Lesson 2.2: Stress & Strain (Hooke’s Law, Poisson’s Ratio, Stress-Strain Curve)

Stress & Strain are fundamental for mechanical and civil engineering. GATE and PSU exams often test tensile, compressive, shear stresses, Hooke’s Law, Poisson’s ratio, and stress-strain behavior of materials.

🔹 1. Stress

• Definition: Internal resistance offered by a material per unit area to deformation.

σ=FA\sigma = \frac{F}{A}σ=AF​

• Types of Stress:

  • Tensile Stress (σt\sigma_tσt​) → Pulling force

  • Compressive Stress (σc\sigma_cσc​) → Pushing force

  • Shear Stress ($\tau$) → Force parallel to area

Applications:

• Shafts, beams, columns, fasteners

Example:
A rod of area 50 mm² is subjected to 10 kN tensile force →
$\sigma =10000/50=200\text{ N/mm²}$

🔹 2. Strain

• Definition: Measure of deformation per unit length.

ϵ=ΔLL\epsilon = \frac{\Delta L}{L}ϵ=LΔL​

• Types of Strain:

  • Tensile/Compressive Strain: Along axis of force

  • Shear Strain ($\gamma$) → Angular deformation

Applications:

• Deflection of beams, elongation of shafts

🔹 3. Hooke’s Law

• Statement: Stress is proportional to strain within elastic limit.

$$\sigma =Eϵ$$

Where E = Young’s Modulus

• Shear Stress-Strain Relation:

$$\tau =G\gamma$$

Where G = Modulus of Rigidity

• Bulk Modulus:

K=Hydrostatic StressVolumetric StrainK = \frac{\text{Hydrostatic Stress}}{\text{Volumetric Strain}}K=Volumetric StrainHydrostatic Stress​

Applications:

• Elastic analysis of machine elements

• Design of rods, bars, springs

🔹 4. Poisson’s Ratio

• Definition: Lateral contraction per unit longitudinal extension.

ν=−ϵlateralϵlongitudinal\nu = – \frac{\epsilon_{\text{lateral}}}{\epsilon_{\text{longitudinal}}}ν=−ϵlongitudinal​ϵlateral​​

• Typical values: 0.25 – 0.35 for metals

Applications:

• Thin-walled pressure vessels

• Beam & shaft design

🔹 5. Stress-Strain Curve

• Elastic Region: Obeys Hooke’s law, linear portion

• Yield Point: Material starts plastic deformation

• Plastic Region: Permanent deformation occurs

• Ultimate Strength: Maximum stress before failure

• Fracture Point: Material breaks

Important Parameters:

• Young’s Modulus $E$ → Slope of linear region

• Yield Stress σy\sigma_yσy​

• Ultimate Stress σu\sigma_uσu​

• % Elongation → Ductility

Applications:

• Material selection

• Designing load-bearing members

🔹 6. Solved Examples (PYQ Style)

Example 1 (GATE ME 2017):
A rod of length 2 m, diameter 20 mm, subjected to 50 kN tensile force. Find tensile stress and elongation if $E=200\text{ GPa}$.
👉 Solution:
σ=F/A=50000/(π∗(10)2)≈159.15 N/mm²\sigma = F/A = 50000/(π*(10)^2) ≈ 159.15 \text{ N/mm²}σ=F/A=50000/(π∗(10)2)≈159.15 N/mm²
$ϵ=\sigma /E=159.15/200000\approx 0.0007957$
$\Delta L=ϵL=0.0007957\ast 2000\approx 1.59\text{ mm}$

Example 2 (PSU Exam):
Poisson’s ratio = 0.3, axial strain = 0.001. Find lateral strain.
👉 Solution: ϵlat=−νϵlong=−0.3∗0.001=−0.0003\epsilon_{\text{lat}} = -\nu \epsilon_{\text{long}} = -0.3*0.001 = -0.0003ϵlat​=−νϵlong​=−0.3∗0.001=−0.0003

🔹 7. Practice Exercises

1. A steel bar 1 m long, area 100 mm², F = 20 kN. Find stress & strain if E = 210 GPa.

2. Calculate lateral contraction for rod with Poisson’s ratio = 0.28 and axial strain = 0.002.

3. Draw typical stress-strain curve and label key points: elastic, yield, ultimate, fracture.

4. A shaft experiences shear force of 5000 N, diameter 50 mm. Find shear stress.

🔹 8. Summary

• Stress: Internal resistance per unit area (tension, compression, shear).

• Strain: Deformation per unit length (axial, shear).

• Hooke’s Law: Stress ∝ Strain (Elastic region).

• Poisson’s Ratio: Lateral vs. longitudinal strain.

• Stress-Strain Curve: Material behavior under load → design & selection.