Young\'s modulus - Citizendia

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 material. Solid mechanics is the branch of Mechanics, Physics, and Mathematics that concerns the behavior of solid matter under external actions (e Stiffness is the resistance of an elastic body to Deformation by an applied Force. It is also known as the Young modulus, modulus of elasticity, elastic modulus (though the Young's modulus is actually one of several elastic moduli such as the bulk modulus and the shear modulus) or tensile modulus. An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear It is defined as the ratio of stress over strain in the region in which Hooke's Law is obeyed for the material. Stress is a measure of the average amount of Force exerted per unit Area. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the ^[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. Slope is used to describe the steepness incline gradient or grade of a straight line. During testing of a material sample the stress–strain curve is a graphical representation of the relationship between stress, derived from measuring the load applied on the Stress is a measure of the average amount of Force exerted per unit Area.

Young's modulus is named after Thomas Young, the 18th century British scientist. Thomas Young (13 June 1773 &ndash 10 May 1829 was an English Polymath who contributed to the scientific understanding of vision, Light 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. Giordano Riccati or Jordan Riccati (fl 1782 was the first experimental Mechanician to study material Elastic moduli as we understand them today ^[2]

1 Units
2 Usage
- 2.1 Linear vs 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. Stress is a measure of the average amount of Force exerted per unit Area. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface

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). The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical In United States customary units, it is expressed as pounds (force) per square inch (psi). US customary units, also known in the United States as English units or Imperial units (in reference to the British Empire) (but see English The pound per square inch or more accurately pound-force per square inch (symbol psi or lbf/in² or lbf/in²) is a unit of

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 buckle under compression. In Engineering, buckling is a failure mode characterized by a sudden failure of a structural member subjected to high Compressive stresses where Some calculations also require the use of other material properties, such as the shear modulus, density, or Poisson's ratio. In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Poisson's ratio ( ν) named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to

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. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the Examples of linear materials include steel, carbon fiber, and glass. Steel is an Alloy consisting mostly of Iron, with a Carbon content between 0 Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many Rubber and soils (except at very small strains) are non-linear materials. Soil, often typeset as SOiL, is a four piece rock band from Chicago Illinois United States founded by Shaun Glass Tom Schofield Tim King and Adam Zadel

Directional materials

Most metals and ceramics, along with many other materials, are isotropic: their mechanical properties are the same in all directions. Isotropy is uniformity in all directions Precise definitions depend on the subject area However metals and ceramics can be treated with certain impurities to give them a “grain”. The grain of these, and other composites of two or more ingredients, is a mechanical structure of various orientations and sizes, which makes the material anisotropic. This means that Young's modulus will change depending on which direction the force is applied from. As a result, these anisotropic materials have different mechanical properties when load is applied in different directions. Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity For example, carbon fiber is much stiffer (higher Young's modulus) when loaded parallel to the fibers (along the grain), and is an example of a material with transverse isotropy. A transversely isotropic material is symmetric about an axis that is normal to a plane of Isotropy. Other such materials include wood and reinforced concrete. Wood is hard fibrous lignified structural tissue produced as secondary Xylem in the stems of Woody plants notably trees but also shrubs Reinforced concrete is Concrete in which reinforcement bars (" Rebars quot or fibers have been incorporated to strengthen a material that would otherwise be Engineers can use this directional phenomenon to their advantage in creating various structures in our environment.

Copper as an excellent electrical conductor is used to transmit electricity over long distance cables, however although copper has a relatively high value for Young's modulus at 130 GPa, it has a very low value for yield strength, and thus easily deforms in tension. When the copper cable is co-wound with hardened steel wire the stretching can largely be prevented, as the steel (with a higher value of Young's modulus in tension and much higher yield strength) 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:

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

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. Stress is a measure of the average amount of Force exerted per unit Area.

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. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the The elastic Potential energy is defined as a work (force x distance needed to compress or expand an elastic body

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. An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear Poisson's ratio ( ν) named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to The values here are approximate.

