Deformation of solids - Physics (9702) - Cambridge A-Level

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Deformation of Solids (AS Level Physics 9702, Topic 6)

Welcome to the chapter on Deformation of Solids! This might sound like a technical topic, but it’s actually physics in action all around you. Every time you stretch a spring, stand on a floor, or even bite into an apple, forces are causing materials to change shape—or deform.

In this chapter, we will look at how materials behave when subjected to stress, introducing fundamental concepts like stress, strain, Hooke's Law, and the all-important Young Modulus. Understanding this is key for engineers who design buildings, bridges, and everything else that needs to withstand forces without collapsing!

6.1 Stress and Strain: The Basics of Changing Shape

What is Deformation?

Deformation simply means a change in the shape or size of a solid object due to an applied force. We only consider deformation in one dimension (1D) in AS Physics—meaning the material gets longer or shorter along the line of the applied force.

There are two main types of forces that cause deformation:

Tensile Forces: Forces that pull the material apart, causing extension (making it longer).
Compressive Forces: Forces that push the material together, causing compression (making it shorter).

Introducing Hooke's Law and the Spring Constant

For many materials (like springs, wires, and rods), when a force (or load) is first applied, the extension is directly proportional to the force. This relationship is called Hooke's Law.

If you hang a 1 N weight on a spring, and it stretches 1 cm, a 2 N weight will stretch it 2 cm (as long as you don't stretch it too far!).

Hooke’s Law: Force is directly proportional to extension.
\[F \propto x\]

The Spring Constant, k

To turn the proportionality above into an equation, we introduce the spring constant, $k$.

\[F = kx\]

Where:

$F$ is the applied force or load (in Newtons, N).
$x$ is the extension or compression (change in length, in metres, m).
$k$ is the spring constant (in $\text{N m}^{-1}$).

Think of $k$ as the stiffness of the spring or wire. A large $k$ means the material is very stiff (hard to stretch) and requires a large force for a small extension.

The Limit of Proportionality

Hooke's Law only applies up to a certain point called the limit of proportionality. Beyond this point, the force and extension are no longer linearly related, and the graph stops being a straight line through the origin (see Section 6.2).

Quick Review: Hooke's Law Key Takeaways

Relationship: $F = kx$
$k$ unit: $\text{N m}^{-1}$
Rule: Only valid up to the limit of proportionality.

Defining Stress ($\sigma$), Strain ($\epsilon$), and Young Modulus ($E$)

While Hooke's Law is great for springs, it only tells us about a specific object. For engineers to compare materials (like steel vs. plastic), they need properties independent of the object's initial size or shape. This is where stress and strain come in.

1. Stress ($\sigma$)

Stress is defined as the force applied per unit cross-sectional area of the material. It tells you how concentrated the force is.

\[\text{Stress } (\sigma) = \frac{\text{Force } (F)}{\text{Cross-sectional Area } (A)}\] \[\sigma = \frac{F}{A}\]

SI Unit of Stress: $\text{N m}^{-2}$ or Pascal (Pa).

Analogy: Imagine pushing your finger into clay. If you push with your whole thumb (large area, low stress), it might not dent. If you push with your fingernail (small area, high stress), it dents easily, even with the same force.

2. Strain ($\epsilon$)

Strain is defined as the extension per unit original length. It is a measure of the fractional change in the material's length.

\[\text{Strain } (\epsilon) = \frac{\text{Extension } (x)}{\text{Original Length } (L)}\] \[\epsilon = \frac{x}{L}\]

SI Unit of Strain: Strain is a ratio of two lengths ($\text{m/m}$), so it is dimensionless (it has no units).

Did you know? Strain is often expressed as a percentage or multiplied by a factor of $10^{-6}$ (microstrain) because most engineering materials only strain very slightly before breaking!

3. The Young Modulus ($E$)

The Young Modulus (sometimes called the modulus of elasticity) is the property that relates stress and strain. It is defined as the ratio of tensile stress to tensile strain, provided the limit of proportionality has not been exceeded.

This value is a constant for a given material and temperature. It is the definitive measure of a material's stiffness.

\[\text{Young Modulus } (E) = \frac{\text{Stress } (\sigma)}{\text{Strain } (\epsilon)}\] \[E = \frac{F/A}{x/L} = \frac{FL}{Ax}\]

SI Unit of Young Modulus: Since strain has no units, the units for $E$ are the same as stress: $\text{N m}^{-2}$ or Pa. It is often very large (e.g., steel is around $200 \times 10^9 \text{ Pa}$).

Memory Tip: Remember which is which!

Stress ($\sigma$): Force per area ($F/A$) – it presses you.
Strain ($\epsilon$): Stretch per length ($x/L$) – it's how much you change.
Young Modulus ($E$): Ratio of the two ($\sigma/\epsilon$) – determines the material's stiffness.

6.2 Elastic and Plastic Behaviour

Elastic Deformation vs. Plastic Deformation

When you apply a load to a material, two things can happen to its structure:

1. Elastic Deformation

Elastic deformation is temporary. The material returns to its original dimensions once the applied force is removed. In an elastic material, the bonds between atoms are stretched or compressed, but they spring back when the stress is gone.

Example: Stretching a new rubber band slightly. Once you release it, it snaps back to its original length.

