1. Chapter Introduction
Mechanical Properties of Solids studies how solids respond when forces, pressure, or temperature changes alter their length, volume, or shape. The chapter links stress, strain, elastic moduli, material behaviour, elastic energy, elastic wires, and thermal effects.
2. Chapter in One View
- Loading produces deformation.
- Stress = internal loading per unit area.
- Strain = relative deformation.
- Elastic moduli = resistance to elastic deformation.
- Identify loading: pulling, compression, shear, pressure, heating.
- Identify change: length, volume, shape, transverse dimension.
- Choose required quantity: stress, strain, extension, volume change, energy.
- Select model: Young’s, bulk, shear, spring, thermal.
- Check validity conditions, units, signs, and whether the answer makes physical sense.
3. Must-Know Concepts
- Elasticity: recovery after unloading.
- Plastic deformation: permanent change.
- Stress: internal force per unit area (Pa).
- Strain: relative deformation, dimensionless.
- Normal stress: tensile or compressive.
- Shear stress: tangential loading that causes layers to slide.
- Uniform pressure: primarily produces volume change.
- Hooke’s law: valid in the proportional region.
- Young’s modulus: axial length deformation.
- Bulk modulus: volume change.
- Shear modulus: shape change.
- Poisson’s ratio: lateral versus longitudinal strain.
- Elastic energy: area under the force–extension graph.
- Wire as a spring: an elastic wire has an effective spring constant.
- Thermal expansion: depends on whether movement is free or restrained.
4. Key Formulas
Use original dimensions and compatible SI units. Unless stated otherwise, elastic relations apply within the linear-elastic range.
| Quantity / model | Formula | Meaning and use |
|---|---|---|
| Normal stress | \(\displaystyle \sigma=\frac{F_{\perp}}{A}\) | Normal component of force divided by cross-sectional area. |
| Shear stress | \(\displaystyle \tau=\frac{F_{\parallel}}{A}\) | Tangential force per area. |
| Longitudinal strain | \(\displaystyle \varepsilon=\frac{\Delta L}{L}\) | Change in length divided by original length; dimensionless. |
| Volumetric strain | \(\displaystyle \varepsilon_V=\frac{\Delta V}{V}\) | Volume change divided by original volume. |
| Young’s modulus | \(\displaystyle Y=\frac{\text{longitudinal stress}}{\text{longitudinal strain}}=\frac{F/A}{\Delta L/L}=\frac{FL}{A\Delta L}\) | Axial deformation of a uniform wire or rod. |
| Bulk modulus | \(\displaystyle K=-\frac{\Delta P}{\Delta V/V}=-\frac{V\Delta P}{\Delta V}\) | Resistance to volume change; the minus sign makes \(K\) positive for ordinary compression. |
| Shear modulus | \(\displaystyle G=\frac{\tau}{\gamma}=\frac{F/A}{\gamma}\) | Resistance to shape change. For small deformation, \(\gamma\approx\tan\theta\approx\theta\), with \(\theta\) in radians. |
| Poisson’s ratio | \(\displaystyle \nu=-\frac{\varepsilon_{\text{transverse}}}{\varepsilon_{\text{longitudinal}}}\) | Negative ratio of transverse strain to longitudinal strain. |
| Elastic-constant relations | \(\displaystyle Y=2G(1+\nu)=3K(1-2\nu)\) | For isotropic, linear-elastic materials. |
| Young’s, bulk and shear moduli | \(\displaystyle Y=\frac{9KG}{3K+G}\) | Alternative relation for isotropic, linear-elastic materials. |
| Extension of a wire | \(\displaystyle \Delta L=\frac{FL}{AY}\) | Uniform wire under axial force in the linear-elastic range. |
| Spring elastic energy | \(\displaystyle U=\frac12 kx^2=\frac12 Fx\) | Ideal spring where force increases linearly from zero to final force. |
| Elastic energy in a wire | \(\displaystyle U=\frac12 F\Delta L\) | Total recoverable energy for linear-elastic stretching. |
| Elastic energy density | \(\displaystyle u=\frac{U}{V}=\frac12\sigma\varepsilon=\frac{\sigma^2}{2Y}=\frac12Y\varepsilon^2\) | Energy per unit volume for uniaxial, linear-elastic deformation. |
| Wire as a spring | \(\displaystyle k=\frac{AY}{L}\) | For a uniform wire: larger area increases stiffness; greater length decreases stiffness. |
| Equivalent spring constant — series | \(\displaystyle \frac{1}{k_{\text{eq}}}=\frac{1}{k_1}+\frac{1}{k_2}+\cdots\) | End-to-end connection. |
| Equivalent spring constant — parallel | \(\displaystyle k_{\text{eq}}=k_1+k_2+\cdots\) | Side-by-side connection. |
| Free thermal expansion | \(\displaystyle \Delta L=\alpha L\Delta T\) | Length change when expansion or contraction is unrestrained. |
| Fully restrained thermal stress | \(\displaystyle |\sigma|=Y\alpha|\Delta T|\) | Magnitude of stress when thermal expansion is completely prevented. |
| Fully restrained thermal force | \(\displaystyle |F|=AY\alpha|\Delta T|\) | Magnitude of axial force for complete restraint. Heating causes compression; cooling causes tension. |
5. Formula Selection Guide
Match the physical change in the question to the corresponding deformation model, then use the relation shown.