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. For other meanings see Giga (disambiguation Giga- (symbol G is a prefix in the SI system of units denoting 109 The pound per square inch or more accurately pound-force per square inch (symbol psi or lbf/in² or lbf/in²) is a unit of 01-0. 1	1,500-15,000
PTFE (Teflon)	0. In Chemistry, poly(tetrafluoroethene or poly(tetrafluoroethylene ( PTFE) is a synthetic Fluoropolymer which finds numerous applications 5	75,000
Low density polyethylene	0. Properties LDPE is defined by a density range of 0910 - 0940 g/cm³ 2	30,000
HDPE	1. High-Density Polyethylene ( HDPE) or PolyEthylene High-Density ( PEHD) is a Polyethylene Thermoplastic made from Petroleum 379	200,000
Polypropylene	1. Polypropylene or polypropene ( PP) is a Thermoplastic Polymer, made by the Chemical industry and used in a wide variety of applications 5-2	217,000-290,000
Bacteriophage capsids	1-3	150,000-435,000
Polyethylene terephthalate	2-2. For the leaf bug see Miridae. A capsid is the protein shell of a virus. Uses PET can be semi-rigid to rigid depending on its thickness and is very lightweight 5 OR 2. 8-3. 1	290,000-360,000
Polystyrene	3-3. Polystyrene ˌpɒliˈstaɪriːn ( IUPAC Polyphenylethene is an aromatic Polymer made from the aromatic Monomer Styrene 5	435,000-505,000
Nylon	3-7	290,000-580,000
MDF (wood composite)	3. Overview Nylon is a Thermoplastic silky material first used commercially in a nylon- Bristled Toothbrush (1938 followed more famously by Medium-density fiberboard ( MDF or MDFB) is an Engineered wood product formed by breaking down Softwood into Wood fibers often in 654	530,000
Pine wood (along grain)	8. Wood is hard fibrous lignified structural tissue produced as secondary Xylem in the stems of Woody plants notably trees but also shrubs 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
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)	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 (^[4]). Wood is hard fibrous lignified structural tissue produced as secondary Xylem in the stems of Woody plants notably trees but also shrubs Concrete is a construction material composed of Cement (commonly Portland cement) as well as other cementitious materials such as Fly ash and Slag Magnesium (mægˈniːziəm is a Chemical element with the symbol Mg, Atomic number 12 Atomic weight 24 The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across Aluminium alloys are Alloys of Aluminium, often with copper zinc manganese silicon or magnesium Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many Brass is any Alloy of Copper and Zinc; the proportions of zinc and copper can be varied to create a range of brasses with varying properties Bronze is any of a broad range of Copper alloys, usually with Tin as the main additive but sometimes with other elements such as Phosphorus Titanium (taɪˈteɪniəm is a Chemical element with the symbol Ti and Atomic number 22 Copper (ˈkɒpɚ is a Chemical element with the symbol Cu (cuprum and Atomic number 29 QtubIronPillarJPG|thumb|right| Iron pillar at Delhi India containing 98% wrought iron]] Wrought iron is commercially pure Iron. Steel is an Alloy consisting mostly of Iron, with a Carbon content between 0 Beryllium (bəˈrɪliəm is a Chemical element with the symbol Be and Atomic number 4 Tungsten (ˈtʌŋstən also known as wolfram (/ˈwʊlfrəm/ is a Chemical element that has the symbol W and Atomic number 74 Silicon carbide ( is a compound of Silicon and Carbon bonded together to form Ceramics but it also occurs in nature as the extremely rare mineral Osmium (ˈɒzmiəm is a Chemical element that has the symbol Os and Atomic number 76 Tungsten carbide, WC, or tungsten semicarbide, W2C, is a chemical compound containing Tungsten and Carbon, similar See also Graphene, Buckypaper Carbon nanotubes (CNTs are Allotropes of carbon with a nanostructure that can have a length-to-diameter In Mineralogy, diamond is the allotrope of carbon where the carbon atoms are arranged in

References

^ International Union of Pure and Applied Chemistry. In Materials science, deformation is a change in the shape or size of an object due to an applied force. Hardness refers to various properties of Matter in the Solid phase that give it high resistance to various kinds of shape change when Force In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear Practical The impulse excitation technique is a Nondestructive test method that uses Natural frequency, dimensions and mass of a test-piece to determine Young's Stress is a measure of the average amount of Force exerted per unit Area. Toughness, in Materials science and Metallurgy, is the resistance to Fracture of a material when stressed. The yield strength or yield point of a Material is defined in Engineering and Materials science as the stress at which a material A Materials property is an intensive, often Quantitative property of a material usually with a unit that may be used as a metric The International Union of Pure and Applied Chemistry ( IUPAC) (aɪjuːpæk or ay-yoo-pec) is an international Non-governmental organization "modulus of elasticity (Young's modulus), E". Compendium of Chemical Terminology Internet edition. Compendium of Chemical Terminology (ISBN 0-86542-684-8 is a book published by IUPAC containing internationally accepted definitions for terms in Chemistry.
^ 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
^ Glassproperties.com

External links

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

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