View source for Young's modulus

{{Short description|Mechanical property that measures stiffness of a solid material}}
{{Use mdy dates|date=May 2017}}
[[File:Stress strain ductile.svg|thumb|upright=1.35|Young's modulus is the slope of the linear part of the [[stress–strain curve]] for a material under tension or compression.]]

'''Young's modulus''' (or '''Young modulus''') is a mechanical property of [[solid]] materials that measures the tensile or compressive [[stiffness]] when the force is applied lengthwise. It is the [[modulus of elasticity]] for [[Tension (physics)|tension]] or axial [[Compression (physics)|compression]]. Young's modulus is defined as the ratio of the [[stress (mechanics)|stress]] (force per unit area) applied to the object and the resulting axial [[strain (mechanics)|strain]] (displacement or deformation) in the [[linear elasticity|linear elastic]] region of the material.

Although Young's modulus is named after the 19th-century British scientist [[Thomas Young (scientist)|Thomas Young]], the concept was developed in 1727 by [[Leonhard Euler]]. The first experiments that used the concept of Young's modulus in its modern form were performed by the Italian scientist [[Giordano Riccati]] in 1782, pre-dating Young's work by 25 years.<ref>''The Rational mechanics of Flexible or Elastic Bodies, 1638–1788'': Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae.  Orell Fussli.</ref>  The term modulus is derived from the [[Latin]] [[Root (linguistics)|root term]] ''[[wiktionary:modus|modus]]'' which means ''measure''.

==Definition==
Young's modulus, <math>E</math>, quantifies the relationship between tensile or compressive [[stress (mechanics)|stress]] <math>\sigma</math> (force per unit area) and axial [[strain (mechanics)|strain]] <math>\varepsilon</math> (proportional deformation) in the [[linear elasticity|linear elastic]] region of a material:<ref>{{cite book |last=Jastrzebski |first=D. |title=Nature and Properties of Engineering Materials |publisher=[[John Wiley & Sons, Inc]] |year=1959 |edition=Wiley International}}</ref>
<math display="block">E = \frac{\sigma}{\varepsilon}</math>

Young's modulus is commonly measured in the [[International System of Units]] (SI) in multiples of the [[Pascal (unit)|pascal]] (Pa) and common values are in the range of [[gigapascal]]s (GPa).

Examples:
* [[Rubber]] (increasing pressure: ''length increases quickly, meaning low <math>E</math>'')
* [[Aluminium]] (increasing pressure: ''length increases slowly, meaning high <math>E</math>'')

===Linear elasticity===
{{Main|Linear elasticity}}
A solid material undergoes [[elastic deformation]] when a small load is applied to it in compression or extension. Elastic deformation is reversible, meaning that the material returns to its original shape after the load is removed.

At near-zero stress and strain, the stress–strain curve is [[linear]], and the relationship between stress and strain is described by [[Hooke's law]] that states stress is proportional to strain. The coefficient of proportionality is Young's modulus. The higher the modulus, the more stress is needed to create the same amount of strain; an idealized [[rigid body]] would have an infinite Young's modulus. Conversely, a very soft material (such as a fluid) would deform without force, and would have zero Young's modulus.

===Related but distinct properties===
Material stiffness is a distinct property from the following:
* [[Strength of materials|Strength]]: maximum amount of stress that material can withstand while staying in the elastic (reversible) deformation regime;
* Geometric stiffness: a global characteristic of the body that depends on its shape, and not only on the local properties of the material; for instance, an [[I beam|I-beam]] has a higher bending stiffness than a rod of the same material for a given mass per length;
* [[Hardness]]: relative resistance of the material's surface to penetration by a harder body;
* [[Toughness]]: amount of energy that a material can absorb before fracture.

==Usage==
Young's modulus enables the calculation of the change in the dimension of a bar made of an [[isotropic]] elastic material under tensile or compressive loads.  For instance, it predicts how much a material sample extends under tension or shortens under compression. The Young's modulus directly applies to cases of uniaxial stress; that is, tensile or compressive stress in one direction and no stress in the other directions. Young's modulus is also used in order to predict the deflection that will occur in a [[statically determinate#Statically determinate|statically determinate]] [[beam (structure)|beam]] when a load is applied at a point in between the beam's supports.

Other elastic calculations usually require the use of one additional elastic property, such as the [[shear modulus]] <math>G</math>, [[bulk modulus]] <math>K</math>, and [[Poisson's ratio]] <math>\nu</math>. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. For homogeneous isotropic materials [[Elastic modulus|simple relations]] exist between elastic constants that allow calculating them all as long as two are known:
:<math>E = 2G(1+\nu) = 3K(1-2\nu).</math>

===Linear versus non-linear===
Young's modulus represents the factor of proportionality in [[Hooke's law]], which relates the stress and the strain. However, Hooke's law is only valid under the assumption of an ''elastic'' and ''linear'' response. Any real material will eventually fail and break when stretched over a very large distance or with a very large force; however, all solid materials exhibit nearly Hookean behavior for small enough strains or stresses. If the range over which Hooke's law is valid is large enough compared to the typical stress that one expects to apply to the material, the material is said to be linear. Otherwise (if the typical stress one would apply is outside the linear range), the material is said to be non-linear.

[[Steel]], [[carbon (fiber)|carbon fiber]] and [[glass]] among others are usually considered linear materials, while other materials such as [[rubber]] and [[soils]] are non-linear. However, this is not an absolute classification: if very small stresses or strains are applied to a non-linear material, the response will be linear, but if very high stress or strain is applied to a linear material, the linear theory will not be enough. For example, as the linear theory implies [[Reversible process (thermodynamics)|reversibility]], it would be absurd to use the linear theory to describe the failure of a steel bridge under a high load; although steel is a linear material for most applications, it is not in such a case of catastrophic failure.

