Elastic modulus - wikipedia Elastic modulus This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. ( November 2010 ) An elastic modulus ( also known as modulus of elasticity ) is a quantity that measures an object or substance 's resistance to being deformed elastically ( i.e., non-permanently ) when a stress 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 : A stiffer material will have a higher elastic modulus. An elastic modulus has the form : λ = def stress strain ( displaystyle lambda ( stackrel ( \text ( def ) ) ( = ) ) ( \frac ( \text ( stress ) ) ( \text ( strain ) ) ) ) where stress is the force causing the deformation divided by the area to which the force is applied and strain is the ratio of the change in some parameter caused by the deformation to the original value of the parameter. If stress is measured in pascals, then since strain is a dimensionless quantity, the units of λ will be pascals as well. 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 ( E ) describes tensile 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 tensile strain. It is often referred to simply as the elastic modulus. The shear modulus or modulus of rigidity ( G or μ ( displaystyle mu , ) ) 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 ( K ) describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions ; it is defined as 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 Poisson 's ratio, 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 can not support shear stress, meaning that the shear modulus is always zero. This also implies that Young 's modulus for this group is always zero. In some English texts the here described quantity is called elastic constant, while the inverse quantity is referred to as elastic modulus. 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 Jump up ^ Askeland, Donald R. ; Phulé, Pradeep P. ( 2006 ). The science and engineering of materials ( 5th ed. ). Cengage Learning. p. 198. ISBN 978 - 0 - 534 - 55396 - 8. Jump up ^ Beer, Ferdinand P. ; Johnston, E. Russell ; Dewolf, John ; Mazurek, David ( 2009 ). Mechanics of Materials. McGraw Hill. p. 56. ISBN 978 - 0 - 07 - 015389 - 9. Further reading Hartsuijker, C. ; Welleman, J.W. ( 2001 ). Engineering Mechanics. Volume 2. Springer. ISBN 978 - 1 - 4020 - 4123 - 5. De Jong, Maarten ; Chen, Wei ( 2015 ). `` Charting the complete elastic properties of inorganic crystalline compounds ''. Scientific Data. 2 : 150009. Bibcode : 2013NatSD... 2E0009D. doi : 10.1038 / sdata. 2015.9. hide Elastic moduli for homogeneous isotropic materials Bulk modulus ( K ( displaystyle K ) ) Young 's modulus ( E ( displaystyle E ) ) Lamé 's first parameter ( λ ( displaystyle lambda ) ) Shear modulus ( G, μ ( displaystyle G, mu ) ) Poisson 's ratio ( ν ( displaystyle \nu ) ) P - wave modulus ( M ( displaystyle M ) ) hide 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. K = ( displaystyle K = , ) E = ( displaystyle E = , ) λ = ( displaystyle lambda = , ) G = ( displaystyle G = , ) ν = ( displaystyle \nu = , ) M = ( displaystyle M = , ) Notes ( K, E ) ( displaystyle ( K, , E ) ) K ( displaystyle K ) E ( displaystyle E ) 3 K ( 3 K − E ) 9 K − E ( displaystyle ( \tfrac ( 3K ( 3K - E ) ) ( 9K - E ) ) ) 3 K E 9 K − E ( displaystyle ( \tfrac ( 3KE ) ( 9K - E ) ) ) 3 K − E 6 K ( displaystyle ( \tfrac ( 3K - E ) ( 6K ) ) ) 3 K ( 3 K + E ) 9 K − E ( displaystyle ( \tfrac ( 3K ( 3K + E ) ) ( 9K - E ) ) ) ( K, λ ) ( displaystyle ( K, , lambda ) ) K ( displaystyle K ) 9 K ( K − λ ) 3 K − λ ( displaystyle ( \tfrac ( 9K ( K - lambda ) ) ( 3K - lambda ) ) ) λ ( displaystyle lambda ) 3 ( K − λ ) 2 ( displaystyle ( \tfrac ( 3 ( K - lambda ) ) ( 2 ) ) ) λ 3 K − λ ( displaystyle ( \tfrac ( lambda ) ( 3K - lambda ) ) ) 3 K − 2 λ ( displaystyle 3K - 2 lambda , ) ( K, G ) ( displaystyle ( K, , G ) ) K ( displaystyle K ) 9 K G 3 K + G ( displaystyle ( \tfrac ( 9KG ) ( 3K + G ) ) ) K − 2 G 3 ( displaystyle K - ( \tfrac ( 2G ) ( 3 ) ) ) G ( displaystyle G ) 3 K − 2 G 2 ( 3 K + G ) ( displaystyle ( \tfrac ( 3K - 2G ) ( 2 ( 3K + G ) ) ) ) K + 4 G 3 ( displaystyle K+ ( \tfrac ( 4G ) ( 3 ) ) ) ( K, ν ) ( displaystyle ( K, , \nu ) ) K ( displaystyle K ) 3 K ( 1 − 2 ν ) ( displaystyle 3K ( 1 - 2 \nu ) , ) 3 K ν 1 + ν ( displaystyle ( \tfrac ( 3K \nu ) ( 1 + \nu ) ) ) 3 K ( 1 − 2 ν ) 2 ( 1 + ν ) ( displaystyle ( \tfrac ( 3K ( 1 - 2 \nu ) ) ( 2 ( 1 + \nu ) ) ) ) ν ( displaystyle \nu ) 3 K ( 1 − ν ) 1 + ν ( displaystyle ( \tfrac ( 3K ( 1 - \nu ) ) ( 1 + \nu ) ) ) ( K, M ) ( displaystyle ( K, , M ) ) K ( displaystyle K ) 9 K ( M − K ) 3 K + M ( displaystyle ( \tfrac ( 9K ( M-K ) ) ( 3K + M ) ) ) 3 K − M 2 ( displaystyle ( \tfrac ( 3K - M ) ( 2 ) ) ) 3 ( M − K ) 4 ( displaystyle ( \tfrac ( 3 ( M-K ) ) ( 4 ) ) ) 3 K − M 3 K + M ( displaystyle ( \tfrac ( 3K - M ) ( 3K + M ) ) ) M ( displaystyle M ) ( E, λ ) ( displaystyle ( E, , lambda ) ) E + 3 λ + R 6 ( displaystyle ( \tfrac ( E + 3 lambda + R ) ( 6 ) ) ) E ( displaystyle E ) λ ( displaystyle lambda ) E − 3 λ + R 4 ( displaystyle ( \tfrac ( E-3 lambda + R ) ( 4 ) ) ) 2 λ E + λ + R ( displaystyle ( \tfrac ( 2 lambda ) ( E+ lambda + R ) ) ) E − λ + R 2 ( displaystyle ( \tfrac ( E - lambda + R ) ( 2 ) ) ) R = E 2 + 9 λ 2 + 2 E λ ( displaystyle R = ( sqrt ( E ^ ( 2 ) + 9 lambda ^ ( 2 ) + 2E lambda ) ) ) ( E, G ) ( displaystyle ( E, , G ) ) E G 3 ( 3 G − E ) ( displaystyle ( \tfrac ( EG ) ( 3 ( 3G - E ) ) ) ) E ( displaystyle E ) G ( E − 2 G ) 3 G − E ( displaystyle ( \tfrac ( G ( E-2G ) ) ( 3G - E ) ) ) G ( displaystyle G ) E 2 G − 1 ( displaystyle ( \tfrac ( E ) ( 2G ) ) - 1 ) G ( 4 G − E ) 3 G − E ( displaystyle ( \tfrac ( G ( 4G - E ) ) ( 3G - E ) ) ) ( E, ν ) ( displaystyle ( E, , \nu ) ) E 3 ( 1 − 2 ν ) ( displaystyle ( \tfrac ( E ) ( 3 ( 1 - 2 \nu ) ) ) ) E ( displaystyle E ) E ν ( 1 + ν ) ( 1 − 2 ν ) ( displaystyle ( \tfrac ( E \nu ) ( ( 1 + \nu ) ( 1 - 2 \nu ) ) ) ) E 2 ( 1 + ν ) ( displaystyle ( \tfrac ( E ) ( 2 ( 1 + \nu ) ) ) ) ν ( displaystyle \nu ) E ( 1 − ν ) ( 1 + ν ) ( 1 − 2 ν ) ( displaystyle ( \tfrac ( E ( 1 - \nu ) ) ( ( 1 + \nu ) ( 1 - 2 \nu ) ) ) ) ( E, M ) ( displaystyle ( E, , M ) ) 3 M − E + S 6 ( displaystyle ( \tfrac ( 3M - E + S ) ( 6 ) ) ) E ( displaystyle E ) M − E + S 4 ( displaystyle ( \tfrac ( M - E + S ) ( 4 ) ) ) 3 M + E − S 8 ( displaystyle ( \tfrac ( 3M + E-S ) ( 8 ) ) ) E − M + S 4 M ( displaystyle ( \tfrac ( E-M + S ) ( 4M ) ) ) M ( displaystyle M ) S = ± E 2 + 9 M 2 − 10 E M ( displaystyle S = pm ( sqrt ( E ^ ( 2 ) + 9M ^ ( 2 ) - 10EM ) ) ) There are two valid solutions. The plus sign leads to ν ≥ 0 ( displaystyle \nu geq 0 ). The minus sign leads to ν ≤ 0 ( displaystyle \nu leq 0 ). ( λ, G ) ( displaystyle ( lambda, , G ) ) λ + 2 G 3 ( displaystyle lambda + ( \tfrac ( 2G ) ( 3 ) ) ) G ( 3 λ + 2 G ) λ + G ( displaystyle ( \tfrac ( G ( 3 lambda + 2G ) ) ( lambda + G ) ) ) λ ( displaystyle lambda ) G ( displaystyle G ) λ 2 ( λ + G ) ( displaystyle ( \tfrac ( lambda ) ( 2 ( lambda + G ) ) ) ) λ + 2 G ( displaystyle lambda + 2G , ) ( λ, ν ) ( displaystyle ( lambda, , \nu ) ) λ ( 1 + ν ) 3 ν ( displaystyle ( \tfrac ( lambda ( 1 + \nu ) ) ( 3 \nu ) ) ) λ ( 1 + ν ) ( 1 − 2 ν ) ν ( displaystyle ( \tfrac ( lambda ( 1 + \nu ) ( 1 - 2 \nu ) ) ( \nu ) ) ) λ ( displaystyle lambda ) λ ( 1 − 2 ν ) 2 ν ( displaystyle ( \tfrac ( lambda ( 1 - 2 \nu ) ) ( 2 \nu ) ) ) ν ( displaystyle \nu ) λ ( 1 − ν ) ν ( displaystyle ( \tfrac ( lambda ( 1 - \nu ) ) ( \nu ) ) ) Cannot be used when ν = 0 ⇔ λ = 0 ( displaystyle \nu = 0 Leftrightarrow lambda = 0 ) ( λ, M ) ( displaystyle ( lambda, , M ) ) M + 2 λ 3 ( displaystyle ( \tfrac ( M + 2 lambda ) ( 3 ) ) ) ( M − λ ) ( M + 2 λ ) M + λ ( displaystyle ( \tfrac ( ( M - lambda ) ( M + 2 lambda ) ) ( M+ lambda ) ) ) λ ( displaystyle lambda ) M − λ 2 ( displaystyle ( \tfrac ( M - lambda ) ( 2 ) ) ) λ M + λ ( displaystyle ( \tfrac ( lambda ) ( M+ lambda ) ) ) M ( displaystyle M ) ( G, ν ) ( displaystyle ( G, , \nu ) ) 2 G ( 1 + ν ) 3 ( 1 − 2 ν ) ( displaystyle ( \tfrac ( 2G ( 1 + \nu ) ) ( 3 ( 1 - 2 \nu ) ) ) ) 2 G ( 1 + ν ) ( displaystyle 2G ( 1 + \nu ) , ) 2 G ν 1 − 2 ν ( displaystyle ( \tfrac ( 2G \nu ) ( 1 - 2 \nu ) ) ) G ( displaystyle G ) ν ( displaystyle \nu ) 2 G ( 1 − ν ) 1 − 2 ν ( displaystyle ( \tfrac ( 2G ( 1 - \nu ) ) ( 1 - 2 \nu ) ) ) ( G, M ) ( displaystyle ( G, , M ) ) M − 4 G 3 ( displaystyle M - ( \tfrac ( 4G ) ( 3 ) ) ) G ( 3 M − 4 G ) M − G ( displaystyle ( \tfrac ( G ( 3M - 4G ) ) ( M-G ) ) ) M − 2 G ( displaystyle M - 2G , ) G ( displaystyle G ) M − 2 G 2 M − 2 G ( displaystyle ( \tfrac ( M - 2G ) ( 2M - 2G ) ) ) M ( displaystyle M ) ( ν, M ) ( displaystyle ( \nu, , M ) ) M ( 1 + ν ) 3 ( 1 − ν ) ( displaystyle ( \tfrac ( M ( 1 + \nu ) ) ( 3 ( 1 - \nu ) ) ) ) M ( 1 + ν ) ( 1 − 2 ν ) 1 − ν ( displaystyle ( \tfrac ( M ( 1 + \nu ) ( 1 - 2 \nu ) ) ( 1 - \nu ) ) ) M ν 1 − ν ( displaystyle ( \tfrac ( M \nu ) ( 1 - \nu ) ) ) M ( 1 − 2 ν ) 2 ( 1 − ν ) ( displaystyle ( \tfrac ( M ( 1 - 2 \nu ) ) ( 2 ( 1 - \nu ) ) ) ) ν ( displaystyle \nu ) M ( displaystyle M ) GND : 4151691 - 6 Retrieved from `` https://en.wikipedia.org/w/index.php?title=Elastic_modulus&oldid=842299072 '' Categories : Elasticity ( physics ) Deformation ( mechanics ) Hidden categories : Articles needing additional references from November 2010 All articles needing additional references Wikipedia articles with GND identifiers Talk About Wikipedia Asturianu Azərbaycanca বাংলা Беларуская Български Boarisch Català Čeština Dansk Deutsch Eesti Español Français 한국어 Հայերեն हिन्दी Hrvatski Bahasa Indonesia Italiano עברית Қазақша Magyar Македонски Nederlands 日本 語 Norsk ភាសា ខ្មែរ Русский සිංහල Simple English Slovenčina Slovenščina Suomi Svenska Türkçe Українська Tiếng Việt 中文 29 more Edit links This page was last edited on 21 May 2018, at 15 : 15 ( UTC ). 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Question: the unit of modulus of elasticity is same as
Answer: slope of its stress–strain curve in the elastic deformation region