Revise Stress and strain for Cambridge 9702 (6.1)

When a force acts on an object, its shape or size often changes – this is called deformation. Forces can either stretch an object, known as tensile forces, or squash it, called compressive forces. Think about pulling a spring (tensile) versus pressing down on a block (compressive).

Extension, often symbolised as $\\Delta L$ or $x$, is simply how much longer an object gets from its original length after a tensile force is applied. It's a direct, absolute measure of the change in length.

Stress tells us how concentrated the internal forces are within a material. It's not just about the total force, but how that force is distributed over a specific cross-sectional area. Imagine pushing a pin with your finger; the force is small, but the stress on the tip is huge!

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

While extension tells us the absolute change in length, strain gives us a relative measure of deformation. It compares the amount of stretch to the object's initial size, making it a dimensionless quantity useful for comparing different objects regardless of their original dimensions.

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

For many elastic materials, especially springs, there's a simple relationship between the applied force and the resulting extension, first observed by Robert Hooke. This law is crucial for understanding how objects behave within their elastic limits, provided temperature remains constant.

$F = k\Delta L$

Materials can deform in two main ways. Elastic deformation is like stretching a rubber band – it returns to its original shape when you let go. But if you pull too hard, it might stretch permanently, entering plastic deformation, where it doesn't fully recover.

When you stretch or compress an elastic object, you do work on it, and this work is stored as elastic potential energy. This energy is released when the object returns to its original shape, for example, a stretched catapult launching a projectile.

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

To compare the inherent stiffness of different materials, we use the Young Modulus. Unlike the spring constant, which depends on the object's dimensions, Young Modulus is a fundamental property of the material itself. It's a ratio that normalises for the object's size and shape.

$E = \frac{\sigma}{\epsilon} = \frac{F L_0}{A \Delta L}$

Materials react differently when stretched to their breaking point. Ductile materials like copper can stretch a lot and undergo significant plastic deformation before fracturing. Brittle materials like glass, however, show very little plastic deformation and fracture suddenly with minimal stretching.

When a material is loaded (stretched) and then unloaded (released), its behaviour can tell us if permanent deformation occurred. If the unloading curve doesn't return to the origin and instead shows residual extension, it signifies that plastic deformation has taken place.