In [[solid mechanics]], the slope of the [[stress–strain curve]] at any point is called the [[tangent modulus]]. It can be experimentally determined from the [[slope]] of a stress–strain curve created during [[tensile test]]s conducted on a sample of the material.

===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 [[isotropy|isotropic]], and 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 [[anisotropy|anisotropic]], and Young's modulus will change depending on the direction of the force vector.<ref>{{Cite journal| last1=Gorodtsov |first1=V.A. |last2=Lisovenko |first2=D.S. |date=2019 |title=Extreme values of Young's modulus and Poisson's ratio of hexagonal crystals|journal=Mechanics of Materials |language=en |volume=134 |pages=1–8 |doi=10.1016/j.mechmat.2019.03.017 |s2cid=140493258 }}</ref> Anisotropy can be seen in many composites as well. For example, [[carbon (fiber)|carbon fiber]] has a much higher Young's modulus (is much stiffer) when force is 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 structures.

===Temperature dependence===
The Young's modulus of metals varies with the temperature and can be realized through the change in the interatomic bonding of the atoms, and hence its change is found to be dependent on the change in the work function of the metal. Although classically, this change is predicted through fitting and without a clear underlying mechanism (for example, the Watchman's formula), the Rahemi-Li model<ref>{{cite journal |last1=Rahemi |first1=Reza |last2=Li |first2=Dongyang |date=April 2015 |title=Variation in electron work function with temperature and its effect on the Young's modulus of metals |journal=[[Scripta Materialia]] |volume=99 |issue=2015 |pages=41–44 |arxiv=1503.08250 |bibcode=2015arXiv150308250R |doi=10.1016/j.scriptamat.2014.11.022 |s2cid=118420968}}</ref> demonstrates how the change in the electron work function leads to change in the Young's modulus of metals and predicts this variation with calculable parameters, using the generalization of the [[Lennard-Jones potential]] to solids. In general, as the temperature increases, the Young's modulus decreases via <math>E(T) = \beta(\varphi(T))^6</math> where the electron work function varies with the temperature as <math>\varphi(T)=\varphi_0-\gamma\frac{(k_BT)^2}{\varphi_0}</math> and <math>\gamma</math> is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). <math>\varphi_0</math> is the electron work function at T=0 and <math>\beta</math> is constant throughout the change.

==Calculation==
Young's modulus ''E'', is calculated by dividing the [[Stress (physics)|tensile stress]], <math>\sigma(\varepsilon)</math>, by the [[Strain (physics)|engineering extensional strain]], <math>\varepsilon</math>, in the elastic (initial, linear) portion of the physical [[stress–strain curve]]:

<math display="block"> E \equiv \frac{\sigma(\varepsilon)}{\varepsilon}= \frac{F/A}{\Delta L/L_0} = \frac{F L_0} {A \, \Delta L} </math>
where
* <math>E</math> is the Young's modulus (modulus of elasticity)
* <math>F</math> is the force exerted on an object under tension;
* <math>A</math> is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force;
* <math>\Delta L</math> is the amount by which the length of the object changes (<math>\Delta L</math> is positive if the material is stretched, and negative when the material is compressed);
* <math>L_0</math> is the original length of the object.

===Force exerted by stretched or contracted material===
Young's modulus of a material can be used to calculate the force it exerts under specific strain.

:<math>F = \frac{E A \, \Delta L} {L_0}</math>
where <math>F</math> is the force exerted by the material when contracted or stretched by <math>\Delta L</math>.

[[Hooke's law]] for a stretched wire can be derived from this formula:
:<math>F = \left( \frac{E A} {L_0} \right) \, \Delta L = k x </math>
where it comes in saturation
:<math>k \equiv \frac {E A} {L_0} \,</math> and <math>x \equiv \Delta L. </math>
But note that the elasticity of coiled springs comes from [[shear modulus]], not Young's modulus. {{citation needed|date=April 2021}}

===Elastic potential energy===
The [[elastic potential energy]] stored in a linear elastic material is given by the integral of the Hooke's law:

:<math>U_e = \int {k x}\, dx = \frac {1} {2} k x^2.</math>

now by explicating the intensive variables:

:<math>U_e = \int \frac{E A \, \Delta L} {L_0}\, d\Delta L = \frac {E A} {L_0} \int \Delta L \, d\Delta L = \frac {E A \, {\Delta L}^2} {2 L_0}</math>

This means that the elastic potential energy density (that is, per unit volume) is given by:
:<math>\frac{U_e} {A L_0} = \frac {E \, {\Delta L}^2} {2 L_0^2} </math>

or, in simple notation, for a linear elastic material: <math display="inline"> u_e(\varepsilon) = \int {E \, \varepsilon}\, d\varepsilon = \frac {1} {2} E {\varepsilon}^2</math>, since the strain is defined <math display="inline">\varepsilon \equiv \frac {\Delta L} {L_0}</math>.

