:''This article is about a physical property. For the computer game, see
Young's Modulus (game).''
In
solid mechanics, 'Young's modulus (E)' is a measure of the
stiffness of a given material. It is also known as the ''Young modulus'', 'modulus of elasticity',
elastic modulus or tensile modulus (the
bulk modulus and
shear modulus are different types of
elastic modulus). It is defined as the ratio, for small strains, of the rate of change of
stress with
strain.
[1] 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 18th Century British scientist. However, the concept was developed in 1727 by
Leonhard Euler and the first experiments that used the concept of Young's modulus in its current form were performed by the Italian scientist
Giordano Riccati in 1782 - predating Young's work by 25 years
[2].
Units
The
SI unit of modulus of elasticity (E, or less commonly Y) 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 and
soils (except at very small
strains) are 'non-linear' materials.
Directional materials
Most metals and ceramics, along with many other materials, are
isotropic - their mechanical properties are the same in all directions, but metals and ceramics can be treated to create different grain sizes and orientations. This treatment makes them anisotropic, meaning that Young's modulus will change depending on which direction the force is applied from.
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.
Engineers can use this directional phenomonon to their advantage in creating various structures in our environment.
Copper is an excellent conductor of electricity and is used to transmit electricity over long distance cables, however copper has a relatively low value for Young's modulus at 130 GPa and it tends to stretch in tension. When the copper cable is bound completely in steel wire around its outside this stretching can be prevented as the steel (with a higher value of Young's modulus in tension) takes up the tension that the copper would otherwise experience.
Calculation
Young's modulus, ''E'', can be calculated by dividing the
tensile stress by the
tensile strain:
:
where
:
E is the Young's modulus (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;
:
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.
:
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:
:
where
:
:
Elastic potential energy
The
elastic potential energy stored is given by the integral of this expression with respect to
L:
:
where
Ue is the elastic potential energy.
The elastic potential energy per unit volume is given by:
:
, where
is the strain in the material.
This formula can also be expressed as the integral of Hooke's law:
:
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.
Approximate Young's moduli of various solids| Material | Young's modulus (E) in GPa | 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 | 1-3 | 150,000-435,000 |
| Polyethylene terephthalate | 2-2.5 | 290,000-360,000 |
| Polystyrene | 3-3.5 | 435,000-505,000 |
| Nylon | 3-7 | 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 (50/50 fibre/matrix, unidirectional, along grain) | 125-150 | 18,000,000 - 22,000,000 |
| Wrought iron and steel | 190-210 | 30,000,000 |
| Beryllium (Be) | 287 | 41,500,000 |
| 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 [1] | 1,000+ | 145,000,000 |
| Diamond (C) | 1,050-1,200 | 150,000,000-175,000,000 |
See also
★
Deflection
★
Deformation
★
Hardness
★
Hooke's law
★
Shear modulus
★
Strain
★
Stress
★
Toughness
★
Yield (engineering)
★
List of materials properties
References
1.
2. ''The Rational Mechanics of Flexible or Elastic Bodies, 1638-1788'': Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.
External links
★
Matweb: free database of engineering properties for over 63,000 materials
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