How solids stretch and fluids flow under forces
When a force deforms a solid, the restoring force per unit area is stress and the fractional change in size is strain. Within the elastic limit, Hooke’s law says stress is proportional to strain, and the constant is a modulus of elasticity. Young’s modulus Y = (F/A)/(ΔL/L) describes stretching, the bulk modulus describes change in volume and the shear modulus describes change in shape. Steel has a higher Young’s modulus than rubber, so it is more elastic in the physics sense.
In fluids, pressure at depth h is P = P₀ + ρgh, and Pascal’s law states that pressure applied to an enclosed fluid is transmitted equally in all directions. For a flowing fluid, the equation of continuity is A₁v₁ = A₂v₂, and Bernoulli’s theorem says P + ½ρv² + ρgh is constant along a streamline. Viscosity is internal friction in fluids, described by Stokes’ law F = 6πηrv. Surface tension makes liquid surfaces act like a stretched film and causes capillary rise, h = 2T cos θ/rρg.
Why it matters
Engineers choose materials for bridges, cranes and lift cables using elastic moduli. Hydraulic brakes and lifts work by Pascal’s law, aircraft wings get lift partly from pressure differences described by Bernoulli’s theorem, and blood flow, raindrop speeds and the rise of water in plants depend on viscosity and surface tension.
Key terms
- Stress
- The internal restoring force per unit area that develops in a deformed body.
- Young’s modulus
- The ratio of tensile stress to tensile strain for a material within its elastic limit.
- Bernoulli’s theorem
- For steady flow, the sum of pressure, kinetic energy per volume and potential energy per volume is constant.
- Surface tension
- The force per unit length acting along a liquid surface that makes it behave like a stretched film.