# Young's modulus

(Redirected from Elastic modulus)

In solid mechanics, Young's modulus (also known as the modulus of elasticity or elastic modulus) is a measure of the stiffness of a given material. It is defined as the limit for small strains of the rate of change of stress with strain. This can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material. Young's modulus is named after Thomas Young the English physicist, physician, and Egyptologist.

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## Units

The SI unit of modulus of elasticity is the pascal. However, given the large values typical of many common materials, figures are often quoted in megapascals or gigapascals for convenience.

The modulus of elasticity can also be measured in other units of pressure, for example pounds per square inch.

## Usage

The Young's modulus allows the behavior of a material under load to be calculated. For instance, it can be used to predict the amount a wire will extend under tension, or to predict the load at which a thin column will buckle under compression. Some calculations also require the use of other material properties, such as the shear modulus, density, or Poisson's ratio.

### Linear vs non-linear

For many materials, Young's modulus is a constant over a range of strains. Such materials are called linear, and are said to obey Hooke's law. Examples of linear materials include steel, carbon fiber, and glass. Rubber is a non-linear material.

### Directional materials

Most metals and ceramics, along with many other materials, are uniform, or isotropic - their mechanical properties are the same in all directions.

However, this is not always the case. Some materials, particularly those which are composites of two or more ingredients have a "grain" or similar mechanical structure. As a result, these anisotropic materials have different mechanical properties when load is applied in different directions. For example, carbon fiber is much stiffer (higher Young's Modulus) when loaded parallel to the fibers (along the grain). Other such materials include wood and reinforced concrete.

## Calculation

The modulus of elasticity, λ, can be calculated by dividing the stress by the strain, i.e.

[itex] \lambda = \frac{stress}{strain} = \frac{F/A}{x/l} = \frac{F l} {A x} [itex]

where (in SI units)

λ is the modulus of elasticity, measured in pascals

F is the force, measured in newtons

A is the cross-sectional area through which the force is applied, measured in square metres

x is the extension, measured in metres

l is the natural length, measured in metres

### Tension

The modulus of elasticity of a material can be used to calculate the tension force it exerts under a specific extension.

[itex]T = \frac{\lambda A x}{l}[itex]

where

T is the tension, measured in newtons

### Elastic potential energy

The elastic potential energy stored is given by the integral of this expression with respect to x, i.e. energy stored E is given by:

[itex]E = \frac{\lambda A x^2}{2 l}[itex]

where

E is the elastic potential energy, measured in joules

## Approximate values

Note that Young's Modulus can vary considerably depending on the exact composition of the material. For example, the value for most metals can vary by 5% or more, depending on the precise composition of the alloy and any heat treatment applied during manufacture. As such, many of the values here are very approximate.

Approximate Young's Moduli of Various Solids
Material Young's modulus (E) in GPa Young's modulus (E) in lbf/in²
Rubber (small strain) 0.01-0.1 1,500-15,000
Polystyrene 3-3.5 435,000-505,000
Nylon 2-4 290,000-580,000
Oak wood (along grain) 11 1,600,000
High-strength concrete (under compression) 30 4,350,000
Magnesium metal 45 6,500,000
Glass 50-90 7,250,000-13,000,000
Aluminium alloys 69 10,000,000
Brasses and bronzes 103-124 17,000,000
Titanium (Ti) 105-120 15,000,000-17,500,000
Carbon fiber reinforced plastic (unidirectional, along grain) 150 21,800,000
Wrought iron and steel 190-210 30,000,000
Tungsten (W) 400-410 58,000,000-59,500,000
Silicon carbide (SiC) 450 65,000,000
Tungsten carbide (WC) 450-650 65,000,000-94,000,000
Diamond 1,050-1,200 150,000,000-175,000,000

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