Elastic Modulus>Poisson’s Ratio

When a wire is stretched or compressed, its dimensions change in multiple directions. Poisson's ratio is a fundamental property that describes the relationship between these changes in different directions. It helps us understand how materials respond to applied forces and is crucial in materials engineering and structural design.

Within elastic limit, the ratio of lateral strain to the linear strain is called the Poisson's ratio.

\[\sigma=\frac{\text{Lateral strain}}{\text{Linear strain}}=\frac{\frac{d}{D}}{\frac{\Delta l}{l}}=\frac{d\cdot l}{D\cdot\Delta l}\]

Where:

• σ = Poisson's ratio
• l = original length of the wire
• ∆l = increase or decrease in length of the wire
• D = original diameter of the wire
• d = corresponding change in diameter of the wire

Important Note: Poisson's ratio has no unit. It is dimensionless.

• Dimensionless quantity – has no units
• Range for most materials – typically between 0.25 and 0.35 for commonly used metals
• Maximum theoretical value – approximately 0.5 if volume remains unchanged
• Practical value – much less than 0.5 in reality because volume increases during stretching
• Valid within the elastic limit – the ratio holds only when the material returns to its original shape after force removal

Linear Strain: The ratio of change in dimensions to original dimensions in the direction of the applied force.

Linear strain = \[\frac {Δl}{l}\]

Lateral Strain: The ratio of change in dimensions to original dimensions in a direction perpendicular to the applied force.

Lateral strain = \[\frac {d}{D}\]

Behavior During Stretching and Compression

When a wire is stretched:

• Length increases (∆l is positive)
• Diameter decreases (d is negative, since diameter reduces)
• The wire becomes longer and thinner

When a wire is compressed:

• Length decreases (∆l is negative)
• Diameter increases (d is positive, since diameter expands)
• The wire becomes shorter and thicker

Key Insight About Volume

In reality, when a wire is stretched, its volume also increases, not remaining constant. The assumption of constant volume is an approximation used for calculations, but it does not reflect the actual behavior of materials.

Observation:

Most commonly used metals have Poisson's ratio values between 0.25 and 0.35, indicating consistent material behavior across a range of metallic substances.