Young’s modulus is defined as the ratio of the uniaxial stress over uniaxial strain in the range of stress where Hooke’s law holds. For those who don’t understand this definition, let’s break it down further. Imagine a sample material – say a one meter two by four made of pine wood. Various engineering sources list the Young’s modulus of pine at about 8.962 Gpa. This may change depending on exact type and moisture content.
Practically speaking, the Young’s modulus is the amount of force it takes to stretch an arbitrary sample material to double its length. This concept however, can initially be a little difficult to understand. A common question is “don’t you need to know the original length and cross section of the sample material?”, and sometimes just plain ol’ “I don’t get it…”.
We understand the uniaxial stress to be the stress, or the force, applied to a sample member along a single axis. The uniaxial strain, also known as the engineering strain or Cauchy strain along a single axis, is the ratio of total deformation to the original length of a sample member (divide the amount it stretches by its original length). And Hooke’s law simply governs the elastic portion or linear portion of the stress-strain curve.
- See my article Behavior of Materials in Tension for a diagram of a stress-strain curve as well as further related information.
Now that we understand the defining terminology everything will begin making a lot more sense. So let’s start from the top of the equation:
- E is the Young’s modulus
- F is the applied force
- Ao is the original cross sectional area of the sample material
- ΔL is the change in length of the sample material
- Lo is the original length of the sample material
Taking the equation above, the tensile stress is equal to the applied force divided by the original cross sectional area, and the tensile strain is equal to the change in length divided by the original length. By multiplying the Young’s modulus by an arbitrary (engineering) strain you can determine the stress required to cause the said strain. In doing so however, it’s important that you input the same unit values.
To ensure we fully grasp this concept, let’s use the above example of the one meter two by four, and say that a force was applied causing it to stretch 1 millimeter:
Here, we are simply following the order of the original equation further above and inputting the values accordingly. Remember that all units are in meters following the pascal (Pa) which is defined as one newton per square meter:
- Multiplying the strain, 0.001, by the known constant E, which is 8.962 Gpa, gives us the applied stress, 8.962 Mpa.
- Multiplying this stress, 8.962 Mpa, by the cross sectional area of a two by four, 0.003382 – 0.089 times 0.038 – gives us the force required to stretch a one meter two by four one millimeter – which is roughly 30,309 newtons.
Isotropy and Anisotropy with respect to Young’s Modulus
Isotropic materials are those whose internal grain is more or less uniform in all directions. Such materials may include ceramics, most metals, stone, etc. These materials will have a uniform Young’s modulus in all directions. However, anisotropic materials such as wood and certain reinforced composites have considerably higher Young’s modulus values in the direction parallel to the grain.
Another thing to keep in mind regarding the Young’s modulus is that it assumes a perfectly homogenous (free from knots and other imperfections) and straight member which is completely free from any prior stress. It is also generally understood that materials that meet this criteria don’t exist in the real world, thus an appropriate factor of safety must be added. For example, the one meter two by four mentioned above will most likely have several knots, causing localized weakness.
- Here’s a list of Young’s moduli from Wikipedia – Approximate Values.