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| '''Young's modulus''', also known as the '''Tensile modulus''' or '''[[elastic modulus]]''', is a measure of the [[stiffness]] of an [[Elasticity (physics)|elastic]] [[Isotropy|isotropic]] material and is a quantity used to characterize materials. It is defined as the ratio of the [[Stress (mechanics)|stress]] along an [[Cartesian coordinate system|axis]] over the [[strain (materials science)|strain]] along that axis in the range of stress in which [[Hooke's law]] holds.<ref>{{GoldBookRef|title=modulus of elasticity (Young's modulus), ''E''|file=M03966}}</ref> In [[solid mechanics]], the slope of the [[stress–strain curve]] at any point is called the [[tangent modulus]]. The tangent modulus of the initial, linear portion of a stress–strain curve is called ''Young's 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. In [[anisotropic]] materials, Young's modulus may have different values depending on the direction of the applied force with respect to the material's structure.
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| Young's modulus is the most common ''[[elastic modulus]]'', sometimes called the ''modulus of elasticity'', but there are other elastic moduli measured, too, such as the [[bulk modulus]] and the [[shear modulus]].
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| It is named after [[Thomas Young (scientist)|Thomas Young]], the 19th 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, 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>
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| A material whose Young's modulus is very high is rigid. Do not confuse:
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| * rigidity and [[Strength of materials|strength]]: the strength of material is characterized by its [[Yield (engineering)|yield strength]] and / or its [[tensile strength]];
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| * rigidity and [[stiffness]]: the beam stiffness (for example) depends on its Young's modulus but also on the ratio of its section at its length. The rigidity characterises the materials, the stiffness regards products and constructions: a massive mechanical plastic part can be much stiffer than a steel spring;
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| * rigidity and [[hardness]]: the hardness of a material defines its relative resistance that its surface opposes to the penetration of a harder body.
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| ==Units==
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| Young's modulus is the ratio of [[stress (physics)|stress]] (which has units of [[pressure]]) to [[strain (materials science)|strain]] (which is [[Dimensionless quantity|dimensionless]]), and so Young's modulus has units of [[pressure]]. Its [[SI]] unit is therefore the [[pascal (unit)|pascal]] (Pa or [[newton (unit)|N]]/m<sup>2</sup> or m<sup>−1</sup>·kg·s<sup>−2</sup>). The practical units used are megapascals (MPa or [[newton (unit)|N]]/mm<sup>2</sup>) or gigapascals (GPa or kN/mm<sup>2</sup>). In [[United States customary units]], it is expressed as [[pounds per square inch|pounds (force) per square inch]] (psi). The abbreviation '''ksi''' refers to thousands of psi.
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| == Usage ==
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| The 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. Young's modulus is 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. Some calculations also require the use of other material properties, such as the [[shear modulus]], [[density]], or [[Poisson's ratio]].
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| ===Linear versus non-linear===
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| The Young's modulus represents the factor of proportionality in [[Hooke's law]], relating the stress and the strain ; but this law is only valid under the assumption of an ''elastic'' or ''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 materials exhibit 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 ; if the typical stress one would apply is outside the linear range, then the material is said to be non-linear.
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| [[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 for this one.
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| ===Directional materials===
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| 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. Anisotropy can be seen in many composites as well. For example, [[carbon (fiber)|carbon fiber]] has 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.
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| == Calculation ==
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| Young's modulus, ''E'', can be calculated by dividing the [[Stress (physics)|tensile stress]] by the [[strain (physics)|extensional strain]] in the elastic (initial, linear) portion of the [[stress–strain curve]]:
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| :<math> E \equiv \frac{\mbox {tensile stress}}{\mbox {extensional strain}} = \frac{\sigma}{\varepsilon}= \frac{F/A_0}{\Delta L/L_0} = \frac{F L_0} {A_0 \Delta L} </math>
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| where
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| :<var>E</var> is the Young's modulus (modulus of elasticity)
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| :<var>F</var> is the force exerted on an object under tension;
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| :<var>A<sub>0</sub></var> is the original cross-sectional area through which the force is applied;
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| :<var>ΔL</var> is the amount by which the length of the object changes;
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| :<var>L<sub>0</sub></var> is the original length of the object.
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| ===Force exerted by stretched or contracted material===
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| The Young's modulus of a material can be used to calculate the force it exerts under specific strain. | |
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| :<math>F = \frac{E A_0 \Delta L} {L_0}</math>
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| where <var>F</var> is the force exerted by the material when contracted or stretched by <var>ΔL</var>.
