Mechanical Properties of Solids: Stress, Strain, and Elasticity

Mechanical Properties of Solids

Introduction

The mechanical properties of solids describe how materials respond to external forces, including stretching, compressing, and bending. While ideal rigid bodies do not deform, real solids can change shape and size under force. The ability of a material to regain its original shape after deformation is called elasticity, while permanent deformation is termed plasticity. Understanding these properties is crucial in engineering and structural design.

Stress and Strain

Definitions and Types

When a force is applied to a solid, 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 Stress: Force applied perpendicular to the cross-section, causing elongation or compression. Also called longitudinal stress.

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

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

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

Example: Stretching a spring or compressing a sphere in a fluid demonstrates these types of stress and strain.

Hooke’s Law

Linear Relationship Between Stress and Strain

For small deformations, stress is proportional to strain, described by Hooke’s law:

• $\text{Stress} \propto \text{Strain}$

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

Here, k is the modulus of elasticity. Hooke’s law is valid only in the linear region of the stress-strain curve.

Stress-Strain Curve

Behavior Under Increasing Load

The stress-strain curve illustrates how materials deform under load. Key points include:

• Elastic Region (O-A): Linear, obeys Hooke’s law. Material returns to original shape when load is removed.

• Yield Point (B): End of elastic region; stress at this point is yield strength ($\sigma_y$).

• Plastic Region (B-D): Permanent deformation occurs; material does not return to original shape.

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

• Fracture Point (E): Material breaks.

Brittle materials have ultimate strength and fracture points close together; ductile materials have them far apart.

Elastic Moduli

Young’s Modulus

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

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

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

Materials with high Young’s modulus (e.g., steel) require large forces for small deformations.

Shear Modulus (Modulus of Rigidity)

Shear modulus (G) measures resistance to shearing deformation:

• $G = \frac{\text{Shearing stress}}{\text{Shearing strain}}$

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

• For small angles: $G = \frac{F}{A \times \theta}$

Shear modulus is generally less than Young’s modulus; for most materials, $G \approx Y/3$.

Bulk Modulus

Bulk modulus (B) describes resistance to uniform compression:

• $B = -\frac{p}{\Delta V/V}$

The negative sign indicates that an increase in pressure results in a decrease in volume. The reciprocal of bulk modulus is compressibility ($k$):

• $k = \frac{1}{B} = -\frac{1}{\Delta p} \times \frac{\Delta V}{V}$

Solids are least compressible, gases are most compressible.

Poisson’s Ratio

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

• $\text{Poisson's ratio} = \frac{\text{Lateral strain}}{\text{Longitudinal strain}}$

• For steel: 0.28–0.30; for aluminium alloys: ~0.33

Elastic Potential Energy

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

• $W = \frac{1}{2} Y \frac{A}{L} l^2$

• Elastic potential energy 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 thickness of steel ropes in cranes is determined by yield strength to prevent permanent deformation. 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	Yes	No	$Y = \frac{F \times L}{A \times \Delta L}$	Young's modulus	Solid
Shearing	Pure shear, $\theta$	Yes	No	$G = \frac{F}{A \times \theta}$	Shear modulus	Solid
Hydraulic	Volume change	No	Yes	$B = -\frac{p}{\Delta V/V}$	Bulk modulus	Solid, liquid, gas

Example: Compression of Thighbone in a Human Pyramid

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

• Weight supported by each thighbone: $1078 \text{ N}$

• Length: $0.5 \text{ m}$, Radius: $2.0 \text{ cm}$, Area: $1.26 \times 10^{-3} \text{ m}^2$

• Young’s modulus for bone: $9.4 \times 10^9 \text{ N m}^{-2}$

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

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 have higher Young’s modulus than alloys and elastomers.

• Materials that stretch less for a given load are more elastic.

• Deforming force in one direction can produce strains in other directions, described by Poisson’s ratio.

Practice Exercises

• Calculate the ratio of Young’s modulus for steel and copper wires stretched by the same load.

• Determine Young’s modulus and yield strength from a stress-strain curve.

• Compare materials based on their stress-strain graphs.

• Compute elongations, strains, and compressional strains for various materials under load.

• Calculate bulk modulus and compressibility for liquids and gases.