A cylindrical specimen of a nickel alloy having an elastic modulus of \(207 \mathrm{GPa}\left(30 \times 10^{6}\right.\) psi) and an original diameter of \(10.2 \mathrm{mm}\) \((0.40 \text { in. })\) will experience only elastic deformation when a tensile load of \(8900 \mathrm{N}\left(2000 \mathrm{lb}_{\mathrm{f}}\right)\) is applied. Compute the maximum length of the specimen before deformation if the maximum allowable elongation is \(0.25 \mathrm{mm}\) \((0.010 \text { in. })\).

Understanding Hooke's Law is fundamental when it comes to studying how materials deform under force. It's a principle that tells us how materials will behave when loads are applied that do not exceed the material's elastic limit.

In simpler terms, think of it as a spring; when you stretch or compress it, it will return to its original shape after the force is removed. This is what we mean by elastic deformation - the material's ability to return to its original shape after being subjected to a force. Hooke's Law is mathematically expressed as:
\( F = kx \)
where \( F \) is the force applied to the material, \( k \) is the spring constant or stiffness of the material, and \( x \) represents the change in length of the material (also known as displacement). It's the fundamental concept when calculating how stresses and strains interact in materials.

The correlation between stress and strain is at the heart of understanding material behavior, particularly when looking at elastic deformation.

• Stress is defined as internal resistance or force per unit area inside a material that arises from externally applied forces.
• Strain is the measure of deformation representing the displacement between particles in the material body relative to a reference length.

When stress causes a material to deform, the extent of the deformation is quantified by the strain. In the elastic region, this relationship can be linear, which leads to a simple proportionality that can be described by Hooke's Law:
\( \text{Stress} (\text{σ}) = \text{Elastic Modulus} (E) \times \text{Strain} (\text{ε}) \)
A graph plotting this relationship produces a straight line where the slope represents the material's Elastic Modulus, and the linear portion of the graph is where Hooke's Law applies. This is of great use in engineering and design, ensuring materials won't fail when used within these calculated parameters.

Elastic Modulus, also known as Young's Modulus, is a material property that measures its stiffness and is a fundamental descriptor in the stress-strain relationship. It's a measure of an object or substance's resistance to being deformed elastically when a force is applied to it.

Mathematically, it's defined as the ratio of stress (force per unit area) over strain (proportional deformation) in the range when they are proportional to each other, or:
\( E = \frac{σ}{ε} \)
The Elastic Modulus value is crucial because it tells us how much a material will deform when a certain stress is applied and, therefore, is used to determine a material's ability to withstand changes in length when under tension or compression. High Elastic Modulus materials are stiffer - with less deformation under the same stress compared to materials with lower Elastic Modulus. When you're choosing materials for construction or manufacturing, understanding the Elastic Modulus helps predict how different materials will react under similar stresses.