In a nonlinear elastic material the Young's modulus is a function of the strain, so the second equivalence no longer holds, and the elastic energy is not a [[quadratic function]] of the strain:

: <math> u_e(\varepsilon) = \int E(\varepsilon) \, \varepsilon \, d\varepsilon \ne \frac {1} {2} E \varepsilon^2</math>

==Examples==
[[Image:SpiderGraph YoungMod.gif|350px|thumb|Influences of selected glass component additions on Young's modulus of a specific base glass]]

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 [[polymer]]s. The values here are approximate and only meant for relative comparison.
{| class="wikitable sortable" style="text-align:center;"
|+Approximate Young's modulus for various materials
!Material
! data-sort-type="number" |Young's modulus ([[Pascal (unit)|GPa]])
! data-sort-type="number" |Megapound per square inch ([[Mega-|M]][[Pounds per square inch|psi]])<ref>{{Cite web|title=Unit of Measure Converter|url=http://www.matweb.com/tools/unitconverter.aspx|access-date=May 9, 2021|website=MatWeb}}</ref>
!Ref.
|-
| style="text-align:left;" |[[Aluminium]] (<sub>13</sub>Al)
|68
|9.86
|<ref>{{Cite web|title=Aluminum, Al|url=http://www.matweb.com/search/DataSheet.aspx?MatGUID=0cd1edf33ac145ee93a0aa6fc666c0e0|access-date=May 7, 2021|website=MatWeb}}</ref><ref name=":02">{{Cite book |last=Weast |first=Robert C. |title=[[CRC Handbook of Chemistry and Physics]] |publisher=[[CRC Press]] |year=1981 |isbn=978-0-84-930740-9 |edition=62nd |location=Boca Raton, FL |doi=10.1002/jctb.280500215}}</ref><ref name=":1">{{Cite book|last=Ross|first=Robert B.|title=Metallic Materials Specification Handbook|publisher=[[Chapman & Hall]]|year=1992|isbn=9780412369407|edition=4th|location=London|doi=10.1007/978-1-4615-3482-2}}</ref><ref name=":2">{{Cite book|last1=Nunes|first1=Rafael|url=http://sme.vimaru.edu.vn/sites/sme.vimaru.edu.vn/files/volume_2_-_properties_and_selection_nonf.pdf|title=Volume 2: Properties and Selection: Nonferrous Alloys and Special-Purpose Materials|last2=Adams|first2=J. H.|last3=Ammons|first3=Mitchell|last4=Avery|first4=Howard S.|last5=Barnhurst|first5=Robert J.|last6=Bean|first6=John C.|last7=Beaudry|first7=B. J.|last8=Berry|first8=David F.|last9=Black|first9=William T.|publisher=[[ASM International]]|year=1990|isbn=978-0-87170-378-1|edition=10th|series=ASM Handbook|display-authors=3}}</ref><ref name=":3">{{Cite book|last=Nayar|first=Alok|title=The Metals Databook|publisher=[[McGraw-Hill]]|year=1997|isbn=978-0-07-462300-8|location=New York, NY}}</ref><ref name=":4">{{Cite book |title=CRC Handbook of Chemistry and Physics|publisher=[[CRC Press]]|year=1999|isbn=978-0-84-930480-4|editor-last=Lide|editor-first=David R.|edition=80th|location=Boca Raton, FL|chapter=Commercial Metals and Alloys}}</ref>
|-
| style="text-align:left;" |[[Amino acid|Amino-acid]] [[Molecular solid|molecular crystals]]
|21–44
|3.05–6.38
|<ref name=":5">{{Cite journal |last1=Azuri |first1=Ido |last2=Meirzadeh |first2=Elena |last3=Ehre |first3=David |last4=Cohen |first4=Sidney R. |last5=Rappe |first5=Andrew M. |last6=Lahav |first6=Meir |last7=Lubomirsky |first7=Igor |last8=Kronik |first8=Leeor |display-authors=3 |date=November 9, 2015 |title=Unusually Large Young's Moduli of Amino Acid Molecular Crystals |url=http://www.sas.upenn.edu/rappegroup/publications/Papers/Azuri15p13566.pdf |journal=[[Angewandte Chemie]] |edition=International |publisher=[[Wiley (publisher)|Wiley]] |volume=54 |issue=46 |pages=13566–13570 |doi=10.1002/anie.201505813 |pmid=26373817 |via=[[PubMed]] |s2cid=13717077}}</ref>
|-
| style="text-align:left;" |[[Aramid]] (for example, [[Kevlar]])
|70.5–112.4
|10.2–16.3
|<ref>{{Cite web|date=2017|title=Kevlar Aramid Fiber Technical Guide|url=https://www.dupont.com/content/dam/dupont/amer/us/en/safety/public/documents/en/Kevlar_Technical_Guide_0319.pdf|access-date=May 8, 2021|website=[[DuPont]]}}</ref>
|-
| style="text-align:left;" |Aromatic peptide-nanospheres
|230–275
|33.4–39.9