2. Plastic Deformation

Plastic deformation is permanent. The material does not return to its original shape or size after the load is removed. This happens when the forces applied cause the atoms inside the material to slide past each other into new, permanent positions.

Example: Bending a metal paperclip until it stays bent.

The Elastic Limit

The elastic limit is the point beyond which the material starts to undergo permanent (plastic) deformation. For most metals, the elastic limit is very close to the limit of proportionality.

Common Mistake to Avoid: Don't confuse the *Limit of Proportionality* (where $F \propto x$ stops) with the *Elastic Limit* (where permanent damage starts). While they are often treated as the same point in simple AS problems, technically, the limit of proportionality occurs slightly before the elastic limit.

Work Done and Elastic Potential Energy

When a material is stretched or compressed, work is done by the applied force. This work done is stored as Elastic Potential Energy ($E_p$) within the material (like energy stored in a spring).

Force-Extension Graphs (F-x Graphs)

The work done ($W$) or energy stored ($E_p$) when a material is deformed is given by the area under the force-extension graph.

If the material is stretched only within the limit of proportionality, the graph is a straight line, forming a triangle.

\[\text{Work Done } (W) = \text{Area under the graph}\]

Since the area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$:

Height $= F$ (force)
Base $= x$ (extension)

Elastic Potential Energy ($E_p$) (within the limit of proportionality):
\[E_p = \frac{1}{2} Fx\]

Using Hooke's Law ($F = kx$), we can substitute $F$:

\[E_p = \frac{1}{2} (kx)x = \frac{1}{2} kx^2\]

What if the deformation is plastic? If the material is stretched past its elastic limit, the energy stored is not fully recovered. The area under the loading curve represents the total work done, but when the force is removed, the material follows a straight line back parallel to the elastic region, leaving a permanent extension. The difference between the work done (loading) and the energy recovered (unloading) is energy dissipated as heat.

Key Takeaway: Energy

Work done/Energy stored = Area under the F-x graph.
Use $E_p = \frac{1}{2} Fx$ or $E_p = \frac{1}{2} kx^2$ only if the deformation is elastic (i.e., within the limit of proportionality).

Experiment: Determining the Young Modulus of a Wire

A crucial skill for this topic is describing the experimental method used to determine the Young Modulus ($E$) of a metal in the form of a wire (6.1, point 6). This is often an assessment question!

Step-by-Step Procedure

The goal is to measure the four quantities needed for the formula $E = \frac{FL}{Ax}$:

1. Setup:

Use a long, thin wire (often 2-3 metres) to maximize extension $x$ and minimize percentage uncertainty in length $L$.
The wire is clamped securely at one end.
A second identical comparison wire (or 'dummy wire') is set up next to the test wire to compensate for any movement in the support structure or changes in temperature.

2. Measuring Original Dimensions (L and A):

Measure the original length ($L$) of the test section of the wire (e.g., from the clamp to the start of the scale).
Measure the diameter ($d$) of the wire using a micrometer screw gauge several times at different points along the wire and average the reading. This allows calculation of the cross-sectional area: $A = \pi (d/2)^2$.

3. Measuring Force and Extension (F and x):

Attach a scale and a vernier marker to the end of the test wire (this measures extension $x$).
Start with a small initial tensioning load (this removes kinks and keeps the wire straight). The initial reading on the vernier scale is the zero reading for extension.
Increase the load ($F$) incrementally by adding known masses.
Record the new vernier reading for each mass. The extension ($x$) is the difference between the current reading and the zero reading.

4. Analysis:

Plot a graph of Load ($F$) on the y-axis against Extension ($x$) on the x-axis.
Identify the linear region (where Hooke's Law holds).
Calculate the gradient of this linear section. The gradient is $\frac{F}{x}$, which is equal to the spring constant $k$.

5. Final Calculation:

Substitute the gradient ($k$) into the rearrangement of the Young Modulus formula: \[E = \frac{F}{x} \times \frac{L}{A}\] \[E = k \times \frac{L}{A}\]

Why use the comparison wire? The comparison wire provides a stable reference point. If the whole apparatus shifts slightly, or if the wire expands due to small temperature changes, the comparison wire's reference mark also moves, ensuring that only the extension due to the applied load is measured by the vernier scale.

Quick Review: The Young Modulus Experiment

Measure L (ruler).
Measure d (micrometer) $\rightarrow$ Calculate A.
Vary F (masses).
Measure x (vernier scale/marker).
Calculate $E = \text{gradient} \times (L/A)$.

Summary of Key Definitions and Formulas

Definitions:

Load (F): The force applied to the material.
Extension (x): The increase in length due to a tensile load.
Limit of Proportionality: The point up to which $F \propto x$.
Elastic Limit: The point beyond which the material deforms permanently.
Elastic Deformation: Reversible deformation.
Plastic Deformation: Permanent deformation.
Stress ($\sigma$): Force per unit area.
Strain ($\epsilon$): Extension per unit original length.

Formulas:

Hooke's Law: $F = kx$

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

Strain: $\epsilon = \frac{x}{L}$

Young Modulus: $E = \frac{\sigma}{\epsilon} = \frac{FL}{Ax}$

Elastic Potential Energy (E.P.E.): $E_p = \frac{1}{2} Fx = \frac{1}{2} kx^2$ (Only valid within the limit of proportionality)