 105 mm) moves the result by −3.9 %, from 70 to 67.27 mm.
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. Lowers the radius of the front: a stronger material yields less of the wall under the same pressure. Raising it by 25 MPa (to 325 MPa) moves the result by −7.96 %, from 70 to 64.43 mm.
Peak autofrettage pressure
The peak internal pressure applied during autofrettage, in MPa. For a partial treatment it must lie between the first-yield pressure and the fully plastic limit. Raises the radius of the front, increasingly steeply as the pressure nears the fully plastic limit. Raising it by 5 MPa (to 182.4 MPa) moves the result by +3.44 %, from 70 to 72.41 mm.
Worked example with the values above
- Define g(c) = ln(c/a) + ½(1 − c²/b²) − PA/σY with PA/σY = 0.59147, and bisect on 50 < c < 100 mm.
- At the root c = 70 mm: ln(c/a) + ½(1 − c²/b²) = 0.59147, which equals PA/σY.
- c = 70 mm, a plastic zone 20 mm thick, which is 0.4 of the wall.
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.
Common mistakes
- Giving a pressure outside the window between PY and PL and expecting a plastic zone.
- Treating the bisection result as measured: it is the idealised front.
Statistics and uncertainty
Since the pressure is proportional to σY for a given front, a ±5 % scatter in yield strength gives a clearly different front for the same pump pressure. The steepness of the relation near the limit pressure PL means that small pressure errors can move the front a long way at high fractions.
Where it is used
- Working backwards from a pump or press pressure to the plastic zone it produces.
- Checking a treatment against its intended plastic fraction.
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
- Bisection: finding a root by repeatedly halving an interval