**Young's modulus** (*E *or* Y*) is a measure of a solid's stiffness or resistance to elastic deformation under load. It 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. In other words:

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

## Equation and Units

The equation for Young's modulus is:

E = σ / ε = (F/A) / (ΔL/L_{0}) = FL_{0} / AΔL

Where:

- E is Young's modulus, usually expressed in Pascal (Pa)
- σ is the uniaxial stress
- ε is the strain
- F is the force of compression or extension
- A is the cross-sectional surface area or the cross-section perpendicular to the applied force
- Δ L is the change in length (negative under compression; positive when stretched)
- L
_{0}is the original length

While the SI unit for Young's modulus is Pa, values are most often expressed in terms of megapascal (MPa), Newtons per square millimeter (N/mm^{2}), gigapascals (GPa), or kilonewtons per square millimeter (kN/mm^{2}). The usual English unit is pounds per square inch (PSI) or mega PSI (Mpsi).

## History

The basic concept behind Young's modulus was described by Swiss scientist and engineer Leonhard Euler in 1727. In 1782, Italian scientist Giordano Riccati performed experiments leading to modern calculations of the modulus. Yet, the modulus takes its name from British scientist Thomas Young, who described its calculation in his *Course of Lectures on Natural Philosophy and the Mechanical Arts* in 1807. It should probably be called Riccati's modulus, in light of the modern understanding of its history, but that would lead to confusion.

## Isotropic and Anisotropic Materials

The Young's modulus often depends on the orientation of a material. Isotropic materials display mechanical properties that are the same in all directions. Examples include pure metals and ceramics. Working a material or adding impurities to it can produce grain structures that make mechanical properties directional. These anisotropic materials may have very 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 values for samples of various materials. Keep in mind, the precise value for a sample may be somewhat different since the test method and sample composition affect the data. In general, most synthetic fibers have low Young's modulus values. Natural fibers are stiffer. Metals and alloys tend to exhibit high values. The highest Young's modulus of all is for carbyne, an allotrope of carbon.

Material | GPa | Mpsi |
---|---|---|

Rubber (small strain) | 0.01–0.1 | 1.45–14.5×10^{−3} |

Low-density polyethylene | 0.11–0.86 | 1.6–6.5×10^{−2} |

Diatom frustules (silicic acid) | 0.35–2.77 | 0.05–0.4 |

PTFE (Teflon) | 0.5 | 0.075 |

HDPE | 0.8 | 0.116 |

Bacteriophage capsids | 1–3 | 0.15–0.435 |

Polypropylene | 1.5–2 | 0.22–0.29 |

Polycarbonate | 2–2.4 | 0.29-0.36 |

Polyethylene terephthalate (PET) | 2–2.7 | 0.29–0.39 |

Nylon | 2–4 | 0.29–0.58 |

Polystyrene, solid | 3–3.5 | 0.44–0.51 |

Polystyrene, foam | 2.5–7x10^{-3} |
3.6–10.2x10^{-4} |

Medium-density fiberboard (MDF) | 4 | 0.58 |

Wood (along grain) | 11 | 1.60 |

Human Cortical Bone | 14 | 2.03 |

Glass-reinforced polyester matrix | 17.2 | 2.49 |

Aromatic peptide nanotubes | 19–27 | 2.76–3.92 |

High-strength concrete | 30 | 4.35 |

Amino-acid molecular crystals | 21–44 | 3.04–6.38 |

Carbon fiber reinforced plastic | 30–50 | 4.35–7.25 |

Hemp fiber | 35 | 5.08 |

Magnesium (Mg) | 45 | 6.53 |

Glass | 50–90 | 7.25–13.1 |

Flax fiber | 58 | 8.41 |

Aluminum (Al) | 69 | 10 |

Mother-of-pearl nacre (calcium carbonate) | 70 | 10.2 |

Aramid | 70.5–112.4 | 10.2–16.3 |

Tooth enamel (calcium phosphate) | 83 | 12 |

Stinging nettle fiber | 87 | 12.6 |

Bronze | 96–120 | 13.9–17.4 |

Brass | 100–125 | 14.5–18.1 |

Titanium (Ti) | 110.3 | 16 |

Titanium alloys | 105–120 | 15–17.5 |

Copper (Cu) | 117 | 17 |

Carbon fiber reinforced plastic | 181 | 26.3 |

Silicon crystal | 130–185 | 18.9–26.8 |

Wrought iron | 190–210 | 27.6–30.5 |

Steel (ASTM-A36) | 200 | 29 |

Yttrium iron garnet (YIG) | 193-200 | 28-29 |

Cobalt-chrome (CoCr) | 220–258 | 29 |

Aromatic peptide nanospheres | 230–275 | 33.4–40 |

Beryllium (Be) | 287 | 41.6 |

Molybdenum (Mo) | 329–330 | 47.7–47.9 |

Tungsten (W) | 400–410 | 58–59 |

Silicon carbide (SiC) | 450 | 65 |

Tungsten carbide (WC) | 450–650 | 65–94 |

Osmium (Os) | 525–562 | 76.1–81.5 |

Single-walled carbon nanotube | 1,000+ | 150+ |

Graphene (C) | 1050 | 152 |

Diamond (C) | 1050–1210 | 152–175 |

Carbyne (C) | 32100 | 4660 |

## Modulii of Elasticity

A modulus is literally a "measure." You may hear Young's modulus referred to as the **elastic modulus**, but there are multiple expressions used to measure 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 like Young's modulus, except in three dimensions. It is a measure of volumetric elasticity, calculated as volumetric stress divided by volumetric strain.
- The shear or modulus of rigidity (G) describes shear when an object is acted upon by opposing forces. It is calculated as shear stress over 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.

## Sources

- ASTM E 111, "Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus". Book of Standards Volume: 03.01.
- G. Riccati, 1782,
*Delle vibrazioni sonore dei cilindri*, Mem. mat. fis. soc. Italiana, vol. 1, pp 444-525. - 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 - 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.