|<ref>{{Cite journal|last1=Adler-Abramovich|first1=Lihi |last2=Kol|first2=Nitzan |last3=Yanai|first3=Inbal |last4=Barlam|first4=David |last5=Shneck|first5=Roni Z. |last6=Gazit|first6=Ehud |last7=Rousso |first7=Itay|display-authors=3 |date=December 17, 2010 |title=Self-Assembled Organic Nanostructures with Metallic-Like Stiffness |journal=Angewandte Chemie |edition=International|publisher=[[Wiley-VCH]] |publication-date=September 28, 2010 |volume=49 |issue=51 |pages=9939–9942 |doi=10.1002/anie.201002037 |pmid=20878815|s2cid=44873277 }}</ref>
|-
| style="text-align:left;" |Aromatic peptide-nanotubes
|19–27
|2.76–3.92
|<ref>{{Cite journal |last1=Kol |first1=Nitzan |last2=Adler-Abramovich |first2=Lihi |last3=Barlam |first3=David |last4=Shneck |first4=Roni Z. |last5=Gazit |first5=Ehud |last6=Rousso |first6=Itay |display-authors=3 |date=June 8, 2005 |title=Self-Assembled Peptide Nanotubes Are Uniquely Rigid Bioinspired Supramolecular Structures |url=https://pubs.acs.org/doi/full/10.1021/nl0505896 |journal=[[Nano Letters]] |location=Israel |publisher=[[American Chemical Society]] |volume=5 |issue=7 |pages=1343–1346 |bibcode=2005NanoL...5.1343K |doi=10.1021/nl0505896 |pmid=16178235 |via=[[ACS Publications]]}}</ref><ref>{{Cite journal |last1=Niu |first1=Lijiang |last2=Chen |first2=Xinyong |last3=Allen |first3=Stephanie |last4=Tendler |first4=Saul J. B. |display-authors=3 |date=June 6, 2007 |title=Using the Bending Beam Model to Estimate the Elasticity of Diphenylalanine Nanotubes |url=https://pubs.acs.org/doi/full/10.1021/la7010106 |journal=[[Langmuir (journal)|Langmuir]] |publisher=[[American Chemical Society]] |volume=23 |issue=14 |pages=7443–7446 |doi=10.1021/la7010106 |pmid=17550276 |via=[[ACS Publications]]}}</ref>
|-
| style="text-align:left;" |[[Bacteriophage]] [[capsid]]s
|1–3
|0.145–0.435
|<ref>{{cite journal|last1=Ivanovska|first1=Irena L.|last2=de Pablo|first2=Pedro J. |last3=Ibarra |first3=Benjamin |last4=Sgalari |first4=Giorgia |last5=MacKintosh |first5=Fred C. |last6=Carrascosa |first6=José L. |last7=Schmidt |first7=Christoph F. |last8=Wuite |first8=Gijs J. L. |display-authors=3 |date=May 7, 2004 |editor-last=Lubensky |editor-first=Tom C. |title=Bacteriophage capsids: Tough nanoshells with complex elastic properties|journal=[[Proceedings of the National Academy of Sciences of the United States of America]] |publisher=[[The National Academy of Sciences]] |volume=101 |issue=20 |pages=7600–7605 |bibcode=2004PNAS..101.7600I |doi=10.1073/pnas.0308198101|pmc=419652|pmid=15133147|doi-access=free}}</ref>
|-
| style="text-align:left;" |[[Beryllium]] (<sub>4</sub>Be)
|287
|41.6
|<ref>{{Cite book|last1=Foley|first1=James C.|title=Powder Materials: Current Research and Industrial Practices III|last2=Abeln|first2=Stephen P.|last3=Stanek|first3=Paul W.|last4=Bartram|first4=Brian D.|last5=Aikin|first5=Beverly|last6=Vargas|first6=Victor D.|publisher= [[Wiley (publisher)|John Wiley & Sons, Inc.]]|year=2010|isbn=978-1-11-898423-9|editor-last=Marquis|editor-first=Fernand D. S.|location=Hoboken, NJ|pages=263|chapter=An Overview of Current Research and Industrial Practices of be Powder Metallurgy|doi=10.1002/9781118984239.ch32|display-authors=3}}</ref>
|-
| style="text-align:left;" |[[Bone]], human cortical
|14
|2.03
|<ref>{{Cite journal|last1=Rho|first1=Jae Young|last2=Ashman|first2=Richard B.|last3=Turner|first3=Charles H.|date=February 1993|title=Young's modulus of trabecular and cortical bone material: Ultrasonic and microtensile measurements |url=https://www.sciencedirect.com/science/article/abs/pii/002192909390042D|journal=Journal of Biomechanics|publisher=[[Elsevier]] |volume=26 |issue=2 |pages=111–119|doi=10.1016/0021-9290(93)90042-d|pmid=8429054|via=[[Elsevier Science Direct]]}}</ref>
|-
| style="text-align:left;" |[[Brass]]
|106
|15.4
|<ref>{{Cite web|title=Overview of materials for Brass|url=http://www.matweb.com/search/DataSheet.aspx?MatGUID=d3bd4617903543ada92f4c101c2a20e5 |access-date=May 7, 2021|website=MatWeb}}</ref>
|-
| style="text-align:left;" |[[Bronze]]
|112
|16.2
|<ref>{{Cite web|title=Overview of materials for Bronze|url=http://www.matweb.com/search/datasheet.aspx?MatGUID=66575ff2cd5249c49d76df15b47dbca4|access-date=May 7, 2021 |website=MatWeb}}</ref>
|-
| style="text-align:left;" |[[Carbon nitride]] (CN<sub>2</sub>)
|822
|119
