# 7.2.1 Elastic & Plastic Behaviour

### Elastic & Plastic Deformation

• Plastic deformation: when the load is removed, the object will not return to its original shape or length. This is beyond the elastic limit
• Elastic limit: the point beyond which the object does not return to its original length when the load is removed
• These regions can be determined from a Force-Extension graph:

Below the elastic limit, the material exhibits elastic behaviour
Above the elastic limit, the material exhibits plastic behaviour

• Elastic deformation occurs in the ‘elastic region’ of the graph. The extension is proportional to the force applied to the material (straight line)
• Plastic deformation occurs in the ‘plastic region’ of the graph. The extension is no longer proportional to the to the force applied to the material (graph starts to curve)
• These regions are divided by the elastic limit

#### Brittle and ductile materials

• Brittle materials have very little to no plastic region e.g. glass, concrete. The material breaks with little elastic and insignificant plastic deformation
• Ductile materials have a larger plastic region e.g. rubber, copper. The material stretches into a new shape before breaking

• To identify these materials on a stress-strain or force-extension graph up to their breaking point:
• A brittle material is represented by a straight line through the origins with no or negligible curved region
• A ductile material is represented with a straight line through the origin then curving towards the x-axis

#### Worked example

• Since the graph is a straight line and the metal fractures, the point after X must be its elastic limit
• The graph starts to curve after this and fractures at point Y
• This curve between X and Y denotes plastic behaviour
• Therefore, the correct answer is C

#### Exam Tip

Although similar definitions, the elastic limit and limit of proportionality are not the same point on the graph. The limit of proportionality is the point beyond which the material is no longer defined by Hooke’s law. The elastic limit is the furthest point a material can be stretched whilst still able to return to its previous shape. This is at a slightly higher extension than the limit of proportionality. Be sure not to confuse them.

### Area under a Force-Extension Graph

• The work done in stretching a material is equal to the force multiplied by the distance moved
• Therefore, the area under a force-extension graph is equal to the work done to stretch the material
• The work done is also equal to the elastic potential energy stored in the material

Work done is the area under the force-extension graph

• This is true for whether the material obeys Hooke’s law or not
• For the region where the material obeys Hooke’s law, the work done is the area of a right angled triangle under the graph
• For the region where the material doesn’t obey Hooke’s law, the area is the full region under the graph. To calculate this area, split the graph into separate segments and add up the individual areas of each

• The force-extension curve for stretching and contraction of a material that has exceeded its elastic limit is shown below

• The curve for contraction is always below the curve for stretching
• The area X represents the net work done or the thermal energy dissipated in the material
• The area X + Y is the minimum energy required to stretch the material to extension e

#### Exam Tip

Make sure to be familiar with the formula for the area of common 2D shapes such as a right angled triangle, trapezium, square and rectangles.

### Elastic Potential Energy

• Elastic potential energy is defined as the energy stored within a material (e.g. in a spring) when it is stretched or compressed
• It can be found from the area under the force-extension graph for a material deformed within its limit of proportionality

### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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