Question 1 of 6: Short-Answer Concepts

Problem 1: Short-Answer Concepts (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Higher yield stress in plane strain

In a plane–strain condition the material is prevented from straining in the third (thickness) direction, i.e. $\varepsilon_z = 0$. To keep that strain at zero, a constraining transverse stress must develop; elastically $\sigma_z = \nu(\sigma_x+\sigma_y)$, and at the onset of yield it approaches $\sigma_z \approx \tfrac{1}{2}(\sigma_x+\sigma_y)$. This added stress makes the local state markedly more triaxial / hydrostatic.

Yielding, however, is governed by the deviatoric (shear) part of the stress tensor only — both the von Mises and Tresca criteria are insensitive to the hydrostatic component. Because the constraint pumps up the hydrostatic part without adding to the shear part, a larger applied stress is required to reach the yield shear. For uniaxial-type loading under full plane-strain constraint, the von Mises criterion predicts first yield at

$$\boxed{\;\sigma_{Y,\text{plane strain}} = \frac{2}{\sqrt{3}}\,S_y \approx 1.155\,S_y\;}$$

This is exactly why the highly constrained plane-strain region ahead of a crack tip, or in the interior of a thick section, can carry a higher nominal stress before yielding than a thin (plane-stress) specimen of the same material.

(b) When solid-film lubricants are chosen over liquids

Solid-film lubricants (graphite, molybdenum disulphide $\text{MoS}_2$, PTFE, boron nitride) are selected wherever a liquid film cannot survive or is undesirable. Two clear cases:

(i) Extreme temperature. At high temperature mineral and even synthetic oils oxidise, char, or evaporate, and at very low temperature they thicken or freeze; $\text{MoS}_2$ and graphite retain low friction over a far wider temperature band. (ii) Vacuum or space service. Liquids evaporate and outgas in vacuum, so satellite and spacecraft mechanisms rely on bonded solid films. Other justifying cases are very high contact pressure with low sliding speed (a boundary regime where a hydrodynamic film cannot form), and clean-process machinery (food, textile, optics, electronics) where oil contamination is unacceptable. The reason they work is intrinsic: their lamellar crystal structure shears along weakly bonded planes, giving low friction with no fluid present.

(c) Fretting corrosion

Fretting corrosion is surface damage that develops at the interface of two nominally clamped or fitted surfaces subjected to small-amplitude oscillatory relative motion (micro-slip of only a few micrometres) under load. It couples mechanical wear with oxidation: freshly exposed metal oxidises, the hard oxide debris is trapped between the surfaces and abrades them further, producing pits, reddish or black powdery debris, and surface micro-cracks that act as fatigue-initiation sites (“fretting fatigue”). It is common at press-fit hubs and bearing seats, bolted and riveted joints, splines, and clamped leaf springs — including the gear-seat interface of Problem 2. It is countered by raising the clamping preload to suppress slip, by lubrication or hard coatings, by surface hardening, and by inducing compressive residual stress (shot peening).

(d) Effect of mean stress on fatigue

For a given alternating stress amplitude, a superimposed tensile mean stress reduces fatigue life (lowers the permissible alternating stress), whereas a compressive mean stress is beneficial. A tensile mean stress holds micro-cracks open and speeds their propagation. The effect is captured by the Goodman, Gerber, Soderberg and ASME-elliptic criteria; the Goodman line, for example, is $\sigma_a/S_e + \sigma_m/S_{ut} = 1$, so the allowable $\sigma_a$ falls linearly as $\sigma_m$ rises. Zero mean stress is the fully-reversed reference case that defines $S_e$ — the same interaction drives the diameter check in Problem 2, where fully-reversed bending combines with a mean (steady) torque.