| See this situation | Think this model | Use this relation |
|---|---|---|
| Force perpendicular to a cross-section; force per area is asked | Normal stress | \(\displaystyle \sigma=\frac{F_{\perp}}{A}\) |
| Tangential force acts along a surface | Shear stress | \(\displaystyle \tau=\frac{F_{\parallel}}{A}\) |
| Change in length relative to original length | Longitudinal strain | \(\displaystyle \varepsilon=\frac{\Delta L}{L}\) |
| Pressure change produces a volume change | Bulk modulus | \(\displaystyle K=-\frac{\Delta P}{\Delta V/V}\) |
| Top layer shifts sideways while the base is fixed | Shear modulus | \(\displaystyle G=\frac{\tau}{\gamma}\), where \(\gamma\approx\theta\) for small angles |
| Axial force stretches a uniform wire; extension is asked | Wire extension | \(\displaystyle \Delta L=\frac{FL}{AY}\) |
| Axial stress and longitudinal strain are known | Young’s modulus | \(\displaystyle Y=\frac{\sigma}{\varepsilon}=\frac{FL}{A\Delta L}\) |
| Wire diameter or width changes during stretching | Poisson’s ratio | \(\displaystyle \nu=-\frac{\varepsilon_{\text{transverse}}}{\varepsilon_{\text{longitudinal}}}\) |
| Force varies as an object is extended | Work / elastic energy | \(\displaystyle W=\int_{x_i}^{x_f}F(x)\,dx\); area under the \(F\)-versus-\(x\) graph |
| Ideal spring is stretched by displacement \(x\) | Spring elastic energy | \(\displaystyle U=\frac12kx^2=\frac12 Fx\) |
| Elastic wire behaves like a spring | Wire spring constant | \(\displaystyle k=\frac{AY}{L}\) |
| Wires or springs are connected end-to-end | Series combination | \(\displaystyle \frac{1}{k_{\text{eq}}}=\sum_i\frac{1}{k_i}\) |
| Wires or springs are connected side-by-side | Parallel combination | \(\displaystyle k_{\text{eq}}=\sum_i k_i\) |
| Heated or cooled rod is free to move | Thermal expansion | \(\displaystyle \Delta L=\alpha L\Delta T\) |
| Rod is fully fixed and cannot expand or contract | Thermal stress | \(\displaystyle |\sigma|=Y\alpha|\Delta T|,\quad |F|=AY\alpha|\Delta T|\) |
6. When to Use / When Not to Use
- Hooke’s law and elastic moduli: use only in the valid elastic range.
- Elastic energy: use area under the F–x graph if force varies.
- Thermal deformation: free body → expansion; restrained body → stress develops.
7. Common Misconceptions
- Easy to deform does not mean less elastic.
- Elastic does not always mean proportional.
- Same force does not mean same stress.
- Same extension does not mean same strain.
- Young’s modulus is not for every type of deformation.
- A stretched wire also contracts laterally.
8. Exam Traps
- Stress is not force; strain is not extension.
- Do not use the wrong dimensions in strain.
- Do not confuse proportional limit with elastic limit.
- Do not apply Hooke’s law outside its valid range.
- Keep the signs in bulk modulus and Poisson’s ratio.
- Do not use the wrong work formula for variable force.
- Distinguish total energy from energy density.
