Mechanical Properties of Solids - Quick Revision
Stress and strain
- Stress = F/A (Pa); normal (tensile/compressive) or tangential (shear).
- Strain = fractional deformation (no units): longitudinal dL/L, shear (angle), volume dV/V.
Hooke's law and moduli
- Within the elastic limit, stress is proportional to strain; the ratio is a modulus of elasticity.
- Young's modulus Y = (F/A)/(dL/L) = FL/(A dL); elongation dL = FL/AY.
- Shear modulus G; Bulk modulus B = -dP/(dV/V); compressibility = 1/B.
- Solids have all three moduli; liquids/gases only B.
Stress-strain curve
- Proportional limit -> elastic limit (yield point) -> plastic region -> fracture point.
- Elastomers (rubber) have no proportional region; large reversible strains.
- Ductile materials have a large plastic region; brittle ones fracture soon after the elastic limit.
Poisson's ratio and elastic energy
- Poisson's ratio = lateral strain / longitudinal strain (typically 0.2-0.4).
- Elastic PE U = (1/2) x stress x strain x volume = (1/2) F dL; energy density u = (1/2) stress x strain.
Applications
- Steel is preferred for beams/cranes (high Y); the cross-section of a beam (I-shape) and the thickness of ropes (safety factor) are designed from these relations; maximum height of a mountain limited by the elastic strength of rock.