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| [[Hooke's law]] can be derived from this formula, which describes the stiffness of an ideal spring:
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| :<math>F = \left( \frac{E A_0} {L_0} \right) \Delta L = k x \,</math>
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| where it comes in saturation
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| :<math>k = \begin{matrix} \frac {E A_0} {L_0} \end{matrix} \,</math> and <math>x = \Delta L. \,</math>
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| ===Elastic potential energy===
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| The [[elastic potential energy]] stored is given by the integral of this expression with respect to <var>L</var>:
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| :<math>U_e = \int {\frac{E A_0 \Delta L} {L_0}}\, d\Delta L = \frac {E A_0} {L_0} \int { \Delta L }\, d\Delta L = \frac {E A_0 {\Delta L}^2} {2 L_0}</math>
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| where <var>U<sub>e</sub></var> is the elastic potential energy.
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| The elastic potential energy per unit volume is given by:
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| :<math>\frac{U_e} {A_0 L_0} = \frac {E {\Delta L}^2} {2 L_0^2} = \frac {1} {2} E {\varepsilon}^2</math>, where <math>\varepsilon = \frac {\Delta L} {L_0}</math> is the strain in the material.M
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| This formula can also be expressed as the integral of Hooke's law:
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| :<math>U_e = \int {k x}\, dx = \frac {1} {2} k x^2.</math>
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| ===Relation among elastic constants===
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| For homogeneous isotropic materials [[Elastic modulus|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:
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| :<math>E = 2G(1+\nu) = 3K(1-2\nu).\,</math>
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| == Approximate values ==
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| [[Image:SpiderGraph YoungMod.gif|350px|thumb|Influences of selected glass component additions on Young's modulus of a specific base glass]]
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| 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 comparison.
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| {| class="wikitable" style="text-align:center;"
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| |+ Approximate Young's modulus for various materials
| |
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| |-
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| ! Material
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| ! [[pascal (unit)|GPa]]
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| ! [[pound-force per square inch|lbf/in²]] (psi)
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| |-
| |
| | style="text-align:left;"| [[Rubber]] (small strain)
| |
| | 0.01–0.1<ref name=etb20120106>
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| {{cite web|title=Elastic Properties and Young Modulus for some Materials |url=http://www.engineeringtoolbox.com/young-modulus-d_417.html |publisher=The Engineering ToolBox |accessdate=2012-01-06 }}</ref>
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| | 1,450–14,503
| |
| |-
| |
| | style="text-align:left;"| [[PTFE]] (Teflon)
| |
| | 0.5 <ref name="etb20120106"/>
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| | 75,000
| |
| |-
| |
| | style="text-align:left;"| [[Low density polyethylene]]<ref>{{cite web|url=http://www.matweb.com/search/datasheet.aspx?MatGUID=557b96c10e0843dbb1e830ceedeb35b0|title=Overview of materials for Low Density Polyethylene (LDPE), Molded|publisher=Matweb|accessdate=Feb 7, 2013}}</ref>
| |
| | 0.11–0.45
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| | 16,000–65,000
| |
| |-
| |
| | style="text-align:left;"| [[HDPE]]
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| | 0.8
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| | 116,000
| |
| |-
| |
