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 (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.International Union of Pure and Applied Chemistry. "modulus of elasticity (Young\'s modulus), E". Compendium of Chemical Terminology Internet edition. 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.The Rational Mechanics of Flexible or Elastic Bodies, 1638-1788: Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.

Contents 1 Units 2 Usage 2.1 Linear vs non-linear 2.2 Directional materials 3 Calculation 3.1 Force exerted by stretched or compressed material 3.2 Elastic potential energy 3.3 Relation among elastic constants 4 Approximate values 5 See also 6 References 7 External links

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:

$E\; \backslash equiv\; \backslash frac\{\backslash mbox\; \{tensile\; stress\}\}\{\backslash mbox\; \{tensile\; strain\}\}\; =\; \backslash frac\{\backslash sigma\}\{\backslash varepsilon\}=\; \backslash frac\{F/A\_0\}\{\backslash Delta\; L/L\_0\}\; =\; \backslash frac\{F\; L\_0\}\; \{A\_0\; \backslash Delta\; L\}$

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.

$F\; =\; \backslash frac\{E\; A\_0\; \backslash Delta\; L\}\; \{L\_0\}$

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:

$F\; =\; \backslash left(\; \backslash frac\{E\; A\_0\}\; \{L\_0\}\; \backslash right)\; \backslash Delta\; L\; =\; k\; x\; \backslash ,$

where

$k\; =\; \backslash begin\{matrix\}\; \backslash frac\; \{E\; A\_0\}\; \{L\_0\}\; \backslash end\{matrix\}\; \backslash ,$

$x\; =\; \backslash Delta\; L\; \backslash ,$

Elastic potential energy

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

$U\_e\; =\; \backslash int\; \{\backslash frac\{E\; A\_0\; \backslash Delta\; L\}\; \{L\_0\}\}\backslash ,\; dL\; =\; \backslash frac\; \{E\; A\_0\; \{\backslash Delta\; L\}^2\}\; \{2\; L\_0\}$

where U_e is the elastic potential energy.

The elastic potential energy per unit volume is given by:

$\backslash frac\{U\_e\}\; \{A\_0\; L\_0\}\; =\; \backslash frac\; \{E\; \{\backslash Delta\; L\}^2\}\; \{2\; L\_0^2\}\; =\; \backslash frac\; \{1\}\; \{2\}\; E\; \{\backslash varepsilon\}^2$, where $\backslash varepsilon\; =\; \backslash frac\; \{\backslash Delta\; L\}\; \{L\_0\}$ is the strain in the material.

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

$U\_e\; =\; \backslash int\; \{k\; x\}\backslash ,\; dx\; =\; \backslash frac\; \{1\}\; \{2\}\; k\; x^2$

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:

$E\; =\; 2G(1+\backslash nu)\; =\; 3K(1-2\backslash nu)\backslash ,$

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)http://www.engineeringtoolbox.com/young-modulus-d_417.html	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

Influences of selected glass component additions on Young\'s modulus of a specific base glass (Glassproperties.com).

See also

• Deflection
• Deformation
• Hardness
• Hooke\'s law
• Shear modulus
• Impulse excitation technique
• Strain
• Stress
• Toughness
• Yield (engineering)
• List of materials properties

References

External links

• Matweb: free database of engineering properties for over 63,000 materials

v • d • e Elastic moduli for homogeneous isotropic materials
Bulk modulus ($K$) • Young\'s modulus ($E$) • Lamé\'s first parameter ($\backslash lambda$) • Shear modulus ($\backslash mu$) • Poisson\'s ratio ($\backslash nu$) • P-wave modulus ($M$)

