Mechanical properties of matter (3.4.2)
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Materials can be deformed from their natural shape.
When a material is deformed within its elastic limit, the work done on the material is stored as elastic potential energy.
On a force–extension graph, a linear relationship shows that the material is following Hooke’s law:
where:
- is the force applied in
- is the extension of the material from its natural length in and
- is the spring constant in units of
A force–extension graph can be used to find the work done to deform a material elastically.
The work done on the material equals the area under the force–extension graph, which is a triangle for extension within the elastic limit.
The triangular area under the graph representing the work done can be found by using the equation:
and is measured in units of which is equivalent to
The work done on an object within the elastic limit is stored completely as elastic potential energy, , and released when the object is allowed to return to its original length.
Force–extension graphs, when deformed within the elastic limit, are linear. The area under the graph is the work done, which is equal to the elastic potential energy stored in the material:
Within the elastic limit, Hooke’s law is obeyed:
Substituting this into the expression for elastic potential energy gives:
The gradient of a force–extension graph (in the elastic limit) is equal to the spring constant, which indicates the stiffness of a material.
A stretchy material like a rubber band has a spring constant whereas a stiff material like a metal rod can have a spring constant of
Question walkthrough
Sketching Elastic PE Against Extension
Sketches how elastic potential energy varies with extension for a stretched elastic band, using E = ½kx² to justify the quadratic shape of the curve.
Stress is defined as the force applied to an object per unit cross-sectional area,
Stress is measured in or the pascal Pascal is also the unit of pressure.
Stress can be referred to as compressive or tensile, depending on the direction of the applied force:
- An object extends under tensile stress.
- An object contracts under compressive stress.
Ultimate tensile strength (UTS) is the maximum stress a material can withstand before it begins to plastically deform, meaning to deform permanently.
Removing the stress when the values exceed the UTS will not cause the material to return to its original length.
UTS is important to consider when designing structures such as bridges and buildings that may be subjected to high stresses.
Strain is defined as the change in length of an object per unit of its original length, under an applied force
Strain has no unit and is sometimes written as a percentage. As a percentage, the strain can be found from the equation:
Tensile strain is positive: it results from the extension of an object under tensile stress.
Compressive strain is negative: it results from the contraction of an object under compressive stress.
Question walkthrough
Compressive and Tensile Strain in a Stress Ball
Calculates the compressive and tensile strain on a squeezed stress ball from the change in diameter along and perpendicular to the squeezing direction.
The Young’s modulus of a material is defined as the ratio of stress to strain:
The Young’s modulus of an object depends only on the material. It is a measure of the material’s ‘stiffness’, independent of the shape and size of the material.
The stress-strain graph for a material deformed in the elastic limit has a constant gradient. The gradient equals the Young’s modulus of the material.
Young’s modulus is also called the elastic modulus.
The equation for Young’s modulus can be written as
where:
- is stress, and
- is strain.
As strain has no units, the units for Young’s Modulus are the same as the units for stress: (or Pascals).
It can be useful further to break down the equation for Young’s modulus using the equations for stress and strain, which leads to
where:
- is the force exerted on the material,
- is the natural length,
- is the cross sectional area, and
- is the extension of the material.
Experimental procedure for determining the Young’s modulus of a metal wire:
- Find the diameter of the wire using a micrometre. The measurement should be taken in multiple places and averaged.
- From the diameter, calculate the cross-sectional area of the wire.
- Clamp the wire at one end and suspend it over a table using a pulley.
- Place a ruler parallel to the wire with a tape marker to measure the wire’s extension.
- Apply different masses onto the wire and calculate the force applied to the wire. using where is the total mass hanging from the wire and is acceleration due to gravity.
- Plot data points of stress () against strain () for at least 5 different masses and calculate the gradient.
- The gradient is the Young’s modulus of the metal.
Question walkthrough
Young's Modulus from Stress-Strain Data
Calculates the Young's modulus of rubber from a table of stress and strain values by finding the gradient of the best-fit stress-strain line.
Tensile testing is a common method to find the relationship between the stress acting on a material and the strain it causes.
Stress-strain curves have characteristic shapes dependent on the type of material.
There are five significant points of interest on this stress-strain curve:
- the proportionality limit – the final point on the curve where the graph is linear.
- the elastic limit – removing the force below this point will allow the material to return to its original shape. Lower values of strain correspond to the elastic region, and higher strains correspond to the plastic region
- the yield point – any additional stress above this point, and the strain will begin to increase rapidly during plastic deformation. An object does not return to its original length after plastic deformation.
- the ultimate tensile strength – the maximum stress value on the curve. Beyond this point, the material will begin necking, which is when a weak point of the material rapidly becomes thinner.
- the fracture point – the material will break.
Glass, cast iron, and ceramics are examples of brittle materials.
Brittle materials follow stress-strain curves with the shape shown below.
Brittle materials can not withstand strain beyond the ultimate tensile strength and do not undergo plastic deformation. The fracture point of the material coincides with the maximum stress point at
Ductile materials include metals like copper and aluminium.
They follow the stress-strain curves shown below.
The plastic region extends to large strain values – ductile materials can withstand large plastic deformation (stretching) before breaking.
Polymeric materials are made of polymers, such as rubber or nylon. They are naturally stretchy materials and have a stress-strain curve, as shown below.
The curve is non-linear. Polymeric materials can undergo large elastic deformation but still return to their original shape when force is removed.
The yield point can vary depending on the material. Some polymeric materials have no yield point at all.
The stress-strain curves of ductile, brittle and polymeric materials have distinct shapes, summarised in the table below.
Note that linear elastic region indicates that the material obeys Hooke’ law.
Elastic deformation is a reversible change of the shape of an object under stress. Removing the force allows the material to return to its original shape.
Elastic deformation occurs within the elastic region, where Hooke’ law holds.
The work done on an object within the elastic region is stored as elastic potential energy.
Examples of materials that significantly elastically deform include springs and elastic bands.
When a relatively small force is applied to a material, the bonds between atoms stretch but return to their initial state when the force is relaxed.
Plastic deformation is the permanent change of an object under stress. The object does not return to its original shape after the force is removed.
A significant rearrangement of the material’s atoms occurs in the plastic region. This requires energy, meaning not all the work done is converted to elastic potential energy.
When an applied force on a material is great enough to cause plastic deformation, the bonds between atoms stretch and planes shear. Deformation remains after the applied force is relaxed. Plastic deformation can be useful, for instance, in metal shaping.