Library / Mechanical / MEC-72
Mechanical
Contact pressure at the elastic–plastic interface
Radial pressure the yielded inner zone passes to the elastic outer ring.
- Ref
- MEC-72
- Rev
- v1
- Field
- Mechanical
- Result
- q [MPa]
At the interface r = c the elastic outer ring acts as a Lamé cylinder with inner radius c and outer radius b, loaded by an internal contact pressure q and at the point of yield at its inner surface, 2qb²/(b² − c²) = σY. Solving gives q = (σY/2)(1 − c²/b²). The rest of the autofrettage pressure, σY ln(c/a), is the pressure used up crossing the plastic zone. This is the key link in the derivation of PA.
Radial pressure the yielded inner zone passes to the elastic outer ring. The ring is a Lamé cylinder that has just reached yield at its inner surface, which fixes q; the rest of the autofrettage pressure is carried by the plastic zone.
Calculator
- Pressure used crossing the plastic zone, σY ln(c/a)100.9 MPa
- Autofrettage pressure, q + σY ln(c/a)177.4 MPa
- Interface radius c70 mm
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.
- Of the 177.4 MPa autofrettage pressure, 76.5 MPa (43.1 %) is carried by the elastic ring and 100.9 MPa by the plastic zone.
- As the front approaches the outside (f → 1) q falls to zero: nothing elastic is left to hold the yielded material, and PA approaches the fully plastic limit.
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 | 25.5 | 382.5 | 357 | +10% | +10 % |
| a Inner radius (bore) | 10 to 150 mm | 2.39 | 118.3 | 115.9 | +10% | −8.41 % |
| f Target plastic fraction of the wall | 0.05 to 0.95 | 7.406 | 108.7 | 101.3 | +10% | −5.57 % |
| b Outer radius | 60 to 300 mm | 28.5 | 112.5 | 84 | +10% | +7.34 % |
Sweep of Target plastic fraction of the wall
| Target plastic fraction of the wall | Interface contact pressure (MPa) |
|---|---|
| 0.05 | 108.7 |
| 0.1625 | 99.32 |
| 0.275 | 89.04 |
| 0.3875 | 77.81 |
| 0.5 | 65.63 |
| 0.6125 | 52.49 |
| 0.725 | 38.41 |
| 0.8375 | 23.38 |
| 0.95 | 7.406 |
Sweep one variable
Variables and how each one contributes
Interface contact 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 q, because c moves inward with the bore. Raising it by 5 mm (to 55 mm) moves the result by −8.41 %, from 76.5 to 70.06 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 q: a thicker elastic ring can carry more. Raising it by 5 mm (to 105 mm) moves the result by +3.88 %, from 76.5 to 79.47 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 q in direct proportion. Raising it by 25 MPa (to 325 MPa) moves the result by +8.33 %, from 76.5 to 82.88 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. Lowers q: the further the front moves out, the thinner the elastic ring that carries it. Raising it by 0.05 (to 0.45) moves the result by −6.99 %, from 76.5 to 71.16 MPa.
Worked example with the values above
- Interface radius: c = 70 mm, so c²/b² = 0.49.
- Ring factor: 1 − c²/b² = 0.51.
- q = (σY/2) × 0.51 = 150 × 0.51 = 76.5 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.
Common mistakes
- Treating q as an extra pressure to be added to PA: it is already included in it.
- Mixing units: a, b and r must share one length unit and the strengths and pressures must all be in MPa.
Statistics and uncertainty
q depends on the yield strength in direct proportion and on the radii through c²/b², so the scatter in yield strength is passed straight on. The split between the ring and the plastic zone is a property of the model and is not measured directly.
Where it is used
- Following the derivation of the autofrettage pressure step by step.
- Checking a finite-element contact pressure at the plastic front.
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