Hooke's Law (Stress-Strain & Elastic Modulus σ = Eε)
Calculate normal stress, linear elastic strain, and Young's modulus (σ = E·ε) for materials within proportional elastic limits.
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Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| \sigma | Normal stress | MPa | psi | The internal normal resistance force per unit area. |
| E | Modulus of elasticity | GPa | Mpsi | Material stiffness property / Young's Modulus. |
| \epsilon | Strain | dimensionless | dimensionless | Relative deformation ratio (change in length divided by original length). |
Step-by-Step Derivation
- 1
Consider a bar under axial tension, where stress \sigma represents internal force intensity and strain \epsilon is the ratio of elongation.
- 2
Robert Hooke observed in 1676 that the extension of a spring is directly proportional to the applied load.
- 3
In materials science, this is normalized as stress being proportional to strain: \sigma \propto \epsilon.
- 4
The constant of proportionality is defined as Young's Modulus (E), which measures the material's stiffness.
- 5
Therefore, the linear elastic relation is written as Hooke's Law: \sigma = E * \epsilon.
Worked Example Calculation
A structural steel bar (E = 200 GPa) undergoes an elastic strain of 0.0008. Calculate the normal stress in the bar.
- •Identify input parameters: E = 200 GPa = 200,000 MPa, \epsilon = 0.0008.
- •Apply Hooke's Law Formula: \sigma = E * \epsilon.
- •Compute: \sigma = 200,000 * 0.0008 = 160 MPa.
Engineering Assumptions
- •Material is homogeneous, isotropic, and elastic.
- •Stress levels do not exceed the proportional elastic limit.
Design Limitations
- •Not valid in the plastic deformation region (requires strain-hardening formulations).
Academic References & Standards
Tension Testing of Metallic Materials
Standard test procedures for metallic materials tensile evaluation.
Shigley's Mechanical Engineering Design - Materials properties
Textbook covering Hooke's Law and elastic curves.
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