# Young’s Modulus Formula and Example Young’s modulus is a modulus of elasticity equal to the compressive stress divided by the axial strain. (image: Nicoguaro. CC 4.0)

Young’s modulus (E) is the modulus of elasticity under tension or compression. In other words, it describes how stiff a material is or how readily it bends or stretches. Young’s modulus relates stress (force per unit area) to strain (proportional deformation) along an axis or line.

The basic principle is that a material undergoes elastic deformation when it is compressed or extended, returning to its original shape when the load is removed. More deformation occurs in a flexible material compared to that of a stiff material.

• A low Young’s modulus value means a solid is elastic.
• A high Young’s modulus value means a solid is inelastic or stiff.

The behavior of a rubber band illustrates Young’s modulus. A rubber band stretches, but when you release the force it returns to its original shape and is not deformed. However, pulling too hard on the rubber band causes deformation and eventually breaks it.

### Young’s Modulus Formula

Young’s modulus compares tensile or compressive stress to axial strain. The formula for Young’s modulus is:

E = σ / ε = (F/A) / (ΔL/L0) = FL0 / AΔL = mgL0/ πr2ΔL

Where:

• E is Young’s modulus
• σ is the uniaxial stress (tensile or compressive), which is force per cross sectional area
• ε is the strain, which is the change in length per original length
• F is the force of compression or extension
• A is the cross-sectional surface area or the cross-section perpendicular to the applied force
• ΔL is the change in length (negative under compression; positive when stretched)
• L0 is the original length
• g is the acceleration due to gravity
• r is the radius of a cylindrical wire

### Young’s Modulus Units

While the SI unit for Young’s modulus is the pascal (Pa). However, the pascal is a small unit of pressure, so megapascals (MPa) and gigapascals (GPa) are more common. Other units include newtons per square meter (N/m2), newtons per square millimeter (N/mm2), kilonewtons per square millimeter (kN/mm2), pounds per square inch (PSI), mega pounds per square inch (Mpsi).

### Example Problem

For example, find the Young’s modulus for a wire that is 2 m long and 2 mm in diameter if its length increases 0.24 mm when stretched by an 8 kg mass. Assume g is 9.8 m/s2.

First, write down what you know:

• L = 2 m
• Δ L = 0.24 mm = 0.00024 m
• r = diameter/2 = 2 mm/2 = 1 mm = 0.001 m
• m = 8 kg
• g = 9.8 m/s2

Based on the information, you know the best formula for solving the problem.

E = mgL0/ πr2ΔL = 8 x 9.8 x 2 / 3.142 x (0.001)2 x 0.00024 = 2.08 x 1011 N/m2

### History

Despite its name, Thomas Young is not the person who first described Young’s modulus. Swiss scientist and engineer Leonhard Euler outlined the principle of the modulus of elasticity in 1727. In 1782, Italian scientist Giordano Riccati’s experiments led to modulus calculations. British scientist Thomas Young described the modulus of elasticity and its calculation in his Course of Lectures on Natural Philosophy and the Mechanical Arts in 1807.

### Isotropic and Anisotropic Materials

The Young’s modulus often depends on the orientation of a material. Young’s modulus is independent of direction in isotropic materials. Examples include pure metals (under some conditions) and ceramics. Working a material or adding impurities forms grain structures that make mechanical properties directional. These anisotopic materials have different Young’s modulus values, depending on whether force is loaded along the grain or perpendicular to it. Good examples of anisotropic materials include wood, reinforced concrete, and carbon fiber.

### Table of Young’s Modulus Values

This table contains representative Young’s modulus values for various materials. Keep in mind, the value depends on the test method. In general, most synthetic fibers have low Young’s modulus values. Natural fibers are stiffer than synthetic fibers. Metals and alloys usually have high Young’s modulus values. The highest Young’s modulus is for carbyne, an allotrope of carbon.

### Modulii of Elasticity

Another name for Young’s modulus is the elastic modulus, but it is not the only measure or modulus of elasticity:

• Young’s modulus describes tensile elasticity along a line when opposing forces are applied. It is the ratio of tensile stress to tensile strain.
• The bulk modulus (K) is the three-dimensional counterpart of Young’s modulus. It is a measure of volumetric elasticity, calculated as volumetric stress divided by volumetric strain.
• The shear modulus or modulus of rigidity (G) describes shear when opposing forces act upon an object. It is shear stress divided by shear strain.

The axial modulus, P-wave modulus, and Lamé’s first parameter are other modulii of elasticity. Poisson’s ratio may be used to compare the transverse contraction strain to the longitudinal extension strain. Together with Hooke’s law, these values describe the elastic properties of a material.

### References

• ASTM International (2017). “Standard Test Method for Young’s Modulus, Tangent Modulus, and Chord Modulus“. ASTM E111-17. Book of Standards Volume: 03.01.
• Jastrzebski, D. (1959). Nature and Properties of Engineering Materials (Wiley International ed.). John Wiley & Sons, Inc.
• Liu, Mingjie; Artyukhov, Vasilii I.; Lee, Hoonkyung; Xu, Fangbo; Yakobson, Boris I. (2013). “Carbyne From First Principles: Chain of C Atoms, a Nanorod or a Nanorope?”. ACS Nano. 7 (11): 10075–10082. doi:10.1021/nn404177r
• Riccati, G. (1782). “Delle vibrazioni sonore dei cilindri”. Mem. mat. fis. soc. Italiana. 1: 444-525.
• Truesdell, Clifford A. (1960). The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788: Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.