Hooke's Law & Linear Elasticity

Hooke's Law describes linear elastic behavior where deformation is fully recoverable and proportional to applied load. For uniaxial tension/compression below the proportional limit, normal stress σ is proportional to normal strain ε through Young's Modulus E: σ = Eε. Strain is ε = ΔL / L. This gives the bar elongation ΔL = F·L / (A·E) and the equivalent axial spring stiffness k = A·E / L, with F = k·x. Linear elasticity assumes small strains, homogeneous isotropic material, constant temperature, and loading below the proportional limit - the point where σ-ε deviates from linearity, distinct from but close to the elastic limit and yield point. Beyond this, the relation is no longer linear. In shear, τ = G·γ where G is shear modulus and γ is engineering shear strain. In hydrostatic loading, volumetric strain relates to pressure via bulk modulus K. For isotropic linear elasticity, E, G, and Poisson's ratio ν are linked: G = E / [2(1+ν)], and K = E / [3(1-2ν)]. Poisson's ratio couples transverse strain: ε_transverse = -ν·ε_axial. Elastic strain energy in a linear spring/bar is U = 1/2·k·x² = 1/2·F·x = F²/(2k). Engineers use Hooke's Law to compute deflections, stiffness, stresses from measured strains, and to provide the constitutive model for linear FEA, beam theory, and vibration analysis. It does not predict yielding, plasticity, creep, or fracture.