3.4.2 Mechanical properties of matter

As we know from 3.4.1, the area under a force-extension or force-compression graph is the work done in extending/compressing the spring. This means that:

W = F x

The work done on the spring is transferred to elastic potential energy within the spring and is fully recoverable because of the elastic behaviour of the spring. This means that:

E = 0.5 F x

Since a spring obeys Hooke's law, we can also substitute F = kx into the equation if we don't know F:

E = 0.5 k x²

From this equation we can see that E (elastic potential energy) is directly proportional to extension squared.

It is important to realise that the extension of a wire depends on the length of the wire, its diameter, the tension in the wire, and the material of the wire...

Tensile stress
Tensile stress is defined as the force applied per unit cross-sectional area of the wire:

Tensile stress = force / cross-sectional area

σ = F / A

Tensile strain
Tensile strain is defined as the ratio of extension to original length of a material:

Tensile strain = extension / original length

Ɛ = x / L

NOTE: Don't forget that tensile strain is written as a percentage and not a decimal.


Stress-strain graphs
A stress-strain graph for a brittle material:
Brittle materials show elastic behaviour until breaking point with plastic deformation only. They obey Hooke's law.



A stress-strain graph for a polymeric material:
Polymeric materials consist of long molecular chains. They behave differently depending on their molecular structure and temperature. For example, rubber and polythene both stretch before breaking but rubber shows elastic behaviour and polythene shows plastic behaviour.
        



A stress-strain graph for a ductile material:

In this graph...

• Stress is proportional to strain from the origin to the limit of proportionality (this is the straight bit of the graph, the bit with uniform gradient). In this region the material obeys Hooke's law.
• The elastic limit sits just past the limit of proportionality. Elastic deformation occurs up to this point.
• Plastic deformation occurs beyond this point.
• The upper and lower bounds of the yield strength bit are the upper and lower yield points (where the material extends rapidly) - this may not occur in all ductile materials.
• The stress at ultimate tensile strength (UTS) represents the materials ultimate tensile strength.....duh. This is the maximum stress that a material can withstand when being stretched before it breaks.
• Beyond the point of UTS, the material may become longer and thinner at its weakest point (necking).
• The material eventually snaps at its weakest point.
• The stress value at the point of breaking is the breaking strength of the material.

The higher the ultimate tensile strength the stronger the material.

Within the limit of proportionality, stress is directly proportional to strain. The ratio of stress to strain for a material is constant - it is known as the Young modulus:

Young modulus = tensile stress / tensile strain

E = σ / Ɛ

The Young modulus is the gradient of the linear region of the stress-strain graph. It depends on the material (e.g not it's shape/size). We need to know how to experimentally determine the Young modulus of a metal (wire)...

• Measure the wires diameter (micrometer) across various points and take an average. Do A = π r².
• Apply various loads
• Measure it's length each time
• Calculate the extension by extended length - original length (x - L)
• Take readings for at least 6 different masses
• Plot a stress-strain graph
• Determine the gradient of the linear region of this graph - this is the Young modulus