A cylindrical specimen of an alloy \(8 \mathrm{~mm}\) (0.31 in.) in diameter is stressed elastically in tension. A force of \(15,700 \mathrm{~N}\) (3530 lb_{f } \()\) produces a reduction in specimen diameter of \(5 \times 10^{-3} \mathrm{~mm}\left(2 \times 10^{-4}\right.\) in.). Compute Poisson's ratio for this material if its modulus of elasticity is \(140 \mathrm{GPa}\left(20.3 \times 10^{6} \mathrm{psi}\right)\).

The modulus of elasticity, also known as Young's modulus, is a measure of the stiffness of a material. It defines the relationship between stress (force per unit area) and strain (deformation) in a material in the elastic region where the material returns to its original shape once the applied stress is removed. The higher the modulus of elasticity, the stiffer the material, meaning it deforms less under a given load.

In practice, if you have a material with a high modulus of elasticity, such as steel, you would expect it to stretch less under tension than a material with a lower modulus of elasticity, like rubber. Mathematically, it can be expressed as:

\[ E = \frac{\text{Stress}}{\text{Strain}} \]
In our exercise, we're given that the alloy has a modulus of elasticity of 140 GPa. To apply this concept in calculations, we often convert this value into a consistent unit system that matches the stress units (N/mm², in this case) to ensure that the strain is dimensionless.

Axial strain is a dimensionless measure of deformation representing the elongation or compression of a material along its axis. Calculated as the change in length divided by the original length, axial strain provides insight into how a material will behave under tensile or compressive stress.

In the cylindrical specimen described in the exercise, traction force leads to stretching, resulting in elongation. Thus, axial strain pertains to the lengthwise deformation due to the applied force. The formula for axial strain (\( \epsilon_a \)) is:

\[ \epsilon_a = \frac{\Delta L}{L_0} \]
where \(\Delta L\) denotes the change in length and \(L_0\) is the original length. However, to calculate this value, we must first compute the actual change in length from the given stress and modulus of elasticity, as described in the step-by-step solution.

Lateral strain refers to the deformation of a material perpendicular to the direction of the applied force. While axial strain measures stretch in the length of a material, lateral strain measures the thinning in the material's cross-sectional area. It can be caused by the Poisson effect, where the material's volume attempts to remain constant when stretched or compressed.

The formula to calculate lateral strain is similar to the one for axial strain, but it applies to the diameter (for a cylindrical specimen) or width/height (for other shapes):

\[ \epsilon_l = \frac{\Delta D}{D_0} \]
where \(\Delta D\) is the change in diameter and \(D_0\) is the original diameter. Understanding both axial and lateral strain is crucial when engineering materials for structures, as it influences the material's ability to withstand multidirectional forces.

The concepts of stress and strain are fundamentally related in materials science and engineering. Stress is the internal distribution of force per unit area within a material that arises from externally applied forces. Strain, on the other hand, is the measure of the material's deformation in response to stress. These two quantities are linked by the material's mechanical properties, such as the modulus of elasticity.

Explicitly, pre-defined material characteristics such as Poisson's ratio (\(u\)) allow us to relate axial and lateral strain. Poisson's ratio is a measure of the Poisson effect that indicates how much a material tends to expand in directions perpendicular to the direction of compression or to contract in directions perpendicular to the direction of stretch.

The relationship is given by the formula:

\[ u = -\frac{\epsilon_l}{\epsilon_a} \]
Negative sign indicates that the lateral strain is opposite in direction to the axial strain under tensile stress. For most materials, the value of Poisson's ratio lies between 0 and 0.5. By combining these relationships, we can describe the complete elastic behaviour of a material under various loading conditions. This interconnectedness is crucial for designing and predicting how materials will perform in real-world applications.