|<ref>{{Cite journal |last1=Chowdhury |first1=Shafiul |last2=Laugier |first2=Michael T. |last3=Rahman |first3=Ismet Zakia |date=April–August 2004 |title=Measurement of the mechanical properties of carbon nitride thin films from the nanoindentation loading curve |journal=[[Diamond and Related Materials]] |volume=13 |issue=4–8 |pages=1543–1548 |bibcode=2004DRM....13.1543C |doi=10.1016/j.diamond.2003.11.063 |via=[[Elsevier Science Direct]]}}</ref>
|-
| style="text-align:left;" |[[Carbon-fiber-reinforced polymers|Carbon-fiber-reinforced plastic]] (CFRP), 50/50 fibre/matrix, biaxial fabric
|30–50
|4.35–7.25
|<ref>{{cite web|last=Summerscales|first=John|date=September 11, 2019|title=Composites Design and Manufacture (Plymouth University teaching support materials)|url=https://www.fose1.plymouth.ac.uk/sme/MATS347/MATS347A2%20E-G-nu.htm#E|access-date=May 8, 2021 |website=Advanced Composites Manufacturing Centre|publisher=[[University of Plymouth]]}}</ref>
|-
| style="text-align:left;" |[[Carbon-fiber-reinforced polymers|Carbon-fiber-reinforced plastic]] (CFRP), 70/30 fibre/matrix, unidirectional, along fibre
|181
|26.3
|<ref>{{Cite web|last=Kopeliovich|first=Dmitri|date=June 3, 2012|title=Epoxy Matrix Composite reinforced by 70% carbon fibers |url=http://www.substech.com/dokuwiki/doku.php?id=epoxy_matrix_composite_reinforced_by_70_carbon_fibers|access-date=May 8, 2021|website=SubsTech}}</ref>
|-
| style="text-align:left;" |[[Cobalt-chrome]] (CoCr)
|230
|33.4
|<ref>{{Cite book|last1=Bose|first1=Susmita|title=Materials for Bone Disorders|last2=Banerjee|first2=Dishary|last3=Bandyopadhyay|first3=Amit|publisher=[[Academic Press]]|year=2016|isbn=978-0-12-802792-9|editor-last=Bose|editor-first=Susmita|pages=1–27|chapter=Introduction to Biomaterials and Devices for Bone Disorders|doi=10.1016/B978-0-12-802792-9.00001-X|editor-last2=Bandyopadhyay|editor-first2=Amit}}</ref>
|-
| style="text-align:left;" |[[Copper]] (Cu), annealed
|110
|16
|<ref>{{Cite web|title=Copper, Cu; Annealed|url=http://www.matweb.com/search/DataSheet.aspx?MatGUID=9aebe83845c04c1db5126fada6f76f7e|access-date=May 9, 2021|website=MatWeb}}</ref>
|-
| style="text-align:left;" |[[Diamond]] (C), synthetic
|1050–1210
|152–175
|<ref>{{Cite book|title=Synthetic Diamond: Emerging CVD Science and Technology|publisher= [[Wiley (publisher)|Wiley]] |year=1994 |isbn=978-0-47-153589-8|editor-last=Spear|editor-first=Karl E.|pages=315|issn=0275-0171|editor-last2=Dismukes|editor-first2=John P.}}</ref>
|-
| style="text-align:left;" |[[Diatom]] [[frustule]]s, largely [[Orthosilicic acid|silicic acid]]
|0.35–2.77
|0.051–0.058
|<ref>{{cite journal |last1=Subhash |first1=Ghatu |last2=Yao |first2=Shuhuai |last3=Bellinger |first3=Brent |last4=Gretz |first4=Michael R. |date=January 2005 |title=Investigation of mechanical properties of diatom frustules using nanoindentation |journal=[[Journal of Nanoscience and Nanotechnology]] |publisher=American Scientific Publishers |volume=5 |issue=1 |pages=50–56 |doi=10.1166/jnn.2005.006 |pmid=15762160 |via=[[Ingenta |Ingenta Connect]]}}</ref>
|-
| style="text-align:left;" |[[Flax]] fiber
|58
|8.41
|<ref name=":6">{{Cite journal |last1=Bodros |first1=Edwin |last2=Baley |first2=Christophe |date=May 15, 2008 |title=Study of the tensile properties of stinging nettle fibres (''Urtica dioica'') |journal=[[Materials Letters]] |volume=62 |issue=14 |pages=2143–2145 |citeseerx=10.1.1.299.6908 |doi=10.1016/j.matlet.2007.11.034 |via=[[Elsevier Science Direct]]}}</ref>
|-
| style="text-align:left;" |[[Float glass]]
|47.7–83.6
|6.92–12.1
|<ref>{{Cite web|date=February 16, 2001|title=Float glass – Properties and Applications|url=https://www.azom.com/properties.aspx?ArticleID=89|access-date=May 9, 2021|website=AZO Materials}}</ref>
|-
| style="text-align:left;" |[[Fiberglass|Glass-reinforced polyester]] (GRP)
|17.2
|2.49
|<ref>{{Cite web|last=Kopeliovich|first=Dmitri|date=March 6, 2012|title=Polyester Matrix Composite reinforced by glass fibers (Fiberglass)|url=http://www.substech.com/dokuwiki/doku.php?id=polyester_matrix_composite_reinforced_by_glass_fibers_fiberglass|access-date=May 7, 2021|website=SubsTech}}</ref>

