08Class 11 Physics

Mechanical Properties of Solids formulas

Stress, strain and elastic moduli formulas for Mechanical Properties of Solids. A short sheet, and a common source of quick numericals.

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All Physics formulas

Stress and strain

Stress: $\displaystyle \sigma = \frac{F}{A}$
Longitudinal strain: $\displaystyle \varepsilon = \frac{\Delta L}{L}$
Hooke's law: $\displaystyle \text{stress} = \text{modulus} \times \text{strain}$
Within the elastic limit.

Moduli of elasticity

Young's modulus: $\displaystyle Y = \frac{FL}{A\,\Delta L}$
Shear modulus: $\displaystyle G = \frac{F/A}{\Delta x / L} = \frac{F/A}{\theta}$
Bulk modulus: $\displaystyle B = -\frac{p}{\Delta V / V}$
Compressibility: $\displaystyle k = \frac{1}{B}$
Elastic energy per unit volume: $\displaystyle u = \tfrac12 \times \text{stress} \times \text{strain}$

Where students slip up

Treating strain as having units. It's a ratio, so stress and modulus share the same units.
Forgetting the area is πr² for a wire, or that doubling a radius quadruples the area.
Using the cross-section's length in shear. In shear, the force is parallel to the area it acts on.

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