# Force Analysis: Cam: Maximum Shear-Stress Failure Criterion

## Contact Stress...'Static-Load Capacity' and 'Low Cycle Fatigue'.

This topic does not explain fatigue that leads to pitting. It is limited to the lower end of the Fatigue-Life : 1 < Number of Cycles ≤103.

#### 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 fails plastically, 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 one-half the Yield-Strength, when yield results from a normal tension test.

 $k=\frac{Y}{2}$ Equation 1 Y = Yield Strength k = Shear Strength Yield-strength usually has the label as Re, Rpor Rp0.2 .

#### Elastic Limit, or 'Onset of Plasticity' (Contact Stress ≈ 1.6×Y)

 When a body initially yields it is at the onset-of-plasticity. We can use Contact Stress with the Maximum Shear Stress failure criterion to determine the onset-of-plasticity. We know that:   ${\tau }_{\mathrm{max}}=0.3×{P}_{\mathrm{max}}$ If maximum contact-stress is increased until maximum shear strength   ${\tau }_{\mathrm{max}}=k$ ... then, it also equals the one half of the Yield Strength, see Equation 1   ${\tau }_{\mathrm{max}}=k=0.3×{P}_{\mathrm{max}}=\frac{Y}{2}$ Then, we state that the Onset-of-Plasticity, for line-contact, occurs when...   $k∕0.3={P}_{\mathrm{max}}=Y∕2×0.3$ or, equivalently ...   ${P}_{\mathrm{max}}=3.3k=1.67Y$ LINE-CONTACT (cylindrical cam-followers) Plasticity starts, below the surface, when: •Maximum Shear-Stress (below surface), τmax = Y/2 or τmax = k•Maximum Contact-Stress (at surface) Pmax = 1.67Y or Pmax = 3.3kPlasticity starts at a depth of •0.78b below the surface•b = 0.5 × Total Contact-width.•Y is the yield strength when under Axial Tension.ELLIPTICAL-CONTACT (barrel, or crown, cam-followers) For Elliptical Contact, the Stress at which plasticity starts is not very different to that with Line-Contact. However, the depth at which Plasticity starts reduces to ~0.33b.

#### Elastic Shakedown  (1.6×Y < Contact Stress <2×Y)

 When = 1.67 × Y (τmax = Y/2) the onset of plasticity (yield begins) 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 occurs beneath the surface and the plastic zone is surrounded by a region in which the stress and strains remain elastic. This limits the plastic deformation since the plastic strains must be similar to adjacent elastic strains. The amount of steel that is flattened (moved) by Plastic Strain under the contact must be equal to the expansion of the surrounding steel with Elastic Strain. The first time that the elastic limit is exceeded at a point on the cam, a small plastic deformation occurs. When the cam-follower is further along the cam, and the load becomes zero again, residual stresses remain in the cam, below the surface. When the cam-follower passes the same point on the cam a second time, the cam stress is the combined action of contact-stress + residual stress from the first pass. Because residual stresses are usually 'protective', the cam is less likely to yield when the cam-follower makes a second pass than the cam was when the cam-follower made its first pass. ( Note: according to theory not presented here: 'if it is possible for residual stresses to occur, they will always occur'). It is possible, that after a few rotations of the cam, the contact becomes elastic, because of the build-up of the protective residual-stresses. This process is called Elastic Shakedown. The new Elastic-Limit after Shakedown can increase to: 4.0 × k , compared to 3.33 × k for the Elastic-Limit before Elastic-Shakedown. Other factors that help Shakedown: Strain-Hardening: where the cam strains, it may 'Strain-Harden'. Thus, its yield-strength will also increase. Note: Strain-Hardening is not Elastic-Shakedown. Steels that already very hard may in fact tend to Strain-Weaken, while those that are soft will tend to Strain Harden Geometric-Changes: if the cam-steel deforms, then the cam and cam-follower become more 'conformal'. A flat cam becomes cup-shaped when the cam-follower is barrel-shaped, for example. The conformity decreases the contact-stress because the Equivalent Radius-of-Curvature, Re, increases. 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: ...then, Elastic Shakedown will occur. Also, as the load F is proportional to , 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, ~103 cycles. The theory that predicts this is 'upper bound' theory, which postulates plastic flow. It equates the rate at which the external load does work to the rate at which energy is absorbed within the material - defined as the upper bound of 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 moves to, 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.