- Do not mix up series and parallel combinations.
- Do not use free expansion as the final outcome for a fully restrained rod.
9. See This → Think This
- Returns after unloading → Elastic.
- Permanent deformation → Plastic.
- Force perpendicular to section → Normal stress.
- Layers slide → Shear stress.
- Length change → Young’s modulus.
- Volume change → Bulk modulus.
- Shape change → Shear modulus.
- Maximum stress → Ultimate tensile strength.
- Local area decreases → Necking.
- Variable force → Area under F–x.
- End-to-end wires → Series.
- Side-by-side wires → Parallel.
- Heated rod free → Expansion.
- Heated rod fixed → Thermal stress.
10. Stress–Strain Curve
- Proportional limit: stress ∝ strain ends.
- Elastic limit: recovery boundary.
- Yielding: large deformation begins.
- Plastic region: permanent deformation.
- Ultimate tensile strength: maximum stress.
- Necking: local area reduction.
- Fracture: specimen breaks.
- Area under stress–strain curve = work per unit volume.
11. Material Properties
| Property | Recognize by |
|---|---|
| Strength | Withstands large stress |
| Ductility | Can be drawn into wire |
| Malleability | Can be made into sheets |
| Brittleness | Breaks with little deformation |
| Toughness | Absorbs energy before breaking |
| Resilience | Stores/releases elastic energy |
| Hardness | Resists indentation/scratch |
| Fatigue | Weakens under repeated loading |
| Creep | Slow deformation under sustained load |
12. Quick Checks
Q: Stress?
Answer: Force per area: σ = F/A.
Q: Strain unit?
Answer: None; strain is dimensionless.
Q: Same force, smaller area?
Answer: Stress increases.
Q: Length change?
Answer: Young’s modulus.
Q: Volume change?
Answer: Bulk modulus.
Q: Sliding deformation?
Answer: Shear modulus.
Q: Elastic limit?
Answer: Boundary for recovery after unloading.
Q: Variable-force work?
Answer: Area under the F–x graph.
Q: Spring energy?
Answer: \(U=\frac12kx^2\).
Q: Wire-making property?
Answer: Ductility.
Q: Sheet-making property?
Answer: Malleability.
Q: End-to-end wires?
Answer: Series combination.
Q: Side-by-side wires?
Answer: Parallel combination.
Q: Heated rod fixed?
Answer: Thermal stress develops.
13. Mini Exam Reasoning
Recognition Pattern 1
Clue: Axial force stretches a wire.
Physics situation: Length deformation.
Model: Young’s modulus / extension of wire.
Relation: \(\Delta L=\frac{FL}{AY}\)
Check: Use original length and cross-sectional area; convert to SI units.
Recognition Pattern 2
Clue: A sideways force shifts the top of a block.
Physics situation: Shape deformation.
Model: Shear modulus.
Relation: \(G=\frac{\tau}{\gamma}\)
Check: For small deformation, γ ≈ θ in radians.
Recognition Pattern 3
Clue: A heated rod is fixed at both ends.
Physics situation: Thermal expansion is prevented.
Model: Thermal stress.
Relation: \(|\sigma|=Y\alpha|\Delta T|,\quad |F|=AY\alpha|\Delta T|\)
Check: Heating gives compression; cooling gives tension.
Recognition Pattern 4
Clue: An ideal spring is stretched by x.
Physics situation: Recoverable elastic energy.
Model: Spring energy.
Relation: \(U=\frac12kx^2\)
Check: Force increases linearly from zero to its final value.
14. Before You Calculate
- Identify the quantity asked.
- Choose the correct deformation model.
- Use original dimensions.
- Convert to SI units.
- Check signs.
- Verify validity conditions.
- Check the magnitude and units of the answer.
15. One-Minute Recall
- Stress \(\sigma=F/A\); strain \(\varepsilon=\Delta L/L\).
- Elasticity = recovery; proportionality = stress ∝ strain.
- Length → Young’s; volume → bulk; shape → shear.
- Poisson’s ratio = − transverse strain / longitudinal strain.
- Stress–strain story: proportional → elastic limit → yield → plastic → ultimate → necking → fracture.
- Spring energy \(U=\frac12kx^2\).
- Wire spring constant \(k=AY/L\).
- Series uses reciprocals; parallel spring constants add.
- Free thermal expansion differs from restrained thermal stress.
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