Bulk Properties of Solids

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Hooke's Law

Hooke's law is a proportional relationship between the force experienced and the extension observed. The relationship exists up to an elastic limit.

Hooke's law

Hooke's law is the relationship that the force F experienced by an extensible object is proportional to its extension ΔL:
- F α ΔL
The constant of proportionality is labelled k and is called the stiffness constant.
This gives the equation:
- F = kΔL
The force acts to restore the object back to its original shape, so it acts in the opposite direction to that of the extension.

Limit of proportionality

Hooke's law is a relationship between two quantities. But, this relationship does not always hold.
- When a force-extension graph is linear, Hooke's law applies.
- When a force-extension graph is non-linear, Hooke's law does NOT apply.

Elastic limit

An object (e.g. a spring) is elastic if it returns back to zero extension when the load on it is removed.
The elastic limit is the maximum force the spring can sustain and then return to zero extension.

Gradient of force-extension graphs

The gradient of a force-extension graph is the value of the constant of proportionality, k.
For springs, this is the familiar spring constant.

Strain and Stress

Summary values of materials are useful for engineers to compare the qualities of different materials. Two examples of summary values are stress and strain.

Effect of forces

Forces can change an object's shape.
The study of stress and strain is a study of how forces change an object's shape.
- Some forces stretch the object.
  - These forces are called tensile forces.
- Some forces squash the object.
  - These forces are called compressive forces.

Stress

Stress can be defined as:
- Stress = $\frac{force\space along\space the\space axis\space of\space the\space object}{cross-sectional\space area \space of \space the \space object}$
- $\sigma = \frac{F}{A}$
If the force is tensile, then the stress is positive.
If the force is compressive, then the stress is negative.
The units are newtons per metre squared (N/m²) or pascals (Pa).

Strain

Strain can be defined as:
- $strain = \frac{extension}{original\space length}$
- $\epsilon = \frac{\Delta L}{L}$
If the force is tensile, then the strain is positive.
If the force is compressive, then the strain is negative.
There are no units for strain because strain is a ratio.

Elastic strain energy

When an elastic object is stretched, energy needs to be supplied to the object to stretch the object.
The supplied energy is the elastic strain energy.
The elastic strain energy can be determined from the area underneath a force-extension graph.

Breaking stress

Breaking stress is the stress required to break the material.
The maximum tensile stress an object can withstand is called the ultimate tensile stress.
Some materials can undergo some strain beyond the point of ultimate tensile stress before breaking.

Plastic and Brittle Behaviour

Objects respond differently to a wide range of stresses and strains.

Plastic behaviour

An object behaves plastically if it undergoes permanent deformation under stress.
Plastic behaviour occurs after the elastic limit.
Stretching strawberry laces is an example of plastic behaviour.
To identify areas of plastic behaviour on a force-extension graph, look to the right of the elastic limit point.

Brittle behaviour

An object is brittle if it breaks suddenly and cracks. There will be very little plastic behaviour shown.
An example of a brittle food is hard sugar sweets.