4.2.2

# Bulk Properties of Solids

Test yourself

## 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.

• 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.

### Fractures

• Fractures happen when the material completely breaks.