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Hooke's Law (Stress-Strain)
Relates stress to strain in the linear-elastic region of a material, with the constant of proportionality being the material's elastic modulus.
Interpretation
Hooke's law in materials: σ = E·ε. Stress is proportional to strain. E is modulus of elasticity. Valid in elastic region. Core relationship for mechanical design and stress analysis.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| sigma | Engineering stress | MPa |
| E | Young's modulus | GPa |
| epsilon | Engineering strain (dimensionless) |
What it means
This is the same as id=53. Hooke’s law establishes a linear relationship between stress and strain in the elastic regime. The modulus of elasticity E, also known as Young’s modulus, is a measure of a material’s stiffness; a higher E means the material deforms less under load. This law is central to the design of structural and mechanical components because it allows engineers to compute deflections and stresses from known loads and geometry. It is also the basis for strain gauges and many experimental stress‑analysis techniques. The law holds only up to the proportional limit; beyond that, plasticity occurs. In addition to normal stress, a similar relationship exists for shear stress (τ = Gγ) with shear modulus G. Understanding Hooke’s law is a prerequisite for mechanics of materials, structural engineering, and many disciplines that deal with solid deformation.
Worked example
Hooke's Law (Stress‑Strain) – Two Examples
Real‑World| Parameter | Value |
|---|---|
| E | 200 GPa |
| ε | 0.001 |
| Parameter | Value |
|---|---|
| E | 70 GPa |
| σ | 140 MPa |
Common mistakes
- Same as ID 53 – see that entry.
- Note: This duplicate uses the symbols sigma and epsilon explicitly – ensure you apply the same considerations.
Applications
Hooke's law (stress‑strain) is the foundational relationship for linear elastic materials, linking stress and strain through the modulus of elasticity. It is used everywhere in structural and mechanical engineering to predict deformation, to ensure that components remain within the elastic range, and to design resilient structures. For example, beams in buildings, shafts in machines, and springs in suspension systems all rely on Hooke's law for their design. In materials testing, it is used to determine Young's modulus from tensile tests. The linearity ensures that the material returns to its original shape when loads are removed, which is crucial for functional performance. By understanding this law, engineers can avoid plastic deformation and ensure that designs meet both strength and stiffness requirements.
- Design of structural members (beams, columns, trusses) for stiffness
- Calculation of deflections and rotations in machine components
- Spring design for force and displacement control
- Material property determination via tensile testing
- Finite element modelling of elastic behaviour
Frequently Asked Questions
Hooke's law in stress‑strain form is σ = E · ε, where σ is the applied stress, ε is the resulting strain, and E is the modulus of elasticity (Young's modulus). It describes the linear elastic response of materials.
The proportional limit is the maximum stress for which stress is proportional to strain (i.e., Hooke's law holds). Beyond this point, the stress‑strain curve deviates from linearity, and the material may still be elastic but not linear.
Young's modulus is the slope of the initial linear portion of the stress‑strain curve. It is calculated as E = Δσ / Δε, taken within the proportional limit.
Elastic deformation is recoverable – when the load is removed, the material returns to its original shape. Plastic deformation is permanent – the material does not return to its original shape after unloading. Hooke's law applies only to the elastic region.
For anisotropic materials (e.g., composites, single crystals), Hooke's law is generalised to a stiffness matrix relating stress and strain tensors: σ_ij = C_ijkl · ε_kl. There are multiple elastic constants.
Engineering strain is defined as ε = (L – L₀) / L₀, where L₀ is the original length and L is the current length. This is valid for small strains; for large strains, true strain is used.
For a rod of length L and cross‑sectional area A, the stiffness (k = F/δ) is given by k = A·E / L. This is directly derived from Hooke's law and is used in finite element analysis.
- Assumes a uniaxial stress state.
- Assumes small strains.
- Assumes linearity; non‑linear materials require more complex models.
- Does not account for time (viscoelasticity) or temperature effects.
The modulus of resilience is the energy per unit volume absorbed by a material up to the elastic limit. It is U_r = σ_y² / (2E) and is derived from Hooke's law. It indicates the material's ability to absorb energy without permanent deformation.
By ensuring that stresses remain below the proportional limit under service loads, engineers can prevent plastic deformation and ensure the structure returns to its original shape after loading. This is the basis of elastic design.