Chapter 6: Deformation of Solids - Stress and Strain
Hello future engineers and physicists! This chapter might seem tough because it introduces a lot of new terms, but it’s incredibly important. We are going to explore how materials respond when you push or pull them. Understanding stress and strain is crucial for designing everything from suspension bridges to skyscrapers!
Don't worry if the formulas look complicated at first. We will break them down piece by piece and use simple analogies to make them stick.
6.1 Stress and Strain: The Basics
Forces Causing Deformation (Tensile and Compressive)
When an object changes its shape or size, we say it has undergone deformation. This change is caused by forces acting on the object.
- Tensile Forces: These forces try to stretch the material, pulling it apart. (Imagine pulling a rubber band). This causes extension.
- Compressive Forces: These forces try to squeeze or crush the material. (Imagine standing on a soda can). This causes compression.
Note: For this syllabus, we only consider deformation happening in one dimension (meaning the force acts along the length of the object).
Load, Extension, and the Limit of Proportionality
The applied force causing the deformation is often called the load.
When you apply a load to a spring or a wire, it extends. The change in length is called the extension (\(x\) or \(\Delta L\)).
Hooke’s Law
The most fundamental relationship in this section is Hooke's Law. It describes the behavior of many materials, especially springs, under small loads.
Definition: Hooke's Law states that the force (\(F\)) needed to extend or compress an object is directly proportional to the extension (\(x\)) or compression, provided the limit of proportionality is not exceeded.
Formula:
\[ F = kx \]
Where:
- \(F\) is the applied force (Load), measured in Newtons (N).
- \(x\) is the extension (or compression), measured in metres (m).
- \(k\) is the spring constant or stiffness constant, measured in Newtons per metre (\(\text{N m}^{-1}\)).
The spring constant, \(k\), tells you how stiff a material is. A large \(k\) means the material is very stiff and hard to stretch.
Limit of Proportionality: This is the point on a Force-Extension graph beyond which the relationship \(F \propto x\) is no longer valid. The line stops being a straight line through the origin.
Hooke’s Law is the linear relationship (\(F=kx\)). It only works up to the Limit of Proportionality.
6.2 Stress, Strain, and the Young Modulus
Force (\(F\)) and Extension (\(x\)) are useful for a specific object (like a specific spring). But if you want to compare materials—say, comparing steel and aluminum—you need measures that don't depend on the object's specific dimensions (length and thickness).
This is why we use Stress and Strain.
1. Stress (\(\sigma\))
Stress is a measure of the force acting per unit cross-sectional area.
Analogy: Imagine standing on ice. If you wear skis (large area), the force (your weight) is spread out, causing low stress. If you wear ice skates (tiny area), the stress is very high.
Definition and Formula:
\[ \text{Stress} \ (\sigma) = \frac{\text{Force} \ (F)}{\text{Cross-sectional Area} \ (A)} \]
\[ \sigma = \frac{F}{A} \]
- Unit: Newtons per square metre (\(\text{N m}^{-2}\)) or Pascals (Pa).
2. Strain (\(\epsilon\))
Strain is a measure of the fractional change in the original dimension.
Analogy: Stretching a 1 cm wire by 1 mm is a huge stretch proportionally. Stretching a 100 m rope by 1 mm is negligible. Strain accounts for the original length.
Definition and Formula:
\[ \text{Strain} \ (\epsilon) = \frac{\text{Extension} \ (x)}{\text{Original Length} \ (L)} \]
\[ \epsilon = \frac{x}{L} \]
- Unit: Strain has no units, as it is a ratio of two lengths (\(\text{m}/\text{m}\)). It is sometimes given as a percentage or just a pure number.
3. The Young Modulus (\(E\))
Just as the spring constant (\(k\)) measures stiffness for a specific object, the Young Modulus measures the stiffness of the material itself.
For a material behaving according to Hooke's Law (up to the limit of proportionality), the ratio of stress to strain is constant. This constant is the Young Modulus.
Definition: The Young Modulus is the ratio of tensile stress to tensile strain for a material when deformation is elastic.
Formula:
\[ \text{Young Modulus} \ (E) = \frac{\text{Stress} \ (\sigma)}{\text{Strain} \ (\epsilon)} \]
Substituting the definitions of \(\sigma\) and \(\epsilon\):
\[ E = \frac{F/A}{x/L} = \frac{FL}{Ax} \]
- Unit: Since strain has no units, the unit of Young Modulus is the same as the unit of stress: Pascals (Pa) or \(\text{N m}^{-2}\). Typically, materials have Young Moduli in the order of GPa (\(10^9 \text{ Pa}\)).
Did you know? Steel has a high Young Modulus (around 200 GPa), meaning it resists stretching greatly, making it perfect for construction. Rubber has a very low Young Modulus.
