Young's modulus

This article is about a physical property. For the computer game, see Young's Modulus (game).

In solid mechanics, Young's Modulus (also known as the Young Modulus, modulus of elasticity, elastic modulus or tensile modulus) is a measure of the stiffness of a given material. It is defined as the ratio, 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.

Units

The SI unit of modulus of elasticity is the pascal. Given the large values typical of many common materials, figures are usually quoted in megapascals or gigapascals. Some use an alternative unit form, kN/mm², which gives the same numeric value as gigapascals.

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 isotropic - their mechanical properties are the same in all directions. However, 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

Young's modulus, Y, can be calculated by dividing the tensile stress by the tensile strain:

Y \equiv \frac{\mbox {tensile stress}}{\mbox {tensile strain}} = \frac{F/A_0}{\Delta l/l_0} = \frac{F l_0} {A_0 \Delta l}

where Y is the modulus of elasticity, measured in pascals; F is the force applied to the object; A₀ is the original cross-sectional area through which the force is applied; Δl is the amount by which the length of the object changes; and l₀ is the original length of the object.

Force exerted by stretched or compressed material

The Young's modulus of a material can be used to calculate the force it exerts under a specific strain.

F = \frac{Y A_0 \Delta l} {l_0}

where F is the force exerted by the material when compressed or stretched by Δl. From this formula can be derived Hooke's law, which describes the stiffness of an ideal spring: $F = k x$ , where $x = \Delta l$ , and $k = \begin{matrix} \frac {Y A_0} {l_0} \end{matrix}$ .

Elastic potential energy

The elastic potential energy stored is given by the integral of this expression with respect to l:

U_e = \int {\frac{Y A_0 \Delta l} {l_0}}\, dl = \frac {Y A_0 {\Delta l}^2} {2 l_0}

where U_e is the elastic potential energy.

The elastic potential energy per unit volume is given by:

\frac{U_e} {A_0 l_0} = \frac {Y {\Delta l}^2} {2 l_0^2} = \frac {1} {2} Y {\varepsilon}^2

, where

\varepsilon = \frac {\Delta l} {l_0}

is the strain in the material.

This formula can also be expressed as the integral of Hooke's law:

U_e = \int {k x}\, dx = \frac {1} {2} k x^2

Approximate values

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 approximate.

style="font-size:larger;font-weight:bold;"|Approximate Young's Moduli of Various Solids
Material	Young's modulus (E) in G Pa	Young's modulus (E) in lbf/in² (psi)
Rubber (small strain)	0.01-0.1	1,500-15,000
Low density polyethylene	0.2	30,000
Polypropylene	1.5-2	217,000-290,000
Bacteriophage capsids (virus)	1-3	150,000-435,000
Polyethylene terephthalate	2-2.5	290,000-360,000
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 (Mg)	45	6,500,000
Aluminium alloy	69	10,000,000
Glass (all types)	72	10,400,000
Brass and bronze	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
Single Carbon nanotube ^*	approx. 1,000+	approx. 145,000,000
Diamond (C)	1,050-1,200	150,000,000-175,000,000

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