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MATERIALS
Prerequisite knowledge
You should be able to recall the definitions for the following terms…
Limit of proportionality — the point beyond which Hooke's law is no longer true when stretching a material.
Elastic limit — The point at which the maximum stress can be exerted on a material before it undergoes plastic deformation, i.e. beyond this point the material no longer behaves elastically.
Plastic deformation — When a material is permanently deformed after an applied stress such that it cannot return to its original size or shape even after the applied force is removed.
Hooke’s law
This law states that a material obeys Hooke’s law if… The extension is directly proportional to the applied force or load up to the limit of proportionality.
> the spring constant is a measure of the stiffness of a material in resisting stretches **> Hooke’s law applies to both extensions and compressions
** materials have a non-linear force-extension graph after the limit of proportionality > the elastic limit occurs after the limit of proportionality
Materials that obey Hooke’s law have a force-extension graph that passes through the origin and remains linear for a range of extension values up to the limit of proportionality.
The gradient of a force-extension graph when linear is equal to the spring constant, 𝑘.
Equations
The linear relationship representing Hooke’s law…
For springs connected in series…
𝑘ᴛ =^
𝑘₁ +^
𝑘₂ +^ ...
For springs connected in parallel…
𝑘ᴛ = 𝑘₁ + 𝑘₂ + ...
Energy stored We can calculate the energy stored in a stretched material that obeys Hooke’s law by considering the work done on it. We define work done (energy stored) as the force multiplied by the distance moved in the direction of the force.
This elastic strain energy is what the area under the force-extension graph corresponds to. If the graph is not linear, this means counting squares, but if the object is within the elastic region where it obeys Hooke’s law the energy stored can be calculated using…
𝑊 = for when the force-extension graph is linear
2 𝐹Δ𝑙
OR
𝐸 =
2 𝐹Δ𝑙
This is not always the most useful form of the formula for answering questions as it is an equation with three variables, e.g. if you change the force, the extension changes and the work done changes. We can write two more versions of this equation by using Hooke’s law to substitute either
for 𝐹 or Δ𝑙, to give us equations with two variables and the spring constant…
𝐸 = for when the force-extension graph is linear
2 𝑘(Δ𝑙)²
𝐸 = “ _______ ” _______ ”
Stress & strain
Forces can be used to change the shape, speed or direction of an object. In materials, we look at the change in shape of an object from the perspective of external forces being exerted.
Yield point — A point reached beyond the elastic limit where small increases in stress cause a large increase in strain. The yield stress refers to the point of the curve that is where the force per unit area at which the material extends plastically is for small incremental values of stress.
Ductile — Refers to the property of a material to be able to be easily and permanently stretched, e.g. copper.
Malleable — Refers to the property of a material to be able to be hammered into shape. Malleable materials are usually tough - gold is very malleable, for instance.
Tough — Requires a large amount of work to fracture, i.e. large area under the stress-strain graph. For tough materials, the UTS is usually higher than the breaking point i.e. the maximum point.
Brittleness — A brittle material will extend obeying Hooke’s Law when a stress is applied to it. It will suddenly fracture with no warning sign of plastic deformation.
Breaking point — The point reached after the maximum stress is applied and the material fractures. The breaking stress is the maximum stress that a material can withstand before it fractures. A material with high breaking stress is considered ductile, which means it can extend more before breaking because of plastic deformation, i.e. relatively large elastic region.
Note that the UTS and breaking stress can vary depending on the conditions that the material is in. You should be able to identify, describe the stress-strain graphs for applying and unloading a load from a metal wire and a rubber band and label the key turning points, as shown above.
The Young modulus is the measure of the ability of a material to withstand changes in length with an added load. It is defined as the ratio of tensile stress to the tensile strain of a material…
𝑌𝑜𝑢𝑛𝑔 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 =
𝐸 =
The Young modulus is equal to the gradient of a stress-strain graph when it is linear. This part of the graph corresponds to when the material obeys Hooke’s law. Since strain is unitless, the unit for the Young Modulus is pascals (Pa).