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.
	$(\backslash lambda,\backslash ,\backslash mu)$	$(E,\backslash ,\backslash mu)$	$(K,\backslash ,\backslash lambda)$	$(K,\backslash ,\backslash mu)$	$(\backslash lambda,\backslash ,\backslash nu)$	$(\backslash mu,\backslash ,\backslash nu)$	$(E,\backslash ,\backslash nu)$	$(K,\backslash ,\; \backslash nu)$	$(K,\backslash ,E)$	$(M,\backslash ,\backslash mu)$
$K=\backslash ,$	$\backslash lambda+\; \backslash frac\{2\backslash mu\}\{3\}$	$\backslash frac\{E\backslash mu\}\{3(3\backslash mu-E)\}$			$\backslash lambda\backslash frac\{1+\backslash nu\}\{3\backslash nu\}$	$\backslash frac\{2\backslash mu(1+\backslash nu)\}\{3(1-2\backslash nu)\}$	$\backslash frac\{E\}\{3(1-2\backslash nu)\}$			$M\; -\; \backslash frac\{8\backslash mu\}\{3\}$
$E=\backslash ,$	$\backslash mu\backslash frac\{3\backslash lambda\; +\; 2\backslash mu\}\{\backslash lambda\; +\; \backslash mu\}$		$9K\backslash frac\{K-\backslash lambda\}\{3K-\backslash lambda\}$	$\backslash frac\{9K\backslash mu\}\{3K+\backslash mu\}$	$\backslash frac\{\backslash lambda(1+\backslash nu)(1-2\backslash nu)\}\{\backslash nu\}$	$2\backslash mu(1+\backslash nu)\backslash ,$		$3K(1-2\backslash nu)\backslash ,$		$\backslash mu\backslash frac\{3M-4\backslash mu\}\{M-\backslash mu\}$
$\backslash lambda=\backslash ,$		$\backslash mu\backslash frac\{E-2\backslash mu\}\{3\backslash mu-E\}$		$K-\backslash frac\{2\backslash mu\}\{3\}$		$\backslash frac\{2\; \backslash mu\; \backslash nu\}\{1-2\backslash nu\}$	$\backslash frac\{E\backslash nu\}\{(1+\backslash nu)(1-2\backslash nu)\}$	$\backslash frac\{3K\backslash nu\}\{1+\backslash nu\}$	$\backslash frac\{3K(3K-E)\}\{9K-E\}$	$M\; -\; 2\backslash mu\backslash ,$
$\backslash mu=\backslash ,$			$3\backslash frac\{K-\backslash lambda\}\{2\}$		$\backslash lambda\backslash frac\{1-2\backslash nu\}\{2\backslash nu\}$		$\backslash frac\{E\}\{2+2\backslash nu\}$	$3K\backslash frac\{1-2\backslash nu\}\{2+2\backslash nu\}$	$\backslash frac\{3KE\}\{9K-E\}$
$\backslash nu=\backslash ,$	$\backslash frac\{\backslash lambda\}\{2(\backslash lambda\; +\; \backslash mu)\}$	$\backslash frac\{E\}\{2\backslash mu\}-1$	$\backslash frac\{\backslash lambda\}\{3K-\backslash lambda\}$	$\backslash frac\{3K-2\backslash mu\}\{2(3K+\backslash mu)\}$					$\backslash frac\{3K-E\}\{6K\}$	$\backslash frac\{M\; -\; 2\backslash mu\}\{2M\; -\; 3\backslash mu\}$
$M=\backslash ,$	$\backslash lambda+2\backslash mu\backslash ,$	$\backslash mu\backslash frac\{4\backslash mu-E\}\{3\backslash mu-E\}$	$3K-2\backslash lambda\backslash ,$	$K+\backslash frac\{4\backslash mu\}\{3\}$	$\backslash lambda\; \backslash frac\{1-\backslash nu\}\{\backslash nu\}$	$\backslash mu\backslash frac\{2-2\backslash nu\}\{1-2\backslash nu\}$	$E\backslash frac\{1-\backslash nu\}\{(1+\backslash nu)(1-2\backslash nu)\}$	$3K\backslash frac\{1-\backslash nu\}\{1+\backslash nu\}$	$3K\backslash frac\{3K+E\}\{9K-E\}$

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