Materials: Density, Deformation, Stress, Strain and Young Modulus
Stress and strain
Tensile stress σ = F/A, where A is the cross-sectional area perpendicular to the force. Tensile strain ε = ΔL/L, using the original length. The Young modulus E = σ/ε applies only in the linear elastic region of a stress–strain graph. Hooke's law F = kΔL is the same linear behaviour.
Energy and deformation
Elastic potential energy stored in a stretched wire is ½FΔL (or ½kΔL²). Beyond the elastic limit the material deforms plastically and does not return to its original length. The ultimate tensile strength is the maximum stress; toughness is the area under the stress–strain curve to fracture. Ductile materials show large plastic strain; brittle materials fracture with little.
Worked example
A steel wire of length 2.00 m and diameter 0.50 mm carries 100 N. Area A = π(0.25 × 10^-3)² = 1.96 × 10^-7 m². Stress = 100/(1.96 × 10^-7) = 5.1 × 10^8 Pa. If the measured extension is 5.1 mm, strain = 5.1 × 10^-3/2.00 = 2.55 × 10^-3. Hence E = 5.1 × 10^8 / 2.55 × 10^-3 = 2.0 × 10^11 Pa. Energy stored = ½ × 100 × 5.1 × 10^-3 = 0.255 J.
Traps
Using diameter as radius; forgetting to square the radius; leaving mm in the area; using the stretched length in strain; applying E beyond the elastic limit; confusing stress with force.