Short Notes: Mechanical Properties of Solids

8.1 Stress and Strain

Stress is the internal restoring force per unit area that develops in a material when an external force acts on it. Stress is denoted by σ and its SI unit is the pascal (Pa), where 1 Pa = 1 N m⁻².

Types of stress: tensile stress (pulling, positive σ), compressive stress (squeezing, negative σ), and shear stress (force tangential to surface). Stress may be uniform or vary over the cross-section; stress defined above is the average (engineering) stress.

Hooke's law (linear elastic region): Within the elastic limit and limit of proportionality, stress is proportional to strain. For simple tensile/compressive loading this gives σ ∝ ε, and the constant of proportionality is Young's modulus.

8.2 Elastic Moduli

Elastic moduli describe the stiffness of a material under different types of loading. They relate stress to corresponding strain in the elastic region (where deformation is reversible).

Common interrelations for isotropic linear elastic materials (useful for converting between moduli):

• Y = 2η(1 + ν) (where η is shear modulus; often η is written as G)
• B = Y / [3(1 - 2ν)] (bulk modulus in terms of Young's modulus and Poisson's ratio)

Units: all three moduli (Y, B, η) have the unit pascal (Pa) or N m⁻². Poisson's ratio ν is dimensionless.

8.3 Elastic Energy

When a material is elastically deformed, mechanical work is stored as elastic potential energy. This energy can be expressed per unit volume or for an entire object.

• Energy density (energy per unit volume): u = \( \displaystyle \tfrac{1}{2}\times\text{stress}\times\text{strain} \).
• For linear tensile deformation using Young's modulus: u = \( \displaystyle \tfrac{1}{2}Y\varepsilon^{2} \) = \( \displaystyle \tfrac{1}{2}\sigma\varepsilon \).
• Elastic potential energy stored in a stretched wire of length L and cross-sectional area A under tensile force F: U = \( \displaystyle \tfrac{1}{2}\frac{F^{2}L}{AY} \) = \( \displaystyle \tfrac{1}{2}F\Delta L \), where ΔL is the extension.

These expressions assume linear elastic behaviour (Hooke's law) so that stress and strain are proportional throughout the deformation considered.

8.4 Important Relations and Additional Notes

• Compressibility (k) is the reciprocal of bulk modulus: k = \( \displaystyle \frac{1}{B} \).
• For an ideally perfectly rigid body, B → ∞ and so k = 0.
• Range of Poisson's ratio for stable, isotropic materials: -1 ≤ ν ≤ 0.5. Most common materials have 0 ≤ ν ≤ 0.5; ν = 0.5 corresponds to an incompressible material (volume preserving under small deformations).
• Notation clarity: stress is commonly denoted by σ, strain by ε, shear modulus by η (or G), bulk modulus by B (or K), Young's modulus by Y (or E), and Poisson's ratio by ν.
• Small-angle approximation for shear: when the angle θ is small, φ = tan θ ≈ θ (θ in radians), so shear strain ≈ angular displacement.
• Elastic limit and strength: elastic moduli describe behaviour only in the elastic region. Beyond the elastic limit (yield point) permanent (plastic) deformation occurs; ultimate tensile strength and fracture strength are separate material properties not given by the elastic moduli.
• Practical use: Young's modulus indicates stiffness in tension/compression (higher Y → stiffer material); shear modulus indicates resistance to shape change at constant volume; bulk modulus indicates resistance to volumetric compression.

Example (illustrative): A steel wire of length L, area A, and Young's modulus Y is stretched by a force F producing extension ΔL. The stress in the wire is σ = \( \displaystyle \frac{F}{A} \), the strain is ε = \( \displaystyle \frac{\Delta L}{L} \), and Young's modulus satisfies Y = \( \displaystyle \frac{\sigma}{\varepsilon} \). The elastic energy stored in the wire is U = \( \displaystyle \tfrac{1}{2}F\Delta L \) = \( \displaystyle \tfrac{1}{2}\frac{F^{2}L}{AY} \).