| | style="text-align:left;"| [[Polypropylene]]
| |
| | 1.5–2<ref name=etb20120106/>
| |
| | 218,000–290,000
| |
| |-
| |
| | style="text-align:left;"| [[Capsid|Bacteriophage capsids]]<ref>{{cite journal|journal=Proc Nat Acad Sci USA. |year=2004|author=Ivanovska IL, de Pablo PJ, Sgalari G, MacKintosh FC, Carrascosa JL, Schmidt CF, Wuite GJL|title=Bacteriophage capsids: Tough nanoshells with complex elastic properties|pmid=15133147|volume=101|issue=20|pages=7600–5|doi=10.1073/pnas.0308198101|pmc=419652|bibcode = 2004PNAS..101.7600I }}</ref>
| |
| | 1–3
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| | 150,000–435,000
| |
| |-
| |
| | style="text-align:left;"| [[Polyethylene terephthalate]] (PET)
| |
| | 2–2.7<ref name=etb20120106/>
| |
| | 290,000–390,000
| |
| |-
| |
| | style="text-align:left;"| [[Polystyrene]]
| |
| | 3–3.5<ref name=etb20120106/>
| |
| | 440,000–510,000
| |
| |-
| |
| | style="text-align:left;"| [[Nylon]]
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| | 2–4
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| | 290,000–580,000
| |
| |-
| |
| | style="text-align:left;"| [[Diatom]] [[frustules]] (largely [[silicic acid]])<ref>{{cite journal|journal=J Nanosci Nanotechnol. |year=2005|author=Subhash G, Yao S, Bellinger B, Gretz MR.|title=Investigation of mechanical properties of diatom frustules using nanoindentation|pmid= 15762160|volume=5|issue=1|pages=50–6|doi=10.1166/jnn.2005.006}}</ref>
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| | 0.35–2.77
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| | 50,000–400,000
| |
| |-
| |
| | style="text-align:left;"| [[Medium-density fiberboard]] (MDF)<ref>[http://www.makeitfrom.com/data/?material=MDF Material Properties Data: Medium Density Fiberboard (MDF)]</ref>
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| | 4
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| | 580,000
| |
| |-
| |
| | style="text-align:left;"| Oak [[wood]] (along grain)
| |
| | 11<ref name=etb20120106/>
| |
| | {{val|1.60|e=6}}
| |
| |-
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| | style="text-align:left;"| Human Cortical [[Bone]]<ref>{{Cite journal | last = Rho | first = JY | title = Young's modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements | journal = Journal of Biomechanics | volume = 26 | issue = 2 | pages = 111–119 | year = 1993 }}</ref>
| |
| | 14
| |
| | {{val|2.03|e=6}}
| |
| |-
| |
| | style="text-align:left;"| Aromatic peptide nanotubes <ref>{{Cite journal | last = Kol | first = N. et al. | title = Self-Assembled Peptide Nanotubes Are Uniquely Rigid Bioinspired Supramolecular Structures | journal = Nano Letters | volume = 5 | issue = 7 | pages = 1343–1346 | date = June 8, 2005| doi = 10.1021/nl0505896|bibcode = 2005NanoL...5.1343K }}</ref><ref>{{Cite journal | last = Niu | first = L. et al. | title = Using the Bending Beam Model to Estimate the Elasticity of Diphenylalanine Nanotubes | journal = Langmuir | volume = 23 | issue = 14 | pages = 7443–7446 | date = June 6, 2007 | doi = 10.1021/la7010106}}</ref>
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| | 19–27
| |
| | {{val|2.76|e=6}}–{{val|3.92|e=6}}
| |
| |-
| |
| | style="text-align:left;"| High-strength [[concrete]]
| |
| | 30<ref name=etb20120106/>
| |
| | {{val|4.35|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Hemp]] fiber <ref>{{Cite journal | doi = 10.1002/(SICI)1098-2329(199924)18:4<351::AID-ADV6>3.0.CO;2-X | last = Nabi Saheb | first = D. | last2 = Jog | first2 = JP. | title = Natural fibre polymer composites: a review | journal = Advances in Polymer Technology | volume = 18 | issue = 4 | pages = 351–363 | year = 1999 }}</ref>
| |
| | 35
| |
| | {{val|5.08|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Magnesium]] [[metal]] (Mg)
| |
| | 45<ref name=etb20120106/>
| |
| | {{val|6.53|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Flax]] fiber <ref>{{Cite journal | doi = 10.1016/S1359-835X(02)00040-4 | last = Bodros | first = E. | title = Analysis of the flax fibres tensile behaviour and analysis of the tensile stiffness increase | journal = Composite Part A | volume = 33 | issue = 7 | pages = 939–948 | year = 2002 }}</ref>
| |
| | 58
| |
| | {{val|8.41|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Aluminum]]
