Mechanics of materials / Sheet 01
Stress and strain
Formula reference for mechanical engineering interviews. Every relation assumes linear elastic, isotropic, homogeneous material unless the note says otherwise.
Basic definitions
True vs engineering
The engineering curve falls after the ultimate point because area shrinks faster than the material hardens, while you keep dividing by the original area. True stress rises all the way to fracture.
Elastic constants
ν approaches 0.5 for rubber and for metal undergoing plastic flow, because plastic deformation conserves volume.
Generalized Hooke's law, 3D
Principal stresses and Mohr's circle
Failure criteria
Hydrostatic stress does not cause yielding, which is exactly why von Mises contains only stress differences.
Applied stress cases
Because J = 2I, the same numerical moment and torque give a bending stress exactly twice the torsional stress.
Thermal
For steel this works out to roughly 2.4 MPa per degree C, so a 100 degree excursion alone can yield mild steel. Watch this in press fits, bonded dissimilar materials, and motor housings.
Strain energy
Stress concentration and measurement
Kt is purely geometric: independent of material and of load magnitude. It matters most in fatigue and in brittle materials, since ductile metals locally yield and blunt the peak under static load. Doubling a fillet radius is usually cheaper than upgrading the alloy.
Worked chain: shaft under bending and torsion
The critical element is on the surface, at the fibre farthest from the neutral axis in bending. Torsional shear is uniform around the circumference, so the point that matters is the one where the bending tensile stress is also at its maximum.
Numbers worth memorizing
| Material | E (GPa) | ρ (kg/m³) | Yield (MPa) | ν | α (µε/°C) |
|---|---|---|---|---|---|
| Steel, all alloys | 200 | 7850 | 250 to 900+ | 0.29 | 12 |
| Aluminium 6061-T6 | 69 | 2700 | 276 | 0.33 | 23 |
| Aluminium 7075-T6 | 71 | 2810 | 503 | 0.33 | 23 |
| Titanium Ti-6Al-4V | 114 | 4430 | 880 | 0.34 | 8.6 |
| ABS and PLA | 2 to 3.5 | 1050 to 1250 | 40 to 60 | 0.35 | 70 to 90 |
Conversions: 1 ksi = 6.895 MPa, 1 GPa = 145 ksi, steel E = 29 Msi, aluminium E = 10 Msi.
Concepts that decide the interview
- Stiffness and strength are independent. Every steel is about 200 GPa, so alloy choice and heat treatment change yield but not deflection. Fix deflection with geometry (A, I); fix yielding with material.
- Specific stiffness E/ρ is roughly 25 MN·m/kg for steel, aluminium, magnesium and titanium alike. A material swap alone cannot lighten an axial member of equal stiffness. Aluminium wins in bending and buckling because you can afford more thickness, and I scales with t³ or d⁴.
- Necking starts at the ultimate point. That is also where εt = ln(1 + ε) stops being valid, because strain is no longer uniform along the gauge length.
- Mild steel shows a distinct upper and lower yield point. Aluminium and most non-ferrous alloys do not, so yield is defined by the 0.2 percent offset.
- Cast iron in torsion fractures on a 45 degree helix, because brittle materials fail on the plane of maximum tensile principal stress.
- Thick sections approach plane strain, are more constrained, and are more prone to brittle fracture.
- Support configuration only exists to produce the bending moment. Once M is given at a section, the layout no longer affects the stress calculation.