## Young Modulus

Glossary Definition for 16-19

#### Description

The Young modulus of a material is an intrinsic, bulk property that describes how the material deforms when subjected to stress. It relates the size of the applied compressive or tensile stress to the corresponding strain. The larger the Young modulus, the less a material deforms to a given stress, i.e. the stiffer the material.

The Young modulus is usually represented by the symbol *E* (sometimes *Y*).
For a material of uniform cross-sectional area

*E* = *σ**ε*

where *σ* is the stress and *ε* the strain

#### Discussion

*Young modulus and stress-strain graphs*

Stress is proportional to strain provided the strain is small; this is sometimes referred to as Hooke’s law, although it is just a description of a linear dependence under particular circumstances, not strictly a physical law. The Young modulus of some material under a particular stress and strain is obtained by dividing the stress by the strain, whether or not the relationship between stress and strain is linear. Note that this is not necessarily the same as the gradient of the graph at that point, although it happens to coincide with the gradient if stress is proportional to strain.

The maximum stress (or strain) for which a linear relationship holds between them is called the limit of proportionality.

The elastic limit of a material is the maximum stress (or strain) for which there is no permanent deformation, so that when the applied stress is reduced to zero, a sample returns to its original size and shape. The elastic limit is usually slightly greater than the limit of proportionality.

*Material properties*

When some materials are subjected to a sufficiently large stress, they yield, undergoing increasing permanent deformation without any increase in applied stress (see **figure 1**).

A material that yields under tensile stress is described as ductile (‘leadable’: can be drawn into wires), and one that yields under compressive stress is described as malleable (‘hammerable’: can be beaten into thin sheets). Samples of many metals, for example copper, can be both ductile and malleable depending on how they have been treated.

A material’s ultimate tensile stress (also called tensile strength) specifies the maximum tensile stress that can be applied a sample of the material without breaking it. Similarly, ultimate compressive stress is the maximum compressive stress that can be applied to a sample of a material without causing it to shatter or buckle.

*Figure 1: Typical stress-strain curves for low-carbon steel (a) and crosslinked natural rubber (b).*

#### SI unit

pascal, Pa

#### Expressed in SI base units

m^{-1} kg s^{-2}

#### Other commonly used unit(s)

pounds per square inch, psi; N m^{-2}

#### Mathematical expressions

- For any material under tensile or compressive stress σ and corresponding strain
*ε*, the Young modulus is given by

*E*=*σ**ε*

#### Related entries

- Strain
- Stress

#### In context

The Young modulus is an important property when selecting a material for a particular purpose. For example, the materials used to replace bone in artificial hip and knee joints should ideally have a Young modulus that is close to that of real bone. If the replacement material has a much lower Young modulus (meaning that it is less stiff than bone) then deformation of the material will cause greater stresses within the remaining bone than would be natural, which may cause it to fracture. If the replacement material has a much higher Young modulus, the remaining natural bone will be subject to lower stresses than would be natural, which, for biological reasons, inhibits its ability to regrow.

Materials have been developed for replacement joints whose Young moduli are a close match to that of bone, but other properties make them undesirable. Ceramic materials such as hydroxyapatite are too brittle, and polymers such as high-density polyethylene deform over time. Currently, good compromise materials are metals such as titanium, although their Young moduli are higher than desirable.

Material | Young modulus E/GPa |

Human bone | 0.3--30 |

Titanium | 110 |

Steel | 200 |

Hydroxyapatite | 80--100 |

High-density polyethylene | 0.8 |