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Mechanical
Elastic–plastic interface radius from a target plastic fraction
Radius to which yielding has spread at peak autofrettage pressure, set by the fraction f of the wall thickness you choose to plastify.
- Ref
- MEC-68
- Rev
- v1
- Field
- Mechanical
- Result
- c [mm]
The radius c marks how far yielding has spread when the autofrettage pressure is at its peak. The zone from the bore (r = a) out to c has yielded and the ring from c to the outside (r = b) is still elastic. f is a fraction of the radial thickness, not a plastic strain and not an area fraction. Geometry and yield strength alone do not fix a pressure: a treatment target such as f must be chosen first.
Radius to which yielding has spread at peak autofrettage pressure, set by the fraction f of the wall thickness you choose to plastify. Inside it the material has yielded; outside it the wall is still elastic.
Calculator
- Plastic zone thickness, c − a20 mm
- Elastic ring thickness, b − c30 mm
- Share of the cross-sectional area that has yielded, (c² − a²)/(b² − a²)32 %
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.
- With a 50 mm bore, a 100 mm outer radius and a target of 40 % of the wall, yielding reaches a radius of 70 mm: 20 mm plastic and 30 mm elastic.
- The same 40 % of the thickness covers 32 % of the cross-sectional area, because area grows with radius. Quoting a plastic fraction without saying which kind is a common source of confusion.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Outer radius | 60 to 300 mm | 54 | 150 | 96 | +10% | +5.71 % |
| a Inner radius (bore) | 10 to 150 mm | 46 | 130 | 84 | +10% | +4.29 % |
| f Target plastic fraction of the wall | 0.05 to 0.95 | 52.5 | 97.5 | 45 | +10% | +2.86 % |
Sweep of Target plastic fraction of the wall
| Target plastic fraction of the wall | Elastic–plastic interface radius (mm) |
|---|---|
| 0.05 | 52.5 |
| 0.1625 | 58.13 |
| 0.275 | 63.75 |
| 0.3875 | 69.38 |
| 0.5 | 75 |
| 0.6125 | 80.63 |
| 0.725 | 86.25 |
| 0.8375 | 91.88 |
| 0.95 | 97.5 |
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 c by (1 − f) of the change, because the bore moves the start of the wall. Raising it by 5 mm (to 55 mm) moves the result by +4.29 %, from 70 to 73 mm.
Outer radius
The outer radius of the cylinder, in mm, in the same length unit as a. It must be larger than a. Raises c by f of the change, because the outside moves the end of the wall. Raising it by 5 mm (to 105 mm) moves the result by +2.86 %, from 70 to 72 mm.
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. Moves c in direct proportion, by (b − a) for each unit of f. Raising it by 0.05 (to 0.45) moves the result by +3.57 %, from 70 to 72.5 mm.
Worked example with the values above
- Wall thickness: b − a = 100 − 50 = 50 mm.
- Plastic part: f (b − a) = 0.4 × 50 = 20 mm.
- c = a + 20 = 70 mm.
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
- Reading f as a plastic strain or as an area fraction.
- Mixing units: a, b and r must share one length unit and the strengths and pressures must all be in MPa.
Statistics and uncertainty
This is pure geometry, so the only scatter is the tolerance on the radii: c moves by (1 − f) of any change in a and by f of any change in b. In practice the pressure that actually produces a chosen front is far less certain than c itself, because it depends on the yield strength of the real material.
Where it is used
- Turning a treatment target into a radius before the autofrettage pressure is calculated.
- Specifying where in the wall the plastic front should end up, for comparison with a measured or finite-element front.
Maths you need
- Fraction: part of a whole
- Thick-walled cylinder: a wall too thick for the thin-wall formula