# Force Analysis: Tresca Failure Criterion, Shakedown, Ratcheting

### Increasing Contact Stress...towards the 'Static-Load Capacity' of the Cam.

#### Tresca Maximum Shear Stress Failure Criterion

As we increase the contact-force/payload between the cam and the cam-follower, the weaker material will eventually deform plastically rather than elastically.

To specify the critical load at which the weaker material 'plastically fails', we need to use a yield criterion. We can find the yield of metals with the 'Tresca Maximum Shear Stress' or the 'von-Mises Strain Energy' failure criteria. We will use the Tresca Maximum Shear Stress Failure Criterion.

Simplified, the Tresca Shear Stress failure criterion tells us that Yield occurs due to Shear Stress when the Shear Stress becomes equal to half the Yield Strength that results from a normal tension test.

 $k=\frac{Y}{2}$ Y = Yield Strength k = Shear Strength Yield strength usually has the label as Re in material tables that yield, or Rp0.2 for brittle materials that do not have a yield point.

#### Elastic Limit, or 'Onset of Plasticity' [Contact Stress ≈ 1.6×Y]

We can use the Contact Stress and Maximum Shear Stress failure criterion to determine the 'onset of plasticity'.

We know that...

${\tau }_{\mathrm{max}}=0.3×{P}_{\mathrm{max}}$

Thus, when...

${\tau }_{\mathrm{max}}=k$

...then...

${\tau }_{\mathrm{max}}=k=0.3×{P}_{\mathrm{max}}=\frac{Y}{2}$

Then, we can state that the 'onset of plasticity', for line contact, occurs when...

$k∕0.3={P}_{\mathrm{max}}=Y∕2×0.3$

or, equivalently, ...

${P}_{\mathrm{max}}=3.3k=1.67Y$

Plasticity

Plasticity starts when the:

 • Maximum Shear Stress, τmax = Y/2 , Y is the yield strength of a part under Axial Tension.

or, when the

 • Maximum Contact Stress (surface) Pmax = 1.67.Y

Plasticity starts at a depth of 0.78b below the surface, for line contact.

Note: If the Cam-Follower is a crown, barrel, or spherical shaped roller, then the stress at which plasticity starts does not change much. However, the depth at which Plasticity starts decreases to approximately 0.33b.

#### Elastic Shakedown  [1.6×Y < Contact Stress <2×Y]

When = 1.67 × Y [or τmax = Y/2] there is an 'onset of plasticity' [yield begins] that starts at 0.78b below the surface.

Even if some yield takes place, there will only be a small change to the shape. This is because the yield has occurred beneath the surface and the plastic zone is surrounded by a region in which the stress and strains are elastic. This limits the extent of plastic deformation since the plastic strains must be of the same order as the adjacent elastic strains. The material displaced by the flattening of the contact is accommodated by an elastic expansion of the surrounding volume.

When the cam-follower passes a point on the cam the first time, and the elastic limit is exceeded, some plastic deformation will occur. When the cam-follower continues along the cam so that load becomes zero again, residual stresses remain in the cam, below the surface.

When the cam-follower passes the same point a second time, the cam is subject to the combined action of the contact-stress and the residual stress from the first pass of the cam-follower. Generally, residual stresses are 'protective', in the sense that further yielding is less likely to occur when the cam-follower makes a second pass than when it made its first pass.

It is possible, that after a few passes of the cam-follower, the contact force can be carried elastically once again, because of the build up of the protective residual stresses. The Elastic Limit has increased because of the protective Residual Stresses. The process is called Elastic Shakedown.

Other factors that help the 'Shakedown' phenomenon are:

 • Strain-Hardening: where the cam steel may become harder and an increase in yield strength

Note: Materials that have a high initial yield strength typically strain-weaken. Materials with a low initial yield strength typically Strain-Harden].

 • Geometric-Changes: if there is any deformation, then it will tend to increase the conformity of the cam and the cam-follower (a 'cup' forms for the roller). This will decrease the contact-stress.

The new elastic-limit after 'shakedown' can increase to: 4.0 × k [the initial Shear Strength]

Importantly, according to theory not presented here, that 'if it is possible for residual stresses to occur, they will always occur'.

It turns out that the Hertzian Contact Stress must be less than or equal to 4.00×k for Elastic Shakedown to occur.

This means that if the contact stress is.such that:

...then, although there is some yield in the first machine cycle, Elastic Shakedown will occur.

Also, as the load, F, is proportional to the square of maximum contact stress, , it follows that the ratio of the Shakedown Force limit, , to the Elastic Force Limit, , is given by

This means that the actual load can be increased by ~50% more than the load that is needed to bring about yield on the first machine cycle.

#### Full Plasticity [Contact Stress > 2.83×Y]

 As the contact stress increases, the plastic zone grows until eventually it reaches the cam surface. Full Plasticity occurs when the Contact Stress reaches the surface. The Contact Stress is then: [The cam will fail with Low Cycle Fatigue - from Rail Paper: The theory that predicts this is 'upper bound' theory, which postulates plastic flow. Equating the rate at which the external load does work to the rate at which energy is absorbed within the material provides an upper bound on the true maximum load the contact can sustain. Personally, I have no idea about this theory]

#### Shakedown, Ratcheting, Traction / Friction between the Cam-Follower and the Cam

 A: Upper Bound to elastic-shakedown limit against alternating plasticity B: Upper Bound to plastic shakedown limit against incremental growth. C: Upper bound to elastic shakedown limit against incremental growth of surface strain E: Elastic limit  - Lower bound to elastic limit. Traction Coefficient, μ, has an effect on the maximum value and the depth of the maximum shear stress. If the cam-follower both slides and rolls over the cam, then the friction force moves the maximum shear stress closer to the surface as well as increasing its value. When traction or friction is zero, the maximum shear-stress is at 0.78×b [contact half width]. At a Coefficient-of-Friction μ = 0.3, the maximum shear stress is at or very near to the surface. Thus, plasticity is not constrained by the elastic hinterland. You can see in the image, that the range between the stress when there is 'first yield' (Blue line) and that for 'shakedown' (Magenta line) becomes narrower as you increase the coefficient of friction and there is sliding at the contact.

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