|-
| style="text-align:left;" |[[Gold]]
|77.2
|11.2
|<ref>{{cite web|url=http://www.matweb.com/search/DataSheet.aspx?MatGUID=d2a2119a08904a0fa706e9408cddb88e|title=Gold material property data|website=MatWeb|accessdate=September 8, 2021}}</ref>

|-
| style="text-align:left;" |[[Graphene]]
|1050
|152
|<ref>{{Cite journal|last1=Liu|first1=Fang|last2=Ming|first2=Pingbing|last3=Li|first3=Ju|date=August 28, 2007|title=''Ab initio'' calculation of ideal strength and phonon instability of graphene under tension|url=http://li.mit.edu/A/Papers/07/Liu07.pdf|journal=Physical Review B|publisher=[[American Physical Society]]|volume=76|issue=6|page=064120|doi=10.1103/PhysRevB.76.064120|bibcode=2007PhRvB..76f4120L|via=[[APS Physics]]}}</ref>
|-
| style="text-align:left;" |[[Hemp]] fiber
|35
|5.08
|<ref>{{Cite journal|last1=Saheb|first1=Nabi|last2=Jog|first2=Jyoti|date=October 15, 1999|title=Natural fibre polymer composites: a review|journal=Advances in Polymer Technology|publisher= [[Wiley (publisher)|John Wiley & Sons, Inc.]]|volume=18|issue=4|pages=351–363|doi=10.1002/(SICI)1098-2329(199924)18:4<351::AID-ADV6>3.0.CO;2-X|doi-access=free}}</ref>
|-
| style="text-align:left;" |[[High-density polyethylene]] (HDPE)
|0.97–1.38
|0.141–0.2
|<ref>{{Cite web|title=High-Density Polyethylene (HDPE)|url=https://polymerdatabase.com/Commercial%20Polymers/HDPE.html|access-date=May 9, 2021|website=Polymer Database|publisher=Chemical Retrieval on the Web}}</ref>
|-
| style="text-align:left;" |[[Types of concrete|High-strength concrete]]
|30
|4.35
|<ref>{{Cite book|last=Cardarelli|first=François|title=Materials Handbook: A Concise Desktop Reference|publisher= [[Springer Publishing|Springer-Verlag]]|year=2008|isbn=978-3-319-38923-3|edition=2nd|location=London|pages=1421–1439|chapter=Cements, Concrete, Building Stones, and Construction Materials|doi=10.1007/978-3-319-38925-7_15}}</ref>
|-
| style="text-align:left;" |[[Lead]] (<sub>82</sub>Pb), chemical
|13
|1.89
|<ref name=":4" />
|-
| style="text-align:left;" |[[Low-density polyethylene]] (LDPE), molded
|0.228
|0.0331
|<ref>{{cite web|title=Overview of materials for Low Density Polyethylene (LDPE), Molded|url=http://matweb.com/search/DataSheet.aspx?MatGUID=557b96c10e0843dbb1e830ceedeb35b0|access-date=May 7, 2021|website=MatWeb}}</ref>
|-
| style="text-align:left;" |[[Magnesium alloy]]
|45.2
|6.56
|<ref>{{Cite web|title=Overview of materials for Magnesium Alloy|url=http://www.matweb.com/search/DataSheet.aspx?MatGUID=4e6a4852b14c4b12998acf2f8316c07c|access-date=May 9, 2021|website=MatWeb}}</ref>
|-
| style="text-align:left;" |[[Medium-density fibreboard|Medium-density fiberboard]] (MDF)
|4
|0.58
|<ref>{{cite web|date=May 30, 2020|title=Medium Density Fiberboard (MDF)|url=http://www.makeitfrom.com/data/?material=MDF|access-date=May 8, 2021|website=MakeItFrom}}</ref>
|-
| style="text-align:left;" |[[Molybdenum]] (Mo), annealed
|330
|47.9
|<ref>{{Cite web|title=Molybdenum, Mo, Annealed|url=http://www.matweb.com/search/datasheet.aspx?matguid=ef57c33963404798ad0301a05692312a|access-date=May 9, 2021|website=MatWeb}}</ref><ref name=":02" /><ref name=":1" /><ref name=":2" /><ref name=":3" /><ref name=":4" />
|-
| style="text-align:left;" |[[Monel]]
|180
|26.1
|<ref name=":4" />
|-
| style="text-align:left;" |[[Nacre|Mother-of-pearl]] (largely [[calcium carbonate]])
|70
|10.2
|<ref>{{cite journal|author=Jackson|first1=Andrew P.|last2=Vincent|first2=Julian F. V.|last3=Turner|first3=R. M.|date=September 22, 1988|title=The mechanical design of nacre|journal=Proceedings of the Royal Society B|publisher=[[Royal Society]]|volume=234|issue=1277|pages=415–440|bibcode=1988RSPSB.234..415J|doi=10.1098/rspb.1988.0056|issn=0080-4649|eissn=2053-9193|via= [[Royal Society|The Royal Society Publishing]]|s2cid=135544277}}</ref>
|-
| style="text-align:left;" |[[Nickel]] (<sub>28</sub>Ni), commercial
|200
|29
|<ref name=":4" />
|-
| style="text-align:left;" |[[Nylon 66]]
|2.93
|0.425
|<ref>{{Cite web|date=2011|title=Nylon® 6/6 (Polyamide)|url=https://www.polytechindustrial.com/products/plastic-stock-shapes/nylon-66|access-date=May 9, 2021|website=Poly-Tech Industrial, Inc.}}</ref>
|-
| style="text-align:left;" |[[Osmium]] (<sub>76</sub>Os)
|525–562
|76.1–81.5
|<ref>{{cite journal|author=Pandey|first1=Dharmendra Kumar|last2=Singh|first2=Devraj|last3=Yadawa|first3=Pramod Kumar|date=April 2, 2009|title=Ultrasonic Study of Osmium and Ruthenium|url=http://www.technology.matthey.com/pdf/91-97-pmr-apr09.pdf|journal=Platinum Metals Review|publisher=[[Johnson Matthey]]|volume=53|issue=4|pages=91–97|doi=10.1595/147106709X430927|access-date=May 7, 2021|via=[[Ingenta Connect]]|doi-access=free}}</ref>
|-
| style="text-align:left;" |[[Osmium]] [[nitride]] (OsN<sub>2</sub>)
|194.99–396.44
|28.3–57.5