- Stress \(\sigma\): Force per unit Area (\(F/A\))
- Strain \(\epsilon\): Extension per unit Original Length (\(x/L\))
- Young Modulus \(E\): Stress / Strain (\(FL/Ax\))
6.3 Elastic and Plastic Behaviour
When you pull on a material, what happens when you let go?
1. Elastic Deformation
If a material returns perfectly to its original size and shape after the deforming load is removed, it is undergoing elastic deformation.
- Example: A perfect spring or rubber band being stretched slightly.
- The relationship \(F \propto x\) holds true within the limit of proportionality.
The Elastic Limit is the maximum stress or force a material can withstand before it begins to deform permanently.
Important Note: The Limit of Proportionality and the Elastic Limit are very close together, but they are not always the same point.
- Limit of Proportionality (P): \(F \propto x\) fails here.
- Elastic Limit (E): Beyond this point, permanent deformation (plastic deformation) occurs.
2. Plastic Deformation
If a material remains permanently stretched or compressed after the deforming load is removed, it is undergoing plastic deformation.
- Example: Bending a metal paperclip. Even when you let go, it stays bent.
- Plastic deformation occurs when the force exceeds the material's elastic limit.
Elastic = Temporary change, returns to original shape.
Plastic = Permanent change, material is ruined or permanently reshaped.
6.4 Energy Stored During Deformation
When you stretch a wire or a spring, you are doing work against the internal forces holding the material together. This work done is stored in the material as Elastic Potential Energy (\(E_p\)).
Work Done from the Force-Extension Graph
Remember from the chapter on Work, Energy, and Power that:
\[ \text{Work Done} = \text{Force} \times \text{Distance moved in direction of force} \]
However, when stretching a spring, the force is not constant (it increases as \(x\) increases, \(F=kx\)).
For a non-constant force, the work done (and thus the stored energy) is found by calculating the area under the Force-Extension (\(F-x\)) graph.
If the material obeys Hooke's Law (i.e., we are within the limit of proportionality), the \(F-x\) graph is a straight line, forming a triangle.
Area of triangle: \(\frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times x \times F\)
Formulas for Elastic Potential Energy
Within the limit of proportionality, the stored elastic potential energy (\(E_p\)) is given by:
\[ E_p = \frac{1}{2} Fx \]
Since we know \(F = kx\), we can substitute this into the equation:
\[ E_p = \frac{1}{2} (kx) x \]
\[ E_p = \frac{1}{2} kx^2 \]
Encouraging Phrase: Don't worry if you forget one formula—if you remember \(F = kx\) and that the work done is \(\frac{1}{2}Fx\), you can derive \(\frac{1}{2}kx^2\) yourself!
6.5 Experiment to Determine the Young Modulus
You need to be able to describe the classic experiment used to determine the Young Modulus (\(E\)) of a metal in the form of a wire.
The goal is to measure the four variables needed for the formula: \(E = \frac{FL}{Ax}\).
Setup and Procedure:
-
Setup: Suspend two long, thin wires vertically from a rigid support.
- Wire 1: The test wire, to which known loads (\(F\)) are added.
- Wire 2: A control wire, often supporting a scale, kept under constant tension. This is crucial as it cancels out effects like thermal expansion that might affect both wires.
- Measuring Original Length (\(L\)): Use a metre rule to measure the original length of the test wire (from the fixed point to the reference mark/gauge).
- Measuring Cross-sectional Area (\(A\)): Measure the diameter (\(d\)) of the test wire in several different places using a micrometer screw gauge. Take the average diameter and calculate the area using \(A = \pi (d/2)^2\).
-
Applying Load (\(F\)) and Measuring Extension (\(x\)):
- Start with a small initial load to straighten the wire.
- Add masses (known weights, \(F=mg\)) to the test wire hanger in small increments.
- For each added load, measure the corresponding extension (\(x\)) using a vernier scale attached to the control wire. The control wire ensures accuracy by minimizing measurement errors.
-
Analysis and Calculation:
- Plot a graph of Force (\(F\)) (y-axis) against Extension (\(x\)) (x-axis).
- Identify the straight line portion (Hooke's Law region).
- Calculate the gradient of this straight line. Recall that \(F = (\text{Gradient}) x\), so the Gradient = k (the spring constant).
- Finally, use the formula derived from \(E = \frac{FL}{Ax}\):
Since \(\frac{F}{x}\) is the gradient (\(k\)), we can write:
\[ E = \frac{k L}{A} \]
Substitute the calculated gradient (\(k\)) and the initial measurements (\(L\) and \(A\)) to find the Young Modulus (\(E\)).
Imagine the lab temperature increases. Both wires will expand slightly. By measuring the extension of the test wire relative to the control wire, the temperature effects are cancelled out, making the reading of extension \(x\) much more accurate!
This chapter links force, geometry, and energy, giving you essential tools to analyze real-world structural integrity. Great job sticking with the definitions—you’ve mastered the language of material science!