| |
| | 69<ref name=etb20120106/>
| |
| | {{val|10.0|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Stinging nettle]] fiber <ref>{{Cite journal | last = Bodros | first = E. | last2 = Baley | first2 = C. | title = Study of the tensile properties of stinging nettle fibres (Urtica dioica) | journal = Materials Letters | volume = 62 | issue = 14 | pages = 2143–2145 | date = 15 May 2008 | doi = 10.1016/j.matlet.2007.11.034 }}</ref>
| |
| | 87
| |
| | {{val|12.6|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Glass]] (see chart)
| |
| | 50–90<ref name=etb20120106/>
| |
| | {{val|7.25|e=6}} – {{val|13.1|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Aramid]]<ref>{{Cite journal|year=2001|page=9|author=DuPont|title=Kevlar Technical Guide}}</ref>
| |
| | 70.5–112.4
| |
| | {{val|10.2|e=6}} – {{val|16.3|e=6}}
| |
| |-
| |
| | style="text-align:left;"| Mother-of-pearl ([[nacre]], largely calcium carbonate) <ref>{{cite journal|journal=Proceedings of the Royal Society B|year=1988|volume=234|pages=415–440|author=A. P. Jackson,J. F. V. Vincent and R. M. Turner|title=The Mechanical Design of Nacre|url=http://rspb.royalsocietypublishing.org/content/234/1277/415.abstract|doi=10.1098/rspb.1988.0056|bibcode = 1988RSPSB.234..415J|issue=1277}}</ref>
| |
| | 70
| |
| | {{val|10.2|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Tooth enamel]] (largely [[calcium phosphate]])<ref>{{cite journal|journal=Journal of Materials Science|year=1981|title=Spherical indentation of tooth enamel|author=M. Staines, W. H. Robinson and J. A. A. Hood|url=http://www.springerlink.com/content/w125706571032231/}}</ref>
| |
| | 83
| |
| | {{val|12.0|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Brass]]
| |
| | 100–125<ref name=etb20120106/>
| |
| | {{val|14.5|e=6}} – {{val|18.1|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Bronze]]
| |
| | 96–120<ref name=etb20120106/>
| |
| | {{val|13.9|e=6}} – {{val|17.4|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Titanium]] (Ti) || 110.3 || {{val|16.0|e=6}}<ref name=etb20120106/>
| |
| |-
| |
| | style="text-align:left;"| [[Titanium alloy]]s
| |
| | 105–120<ref name=etb20120106/>
| |
| | {{val|15.0|e=6}} – {{val|17.5|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Copper]] (Cu)
| |
| | 117
| |
| | {{val|17.0|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Glass-reinforced plastic|Glass-reinforced polyester matrix]] <ref>[http://www.substech.com/dokuwiki/doku.php?id=polyester_matrix_composite_reinforced_by_glass_fibers_fiberglass Polyester Matrix Composite reinforced by glass fibers (Fiberglass)]. [SubsTech] (2008-05-17). Retrieved on 2011-03-30.</ref>
| |
| | 17.2
| |
| | {{val|2.49|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Carbon fiber reinforced plastic]] (50/50 fibre/matrix, biaxial fabric)
| |
| | 30–50<ref>{{cite web|url=http://www.tech.plym.ac.uk/sme/MATS324/MATS324A2 E-G-nu.htm|title=Composites Design and Manufacture (BEng) – MATS 324}}</ref>
| |
| | {{val|4.35|e=6}} – {{val|7.25|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Carbon fiber reinforced plastic]] (70/30 fibre/matrix, unidirectional, along grain)<ref>[http://www.substech.com/dokuwiki/doku.php?id=epoxy_matrix_composite_reinforced_by_70_carbon_fibers Epoxy Matrix Composite reinforced by 70% carbon fibers [SubsTech]]. Substech.com (2006-11-06). Retrieved on 2011-03-30.</ref>
| |
| | 181
| |
| | {{val|26.3|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Silicon]] Single crystal, different directions <ref>[http://www.ioffe.ru/SVA/NSM/Semicond/Si Physical properties of Silicon (Si)]. Ioffe Institute Database. Retrieved on 2011-05-27.</ref><ref>{{cite journal|author=E.J. Boyd et al.|journal=Journal of Microelectromechanical Systems|volume=21|issue=1|pages=243–249|date=February 2012|doi=10.1109/JMEMS.2011.2174415|title=Measurement of the Anisotropy of Young's Modulus in Single-Crystal Silicon}}</ref>
| |
| | 130–185
| |
| | {{val|18.9|e=6}} – {{val|26.8|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Wrought iron]]
| |
| | 190–210<ref name=etb20120106/>
| |
| | {{val|27.6|e=6}} – {{val|30.5|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Steel]] (ASTM-A36)
| |
| | 200<ref name=etb20120106/>
