Question

state hookeslaw

Answer 1

Hooke’s law (basic statement)
- For a linear elastic spring: the restoring force is proportional to displacement:
F = −k x
where F is the force (N), x the displacement from equilibrium (m), and k the spring constant or stiffness (N/m). The negative sign indicates the force opposes the displacement.

Continuum (material) form
- In linear elasticity for uniaxial loading:
σ = E ε
where σ is stress (Pa), ε is (engineering) strain (dimensionless), and E is Young’s modulus (Pa). Analogous relations: shear τ = G γ (G = shear modulus) and volumetric p = −K ΔV/V (K = bulk modulus).

Conditions and limitations
- Valid only within the elastic (proportional) region up to the material’s proportional/elastic limit. Beyond that, deformation may be nonlinear and/or plastic (permanent).
- Assumes small deformations; for large strains geometric nonlinearities appear.
- Material must behave (approximately) linearly and reversibly over the range considered.

Energy stored
- Elastic potential energy in a spring: U = 1/2 k x^2.
- Strain energy density in a linear elastic element: u = 1/2 σ ε.

That is the essence of Hooke’s law and its common continuum extensions.