|<ref>{{Cite web|last1=Gaillac|first1=Romain|last2=Coudert|first2=François-Xavier|date=July 26, 2020|title=ELATE: Elastic tensor analysis|url=http://progs.coudert.name/elate/mp?query=mp-973935|access-date=May 9, 2021|website=ELATE}}</ref>
|-
| style="text-align:left;" |[[Polycarbonate]] (PC)
|2.2
|0.319
|<ref>{{Cite web|title=Polycarbonate|url=https://designerdata.nl/materials/plastics/thermo-plastics/polycarbonate|access-date=May 9, 2021|website=DesignerData}}</ref>
|-
| style="text-align:left;" |[[Polyethylene terephthalate]] (PET), unreinforced
|3.14
|0.455
|<ref>{{Cite web|title=Overview of materials for Polyethylene Terephthalate (PET), Unreinforced|url=http://www.matweb.com/search/DataSheet.aspx?MatGUID=a696bdcdff6f41dd98f8eec3599eaa20|access-date=May 9, 2021|website=MatWeb}}</ref>
|-
| style="text-align:left;" |[[Polypropylene]] (PP), molded
|1.68
|0.244
|<ref>{{Cite web|title=Overview of Materials for Polypropylene, Molded|url=http://www.matweb.com/search/DataSheet.aspx?MatGUID=08fb0f47ef7e454fbf7092517b2264b2|access-date=May 9, 2021|website=MatWeb}}</ref>
|-
| style="text-align:left;" |[[Polystyrene]], crystal
|2.5–3.5
|0.363–0.508
|<ref>{{Cite web|title=Young's Modulus: Tensile Elasticity Units, Factors & Material Table|url=https://omnexus.specialchem.com/polymer-properties/properties/young-modulus|access-date=May 9, 2021|website=Omnexus|publisher=SpecialChem}}</ref>
|-
| style="text-align:left;" |[[Polystyrene]], foam
|0.0025–0.007
|0.000363–0.00102
|<ref>{{cite web|date=August 2019|title=Technical Data – Application Recommendations Dimensioning Aids|url=https://www.styrodur.com/portal/streamer?fid=1225078|access-date=May 7, 2021|website=Stryodur|publisher=[[BASF]]}}</ref>
|-
| style="text-align:left;" |[[Polytetrafluoroethylene]] (PTFE), molded
|0.564
|0.0818
|<ref>{{Cite web|title=Overview of materials for Polytetrafluoroethylene (PTFE), Molded|url=http://www.matweb.com/search/datasheet_print.aspx?matguid=4d14eac958e5401a8fd152e1261b6843|access-date=May 9, 2021|website=MatWeb}}</ref>
|-
| style="text-align:left;" |[[Natural rubber|Rubber]], small strain
|0.01–0.1
|0.00145–0.0145
|<ref name=":5" />
|-
| style="text-align:left;" |[[Silicon]], single crystal, different directions
|130–185
|18.9–26.8
|<ref>{{cite journal|author=Boyd|first1=Euan J.|last2=Uttamchandani|first2=Deepak|year=2012|title=Measurement of the Anisotropy of Young's Modulus in Single-Crystal Silicon|journal=Journal of Microelectromechanical Systems|publisher=[[Institute of Electrical and Electronics Engineers]]|volume=21|issue=1|pages=243–249|doi=10.1109/JMEMS.2011.2174415|issn=1057-7157|eissn=1941-0158|via=[[IEEE Xplore]]|s2cid=39025763}}</ref>
|-
| style="text-align:left;" |[[Silicon carbide]] (SiC)
|90–137
|13.1–19.9
|<ref>{{Cite web|date=February 5, 2001|title=Silicon Carbide (SiC) Properties and Applications|url=https://www.azom.com/properties.aspx?ArticleID=42|access-date=May 9, 2021|website=AZO Materials}}</ref>
|-
| style="text-align:left;" |Single-walled [[carbon nanotube]]
|data-sort-value="1000"|<math>></math>1000
|data-sort-value="140"|<math>></math>140
|<ref>{{Cite journal |last1=Forró |first1=László |last2=Salvetat |first2=Jean-Paul |last3=Bonard |first3=Jean-Marc |last4=Bacsa |first4=Revathi Ramachandran |last5=Thomson |first5=Neil H. |last6=Garaj |first6=Slaven |last7=Le |first7=Thien-Nga |last8=Gaál |first8=Richard |last9=Kulik |first9=Andrzej J. |last10=Ruzicka |first10=Barbara |last11=Degiorgi |first11=Leonardo |display-authors=3 |date=January 2002 |editor-last=Thorpe |editor-first=Michael F. |editor2-last=Tománek |editor2-first=David |editor2-link=David Tománek |editor3-last=Enbody |editor3-first=Richard J. |title=Electronic and Mechanical Properties of Carbon Nanotubes |url=https://www.researchgate.net/publication/226537355 |journal=Science and Application of Nanotubes |series=Fundamentals Materials Research |location=Boston, MA |publisher=[[Springer Publishing|Springer]] |pages=297–320 |doi=10.1007/0-306-47098-5_22 |isbn=978-0-306-46372-3 |via=[[ResearchGate]]}}</ref><ref>{{cite journal|author=Yang|first1=Yi-Hsuan|last2=Li|first2=Wenzhi|date=January 24, 2011|title=Radial elasticity of single-walled carbon nanotube measured by atomic force microscopy|journal=Applied Physics Letters|publisher=[[American Institute of Physics]]|volume=98|issue=4|page=041901|bibcode=2011ApPhL..98d1901Y|doi=10.1063/1.3546170}}</ref>
|-
| style="text-align:left;" |[[Steel]], [[A36 steel|A36]]
|200
|29
|<ref>{{Cite web|date=July 5, 2012|title=ASTM A36 Mild/Low Carbon Steel|url=https://www.azom.com/article.aspx?ArticleID=6117|access-date=May 9, 2021|website=AZO Materials}}</ref>
|-
| style="text-align:left;" |[[Urtica dioica|Stinging nettle]] fiber
|87
|12.6
|<ref name=":6" />
|-
| style="text-align:left;" |[[Titanium]] (<sub>22</sub>Ti)
|116
|16.8
