- 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 stiffness. 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
Young's modulus is the ratio of stress, which has units of pressure, to strain, which is dimensionless, and therefore itself has units of pressure.
The SI unit of modulus of elasticity (E, or less commonly Y) is the pascal; the practical units are megapascals (MPa) or gigapascals (GPa or kN/mm²). In English units, it is expressed as pounds (force) per square inch (psi).
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 essentially 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 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 phenomenon 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): E \equiv \frac{\mbox {tensile stress}}{\mbox {tensile strain}} = \frac{\sigma}{\varepsilon}= \frac{F/A_0}{\Delta L/L_0} = \frac{F L_0} {A_0 \Delta L}
where
- E is the Young's modulus (modulus of elasticity)
- 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.
- Failed to parse (Missing texvc executable; please see math/README to configure.): F = \frac{E 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): F = \left( \frac{E A_0} {L_0} \right) \Delta L = k x \,
where
- Failed to parse (Missing texvc executable; please see math/README to configure.): k = \begin{matrix} \frac {E A_0} {L_0} \end{matrix} \,
- Failed to parse (Missing texvc executable; please see math/README to configure.): x = \Delta L \,
Elastic potential energy
The elastic potential energy stored is given by the integral of this expression with respect to L:
- Failed to parse (Missing texvc executable; please see math/README to configure.): U_e = \int {\frac{E A_0 \Delta L} {L_0}}\, dL = \frac {E A_0 {\Delta L}^2} {2 L_0}
where Ue is the elastic potential energy.
The elastic potential energy per unit volume is given by:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{U_e} {A_0 L_0} = \frac {E {\Delta L}^2} {2 L_0^2} = \frac {1} {2} E {\varepsilon}^2
, where Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): U_e = \int {k x}\, dx = \frac {1} {2} k x^2
Relation among elastic constants
For homogeneous isotropic materials simple relations exist between elastic constants (Young's modulus E, shear modulus G, bulk modulus K, and Poisson's ratio ν) that allow calculating them all as long as two are known:
- Failed to parse (Missing texvc executable; please see math/README to configure.): E = 2G(1+\nu) = 3K(1-2\nu)\,
Approximate values
Young's modulus can vary somewhat due to differences in sample composition and test method. 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 |
| PTFE (Teflon) |
0.5 |
75000 |
| Low density polyethylene |
0.2 |
30,000 |
| HDPE |
1.379 |
200000 |
| 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 |
| Pine wood (along grain) |
8.963 |
1,300,000 |
| MDF (wood composite) |
3.654 |
530,000 |
| High-strength concrete (under compression) |
30 |
4,350,000 |
| Magnesium metal (Mg) |
45 |
6,500,000 |
| Aluminium alloy |
69 |
10,000,000 |
| Glass (see also diagram below table) |
65-90 |
9,400,000-13,000,000 |
| Brass and bronze |
103-124 |
17,000,000 |
| Titanium (Ti) |
105-120 |
15,000,000-17,500,000 |
| Copper (Cu) |
110-130 |
16,000,000-19,000,000 |
| 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 |
| Osmium (Os)[3] |
550 |
|
| 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
References
External links
|
Elastic moduli for homogeneous isotropic materials |
| Bulk modulus (Failed to parse (Missing texvc executable; please see math/README to configure.): K
) • Young's modulus (Failed to parse (Missing texvc executable; please see math/README to configure.): E ) • Lamé's first parameter (Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda ) • Shear modulus (Failed to parse (Missing texvc executable; please see math/README to configure.): \mu ) • Poisson's ratio (Failed to parse (Missing texvc executable; please see math/README to configure.): \nu ) • P-wave modulus (Failed to parse (Missing texvc executable; please see math/README to configure.): M
) |
| Conversion formulas |
| Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas. |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): (\lambda,\,\mu) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (E,\,\mu) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (K,\,\lambda) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (K,\,\mu) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (\lambda,\,\nu) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (\mu,\,\nu) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (E,\,\nu) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (K,\, \nu) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (K,\,E) |
Failed to parse (Missing texvc executable; please see math/README to configure.): (M,\,\mu) |
| Failed to parse (Missing texvc executable; please see math/README to configure.): K=\, |
Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda+ \frac{2\mu}{3}
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{E\mu}{3(3\mu-E)} |
|
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda\frac{1+\nu}{3\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{2\mu(1+\nu)}{3(1-2\nu)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{E}{3(1-2\nu)} |
|
|
Failed to parse (Missing texvc executable; please see math/README to configure.): M - \frac{8\mu}{3} |
| Failed to parse (Missing texvc executable; please see math/README to configure.): E=\, |
Failed to parse (Missing texvc executable; please see math/README to configure.): \mu\frac{3\lambda + 2\mu}{\lambda + \mu} |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): 9K\frac{K-\lambda}{3K-\lambda} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{9K\mu}{3K+\mu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\lambda(1+\nu)(1-2\nu)}{\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 2\mu(1+\nu)\, |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): 3K(1-2\nu)\, |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \mu\frac{3M-4\mu}{M-\mu} |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda=\, |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \mu\frac{E-2\mu}{3\mu-E} |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): K-\frac{2\mu}{3} |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{2 \mu \nu}{1-2\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{E\nu}{(1+\nu)(1-2\nu)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{3K\nu}{1+\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{3K(3K-E)}{9K-E} |
Failed to parse (Missing texvc executable; please see math/README to configure.): M - 2\mu\, |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \mu=\, |
|
|
Failed to parse (Missing texvc executable; please see math/README to configure.): 3\frac{K-\lambda}{2} |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda\frac{1-2\nu}{2\nu} |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{E}{2+2\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 3K\frac{1-2\nu}{2+2\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{3KE}{9K-E} |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \nu=\, |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\lambda}{2(\lambda + \mu)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{E}{2\mu}-1 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\lambda}{3K-\lambda} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{3K-2\mu}{2(3K+\mu)} |
|
|
|
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{3K-E}{6K} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{M - 2\mu}{2M - 3\mu} |
| Failed to parse (Missing texvc executable; please see math/README to configure.): M=\, |
Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda+2\mu\, |
Failed to parse (Missing texvc executable; please see math/README to configure.): \mu\frac{4\mu-E}{3\mu-E} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 3K-2\lambda\, |
Failed to parse (Missing texvc executable; please see math/README to configure.): K+\frac{4\mu}{3} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda \frac{1-\nu}{\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \mu\frac{2-2\nu}{1-2\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): E\frac{1-\nu}{(1+\nu)(1-2\nu)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 3K\frac{1-\nu}{1+\nu} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 3K\frac{3K+E}{9K-E} |
bg:Модул на еластичност
ca:Mòdul elàstic da:E-modul de:Elastizitätsmodul es:Módulo de elasticidad fr:Module de Young gl:Módulo de Young it:Modulo di elasticità he:מודול האלסטיות lv:Junga modulis lt:Jungo modulis hu:Rugalmassági modulus nl:Elasticiteitsmodulus ja:ヤング率 no:E-modul pl:Moduł Younga pt:Módulo de Young ru:Модуль Юнга sk:Youngov modul sl:Prožnostni modul fi:Elastiset kertoimet sv:Elasticitetsmodul th:มอดุลัสของยัง tr:Young katsayısı uk:Модуль Юнга
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