Revise Elastic and plastic behaviour for Cambridge 9702 (6.2)

Deformation is simply any change in an object's shape or size when a force acts on it. When we pull on something, it experiences tensile deformation (tension). If we push or squeeze it, that's compressive deformation (compression). These basic forces lead to very different responses from materials.

Imagine pulling a rubber band. It stretches, but let it go, and it snaps back. This is elastic deformation: the change is temporary, and the object returns to its original form once the forces are removed. Now, consider bending a paperclip. It stays bent, right? That's plastic deformation: the change is permanent, and the material doesn't recover its original shape.

For many elastic materials, especially springs, there's a simple relationship between the force applied and the resulting extension. Hooke's Law states that the extension ($\Delta L$) is directly proportional to the applied force (F), as long as you don't stretch it too far!

$F = k\Delta L$

Here, $k$ is the spring constant, which tells you how stiff the object is. A higher $k$ means a stiffer spring, requiring more force to produce the same extension. The unit for $k$ is Newtons per metre (N m⁻¹).

Even elastic materials have their limits. The limit of proportionality is the point on a force-extension graph where the straight line relationship (Hooke's Law) ends. Beyond this, the material might still return to its original shape, but $F$ is no longer directly proportional to $\Delta L$. The elastic limit is the absolute maximum force an object can withstand before it starts to deform permanently. If you exceed this, plastic deformation begins.

Plotting force against extension gives us a powerful visual tool. For materials obeying Hooke's Law, the graph is a straight line through the origin, and its gradient gives you the spring constant, $k$. As you deform an elastic object, you do work on it, and this energy is stored as elastic potential energy (EPE). Within the limit of proportionality, EPE is calculated as the area under the force-extension graph.

$EPE = \frac{1}{2}F\Delta L$ or $EPE = \frac{1}{2}k(\Delta L)^2$

While force and extension describe a specific object, stress and strain describe the material itself, regardless of its dimensions. This allows us to compare different materials directly.

Stress is the force applied per unit cross-sectional area. It measures how concentrated the internal forces are within the material. The unit for stress is Pascals (Pa), which is equivalent to N m⁻².

$\sigma = \frac{F}{A}$

Strain is the fractional change in length. It's the extension divided by the original length, making it a dimensionless quantity (no unit). It tells you how much the material has stretched relative to its initial size.

$\epsilon = \frac{\Delta L}{L_0}$

The Young Modulus (E) is a fundamental property of a material, representing its stiffness or resistance to elastic deformation. It's defined as the ratio of stress to strain within the elastic region. A high Young Modulus means the material is very stiff.

$E = \frac{\sigma}{\epsilon}$ or, by substituting the formulas for stress and strain, $E = \frac{F L_0}{A \Delta L}$

Like stress, the Young Modulus is measured in Pascals (Pa). On a stress-strain graph, the gradient of the linear region gives you the Young Modulus for that specific material.

Materials can be broadly classified by how they behave under stress. The two main categories are ductile and brittle.

• A ductile material, like copper or mild steel, can undergo significant plastic deformation before it fractures. It can be drawn into wires. On a stress-strain graph, it shows a long plastic region after the elastic limit.

• A brittle material, like glass or cast iron, shows very little or no plastic deformation. It fractures suddenly once its elastic limit is reached, often with minimal warning. Its stress-strain graph is almost entirely a straight line up to the point of fracture.

Stress-strain graphs are characteristic of a material. Key features to identify are:

• Ductile Material (e.g., Copper): The graph starts with a linear elastic region (obeying Hooke's Law), where the gradient is the Young Modulus. It passes the limit of proportionality (P) and the elastic limit (E). It then enters a plastic region, reaching the Ultimate Tensile Stress (UTS), which is the maximum stress the material can handle. After the UTS, the material starts to 'neck' (become thinner at one point) and fractures at the breaking point (X).

• Brittle Material (e.g., Glass): The graph is a straight line, showing elastic behaviour, right up until it fractures suddenly. There is no plastic deformation.

• Polymeric Material (e.g., Rubber): The graph is non-linear and shows very large strain for a small stress. The material is elastic, but does not obey Hooke's law. The loading and unloading curves form a hysteresis loop, indicating that energy is dissipated as heat during the deformation cycle.

If you deform a material plastically and then remove the force, it won't return to its original length, resulting in a permanent extension. On a force-extension graph, the unloading curve will be parallel to the initial elastic loading line. The area enclosed between the loading and unloading curves represents the work done to cause the permanent deformation, which is dissipated as internal energy (heat). This phenomenon, where the unloading path differs from the loading path, is known as hysteresis.