|<ref>{{Cite web|title=Titanium, Ti|url=http://www.matweb.com/search/datasheet.aspx?MatGUID=66a15d609a3f4c829cb6ad08f0dafc01|access-date=May 7, 2021|website=MatWeb}}</ref><ref>{{Cite book|title=Materials Properties Handbook: Titanium Alloys|publisher=[[ASM International]]|year=1994|isbn=978-0-87-170481-8|editor-last=Boyer|editor-first=Rodney|location=Materials Park, OH|editor-last2=Welsch|editor-first2=Gerhard|editor-last3=Collings|editor-first3=Edward W.}}</ref><ref name=":02" /><ref name=":2" /><ref name=":1" /><ref name=":4" /><ref name=":3" />
|-
| style="text-align:left;" |[[Titanium alloy]], Grade 5
|114
|16.5
|<ref>{{Cite web|last=U.S. Titanium Industry Inc.|date=July 30, 2002|title=Titanium Alloys – Ti6Al4V Grade 5|url=https://www.azom.com/article.aspx?ArticleID=1547|access-date=May 9, 2021|website=AZO Materials}}</ref>
|-
| style="text-align:left;" |[[Tooth enamel]], largely [[calcium phosphate]]
|83
|12
|<ref>{{cite journal|author=Staines|first1=Michael|last2=Robinson|first2=W. H.|last3=Hood|first3=J. A. A.|date=September 1981|title=Spherical indentation of tooth enamel|journal=Journal of Materials Science|publisher=[[Springer Publishing|Springer]]|volume=16|issue=9|pages=2551–2556|bibcode=1981JMatS..16.2551S|doi=10.1007/bf01113595|via=[[Springer Link]]|s2cid=137704231}}</ref>
|-
| style="text-align:left;" |[[Tungsten carbide]] (WC)
|600–686
|87–99.5
|<ref>{{Cite web|date=January 21, 2002|title=Tungsten Carbide – An Overview|url=https://www.azom.com/properties.aspx?ArticleID=1203|access-date=May 9, 2021|website=AZO Materials}}</ref>
|-
| style="text-align:left;" |[[Wood]], [[Fagus grandifolia|American beech]]
|9.5–11.9
|1.38–1.73
|<ref name=":7">{{Cite book|last1=Green|first1=David W.|url=https://www.fpl.fs.fed.us/documnts/fplgtr/fplgtr113/ch04.pdf|title=Wood Handbook: Wood as an Engineering Material|last2=Winandy|first2=Jerrold E.|last3=Kretschmann|first3=David E.|publisher=[[Forest Products Laboratory]]|year=1999|location=Madison, WI|pages=4–8|chapter=Mechanical Properties of Wood|archive-url=https://web.archive.org/web/20180720153345/https://www.fpl.fs.fed.us/documnts/fplgtr/fplgtr113/ch04.pdf|archive-date=2018-07-20}}</ref>
|-
| style="text-align:left;" |[[Wood]], [[Prunus serotina|black cherry]]
|9–10.3
|1.31–1.49
|<ref name=":7" />
|-
| style="text-align:left;" |[[Wood]], [[Acer rubrum|red maple]]
|9.6–11.3
|1.39–1.64
|<ref name=":7" />
|-
| style="text-align:left;" |[[Wrought iron]]
|193
|28
|<ref>{{Cite web|date=August 13, 2013|title=Wrought Iron – Properties and Applications|url=https://www.azom.com/article.aspx?ArticleID=9555|access-date=May 9, 2021|website=AZO Materials}}</ref>
|-
| style="text-align:left;" |[[Yttrium iron garnet]] (YIG), polycrystalline
|193
|28
|<ref>{{Cite journal|last1=Chou|first1=Hung-Ming|last2=Case|first2=E. D.|date=November 1988|title=Characterization of some mechanical properties of polycrystalline yttrium iron garnet (YIG) by non-destructive methods|journal=Journal of Materials Science Letters|volume=7|issue=11|pages=1217–1220|doi=10.1007/BF00722341|via=[[SpringerLink]]|s2cid=135957639}}</ref>
|-
| style="text-align:left;" |[[Yttrium iron garnet]] (YIG), single-crystal
|200
|29
|<ref>{{Cite web|title=Yttrium Iron Garnet|url=http://deltroniccrystalindustries.com/deltronic_crystal_products/yttrium_iron_garnet|access-date=May 7, 2021|website=Deltronic Crystal Industries, Inc.|date=December 28, 2012}}</ref>
|-
| style="text-align:left;" |[[Zinc]] (<sub>30</sub>Zn)
|108
|15.7
|<ref>{{Cite web|date=July 23, 2001|title=An Introduction to Zinc|url=https://www.azom.com/properties.aspx?ArticleID=602|access-date=May 9, 2021|website=AZO Materials}}</ref>
|-
| style="text-align:left;" |[[Zirconium]] (<sub>40</sub>Zr), commercial
|95
|13.8
|<ref name=":4" />
|}

==See also==
* [[Bending stiffness]]
* [[Deflection (engineering)|Deflection]]
* [[Deformation (engineering)|Deformation]]
* [[Flexural modulus]]
* [[Impulse excitation technique]]
* [[List of materials properties]]
* [[Yield (engineering)]]

==References==
{{Reflist|30em}}

==Further reading==
* [[ASTM]] E 111, [http://www.astm.org/Standards/E111.htm "Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus"] 
* The ''[[ASM Handbook]]'' (various volumes) contains Young's Modulus for various materials and information on calculations. [http://products.asminternational.org/hbk/index.jsp Online version] {{subscription required}}

==External links==
* [http://www.matweb.com/ Matweb: free database of engineering properties for over 175,000 materials]
* [http://www-materials.eng.cam.ac.uk/mpsite/interactive_charts/stiffness-cost/NS6Chart.html Young's Modulus for groups of materials, and their cost]

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[[Category:Elasticity (physics)]]
[[Category:Physical quantities]]
[[Category:Structural analysis]]