Elasticity Formulas
Elasticity is a crucial topic to know during your academics. If you are stuck at some point while dealing the problems on Elasticity make use of the Elasticity Formulas provided. The Formulae List for Elasticity is quite effective to revise all the topics included in a smart way. Grasp all the fundamentals involved and learn when and where to apply the formulas using the Elasticity Formulae Sheet over here. Don't limit yourself by learning this concept alone and be aware of the Physics Formulas from our article.
Complete List of Elasticity Formulae
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 Φ by which a line perpendicular to the fixed face turns.
3. Hooke’s law
Stress ∝ Strain
E is modulus of elasticity = \(\frac{\text { stress }}{\text { strain }}\) N/m2
4. Young’s 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 = πr2 (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 KT = P
(c) Adiabatic modulus of elasticity of gas
Ks = γP, γ = \(\frac{C_{p}}{C_{v}}\)
(d) Compressibility is reciprocal of bulk modulus i.e., χ = 1/K
6. Modulus of rigidity
η = \(\frac{\text { stress }}{\text { shear strain }}\)
= \(\frac{\mathrm{F} / \mathrm{A}}{\phi}=\frac{\mathrm{F}}{\mathrm{A} \phi}=\frac{\mathrm{FL}}{\mathrm{A} \ell}\)
7. Poisson’s ratio
- σ = \(\frac{\text { transverse or lateral strain }}{\text { longitudinal strain }}=\frac{\beta}{\alpha}\)
- The valve of σ lies between -1 and 0.5.
- σ = \(\frac{1}{2}\left[1-\left(\frac{\Delta \mathrm{V}}{\mathrm{V}}\right)\left(\frac{\mathrm{L}}{\Delta \mathrm{L}}\right)\right]=\frac{1}{2}\left[1-\frac{\Delta \mathrm{V}}{\mathrm{A} \Delta \mathrm{L}}\right]\)
8. Relations amongst various elastic constants (Y, K, η)
- Y = 3K(1 – 2σ)
- Y = 2η(1 + σ)
- Y = \(\left(\frac{9 \eta K}{\eta+3 K}\right)\)
- σ = \(\frac{Y}{2 \eta}-1\)
- σ = \(\frac{3 K-2 \eta}{6 K+2 \eta}\)
- \(\frac{9}{\mathrm{K}}=\frac{3}{\eta}+\frac{1}{\mathrm{K}}\)
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 × strain = Y α (t2 – t1) = Y α Δt
Thermal Tension = YA α (t2 – t1) = YA α Δt
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 = \(\frac{1}{2}\)Cθ2
13. Frequency of vertical oscillations of loaded wire
n = \(\frac{1}{2 \pi} \sqrt{\frac{\mathrm{YA}}{\mathrm{mL}}}\)
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