Elasticity Formulas

1. Stress

Stress is internal force of reaction per unit area.
Numerically stress = \(\frac{\text { External force }}{\text { Area of cross sec tion }}=\frac{\mathrm{F}}{\mathrm{A}}\) N/m²

2. Strain

• Longitudinal strain = \(\frac{\text { Change in length }}{\text { Original length }}=\frac{\ell}{\mathrm{L}}\)
• Volume strain = \(\frac{\text { Change in volume }}{\text { Original volume }}=\frac{\Delta \mathrm{V}}{\mathrm{V}}\)
• Shear stain = the angel ╬ª by which a line perpendicular to the fixed face turns.

3. HookeΓÇÖs law

Stress ∝ Strain
E is modulus of elasticity = \(\frac{\text { stress }}{\text { strain }}\) N/m²

4. YoungΓÇÖs modulus of elasticity

Y = \(\frac{\text { stress }}{\text { longitudinal strain }}\)
= \(\frac{F / A}{\ell / L}=\frac{F L}{A \ell}\) N/m²
If F = Mg, A = ╧Çr² (loaded wire)
then Y = \(\frac{\mathrm{MgL}}{\pi \mathrm{r}^{2} \ell}\)

5. Bulk modulus of elasticity

K = \(\frac{\text { stress }}{\text { volume strain }}=\frac{\mathrm{F} / \mathrm{A}}{\Delta \mathrm{V} / \mathrm{V}}=\frac{\mathrm{FV}}{\mathrm{A} \Delta \mathrm{V}}\) N/m²
If \(\frac{\mathrm{F}}{\mathrm{A}}\) = pressure P then
K = \(\frac{P V}{\Delta V}\)

(a) If by a change of pressure dP the change in volume is dV then
K = -V\(\left(\frac{\mathrm{dP}}{\mathrm{dV}}\right)\)

(b) Isothermal modulus of elasticity of a gas K_T = P

(c) Adiabatic modulus of elasticity of gas
K_s = ╬│P, ╬│ = \(\frac{C_{p}}{C_{v}}\)

(d) Compressibility is reciprocal of bulk modulus i.e., χ = 1/K

6. Modulus of rigidity

╬╖ = \(\frac{\text { stress }}{\text { shear strain }}\)
= \(\frac{\mathrm{F} / \mathrm{A}}{\phi}=\frac{\mathrm{F}}{\mathrm{A} \phi}=\frac{\mathrm{FL}}{\mathrm{A} \ell}\)

7. PoissonΓÇÖs ratio

• ╧â = \(\frac{\text { transverse or lateral strain }}{\text { longitudinal strain }}=\frac{\beta}{\alpha}\)
• The valve of ╧â lies between -1 and 0.5.
• ╧â = \(\frac{1}{2}\left[1-\left(\frac{\Delta \mathrm{V}}{\mathrm{V}}\right)\left(\frac{\mathrm{L}}{\Delta \mathrm{L}}\right)\right]=\frac{1}{2}\left[1-\frac{\Delta \mathrm{V}}{\mathrm{A} \Delta \mathrm{L}}\right]\)

8. Relations amongst various elastic constants (Y, K, ╬╖)

• Y = 3K(1 – 2╧â)
• Y = 2╬╖(1 + ╧â)
• Y = \(\left(\frac{9 \eta K}{\eta+3 K}\right)\)
• ╧â = \(\frac{Y}{2 \eta}-1\)
• ╧â = \(\frac{3 K-2 \eta}{6 K+2 \eta}\)
• \(\frac{9}{\mathrm{K}}=\frac{3}{\eta}+\frac{1}{\mathrm{K}}\)

9. Work done in stretching a wire

The work done = Average force × change in length
or W = \(\frac{1}{2}\)Fl

10. Elastic potential energy

(a) U = W = \(\frac{1}{2}\)Fl = \(\frac{1}{2}\left(\frac{F}{A}\right)\left(\frac{l}{L}\right)\) (LA)
= \(\frac{1}{2}\) (stress × strain × volume of wire)

(b) Energy density or elastic energy per unit volume,
u = \(\frac{1}{2}\) (stress × strain)
= \(\frac{1}{2}\) Y(strain)Y(strain)²
= \(\frac{(\text { stress })^{2}}{Y}\)

11. Thermal stress

Thermal stress = Y ├ù strain = Y ╬▒ (t₂ – t₁) = Y ╬▒ ╬öt
Thermal Tension = YA ╬▒ (t₂ – t₁) = YA ╬▒ ╬öt

12. Torsion constant of wire

C = \(\frac{\pi \eta r^{4}}{2 \ell}\)
(a) Torque required for twisting,
τ = Cθ

(b) Work done in twisting by an angle
W = \(\frac{1}{2}\)C╬╕²

13. Frequency of vertical oscillations of loaded wire

n = \(\frac{1}{2 \pi} \sqrt{\frac{\mathrm{YA}}{\mathrm{mL}}}\)

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