Class 11 Physics: Mechanical Properties of Solids Notes

---

1. Introduction

• Solids resist changes in shape and volume due to intermolecular forces.
• Mechanical properties describe how solids deform under forces.

---

2. Stress

• Definition: Force applied per unit area.

$$\text{Stress} = \frac{F}{A}$$Stress=AF​

Where:

• $F$F = applied force (N)
• $A$A = cross-sectional area (m²)

Types of Stress:

1. Tensile stress: Pulling force
2. Compressive stress: Pushing force
3. Shear stress: Force tangential to the surface

---

3. Strain

• Definition: Fractional deformation of a solid.

$$\text{Strain} = \frac{\Delta L}{L} \quad \text{(for length change)}$$Strain=LΔL​(for length change)

Where:

• $\Delta L$ΔL = change in length
• $L$L = original length

Types of Strain:

1. Tensile/compressive strain: $\frac{\Delta L}{L}$LΔL​
2. Shear strain: $\text{angle of deformation} \, (\theta)$angle of deformation(θ)

---

4. Elasticity

• Definition: Ability of a solid to regain its original shape after removing force.
• Elastic limit: Maximum stress up to which solid returns to original shape.
• Plastic deformation: Permanent change beyond elastic limit.

---

5. Hooke’s Law

• Within the elastic limit:

$$F \propto \Delta L \quad \text{or} \quad \text{Stress} \propto \text{Strain}$$F∝ΔLorStress∝Strain

• Mathematically:

$$\sigma = Y \epsilon$$σ=Yϵ

Where:

• $\sigma$σ = stress
• $\epsilon$ϵ = strain
• $Y$Y = Young’s modulus

---

6. Elastic Moduli

1. Young’s Modulus (Y): Measures stiffness under tensile or compressive stress.

$$Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} = \frac{F/A}{\Delta L / L}$$Y=Tensile StrainTensile Stress​=ΔL/LF/A​

2. Bulk Modulus (K): Measures resistance to uniform compression.

$$K = \frac{\text{Hydrostatic Pressure}}{\text{Volumetric Strain}} = -\frac{\Delta P}{\Delta V / V}$$K=Volumetric StrainHydrostatic Pressure​=−ΔV/VΔP​

3. Shear Modulus (η): Measures resistance to shear stress.

$$\eta = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\tan \theta}$$η=Shear StrainShear Stress​=tanθF/A​

---

7. Relation Between Elastic Constants

For isotropic solids:$$Y = 2 \eta (1 + \sigma), \quad K = \frac{Y}{3(1-2\sigma)}$$Y=2η(1+σ),K=3(1−2σ)Y​

Where $\sigma$σ = Poisson’s ratio:$$\sigma = -\frac{\text{lateral strain}}{\text{longitudinal strain}}$$σ=−longitudinal strainlateral strain​

---

8. Stress-Strain Curve

• Proportional limit: Up to which Hooke’s law is valid.
• Elastic limit: Beyond which plastic deformation occurs.
• Yield point: Stress at which plastic deformation starts.
• Ultimate stress: Maximum stress before breaking.

---

9. Key Points

• Solids resist both change in shape and change in volume.
• Stress/Strain relations define elastic behavior.
• Elastic moduli quantify stiffness of solids.
• Hooke’s law is valid only within elastic limit.