Linear elastic constitutive relation
Hooke's Law
Within an elastic system's linear range, deformation is proportional to the load that produces it; remove the load and the system returns to its original configuration.
F_s = -k x
For an ideal one-dimensional spring, x is displacement from equilibrium and k is stiffness. The minus sign says the restoring force points opposite the displacement. In solids, the corresponding linear relation connects stress and strain.
The bench keeps stiffness adjustable and separates static extension from free oscillation. Pull and release to see stored elastic energy exchange with kinetic energy.
(cm)
The plot, diagram, and calculated result share the same state. Animation runs only when it adds explanatory value.
- CHANGE
- Applied force
- WATCH
- extension + oscillation
- MEANING
- The bench keeps stiffness adjustable and separates static extension from free oscillation. Pull and release to see stored elastic energy exchange with kinetic energy.
A straight force-extension line is a regime, not a promise.
The current operating point moves along the linear region; the shaded end zone marks where a real specimen may depart from Hooke behavior.
What it actually says
Hooke's law is a local approximation around an equilibrium configuration. It says that doubling a sufficiently small deformation doubles the restoring load, provided material, geometry, temperature, loading rate, and prior history remain within the same regime.
The scalar spring equation is one member of a wider family. Axial rods use stress proportional to strain, torsion uses torque proportional to angle, and three-dimensional elasticity uses a stiffness tensor. The proportionality constant belongs to the system and loading mode, not to nature as a universal number.
"A useful law compresses a pattern. It does not erase the conditions that make the pattern true."
How the idea developed
The modern form emerged through observation, argument, and later refinement. The timeline separates the first insight from the version now used in textbooks and practice.[1]
Robert Hooke experiments with springs and hanging weights.
Hooke publishes the anagram and phrase ut tensio, sic vis: as the extension, so the force.
Young, Cauchy, Navier, and others develop modern elasticity and stress-strain descriptions.
Linear elasticity supports sensors, structures, vibration systems, and material testing.
How the pattern works
The relation becomes useful only when its mechanism, measurement process, and operating range are visible.
Small displacement raises potential energy approximately quadratically.
The energy gradient points back toward equilibrium, giving F = -kx.
Geometry and material response determine the slope k.
Microstructure, damage, contact, and large geometry changes bend or break the line.
For an ideal one-dimensional spring, x is displacement from equilibrium and k is stiffness. The minus sign says the restoring force points opposite the displacement. In solids, the corresponding linear relation connects stress and strain.
Where it earns its keep
Applications are strongest when the law changes a decision, measurement, model, or experiment rather than merely providing an analogy.
Calibrate force and displacement
ApplicationLoad cells and spring scales use a measured stiffness.
Include hysteresis, creep, temperature, and calibration uncertainty.
Estimate small deflections
ApplicationLinear finite-element models relate loads to displacement.
Check yielding, buckling, joints, and geometric nonlinearity.
Predict natural frequency
ApplicationA mass-spring oscillator has frequency set by k and mass.
Damping and distributed mass change the response.
Where it stops working
Rubber, biological tissue, granular contacts, plastics, and many composites can be nonlinear, rate-dependent, anisotropic, or history-dependent.
A return path may show hysteresis even before visible failure; linearity and perfect reversibility are distinct assumptions.
"The law applies until a spring breaks"
Better: Proportionality can fail well before fracture."k is a material constant"
Better: Spring stiffness also depends on geometry and boundary conditions."The minus sign means negative force magnitude"
Better: It encodes direction relative to displacement."No force means no motion"
Better: With inertia, a body can pass equilibrium with nonzero velocity.Sources and further reading
Original publications and serious secondary scholarship are prioritized over summaries.
- Hooke - Lectures de Potentia RestitutivaDigitized edition of Hooke's work on spring force.https://archive.org/details/lecturesdepoten00hookgoog
- OpenStax - Elasticity and PlasticityUniversity treatment of stress, strain, and elastic moduli.https://openstax.org/books/university-physics-volume-1/pages/12-4-elasticity-and-plasticity
- NIST - Force MetrologyMeasurement and calibration context for force standards.https://www.nist.gov/programs-projects/force-metrology
- ASTM E8/E8MStandard test method for tension testing metallic materials.https://www.astm.org/e0008_e0008m.html