Young's modulus

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In solid mechanics, Young's modulus (E) is a measure of the stiffness of an isotropic elastic material. It is also known as the Young modulus, modulus of elasticity, elastic modulus (though Young's modulus is actually one of several elastic moduli such as the bulk modulus and the shear modulus) or tensile modulus. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds.^[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]

1 Units
2 Usage
- 2.1 Linear versus non-linear
- 2.2 Directional materials
3 Calculation
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; therefore Young's modulus itself has units of pressure. A purist might argue that the units are fine if the dimensionless ratio is left in (i.e. lbs/in^2 over in/in). Carrying the full expression can be helpful for canceling units, in dimensional analysis.

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 United States customary units, it is expressed as pounds (force) per square inch (psi).

Usage

The Young's modulus allows the behavior of a bar made of an isotropic elastic material to be calculated under tensile or compressive loads. For instance, it can be used to predict the amount a wire will extend under tension or buckle under compression. Some calculations also require the use of other material properties, such as the shear modulus, density, or Poisson's ratio.

Linear versus 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

Young's modulus is not always the same in all orientations of a material. Most metals and ceramics, along with many other materials, are isotropic: Their mechanical properties are the same in all orientations. However, metals and ceramics can be treated with certain impurities, and metals can be mechanically 'worked,' to make their grain structures directional. These materials then become anisotropic, and Young's modulus will change depending on which direction the force is applied from. Anisotropy can be seen in many composites as well. For example, carbon fiber has a much higher Young's modulus (is much stiffer) when force is loaded parallel to the fibers (along the grain), and is an example of a material with transverse isotropy. Other such materials include wood and reinforced concrete. Engineers can use this directional phenomenon to their advantage in creating various structures in our environment.

Calculation

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

$E \equiv \frac{\mbox {tensile stress}}{\mbox {tensile strain}} = \frac{\sigma}{\varepsilon}= \frac{F/A_0}{\Delta L/L_0} = \frac{F L_0} {A_0 \Delta L}$

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.

$F = \frac{E A_0 \Delta L} {L_0}$

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:

$F = \left( \frac{E A_0} {L_0} \right) \Delta L = k x \,$

where

$k = \begin{matrix} \frac {E A_0} {L_0} \end{matrix} \,$

$x = \Delta L. \,$

Elastic potential energy

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

$U_e = \int {\frac{E A_0 \Delta L} {L_0}}\, dL = \frac {E A_0} {L_0} \int { \Delta L }\, dL = \frac {E A_0 {\Delta L}^2} {2 L_0}$

where U_e is the elastic potential energy.

The elastic potential energy per unit volume is given by:

$\frac{U_e} {A_0 L_0} = \frac {E {\Delta L}^2} {2 L_0^2} = \frac {1} {2} E {\varepsilon}^2$ , 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:

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

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:

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

Approximate values

Young's modulus can vary somewhat due to differences in sample composition and test method. The rate of deformation has the greatest impact on the data collected, especially in polymers. The values here are approximate and only meant for relative comparisons.

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
PTFE (Teflon)	0.5	75,000
Low density polyethylene	0.2	30,000
HDPE	1.379	200,000
Polypropylene	1.5-2	217,000-290,000
Bacteriophage capsids	1-3	150,000-435,000
Polyethylene terephthalate	2-2.5 OR 2.8-3.1	290,000-360,000
Polystyrene	3-3.5	435,000-505,000
Nylon	2-4	290,000-580,000
MDF (wood composite)	3.654	530,000
Pine wood (along grain)	8.963	1,300,000
Oak wood (along grain)	11	1,600,000
High-strength concrete (under compression)	30-100	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
Sapphire (Al2O3) along C-axis	435	63,000,000
Silicon carbide (SiC)	450	65,000,000
Osmium (Os)^[3]	550	79,800,000
Tungsten carbide (WC)	450-650	65,000,000-94,000,000
Single carbon nanotube [1]	1,000+	145,000,000+
Diamond (C)	1220(^[4])	150,000,000-175,000,000
Yttrium iron garnet (YIG)	2000(^[5])	300,000,000

Influences of selected glass component additions on Young's modulus of a specific base glass (^[6]).

References

^ International Union of Pure and Applied Chemistry. "modulus of elasticity (Young's modulus), E". Compendium of Chemical Terminology Internet edition.
^ The Rational Mechanics of Flexible or Elastic Bodies, 1638-1788: Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.
^ http://www.engineeringtoolbox.com/young-modulus-d_417.html
^ ['Synthetic Diamond - Emerging CVD Science and Technology', Spear and Dismukes, Wiley, NY, 1994]
^ http://www.isowave.com/pdf/materials/Yttrium_Iron_Garnet.pdf YIG properties
^ Glassproperties.com

External links

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

Elastic moduli for homogeneous isotropic materials

Bulk modulus ( $K$ ) • Young's modulus ( $E$ ) • Lamé's first parameter ( $λ$ ) • Shear modulus ( $μ$ ) • Poisson's ratio ( $ν$ ) • 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.
	$(\lambda,\,\mu)$	$(E,\,\mu)$	$(K,\,\lambda)$	$(K,\,\mu)$	$(\lambda,\,\nu)$	$(\mu,\,\nu)$	$(E,\,\nu)$	$(K,\, \nu)$	$(K,\,E)$	$(M,\,\mu)$
$K=\,$	$\lambda+ \frac{2\mu}{3}$	$\frac{E\mu}{3(3\mu-E)}$			$\lambda\frac{1+\nu}{3\nu}$	$\frac{2\mu(1+\nu)}{3(1-2\nu)}$	$\frac{E}{3(1-2\nu)}$			$M - \frac{4\mu}{3}$
$E=\,$	$\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$		$9K\frac{K-\lambda}{3K-\lambda}$	$\frac{9K\mu}{3K+\mu}$	$\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$2\mu(1+\nu)\,$		$3K(1-2\nu)\,$		$\mu\frac{3M-4\mu}{M-\mu}$
$\lambda=\,$		$\mu\frac{E-2\mu}{3\mu-E}$		$K-\frac{2\mu}{3}$		$\frac{2 \mu \nu}{1-2\nu}$	$\frac{E\nu}{(1+\nu)(1-2\nu)}$	$\frac{3K\nu}{1+\nu}$	$\frac{3K(3K-E)}{9K-E}$	$M - 2\mu\,$
$\mu=\,$			$3\frac{K-\lambda}{2}$		$\lambda\frac{1-2\nu}{2\nu}$		$\frac{E}{2+2\nu}$	$3K\frac{1-2\nu}{2+2\nu}$	$\frac{3KE}{9K-E}$
$\nu=\,$	$\frac{\lambda}{2(\lambda + \mu)}$	$\frac{E}{2\mu}-1$	$\frac{\lambda}{3K-\lambda}$	$\frac{3K-2\mu}{2(3K+\mu)}$					$\frac{3K-E}{6K}$	$\frac{M - 2\mu}{2M - 2\mu}$
$M=\,$	$\lambda+2\mu\,$	$\mu\frac{4\mu-E}{3\mu-E}$	$3K-2\lambda\,$	$K+\frac{4\mu}{3}$	$\lambda \frac{1-\nu}{\nu}$	$\mu\frac{2-2\nu}{1-2\nu}$	$E\frac{1-\nu}{(1+\nu)(1-2\nu)}$	$3K\frac{1-\nu}{1+\nu}$	$3K\frac{3K+E}{9K-E}$

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