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An '''elastic modulus''', or '''modulus of elasticity''', is the mathematical description of an object or substance's tendency to be deformed elastically (i.e., non-permanently) when a [[force]] is applied to it.  The elastic modulus of an object is defined as the [[slope]] of its [[stress–strain curve]] in the elastic deformation region:<ref>{{cite book | last = Askeland | first = Donald R. | last2 = Phulé | first2 = Pradeep P. | title = The science and engineering of materials | year = 2006 | publisher = Cengage Learning | page = 198 | edition = 5th | url = http://books.google.com/books?id=fRbZslUtpBYC&pg=PA198 | isbn = 978-0-534-55396-8}}</ref> As such, a stiffer material will have a higher elastic modulus.


:<math>\lambda \ \stackrel{\text{def}}{=}\  \frac {\text{stress}} {\text{strain}}</math>
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where lambda (<var>λ</var>) is the elastic modulus; <var>[[stress (physics)|stress]]</var> is the restoring force caused due to the deformation divided by the area to which the force is applied; and <var>[[strain (materials science)|strain]]</var> is the ratio of the change caused by the stress to the original state of the object.  If stress is measured in [[pascal (unit)|pascal]]s, since strain is a dimensionless quantity, then the units of <var>λ</var> are pascals as well.<ref>{{cite book | last = Beer | first = Ferdinand P. | last2 = Johnston | first2 = E. Russell | last3 = Dewolf | first3 = John | last4 = Mazurek | first4 = David | title = Mechanics of Materials | year = 2009 | publisher = McGraw Hill | page = 56 | isbn = 978-0-07-015389-9}}</ref>
 
Since the denominator turns into unity if length is doubled, the elastic modulus becomes the stress induced in the material, when the sample of the material turns double of its original length on applying external force. While this endpoint is not realistic because most materials will fail before reaching it, it is practical, in that small fractions of the defining load will operate in exactly the same ratio. Thus, for steel with a [[Young's modulus]] of 30 million psi, a 30 thousand psi load will elongate a 1&nbsp;inch bar by one thousandth of an inch; similarly, for metric units, a load of one-thousandth of the modulus (now measured in gigapascals) will change the length of a one-meter rod by a millimeter.
 
Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined.
The three primary ones are:
 
* ''[[Young's modulus]]'' (<var>E</var>) describes tensile [[Elasticity (physics)|elasticity]], or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of [[tensile stress]] to [[Tension (physics)|tensile]] strain. It is often referred to simply as the ''elastic modulus''.
* The ''[[shear modulus]]'' or ''modulus of rigidity'' (<var>G</var> or <math>\mu \,</math>) describes an object's tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as [[shear stress]] over [[shear strain]].  The shear modulus is part of the derivation of [[viscosity]].
* The ''[[bulk modulus]]'' (<var>K</var>) describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions; it is defined as [[Stress (physics)#Stress deviator tensor|volumetric stress]] over volumetric strain, and is the inverse of [[compressibility]]. The bulk modulus is an extension of Young's modulus to three dimensions.
 
Three other elastic moduli are [[Axial Modulus]], [[Lamé's first parameter]], and [[P-wave modulus]].
 
Homogeneous and [[isotropic]] (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair. Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below at the end of page.
 
[[Inviscid fluids]] are special in that they cannot support shear stress, meaning that the shear modulus is always zero. This also implies that Young's modulus is always zero.
 
==Estimating value==
E=33w^1.5*sqrt(f'c)
w=density of concrete
f'c=fracture stress
 
E=57000sqrt(f'c)
f'c=fracture stress
(these are both only good up to 6000psi)
 
==See also==
* [[Bending stiffness]]
* [[Dynamic modulus]]
* [[Elastic limit]]
* [[Elastic wave]]
* [[Flexural modulus]]
* [[Hooke's Law]]
* [[Impulse excitation technique]]
* [[Proportional limit]]
* [[Stiffness]]
* [[Tensile strength]]
* [[Transverse isotropy]]
 
==References==
{{reflist}}
 
==Further reading==
*{{cite book
| last = Hartsuijker
| first = C.
| last2 = Welleman
| first2 = J. W.
| title = Engineering Mechanics
| series = Volume 2
| year = 2001
| publisher = Springer
| isbn = 978-1-4020-4123-5}}
 
{{Elastic moduli}}
 
{{DEFAULTSORT:Elastic Modulus}}
[[Category:Elasticity (physics)]]
[[Category:Deformation]]
 
[[de:Elastizitätsmodul]]
[[simple:Elastic modulus]]

Latest revision as of 17:38, 15 September 2014


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