| |
| | {{val|29.0|e=6}}
| |
| |-
| |
| | style="text-align:left;"| polycrystalline [[Yttrium iron garnet]] (YIG)<ref>{{Cite journal | last = Chou | first = H. M. | last2 = Case | first2 = E. D. | 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 | date = November 1988 | doi = 10.1007/BF00722341}}</ref>
| |
| | 193
| |
| | {{val|28.0|e=6}}
| |
| |-
| |
| | style="text-align:left;"| single-crystal [[Yttrium iron garnet]] (YIG)<ref>[http://www.isowave.com/pdf/materials/Yttrium_Iron_Garnet.pdf YIG properties]</ref>
| |
| | 200
| |
| | {{val|29.0|e=6}}
| |
| |-
| |
| | style="text-align:left;"| Aromatic peptide nanospheres <ref>{{Cite journal | last = Adler-Abramovich | first = L. et al. | title = Self-Assembled Organic Nanostructures with Metallic-Like Stiffness | journal = Angewandte Chemie International Edition | volume = 49 | issue = 51 | pages = 9939–9942 | date = December 17, 2010| doi = 10.1002/anie.201002037}}</ref>
| |
| | 230–275
| |
| | {{val|33.4|e=6}} – {{val|39.9|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Beryllium]] (Be){{cn|date=January 2014}}
| |
| | 287
| |
| | {{val|41.6|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Molybdenum]] (Mo)
| |
| {{cn|date=January 2014}}
| |
| | 329
| |
| | {{val|47.7|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Tungsten]] (W)
| |
| | 400–410<ref name=etb20120106/>
| |
| | {{val|58.0|e=6}} – {{val|59.5|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Silicon carbide]] (SiC)
| |
| | 450<ref name=etb20120106/>
| |
| | {{val|65.3|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Osmium]] (Os){{cn|date=January 2014}}
| |
| | 550
| |
| | {{val|79.8|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Tungsten carbide]] (WC)
| |
| | 450–650<ref name=etb20120106/>
| |
| | {{val|65.3|e=6}} – {{val|94.3|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Single-walled carbon nanotube]]<ref>{{cite web|url=http://ipn2.epfl.ch/CHBU/papers/ourpapers/Forro_NT99.pdf|title=Electronic and mechanical properties of carbon nanotubes|author=L. Forro et al.}}</ref><ref>{{cite journal|author=Y.H.Yang et al.|journal=Applied Physics Letters|volume=98|page=041901|year=2011|doi=10.1063/1.3546170|title=Radial elasticity of single-walled carbon nanotube measured by atomic force microscopy|bibcode = 2011ApPhL..98d1901Y|last2=Li|first2=W. Z.|issue=4 }}</ref>
| |
| | 1,000+
| |
| | {{val|145|e=6}}+
| |
| |-
| |
| | style="text-align:left;"| [[Graphene]]{{cn|date=January 2014}}
| |
| | 1,000
| |
| | {{val|145|e=6}}
| |
| |-
| |
| | style="text-align:left;"| [[Diamond]] (C)<ref>{{cite book |title=Synthetic Diamond – Emerging CVD Science and Technology |author=Spear and Dismukes |publisher=Wiley, NY |year=1994 |isbn=978-0-471-53589-8}}</ref>
| |
| | 1,220
| |
| | {{val|150|e=6}} – {{val|175|e=6}}
| |
| |-
| |
| | style="text-align:left;" | [[Linear acetylenic carbon|Carbyne]] (C)<ref>{{Cite web | url=http://phys.org/news/2013-08-carbyne-stronger-material.html | title=Carbyne is stronger than any known material |last=Owano | first1=Nancy |date=Aug 20, 2013 |website=phys.org }}</ref>
| |
| | 32,700
| |
| | {{val|5388|e=6}} – {{val|5402|e=6}}
| |
| |}
| |
| | |
| == See also ==
| |
| * [[Deflection (engineering)|Deflection]]
| |
| * [[Deformation (engineering)|Deformation]]
| |
| * [[Hardness]]
| |
| * [[Hooke's law]]
| |
| * [[Shear modulus]]
| |
| * [[Bulk Modulus]]
| |
| * [[Bending stiffness]]
| |
| * [[Impulse excitation technique]]
| |
| * [[Toughness]]
| |
| * [[Yield (engineering)]]
| |
| * [[List of materials properties]]
| |
| | |
| == References ==
| |
| {{reflist|30em}}
| |
| | |
| ==Further reading==
| |
| * [[ASTM]] E 111, "Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus," [http://www.astm.org/Standards/E111.htm]
| |
| * 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 63,000 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]
| |
| | |
| {{Physics-footer}}
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| {{Elastic moduli}}
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| | |
| [[Category:Elasticity (physics)]]
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| [[Category:Physical quantities]]
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