What Is Strain? Deformation & Hooke's Law

Strain turns deformation into a material property — independent of how long or thick the specimen is.

Mechanics of MaterialsMechanical Engineering Year 1Free preview
⏱️ About 16 min

Stretch a rubber band and it gets longer. Stretch a steel rod by the same force and the change is almost too small to see. But if you measure the *fractional* change in length — the elongation divided by the original length — you get a number that describes the material itself, not the size of the bar. That fractional change is strain, and it is the bridge between stress and deformation.

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The big idea: Strain is normalized deformation: change in dimension divided by original dimension. Because it is a ratio, strain is independent of the specimen's size. In the elastic range, stress is proportional to strain through Hooke's law, with Young's modulus E as the constant of proportionality. Shear stress and shear strain are linked by the shear modulus G.
🎯 By the end, you'll be able to
  • Define normal strain and shear strain
  • State Hooke's law for normal and shear stress
  • Compute Young's modulus E from stress and strain data
  • Relate the shear modulus G to E and Poisson's ratio
📎 Helpful to know first

Normal strain: the fractional change in length

When you pull on a bar, it lengthens. The absolute elongation ΔL depends on the original length — a 2 m bar stretches twice as much as a 1 m bar under the same strain. To remove that size dependence, engineers use normal strain:

\[ \varepsilon = \frac{\Delta L}{L} \]
Normal strain is dimensionless (m/m or mm/mm). It is often expressed as microstrain (10⁻⁶).

Shear strain: the change in angle

Shear stress does not change the length of a material element; it distorts its shape. Imagine a square element becoming a rhombus: the angle between two originally perpendicular edges changes by a small amount γ. That angle change, in radians, is the shear strain.

Normal strain is the change in length divided by original length. Shear strain is the change in angle between two originally perpendicular lines. Original length L ε = ΔL / L Normal strain Right angle γ (shear strain) γ ≈ tan γ Shear strain
Normal strain is a length ratio; shear strain is an angle change (radians).
\[ \sigma = E\,\varepsilon \qquad \tau = G\,\gamma \]
Hooke's law: in the elastic range, normal stress is proportional to normal strain through Young's modulus E, and shear stress is proportional to shear strain through the shear modulus G.
\[ G = \frac{E}{2(1+\nu)} \]
For an isotropic material, the shear modulus G is determined by Young's modulus E and Poisson's ratio ν. You only need two of the three constants.
✨ Why Hooke's law is a local law

Hooke's law does not say 'the bar stretches by 2 mm.' It says 'at every point in the bar, the stress is 200 times the strain.' That local relationship is what makes it powerful: you can integrate it over a complex shape, apply it to a tiny finite element, or use it in a simple hand calculation. But it is only valid in the elastic range — once the material yields or cracks, the linear relationship breaks down.

📝 Worked example: A 2.0 m steel rod (A = 250 mm², E = 200 GPa) carries an axial load of 50 kN. Find the normal stress, the elastic strain, and the total elongation. Also compute the shear modulus G if ν = 0.3.
  1. Normal stress: σ = P/A = 50 000 N / 250 mm² = 200 MPa.
  2. Elastic strain: ε = σ/E = 200 MPa / 200 000 MPa = 0.001 = 1000 × 10⁻⁶.
  3. Elongation: ΔL = ε × L = 0.001 × 2000 mm = 2.0 mm.
  4. Shear modulus: G = E / [2(1+ν)] = 200 000 / [2(1.3)] = 200 000 / 2.6 = 76 923 MPa ≈ 76.9 GPa.
  5. Sanity check: the elongation is small (0.1% of the length), consistent with elastic deformation of steel.
✓ σ = 200 MPa, ε = 1000 × 10⁻⁶, ΔL = 2.0 mm, G ≈ 76.9 GPa.
✏️ Practice: A 1.5 m metal bar elongates by 0.75 mm under tension. What is the normal strain, expressed as ε × 10⁶?
×10⁻⁶
Solution
  1. ε = ΔL / L = 0.75 mm / 1500 mm = 0.0005 = 500 × 10⁻⁶.
✏️ Practice: A steel bar (E = 200 GPa) is loaded elastically to a normal stress of 120 MPa. What is the strain, expressed as ε × 10⁶?
×10⁻⁶
Solution
  1. ε = σ / E = 120 / 200 000 = 0.0006 = 600 × 10⁻⁶.

Check your understanding

1. Strain is defined as:
Strain is the normalized deformation: ΔL/L for normal strain, or angle change for shear strain.
2. In the elastic range, Hooke's law states that stress is proportional to strain. The constant of proportionality is called:
Young's modulus E is the slope of the linear elastic region in a stress-strain diagram.
3. A steel specimen (E = 200 GPa) is stressed to 100 MPa in the elastic range. The strain is approximately:
ε = σ/E = 100 / 200 000 = 0.0005 = 500 × 10⁻⁶.
🔎 Related lessons
What Is Stress? Normal & Shear Stress The Stress-Strain Diagram · soon
✅ Key takeaways
  • Normal strain ε = ΔL/L is a dimensionless measure of elongation.
  • Shear strain γ is the angle change (radians) caused by shear stress.
  • Hooke's law: σ = Eε and τ = Gγ, valid only in the elastic range.
  • For isotropic materials, G = E / [2(1+ν)].
➡️ With stress and strain defined, the next question is: how much stress is safe? Engineers answer that with the factor of safety — a deliberate margin between expected load and failure.
Want to test yourself on this? Try the Mechanical Aptitude test →
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