How To Calculate Elastic Modulus

If you push the ends of a rubber rod toward each other, you are applying a ​compression​ force and can shorten the rod by some amount. If you pull the ends away from each other, the force is called ​tension,​ and you can stretch the rod lengthwise. If you tug one end toward you and the other end away from you, using what is called a ​shear​ force, the rod stretches diagonally.

Elastic modulus (​E​) is a measure of the stiffness of a material under compression or tension, although there is also an equivalent shear modulus. It is a property of the material and does not depend on the shape or size of the object.

A small piece of rubber has the same elastic modulus as a large piece of rubber. ​Elastic modulus​, also known as Young's modulus, named after British scientist Thomas Young, relates the force of squeezing or stretching an object to the resulting change in length.

​Stress​ (​σ​) is the compression or tension per unit area and is defined as:

\(\sigma=\frac{F}{A}\)

Here F is force, and A is the cross-sectional area where the force is applied. In the metric system, stress is commonly expressed in units of pascals (Pa), newtons per square meter (N/m²) or newtons per square millimeter (N/mm²).

When stress is applied to an object, the change in shape is called ​strain.​ In response to compression or tension, ​normal strain​ (​ε​) is given by the proportion:

\(\epsilon=\frac{\Delta L}{L}\)

In this case Δ​L​ is the change in length and ​L​ is the original length. Normal strain, or simply ​strain​, is dimensionless.

As long as the deformation isn't too great, a material like rubber can stretch, then spring back to its original shape and size when the force is removed; the rubber has experienced ​elastic​ deformation, which is a reversible change of shape. Most materials can sustain some amount of elastic deformation, although it may be tiny in a tough metal like steel.

If the stress is too large, however, a material will undergo ​plastic​ deformation and permanently change shape. Stress can even increase to the point where a material breaks, such as when you pull a rubber band until it snaps in two.

The modulus of elasticity equation is used only under conditions of elastic deformation from compression or tension. The modulus of elasticity is simply stress divided by strain:

\(E=\frac{\sigma}{\epsilon}\)

with units of pascals (Pa), newtons per square meter (N/m²) or newtons per square millimeter (N/mm²). For most materials, elastic modulus is so large that it is normally expressed as megapascals (MPa) or gigapascals (GPa).

To test the strength of materials, an instrument pulls on the ends of a sample with greater and greater force and measures the resulting change in length, sometimes until the sample breaks. The sample's cross-sectional area must be defined and known, allowing the calculation of stress from the applied force. Data from a test on mild steel, for example, can be plotted as a stress‑strain curve, which can then be used to determine the modulus of elasticity of steel.

Elastic deformation occurs at low strains and is proportional to stress. On a stress-strain curve, this behavior is visible as a straight-line region for strains less than about 1 percent. So 1 percent is the elastic limit or the limit of reversible deformation.

To determine the modulus of elasticity of steel, for example, first identify the region of elastic deformation in the stress-strain curve, which you now see applies to strains less than about 1 percent, or ​ε​ = 0.01. The corresponding stress at that point is ​σ​ = 250 N/mm². Therefore, using the modulus of elasticity formula, the modulus of elasticity of steel is

\(E=\frac{\sigma}{\epsilon}=\frac{250}{0.01}=25,000\text{ N/mm}^2\)