Mechanical Properties of Solids: Stress, Strain, and Elasticity

Mechanical Properties of Solids

Introduction

Solids possess definite shape and size, but they are not perfectly rigid. When subjected to external forces, solids can be stretched, compressed, or bent. The ability of a material to regain its original shape and size after the removal of the deforming force is called elasticity, while permanent deformation is referred to as plasticity. Understanding the elastic behavior of materials is crucial in engineering design, such as in buildings, bridges, and machinery.

Stress and Strain

Definitions and Types

When a force is applied to a solid body, it develops a restoring force to maintain equilibrium. The restoring force per unit area is called stress, and the fractional change in dimension is called strain. The SI unit of stress is N m–2 (Pascal, Pa).

• Tensile/Compressive (Longitudinal) Stress: Force applied perpendicular to cross-section, causing elongation or compression.

• Shearing Stress: Force applied parallel to cross-section, causing displacement between faces.

• Hydraulic Stress: Uniform force applied by a fluid, causing volume change without shape change.

Formulas:

• Stress: $\text{Stress} = \frac{F}{A}$

• Longitudinal Strain: $\text{Longitudinal strain} = \frac{\Delta L}{L}$

• Shearing Strain: $\text{Shearing strain} = \frac{\Delta x}{L} = \tan \theta \approx \theta$

• Volume Strain: $\text{Volume strain} = \frac{\Delta V}{V}$

Hooke’s Law

Linear Relationship Between Stress and Strain

For small deformations, stress is proportional to strain, a relationship known as Hooke’s law. The constant of proportionality is called the modulus of elasticity.

• $\text{Stress} = k \times \text{Strain}$

• Valid only in the linear region of the stress-strain curve.

Stress-Strain Curve

Behavior Under Tensile Stress

The stress-strain curve illustrates how a material deforms under increasing load. Key points include:

• Elastic Region (O to A): Linear, obeys Hooke’s law, material returns to original shape.

• Yield Point (B): End of elastic region; yield strength $\sigma_y$.

• Plastic Deformation (B to D): Permanent deformation occurs.

• Ultimate Strength (D): Maximum stress before fracture.

• Fracture Point (E): Material breaks.

• Brittle vs. Ductile: If D and E are close, material is brittle; if far apart, ductile.

Elastomers (e.g., rubber, aorta tissue) can undergo large strains but do not obey Hooke’s law over most of their range.

Elastic Moduli

Young’s Modulus

Young’s modulus (Y) quantifies the stiffness of a material under tensile or compressive stress:

• $Y = \frac{\text{Tensile (or compressive) stress}}{\text{Longitudinal strain}} = \frac{\sigma}{\epsilon}$

• $Y = \frac{F/A}{\Delta L/L} = \frac{F \times L}{A \times \Delta L}$

• Unit: N m–2 (Pa)

Metals have high Young’s moduli, making them suitable for structural applications.

Example: Structural Steel Rod

• Radius: 10 mm, Length: 1.0 m, Force: 100 kN, Young’s modulus: $2.0 \times 10^{11}$ N m–2

• Stress: $3.18 \times 10^8$ N m–2

• Elongation: $1.59 \times 10^{-3}$ m

• Strain: $1.59 \times 10^{-3}$ (0.16%)

Shear Modulus (Modulus of Rigidity)

Shear modulus (G) measures resistance to shearing deformation:

• $G = \frac{\text{Shearing stress}}{\text{Shearing strain}} = \frac{F/A}{\Delta x/L} = \frac{F \times L}{A \times \Delta x}$

• Unit: N m–2 (Pa)

• For most materials, $G \approx Y/3$

Bulk Modulus

Bulk modulus (B) describes resistance to uniform compression:

• $B = -\frac{p}{\Delta V/V}$

• Unit: N m–2 (Pa)

• Solids have much higher bulk moduli than liquids and gases.

• Compressibility: $k = \frac{1}{B}$

Poisson’s Ratio

Poisson’s ratio is the ratio of lateral strain to longitudinal strain in a stretched wire:

• $\text{Poisson’s ratio} = \frac{\Delta d/d}{\Delta L/L}$

• Dimensionless, depends on material.

Elastic Potential Energy

Work done in stretching a wire is stored as elastic potential energy:

• $U = \frac{1}{2} \times \text{Young’s modulus} \times \text{strain}^2 \times \text{volume}$

• Per unit volume: $u = \frac{1}{2} \sigma \epsilon$

Applications of Elastic Behaviour

Engineering and Structural Design

Elastic properties are essential in designing buildings, bridges, cranes, and artificial limbs. For example, the cross-sectional area of a steel rope in a crane must be sufficient to avoid permanent deformation under load:

• $A \geq \frac{W}{\sigma_y}$

Beams and columns are designed to minimize bending and buckling, using materials with high Young’s modulus and optimal cross-sectional shapes.

Summary Table: Types of Stress, Strain, and Elastic Moduli

Type of Stress	Strain	Change in Shape	Change in Volume	Elastic Modulus	Name of Modulus	State of Matter
Tensile/Compressive	Elongation/Compression (ΔL/L)	Yes	No	$Y = \frac{F \times L}{A \times \Delta L}$	Young’s modulus	Solid
Shearing	Pure shear, θ	Yes	No	$G = \frac{F}{A \times \theta}$	Shear modulus	Solid
Hydraulic	Volume change (ΔV/V)	No	Yes	$B = -\frac{p}{\Delta V/V}$	Bulk modulus	Solid, liquid, gas

Example: Compression in Human Thighbone

In a circus human pyramid, the performer at the bottom supports the weight of the group. The compression in each thighbone can be calculated using Young’s modulus for bone:

• Weight supported: 1078 N per thighbone

• Length: 0.5 m, Radius: 2.0 cm, Area: $1.26 \times 10^{-3}$ m2

• Young’s modulus: $9.4 \times 10^9$ N m–2

• Compression: $\Delta L = \frac{F \times L}{Y \times A} = 4.55 \times 10^{-5}$ m

Points to Ponder

• Stress is not a vector quantity; it cannot be assigned a specific direction like force.

• Hooke’s law is valid only in the linear region of the stress-strain curve.

• Young’s modulus and shear modulus are relevant only for solids; bulk modulus applies to solids, liquids, and gases.

• Metals are more elastic than alloys and elastomers.

• Poisson’s ratio describes proportionality between lateral and longitudinal strain.

Practice Exercises

• Calculate ratios of Young’s modulus, yield strength, and elongation for various materials.

• Apply concepts of stress, strain, and elastic moduli to real-world engineering problems.